Lattice simulation of the Gross–Neveu model
Diplomarbeit
von
Markus Limmer
aus
Kelheim
durchgeführt am Institut für Theoretische Physik
der Universität Regensburg
und am Institut für Physik, Fachbereich Theoretische Physik
der Karl–Franzens–Universität Graz
unter Anleitung von
Prof. Dr. Andreas Schäfer
Prof. Dr. Christof Gattringer
Dezember 2006
Abstract
The lattice formulation of the 2–dimensional Gross–Neveu model with one flavor
is investigated, using Monte–Carlo methods. We write the four–Fermi interaction
as a Gaussian integral with an auxiliary field and use a reweighting technique,
i.e., we calculate the fermion determinant explicitly and include it as a weight
factor.
This work is related to a second investigation of the same model using a loop
representation. In this formulation the fermion sign problem can be eliminated
completely. A main part of this work consists of a comparison of the two different approaches, in order to assess the advantages and disadvantages of the two
methods.
Results for bulk observables, such as condensates or susceptibilities, as well
as n–point correlation functions are presented. We analyze the Dirac matrix, in
particular the distribution of its determinant and the eigenvalue spectrum. This
allows us to come up with an estimate for the critical mass parameter of the
theory. Various correlators are used in the variational method to obtain the mass
spectrum of the model. Finally, results for the different quantities of the standard
approach and the loop representation are compared to each other.
iii
Contents
1 Introduction
1
2 Gross–Neveu–type models in the continuum
2.1 The partition function of the Gross–Neveu model . . . . . . . . .
2.2 Symmetries of the model . . . . . . . . . . . . . . . . . . . . . . .
2.3 Euclidean correlators in the continuum . . . . . . . . . . . . . . .
5
5
7
10
3 The
3.1
3.2
3.3
3.4
3.5
.
.
.
.
.
13
13
14
17
18
20
.
.
.
.
23
23
25
26
27
.
.
.
.
29
29
31
32
33
Gross–Neveu model on the lattice
Basic definitions for lattice calculations . . . . . .
Wilson form of the lattice action . . . . . . . . . .
Key formulas of a Grassmann algebra . . . . . . .
The partition function on the lattice . . . . . . .
A loop representation for the Gross–Neveu model
.
.
.
.
.
4 Numerical simulation with Monte–Carlo methods
4.1 Fundamental ideas of a Monte Carlo simulation . .
4.2 The algorithm for our system . . . . . . . . . . . .
4.3 Random numbers and Gaussian distributions . . . .
4.4 Error analysis with the Jackknife method . . . . . .
5 Observables
5.1 Wick’s theorem . . . . . . . . . . .
5.2 Derivatives of the partition function
5.3 n–point correlation functions . . . .
5.4 Numerical aspects . . . . . . . . . .
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
Contents.
6 Tests for the free case
6.1 Testing the entries of the Dirac matrix and its inverse . . . . . . .
6.2 Tests for the derivative of the partition function . . . . . . . . . .
6.3 The correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
35
36
37
7 Analysis of the Dirac matrix M
7.1 The distribution of det(M ) . . . . . . . . . . . . . . . . . . . . . .
7.2 Eigenvalue spectrum of M . . . . . . . . . . . . . . . . . . . . . .
7.3 A rough estimate for the critical mass . . . . . . . . . . . . . . . .
41
41
44
46
8 Correlation functions and the mass spectrum
8.1 Short review of correlation functions . . . . . . .
8.2 How to extract masses from correlation functions
8.3 Excited state masses via the variational method .
8.4 Lines of constant physics . . . . . . . . . . . . . .
.
.
.
.
51
51
52
56
62
.
.
.
.
.
67
67
69
75
75
82
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
9 A comparison between the standard and the loop approach
9.1 The basic formulas . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Results for bulk observables in the free case . . . . . . . . . .
9.3 Results for bulk observables at nonvanishing coupling . . . . .
9.4 n–point correlators in both formalisms . . . . . . . . . . . . .
9.5 Standard techniques vs. loop approach – pros and cons . . . .
10 Summary and outlook
.
.
.
.
.
.
.
.
.
85
Appendix
A The loop representation of the Gross–Neveu model
89
B Fourier transformation on the lattice
93
C γ5 –hermiticity of the lattice Dirac operator
95
D Discussion of the correlation matrix
97
References
99
Chapter 1
Introduction
In the last 20 years lattice QCD has developed into a powerful tool, which started
to produce reliable quantitative results. However, many conceptual and technical
issues can and should be improved. A technical point of enormous importance
is the treatment of fermions. In the usual approach, the fermions are formally
integrated out and the fermion determinant is evaluated. This fermion determinant is then taken into account as an additional weight factor in the Monte–Carlo
calculation.
Even for the case that the fermion determinant is real and positive, i.e., one
does not consider the case of a chemical potential, the numerical treatment of
this determinant is very costly. The underlying reason is, that in a determinant an equal number of positive and negative terms enter, such that a huge
amount of cancellations is necessary to obtain the final number (fermion sign
problem). Finding a representation for the fermions where these cancellations
can be avoided, would be a tremendous progress for the simulation of lattice
QCD. Developing, however, such a new representation for the fermion determinant of QCD is a rather elusive goal at the moment.
The idea of studying difficult problems in low dimensional models has a long
tradition in theoretical physics. The hope is that one can elaborate on new ideas
in a simpler setting, and that those ideas can later be transferred to the “real
1
2
Chapter 1. Introduction
world”. In this spirit we here consider the lattice version of the 2–dimensional
Gross–Neveu (GN) model [1]. This model is a renormalizable and asymptotically
free theory. It has chiral symmetry and dynamical mass generation, as well
as a nontrivial mass spectrum. Since it is a theory containing fermions, it is
plagued with the fermion sign problem, which increases the needed computer
time drastically, compared to bosonic models. For the GN–model with Wilson
fermions this difficulty may be overcome by using a fermion loop representation
for the model [2].
With such a representation, the fermion sign problem can be eliminated completely. The partition function turns into a sum over closed contours, each having
a positive weight, and no cancellations appear. Due to that reason, and due to
the reduced memory requirements of the loop formalism, the available lattice volumes can be increased by a factor of roughly 103 . A simulation of the one–flavor
GN–model, using a fermion loop representation, was done in [3].
The goal of this work is to perform a benchmark simulation for the loop
approach, and to compute different observables on several lattice volumes at
various couplings with the standard approach. These data are then compared to
the loop approach. The agreement of the two approaches is analyzed and finite
size effects are studied. Furthermore an exploratory study of the spectrum of
excitations is presented.
This work is divided into three main parts. In the first part the theoretical
framework is discussed. The GN–model in the continuum and on the lattice is
presented (Chaps. 2 and 3). Then Monte–Carlo methods are explained (Chap. 4),
with special focus on the algorithm used here. To finish this part, we give a short
overview of the considered observables in Chap. 5.
The second part contains results for several observables, obtained from calculations with the previously presented standard approach. We start by presenting
a couple of tests for the correct implementation of the Dirac matrix and the
observables (Chap. 6). Then in Chap. 7 the Dirac matrix is analyzed in great
detail, concerning its determinant and the eigenvalue spectrum. Therefrom the
critical mass parameter of the system can be estimated. Part two closes with the
the analysis of correlation functions in Chap. 8. This chapter also contains the
discussion of the preliminary analysis of the excitations. Here interesting lessons
for the corresponding problem in QCD are to be learned.
3
The third part, Chap. 9, is devoted to assessing the two different approaches
of the Gross–Neveu model. We compare results for bulk observables and scalar
correlation functions, obtained from the two approaches, and finish with the discussion of advantages and disadvantages for working with the two representations.
In Chap. 10 we conclude with a summary of results and give a brief outlook
to possible future directions of research.
Chapter 2
Gross–Neveu–type models in the
continuum
The model we consider here is of the Gross–Neveu–type (GN), i.e., a quantum
field theory of 2–dimensional Dirac fermions interacting through a four–fermion
term [1]. In this chapter we present the model in the continuum and discuss
its symmetries. To set the ground for later discussions, a short introduction of
Euclidean correlation functions is also given.
2.1
The partition function of the Gross–Neveu
model
The model is described by the following action S and Lagrangian density L,
S[ψ, ψ, φ] =
L[ψ, ψ, φ] =
Z
d2 x L[ψ, ψ, φ] ,
Nf
X
f =1
ψ
(f )
(2.1)
1
√
(x) γµ ∂µ + m(f ) + g φ(x) ψ (f ) (x) + φ2 (x) . (2.2)
2
5
6
Chapter 2. Gross–Neveu–type models in the continuum
The index f runs over the Nf different flavors. For later use we define the flavor
mass matrix m as
m = diag m(1) , m(2) , . . . , m(Nf ) .
(2.3)
g ∈ R+
0 is the coupling constant. The action is built from the 2–component
fermion fields
(f )
(f )
(f )
ψ α ≡ ψ 1 (x), ψ2 (x) ,
(2.4)
T
(f )
(f )
ψα(f ) ≡ ψ1 (x), ψ2 (x)
,
(2.5)
and the scalar field
φ ≡ φ(x) ∈ R .
(2.6)
The Dirac index α in the fermion fields will be suppressed in the following expressions. The vector x is a space–time vector with one entry for the position
in space and one for the Euclidian time. For two dimensions, the γ–matrices are
simply the Pauli matrices
!
!
!
0 1
0 −i
1
0
γ1 =
,
γ2 =
,
γ5 = iγ1 γ2 =
.
(2.7)
1 0
i
0
0 −1
The partition function for our system then reads
Z
ZGN = D[ψ, ψ, φ] e−S[ψ,ψ,φ] ,
(2.8)
with the integration measure formally defined as
D[ψ, ψ, φ] =
Y
x
dφ(x)
Nf
Y
dψ (f ) (x)dψ
(f )
(x) .
(2.9)
f =1
We can integrate out the fields φ(x) in Eq. (2.8) (Hubbard–Stratonovich transformation [4, 5]) and obtain an alternative formulation with the effective action
Seff [ψ, ψ] =
Z
2
dx
X
Nf
f =1
N
ψ
(f )
f
i2
(f )
g hX (f )
(f )
(f )
(x) γµ ∂µ + m
ψ (x)−
ψ (x)ψ (x)
2 f =1
.
(2.10)
√
We remark, that replacing g in (2.2) by i g, changes the sign of g in the
effective action such that the sign for the four–Fermi interaction is positive.
√
2.2. Symmetries of the model
7
In terms of only the fermion fields the partition function reads
ZGN =
Z
Z
Nf
X
(f )
2
D[ψ, ψ] exp − d x
ψ (x) γµ ∂µ + m(f ) ψ (f ) (x)
f =1
+
Z
Nf
i2
g hX (f )
(f )
dx
ψ (x)ψ (x)
2
2
f =1
.
(2.11)
The vacuum expectation value of an observable O is given by
1
hOi =
ZGN
2.2
Z
D[ψ, ψ, φ] e−S[ψ,ψ,φ] O[ψ, ψ, φ] .
(2.12)
Symmetries of the model
The Z 2 –symmetry
The Z2 –symmetry is defined by the following discrete transformation,
ψ (f ) → ψ (f )′ = γ5 ψ (f ) ,
ψ
(f )
→ψ
(f )′
= −ψ
(f )
γ5 .
(2.13)
Our effective Lagrangian
Leff [ψ, ψ] =
Nf
X
f =1
ψ
(f )
Nf
i2
(f )
g hX (f )
(f )
(x) γµ ∂µ + m
ψ (x) −
ψ (x)ψ (f ) (x)
(2.14)
2
f =1
is invariant under this transformation only for the massless case m = 0. This
comes from the fact that the mass term would obtain an extra minus sign,
X
f
ψ
(f )
m(f ) ψ (f ) →
X
f
−ψ
(f )
γ5 m(f ) γ5 ψ (f ) = −
X
ψ
(f )
m(f ) ψ (f ) ,
(2.15)
f
and thus break the symmetry explicitly. Here we used the fact that γ52 = 1. On
the other hand, it is obvious that the part with the derivative does not change
due to the anti–commutation relation for the γµ –matrices,
{γµ , γ5} ≡ γµ γ5 + γ5 γµ = 0 .
(2.16)
8
Chapter 2. Gross–Neveu–type models in the continuum
The interaction term is invariant under (2.13),
hX
ψ
(f )
ψ (f )
f
=
hX
f
=
hX
f
−ψ
ψ
(f )
(f )
i2
→
hX
f
γ5 γ5 ψ (f )
ψ (f )
i2
i2
ψ
(f )′
ψ (f )′
i2
i2
h X (f )
ψ ψ (f )
= −
f
.
(2.17)
We remark that since the symmetry (2.13) is discrete, it can also be broken
spontaneously in two dimensions.
The chiral symmetry
Another, more general, continuous symmetry, is defined by the following transformation,
ψ (f ) → ψ (f )′ = eiαγ5 ψ (f ) ,
ψ
(f )
→ψ
(f )′
=ψ
(f ) iαγ5
e
.
(2.18)
This is a continuous chiral rotation of the fermion fields and thus the corresponding symmetry is called chiral symmetry. To get a more usable form of (2.18) for
calculations, we expand the exponential in a power series and obtain
eiαγ5 = 1 cos α + i γ5 sin α .
(2.19)
Again the kinetic term is invariant under such a rotation only for m = 0. The
first part does not change because of (2.16), but the mass term turns over into
ψ
(f )
m(f ) ψ (f ) → ψ
(f ) i2αγ5
e
m(f ) ψ (f ) .
(2.20)
To construct an invariant expression for the interaction we have to add another
four–Fermi term and thus acquire a Nambu–Jona–Lasinio–type model [6, 7]. The
kinetic term is the same as in the GN-model, but its interaction reads
Nf
Nf
i2 g hX (f )
i2
g hX (f )
5
ψ (x)ψ (f ) (x) +
ψ (x)γ5 ψ (f ) (x) .
−
2 f =1
2 f =1
(2.21)
2.2. Symmetries of the model
9
Let us look now at the transformation behavior of the first part,
hX (f )
i2
X (f )′
(g)′
(f )
ψ ψ
ψ ψ (f )′ ψ ψ (g)′
→
f
f,g
X
=
ψ
(f )
(g)
[cos 2α + i γ5 sin 2α]ψ (f ) ψ [cos 2α + i γ5 sin 2α]ψ (g)
f,g
Xh (f )
(g)
(f )
(g)
ψ ψ (f ) ψ ψ (g) cos2 2α − ψ γ5 ψ (f ) ψ γ5 ψ (g) sin2 2α
=
f,g
+ 2i cos 2α sin 2α ψ
(f )
i
(g)
γ5 ψ (f ) ψ ψ (g) .
In combination with the transformed second part,
hX (f )
i2
X (f )′
(g)′
ψ γ5 ψ (f ) →
ψ γ5 ψ (f )′ ψ γ5 ψ (g)′
f
(2.22)
f,g
=
X
ψ
(f )
(g)
[γ5 cos 2α + i sin 2α]ψ (f ) ψ [γ5 cos 2α + i sin 2α]ψ (g)
f,g
Xh (f )
(g)
(f )
(g)
=
ψ γ5 ψ (f ) ψ γ5 ψ (g) cos2 2α − ψ ψ (f ) ψ ψ (g) sin2 2α
f,g
+ 2i cos 2α sin 2α ψ
(f )
γ5 ψ
(f )
(g)
ψ ψ
(g)
i
,
(2.23)
the interaction is evidently invariant for g = g5 . If we consider the case of only one
flavor and g = g5 , it is easy to show the equivalence of the Nambu–Jona–Lasinio
model and the Gross–Neveu model. For the term (2.21) we obtain
i2
i2 g h
gh
− ψ(x)ψ(x) + ψ(x)γ5 ψ(x)
2
2
i2 g h
i2
gh
= − ψ 1 (x)ψ1 (x) + ψ 2 (x)ψ2 (x) + ψ 1 (x)ψ1 (x) − ψ 2 (x)ψ2 (x)
2
2
g
g
= − 2 ψ1 (x)ψ1 (x)ψ 2 (x)ψ2 (x) − 2 ψ1 (x)ψ1 (x)ψ 2 (x)ψ2 (x)
2
2
= −2g ψ 1 (x)ψ1 (x)ψ 2 (x)ψ2 (x)
h
i2
= −g ψ(x)ψ(x) .
(2.24)
Because we are dealing with fermions, we applied the rules for Grassmann numbers. In the third line in particular their nilpotency. This rules will be explained
later in Sec. 3.3.
A third model with a four–Fermi interaction and chiral symmetry, known as
the Thirring model [8], also exists. Again we have the same kinetic term, but the
10
Chapter 2. Gross–Neveu–type models in the continuum
interaction term is
ih
i
g Xh
−
ψ(x)γµ ψ(x) ψ(x)γµ ψ(x) .
2 µ=1
2
(2.25)
For the case of only one flavor it can be shown that the Thirring model and the
GN–model are equal,
ih
i
gh
− ψ(x)γµ ψ(x) ψ(x)γµ ψ(x)
2
i2 g h
i2
gh
= − ψ 1 (x)ψ2 (x) + ψ 2 (x)ψ1 (x) −
− i ψ1 (x)ψ2 (x) + i ψ 2 (x)ψ1 (x)
2
2
i
gh
= − 2 ψ1 (x)ψ2 (x)ψ 2 (x)ψ1 (x) + 2 ψ1 (x)ψ2 (x)ψ 2 (x)ψ1 (x)
2
= 2g ψ 1 (x)ψ1 (x)ψ 2 (x)ψ2 (x)
i2
h
(2.26)
= g ψ(x)ψ(x) .
Here we used again the rules for Grassmann numbers. The interaction term is
invariant under chiral rotation because
h
i2
i2
h
ψ(x)γµ ψ(x)
→ ψ(x)eiαγ5 γµ eiαγ5 ψ(x)
i2
h
iαγ5 −iαγ5
γµ ψ(x)
= ψ(x)e e
h
i2
= ψ(x)γµ ψ(x) .
(2.27)
2.3
Euclidean correlators in the continuum
Since we are dealing with a quantum field theory, we want to explain in this
section how to extract the energy spectrum of such a theory by using Euclidean
correlators. The theoretical background we need for that purpose can, for example, be found in [9, 10].
Let us start with the definition of a correlator. It is defined as
h
i
1
†
−(T −t)H b
−tH b
Tr e
O2 e
O1 ,
(2.28)
h O2 (t) O1(0) iT =
ZT
where the normalization factor ZT is the partition function given by
ZT = Tr e−T H .
(2.29)
2.3. Euclidean correlators in the continuum
11
b1 and O
b2 can be operators that create or annihilate states, operators measuring
O
observables or combinations of all of these. The self–adjoint operator H is the
Hamiltonian of the system which measures the energy in the system and also
governs its time evolution. The arguments t and T are real, non–negative numbers
denoting Euclidean time distances of propagation. In particular t is the actual
distance between the operator insertations, and T is a number which is taken to
infinity to project to the vacuum.
We now study the partition function ZT in more detail. The trace in Eq. (2.29)
can be understood as a sandwich of the exponential, with eigenstates |ni of the
Hamiltonian. These eigenstates obey the following eigenvalue equation:
H|ni = En |ni .
(2.30)
Since En ∈ R we can order the eigenvalues in the way that
E0 6 E1 6 E2 6 . . .
(2.31)
What we find for ZT then is1
ZT =
∞
X
X
hn|e−T H |ni =
e−T En .
n=0
(2.32)
n
In the second step we used Eq. (2.30), and the fact that the eigenstates |ni form
an orthonormal basis, more precisely we used hn|ni = 1.
The Euclidean correlator can be evaluated in the same way. We write the
trace as sum over m, insert the identity in the form
1=
X
n
|nihn|
(2.33)
between the two operators in Eq. (2.28) and obtain
1 X
b2 |ni hn|e−tH O
b1 |mi
hm|e−(T −t)H O
ZT m,n
1 X −(T −t)Em
b2 |ni e−tEn hn|O
b1 |mi .
e
hm|O
=
ZT m,n
h O2 (t) O1† (0) iT =
(2.34)
1
The exponential of the Hamilton operator is defined by its power series expansion e−T H =
P∞
k
k=0 (−T H) /k!.
12
Chapter 2. Gross–Neveu–type models in the continuum
Now one can use (2.32) and pull out a factor of e−T E0 in both the numerator and
the denominator, which then cancel out. This results in
P
−tEn −(T −t)Em
b
b
e
m,n hm|O2 |ni hn|O1 |mi e
†
h O2(t) O1 (0) iT =
.
(2.35)
1 + e−T E1 + e−T E2 + . . .
In the last formula the energies En are now measured relative to the energy E0
of the vacuum, i.e., we have replaced
En − E0 → En .
(2.36)
When we analyze Eq. (2.35) in the limit T → ∞, the denominator is equal to 1.
In that limit the numerator is reduced to terms where Em = 0, in other words
terms where |mi = 0. What we finally achieve is
X
b2 |ni hn|O
b1|0i e−tEn .
h0|O
lim h O2 (t) O1† (0) iT =
(2.37)
T →∞
n
What can be done with this expression now? As mentioned before, the energy
spectrum of a particle can now be obtained.
b1 and O
b2 the operators O
bp† and O
bp , which
Therefore we have to insert for O
create a particle p from the vacuum and annihilate it again later in time. It
means we have to choose operators that create or annihilate a state with the
quantum numbers of the particle p that we want to observe. Then the matrix
elements in (2.37) are only non–vanishing if the states hn| have an overlap with
b † |0i and vice versa. In the end for hn| only the ground state hp| and excited
O
p
b † |0i, survive. We thus
states hp′ |, hp′′ |, . . ., which also have an overlap with O
p
find
b † 2 −tEp ′ b † 2 −tEp′
†
+ ... .
(2.38)
+ hp |Op′ |0i e
lim hOp (t) Op (0) iT = hp|Op|0i e
T →∞
For sufficiently large t the sub–leading terms are strongly suppressed since Ep′ >
Ep and so we are able to observe Ep (and thus by projection to zero momentum
also mp ) from the exponential decay of the correlator.
Later on in Chap. 8 we use that technique to extract masses of particles for
the ground state, as well as for excited states via measuring correlation functions
and using them in the variational method.
Chapter 3
The Gross–Neveu model on the
lattice
In this chapter we want to formulate the Lagrangian and the partition function,
presented in Chap. 2, using the language of lattice field theory. Lattice field theory
provides a mathematically sound definition of the path integral, and allows to go
beyond perturbative expansion. Such a framework is obtained by introducing a
space–time lattice and discretizing the fields and the action. The lattice cuts off
the high frequencies and makes the theory completely finite (on a finite lattice).
We are only giving a very compact introduction of the basic concepts. For a
more detailed description the reader is referred to [11, 12, 13, 14, 15, 16].
3.1
Basic definitions for lattice calculations
The main idea is to replace the (Euclidean) R2 by a discretized space–time lattice.
More precisely, we replace the vector x ∈ R2 by an integer valued vector na,
n ∈ Λ, with the lattice
Λ = {n = (n1 , n2 ) | n1 = 0, 1, . . . , L1 − 1 ; n2 = 0, 1, . . . , L2 − 1} .
(3.1)
The a denotes the lattice spacing, which has the dimension of length. The fermion
fields ψ(na), ψ(na) (the notation here is for one flavor and we drop the flavor
13
14
Chapter 3. The Gross–Neveu model on the lattice
index) and the auxiliary field φ(na), now live on the lattice sites. For notational
convenience we suppress the factor a in the arguments of ψ, ψ and φ from now
on. The total number of lattice points is the volume V = L1 L2 .
One has to decide what to do with the boundaries. Either periodic boundary
conditions (p.b.c.) or antiperiodic boundary conditions (a.p.b.c.) can be used.
This means that we set for an arbitrary function f , defined on the lattice,
f (n + µ̂Lµ ) = e2πiϑµ f (n) ,
(3.2)
where ϑµ = 0 for p.b.c. and ϑµ = 1/2 for a.p.b.c. µ̂ denotes the unit vector in µ–
direction. In our calculations we use p.b.c. for space (1̂–direction), and a.p.b.c. for
time.
It is also necessary to discretize the derivatives that occur in Eq. (2.2). To
get the correct continuum limit up to an order of O(a2 ) we replace
∂µ ψ(x) →
ψ(n + µ̂) − ψ(n − µ̂)
.
2a
(3.3)
The integrals in the continuum are transformed also, namely into sums over all
lattice points
Z
X
d2 x . . . → a2
... .
(3.4)
n∈Λ
The integration measure for the path integrals is now rigorously defined as a
product over all lattice points,
Y
D[ψ, ψ, φ] =
dψ(n)dψ(n)dφ(n) .
(3.5)
n
3.2
Wilson form of the lattice action
Naive discretization of the action
After applying the rules from Sec. 3.1 the action can be split into a fermionic and
a scalar part,
X
SF [ψ, ψ, φ] =
ψ(n)M(n, m)ψ(m) ,
(3.6)
n,m∈Λ
SS [φ] =
1X 2
φ (n) ,
2 n∈Λ
(3.7)
3.2. Wilson form of the lattice action
15
where M is the Dirac matrix given by
M(n, m)αβ = [m +
√
g φ(n)] δn,m δα,β −
2
X
(γµ )αβ
µ=1
δn+µ̂,m − δn−µ̂,m
.
2a
(3.8)
To keep the notation simple, we subsequently suppress the Dirac indices α, β and
use matrix/vector notation.
The doubling problem
Let us now Fourier–transform the Dirac matrix M(n, m) for the free case g = 0
(cf. App. B). When transforming the second index m with the complex conjugate
phase we find
FT M(n, m) = M̃ (p, q) = a4 δ(p − q)M̂ (p) ,
(3.9)
2
M̂ (p) = m +
iX
γµ sin pµ a .
a µ=1
(3.10)
Because M̃(p, q) is diagonal in momentum space, it is easy to invert this expression to derive the inverse Dirac operator, the so–called fermion propagator. First
we have to invert M̂ (p), then use inverse Fourier transformation back to position
space. The result is
1 X ip(n−m)a −1
e
M̂ (p) ,
V
p∈Λ̃
P
−1
m
−
ia
µ γµ sin pµ a
P
.
M̂ −1 (p) =
m2 + a−2 µ sin2 pµ a
M −1 (n, m) =
(3.11)
(3.12)
To get Eq. (3.12) we used the following identity:
1r + i
2
X
µ=1
γ µ sµ
!−1
=
1r − i
r2 +
P2
µ=1
γ µ sµ
µ=1
s2µ
P2
.
(3.13)
Let us now study the momentum space propagator for the massless case m =
0. The propagator has the correct continuum limit,
P
P
−1
−i
−ia
γ
sin
p
a
µ
µ
a→0
µ γ µ pµ
Pµ 2
.
(3.14)
−→
=
M̂ −1 (p)
−2
p·p
m=0
a
µ sin pµ a
16
Chapter 3. The Gross–Neveu model on the lattice
The right–hand side of Eq. (3.14) has a pole at
p = (0, 0) .
(3.15)
We encounter, however, a different situation on the lattice as long as a > 0. The
massless propagator (center term in (3.14)) has additional poles at
π
π π
π
p=
,
.
(3.16)
, 0 , 0,
,
a
a
a a
These three unphysical poles give rise to unwanted degrees of freedom, the so–
called doublers. We now present a way to eliminate them.
The Wilson term
One way to evade the doubling problem was suggested by K. G. Wilson [11].
He proposed to add an extra term to the momentum space propagator (3.10),
attaining
2
2
1X
iX
γµ sin pµ a +
(1 − cos pµ a) .
(3.17)
M̂ (p) = m +
a µ=1
a µ=1
|
{z
}
Wilson term
This extra Wilson term fulfills all conditions that we claim. For components with
pµ = 0 it vanishes, for components with pµ = π/a it contributes a term 2/a. This
term can be understood as an extra mass term, which diverges in the limit a → 0.
The doublers then become very heavy and decouple from the theory since they
can no longer be excited. Applying inverse Fourier transformation on the last
term in (3.17) one obtains for the Wilson term in real space
2
a X δn+µ̂,m − 2δn,m + δn−µ̂,m a→0
a
−
−→ − ∂µ ∂µ ,
2
2 µ=1
a
2
(3.18)
and we see that the Wilson term also disappears in the continuum limit. We find
for the final form of the kernel of the fermion action
±2
X
√
Γµ δn+µ̂,m .
(3.19)
M(n, m) = [2 + m + g φ(n)] δn,m −
µ=±1
The γµ –matrices were replaced by Γµ , defined as
1
Γ±µ = (1 ∓ γµ ) ,
(3.20)
2
and we obtain an extra contribution of 2 for the on–site term. We remark that
this extra term breaks the chiral symmetry (2.18) explicitly, even when m ≡ 0.
3.3. Key formulas of a Grassmann algebra
3.3
17
Key formulas of a Grassmann algebra
Since we are dealing with fermions, we have to take into account the Pauli principle. This is done by requiring anti-commutation rules for the fermion fields,
turning them into so–called Grassmann variables. The formulas presented below
can all be found in [15, 16].
A Grassmann algebra is constructed from a set of generators ηi which obey
anticommuting products,
{ηi , ηj } = ηi ηj + ηj ηi = 0
∀ i, j .
(3.21)
We want to remark that if the number n of generators is finite, they form a 2n –
dimensional vector space over R or C. An obvious consequence from Eq. (3.21) is
the nilpotency of Grassmann numbers, i.e., ηi2 = 0. Therefore the power series for
an arbitrary function P (η1 , . . . , ηn ), only invokes a finite number of terms. Thus
we only need to look at polynomials,
P = p0 +
X
pi ηi +
i
X
pij ηi ηj +
i<j
X
pijk ηi ηj ηk + . . . + p12...n η1 η2 . . . ηn . (3.22)
i<j<k
The coefficients p0 , pi , . . . , p12...n are ordinary complex numbers. The polynomials
(3.22) form an algebra, the so–called Grassmann algebra.
We often need to differentiate elements of the Grassmann algebra with respect to the generators. To get consistent results the derivatives also have to
anticommute with the generators,
∂
∂
ηj = − ηj
∂ηi
∂ηi
∀ i 6= j .
(3.23)
In addition to the above relation the following rules hold:
∂
1 = 0,
∂ηi
∂
ηi = 1,
∂ηi
∂ ∂
∂ ∂
=−
∂ηi ∂ηj
∂ηj ∂ηi
∀ i, j .
(3.24)
For the path integral we also need to integrate over Grassmann numbers, i.e.,
solve integrals of the form
Z
dn ηP (η1 , . . . , ηn ) .
(3.25)
18
Chapter 3. The Gross–Neveu model on the lattice
Due to the nilpotency these integrals are already specified by the following expressions,
Z
dηi 1 = 0 ,
(3.26)
Z
dηi ηi = 1 ,
(3.27)
dηi dηj = −dηj dηi .
(3.28)
Last but not least we have to discuss the transformation behavior of the
(ordered) product measure dn η = dηn dηn−1 . . . dη1 . Let us apply a linear transformation with a complex n × n matrix L,
n
X
′
ηi =
Lij ηj .
(3.29)
j=1
For the integral we find
Z
Z
n
d η η1 . . . ηn =
dn η ′ η1′ . . . ηn′
Z
X
L1j1 . . . Lnjn ηj1 . . . ηjn
=
dn η ′
=
Z
j1 ,...,jn
dn η ′
= det(L)
X
εj1 j2 ...jn L1j1 . . . Lnjn η1 . . . ηn
j1 ,...,jn
Z
dn η ′ η1 . . . ηn .
(3.30)
The product in the second line vanishes when two of the indices are the same.
We reorder the Grassmann numbers and obtain the completely anti–symmetric
tensor εj1j2 ...jn . This tensor, together with the summation over the matrix elements, gives rise to the determinant det(L). To summarize, we have the following
transformation property of the integration measure,
1
dn η ′ =
dn η .
(3.31)
det(L)
3.4
The partition function on the lattice
On the lattice we formally have the same partition function as in Sec. 2.1, using
the splitted action from Eqs. (3.6) and (3.7),
Z
ZGN = D[ψ, ψ, φ] e−SF [ψ,ψ,φ]−SS [φ] .
(3.32)
3.4. The partition function on the lattice
19
However, now the action is discretized and the measure of the path integral is
rigorously defined as a product. We can now write the partition function in two
different ways. One possibility is to integrate out φ(n) in (3.32) to achieve an
effective action on the lattice
i2
X
g Xh
Seff [ψ, ψ] =
ψ(n)M[φ = 0](n, m)ψ(m) −
ψ(n)ψ(n) .
(3.33)
2
n,m∈Λ
n∈Λ
Here ψ, ψ are vectors also in flavor space. M[φ = 0] is the Dirac matrix in flavor
space with the nontrivial flavor dependence coming from the masses m(f ) , i.e.,
we use the mass matrix m (compare Eq. (2.3)). M[φ = 0] indicates that the
expression (3.19) is used for φ = 0. The corresponding partition function is
Z
(3.34)
ZGN = D[ψ, ψ] e−Seff [ψ,ψ] .
The alternative form for ZGN is obtained when we integrate out the fermion
fields in (3.32), this means we evaluate only the fermionic part of (3.32),
X
Z
(F)
ψ(n)M(n, m)ψ(m) .
(3.35)
ZGN = D[ψ, ψ] exp −
n,m∈Λ
When we understand one of the sums as a linear transformation, more precisely
X
ψ ′ (n) =
M(n, m)ψ(m) ,
(3.36)
m
we get for the fermionic partition function
Z Y
P
′
(F)
′
ZGN = det(M)
dψ(n)dψ (n) e− n ψ(n)ψ (n)
= det(M)
= det(M)
n
Z Y
n
Z Y
n
= det(M) ,
′
dψ(n) dψ ′ (n) e−ψ(n)ψ (n)
dψ(n) dψ ′ (n) 1 + ψ ′ (n)ψ(n)
(3.37)
where we used in the third line the nilpotency and Eq. (3.21). The final version
of ZGN is
Z Y
dφ(n) −φ2 (n)/2
√
ZGN =
e
det M[φ] ,
(3.38)
2π
n∈Λ
20
Chapter 3. The Gross–Neveu model on the lattice
√
in which we have normalized the measure by a factor of 1/ 2π. Such a factor will
cancel when calculating expectation values of observables. Here it is introduced
for reasons that will become clear in Sec. 4.3. The form (3.38) of the partition
function is now particularly suitable for a Monte–Carlo simulation (cf. Chap. 4).
3.5
A loop representation for the Gross–Neveu
model
The evaluation of the fermion determinant in lattice field theories is a rather
extensive project. It can be shown [17] that the number of cancellations in the
calculation of the partition function grows exponentially with the lattice volume
and the inverse temperature. One way to handle this so–called fermion sign–
problem is to write the fermion determinant as the exponential of a sum over
closed loops. This approach has been explored in several papers, see, e.g., [2, 18,
19, 20, 21].
A main part of this work consists of a comparison between the standard
approach to the GN–model and the loop formalism for it [3]. Thus we briefly
state the partition function in terms of “loop variables”. It reads
ZL =
X
L1 ,...,L2Nf
1
√
2
c(L1 )+...+c(L2N
f
)
Y
n∈Λ
G[2Nf − On (L1 , . . . , L2Nf )] .
(3.39)
The sum runs over configurations of 2Nf sets of self avoiding loops. The function
G is defined as
Z
√
dφ(n) −φ2 (n)/2
√
e
[2 + m + g φ(n)]x , x = 1, . . . , 2Nf .
(3.40)
G[x] =
2π
On (L1 , . . . , L2Nf ) denotes the number of loops occupying the lattice point n.
Finally, c(Li ) is the number of corners in the configuration of the i–th set of
loops. Altogether the partition function is a sum with a weight factor according
to the total number of corners multiplied by a simple occupation number function.
The integrals (3.40) can be solved in closed form (see e.g. [2]).
One can compute some simple observables, like the chiral condensate or the
chiral susceptibility, by differentiating ln Z with respect to m. n–point functions
of fermions correspond to correlators of occupation numbers in the loop model.
3.5. A loop representation for the Gross–Neveu model
21
A more detailed account of the equivalence between the standard and the loop
representation is given in Appendix A.
Chapter 4
Numerical simulation with
Monte–Carlo methods
Monte Carlo methods are a class of algorithms for studying mathematical or
physical systems characterized by a probability distribution. They differ from
other simulation methods by being stochastic, in other words they use random
numbers. Thus, name is derived from the famous casino in Monte Carlo the .
These methods are very suitable for high dimensional integrals as they appear in
lattice field theory.
4.1
Fundamental ideas of a Monte Carlo simulation
To motivate the use of Monte Carlo methods let us start with an example from
statistical mechanics, the 2–d Ising model. Consider a spin system on a small
lattice, e.g., 100 × 100 lattice points. Each spin can only be up or down. When
we want to compute the partition function with brute force we have to sum over
all 2100·100 ≈ 103000 possible configurations. Comparing this number to the age of
the universe, which is approximately 4 · 1026 nanoseconds, we conclude that no
PC can ever complete such a calculation.
23
24
Chapter 4. Numerical simulation with Monte–Carlo methods
How can we overcome this problem? The main idea is to replace the sum over
all configurations by a small but representative set. These sample configurations
can be chosen in two different ways:
Simple sampling: When walking through configuration space and picking configurations randomly the procedure is called simple sampling. The disadvantage here is that most of the configurations have a very small weight,
i.e., they are heavily suppressed by the exponential factor, and therefore
they could normally be neglected.
Importance sampling: A better way is to choose the configurations biased.
More precisely, we “include” the exponential factor in the choice of the configurations, that means we prefer those with a small action. This approach
is called importance sampling.
When using importance sampling, the estimate for expectation values is
N
1 X
hOiN =
O(n) .
N n=1
(4.1)
Here O(n) is the value of some operator O for the n–th configuration and N the
total number of configurations. The expectation value hOi is approached as
σO
(4.2)
hOi = hOiN ± √
N
where σO is the standard deviation of the observable from its mean value hOiN .
The obvious question is how to create configurations with a distribution proportional to exp(−S). One starts with an initial configuration and then stochastically creates a series of configurations in configuration space. This series of
configurations is called a Markov chain. The transition from a configuration to a
new one is a so–called Monte Carlo step or an update.
We denote the configurations in the Markov chain by Cn , n = 1, 2, . . . A
Markov chain is characterized by a transition probability,
W (Cn+1 = C ′ | Cn = C) = W (C → C ′ ) .
(4.3)
This probability does not depend on n, but rather on the configurations C and
C ′ . To be a probability, it has to obey
X
W (C → C ′ ) > 0 ,
W (C → C ′ ) = 1 .
(4.4)
C′
4.2. The algorithm for our system
25
When our Markov process is in equilibrium, there are no possibilities for sinks or
sources of probability. This is expressed in the balance equation
X
X
!
P (C) · W (C → C ′ ) =
P (C ′ ) · W (C ′ → C) .
(4.5)
C
C
On the left–hand side we sum over all steps that lead from some configuration
C to a configuration C ′ , including the weight factor P (C). The right–hand side
describes the situation when hopping out of C ′ . The two probabilities have to
equal each other. A sufficient solution of (4.5) is obtained by dropping the sums
over C:
P (C) · W (C → C ′ ) = P (C ′ ) · W (C ′ → C) .
(4.6)
This expression is known as the detailed balance condition. A well known algorithm which solves the detailed balance condition is the Metropolis algorithm
[22]. It consists of three steps:
Step 1: Create a trial configuration C̃ out of C randomly.
Step 2: Accept C̃ as the new configuration with the probability
!
P (C̃)
p = min 1,
.
P (C)
(4.7)
Step 3: Go back to step 1.
In our case importance sampling is combined with reweighting. We discuss this
strategy in the next section.
4.2
The algorithm for our system
As soon as the fermions are integrated out, vacuum expectation values are high
dimensional integrals over the scalar fields φ(n),
R Q dφ(n) −φ2 (n)/2
√
e
det(M[φ]) O[M[φ], φ]
n
2π
.
(4.8)
hOiφ =
R Q dφ(n)
−φ2 (n)/2 det(M[φ])
√
e
n
2π
In our approach we generate configurations according to the weight factor
1
2
P ∝ √ e−φ /2 ,
2π
(4.9)
26
Chapter 4. Numerical simulation with Monte–Carlo methods
and approximate the expectation values by
(MCS)
hOiφ
=
PN
i=1
det(M[φ])i O[M[φ], φ]i
.
PN
i=1 det(M[φ])i
(4.10)
In other words we re–weight the Gaussian distributed scalar fields with the determinant. We remark that this strategy is successful only in 2 dimensions [23]
or for small lattice volumes, respectively. In 4 dimensions the effect of the determinant on the distribution is too strong due to the bigger lattice volume. In
order to study the distribution of the weights in the reweighting step we analyze
the effective action defined as
det(M[φ]) ≡ e−Seff [φ] ,
Seff [φ] = − ln det(M[φ]) .
(4.11)
(4.12)
The corresponding numerical results will be presented in Sec. 7.1.
4.3
Random numbers and Gaussian distributions
In this section we discuss how to generate the fields φ distributed according to
(4.9). We use the Box–Muller method for generating random deviates with a
normal distribution [24].
Assume that we have uniformly distributed numbers x with a normalized
probability p(x), and a function y(x). From the transformation law of probabilities we get
!
|p(y)dy| = |p(x)dx|
dx =⇒
p(y) = p(x) .
dy
(4.13)
(4.14)
If the transformation is generalized to more than one dimension, one has to calculate the Jacobian |∂(x1 , x2 , . . .)/∂(y1 , y2 , . . .)| instead of the derivative |dx/dy|.
Now we choose a Gaussian for p(y),
1
2
p(y)dy = √ e−y /2 dy ,
2π
(4.15)
4.4. Error analysis with the Jackknife method
27
and consider a transformation between two uniform deviates on (0, 1), x1 , x2 , and
two values y1 , y2 ,
p
−2 ln x1 cos 2πx2 ,
p
=
−2 ln x1 sin 2πx2 .
y1 =
(4.16)
y2
(4.17)
We also can write for x1 , x2
2
2
x1 = e−(y1 +y2 )/2 ,
y2
1
arctan
.
x2 =
2π
y1
(4.18)
(4.19)
The Jacobian is then
∂(x1 , x2 )
1 −y12 /2
1 −y22 /2
=− √ e
× √ e
.
∂(y1 , y2 )
2π
2π
(4.20)
This term is a product of two functions of either y1 only or y2 only. We see that
both functions are Gaussians.
A useful trick for programming is to pick n1 and n2 as the ordinate and
abscissa of a random point inside the unit circle, instead of x1 and x2 in (0, 1).
The sum of the squares, r 2 = n21 + n22 , can be used for x1 (because it is also
a uniform deviate). The angle between (n1 , n2 ) and the n1 –axis replaces 2πx2 .
√
√
Why is it a “trick”? Because now the sine and cosine are n2 / r 2 and n1 / r 2 –
and don’t have to be calculated explicitly! For the uniformly distributed numbers
we used version 2.1 of Lüscher’s random number generator “ranlux” [25, 26].
A test of our implementation of the Gaussian distribution is shown in Fig. 4.1,
where we compare a histogram of our numerical data to an exact Gaussian.
4.4
Error analysis with the Jackknife method
Having created data sets for an observable concerning N measurements, the average values of our observables will have statistical errors. We estimate these errors
with the Jackknife method [27].
Let us denote the set of N different configurations by Φ,
Φ = φ(k) k = 1, . . . , N ,
(4.21)
28
Chapter 4. Numerical simulation with Monte–Carlo methods
exact
computed points
1
2
exp(-r /2)
0.8
0.6
0.4
0.2
0
-4
-2
0
r
2
4
Figure 4.1: A Gaussian and the distribution of 107 computed points
where φ(k) contains the scalar fields for all lattice points. We also define subsets
Φ(j) of N − 1 configurations, so–called Jackknife blocks,
Φ(j) = φ(k) k = 1, . . . , N; k 6= j .
(4.22)
The first step is to compute the expectation value of some observable O for the
whole set of measurements,
O = O[Φ] .
(4.23)
To determine the variance we need the observables also on the N different Jackknife blocks,
O (j) = O Φ(j) .
(4.24)
For the variance we get
σO2
= (N − 1)
N
X
j=1
O (j) − O
and the final result including the statistical error is
σO
O=O± √ .
N
2
,
(4.25)
(4.26)
Chapter 5
Observables
In this chapter we derive and discuss the observables that will be studied later on.
These observables are bulk observables such as condensates and susceptibilities,
as well as n–point functions that have been introduced already in Sec. 2.3.
But first of all we begin with deriving Wick’s theorem [28], which will be used
later.
5.1
Wick’s theorem
The formula for the evaluation of observables in our numerical approach is
Eq. (4.8). To get this form we had to integrate out the fermionic part of the
partition function, namely Eq. (3.35). Our observables are mostly composed of
products of the fermion fields and we want to stress again that these fields are
Grassmann variables. Here we discuss how an observable built from fermions is
transformed when integrating out the Grassmann variables.
We define a functional F of 4N Grassmann numbers ηi , ηi , θi , θi , i =
1, 2, . . . , N. In this so called generating functional for fermions the ηi , ηi will
be integrated out and the θi , θi serve as source terms. This generating functional
29
30
Chapter 5. Observables
is defined as
F [θ, θ] =
Z Y
N
dηk dηk exp
N
X
η i Mij ηj +
θ i ηi +
i=1
i,j=1
k=1
N
X
N
X
ηi θi
i=1
!
.
(5.1)
The next steps are rewriting the argument of the exponent in the following way
(we use summation convention):
−1
−1
η i + θj Mji−1 Mik ηk + Mkl
θm ,
θl − θn Mnm
(5.2)
and then to make a transformation of variables,
−1
ηk′ = ηk + Mkl
θl ,
η ′i = η i + θj Mji−1 .
The integration measure stays the same and we get
!Z N
N
X
Y
−1
dηk dη k exp
θn Mnm
θm
F [θ, θ] = exp −
n,m=1
k=1
= det(M) exp −
N
X
−1
θn Mnm
θm
n,m=1
!
(5.3)
N
X
η ′i Mij ηj′
i,j=1
.
!
(5.4)
In the last step (3.37) was used again.
Now we are able to compute the above mentioned n–point functions of fermion
fields. With Eq. (5.4) we come to the Wick theorem (h. . .iF denotes only the
fermionic part of the path integral):
ηi1 ηj1 . . . ηin η jn F =
1
det(M)
= (−1)n
Z Y
N
k=1
X
dηk dη k exp
N
X
l,m=1
!
η l Mlm ηm ηi1 η j1 . . . ηin η jn
. . . Mi−1
.
Mi−1
sgn(π)Mi−1
n jπn
2 jπ2
1 jπ1
(5.5)
π(1,2,...,n)
Here the sum runs over all permutations π(1, 2, . . . , n) of the numbers 1, 2, . . . , n
and the sign of the permutation is sgn(π). To prove this result we apply several
derivatives to F [θ, θ],
∂
∂
∂
∂
1
ηi1 ηj1 . . . ηin η jn = (F)
F [θ, θ] ···
.
(5.6)
∂θ
∂θ
j1 ∂θ i1
jn ∂θ in
ZGN
θ,θ=0
5.2. Derivatives of the partition function
31
We are now prepared to discuss several observables and calculate their vacuum
expectation values with
hOiφ =
PN
i=1
det(M[φ])i O[M[φ], φ]i
,
PN
i=1 det(M[φ])i
(5.7)
where the index i denotes the field configuration number i. It is stressed again,
that these configurations are distributed according to
1X 2
φ (n)
P ∝ exp −
2 n∈Λ
5.2
!
.
(5.8)
Derivatives of the partition function
As already stated in Sec. 3.5, we want to compare results for different observables,
calculated in the standard formalism, with the results from the loop approach.
Therefore we define three more or less simple observables for a first test. We
compute derivatives of the partition function Z to obtain so–called bulk observables. The first object we want to look at is the chiral condensate χ, defined as
(repeated Dirac indices are summed)
1 ∂
1 X
ψ(n)ψ(n)
ln Z =
V ∂m
V n
D
E
X
1
M −1 (n, n)αα .
= −
V
φ
n
χ ≡ −
(5.9)
(5.10)
One gets to the second line using Wick’s theorem from Sec. 5.1.
The next observable to be looked at is the chiral susceptibility, obtained by
another derivative with respect to m,
1 ∂2
1 ∂ X
ψ(n)ψ(n)
ln
Z
=
2
V ∂m
V ∂m n
D
E2
2 E
1 DX −1
1 X −1
M (n, n)αα −
M (n, n)αα
=
V n
V
φ
φ
n
D
E
1 X −1
+
M (n, m)αβ M −1 (m, n)βα .
V n,m
φ
Cχ ≡ −
(5.11)
(5.12)
32
Chapter 5. Observables
The last observable we consider is the interaction density ρ, obtained by a
derivative of ln Z with respect to g. We find
2 2 ∂
1 X
ψ(n)ψ(n)
(5.13)
ρ ≡
ln Z =
V ∂g
V n
E
1 DX −1
=
M (n, n)αα M −1 (n, n)ββ
V n
φ
D
E
1 X −1
−1
M (n, n)αβ M (n, n)βα .
(5.14)
−
V n
φ
The expectation values h. . .iφ are computed with Eq. (5.7).
5.3
n–point correlation functions
An important class of observables are n–point correlators. In particular one
can use 2–point correlators of fermionic bilinears for computing masses. In two
dimensions we can construct four different bilinears, such that one has a total
amount of 16 different 2–point functions
L2 −1
L1 −1 h
†
i
1 X
1 X
†
Cij (t) ≡
qi (x, t + τ )qj (y, τ ) − qi (x, t + τ ) qj (y, τ ) , (5.15)
L2 τ =0 L21 x,y=0
where
qi (x, t) ≡ ψ α (x, t) (Ωi )αβ ψβ (x, t) .
(5.16)
The Ωi can be the unit–matrix or one of the γµ –matrices,
Ωi = {1, γ1 , γ2, γ5 } .
(5.17)
The sum over τ is a sliding average to improve the statistics. When we evaluate the expectation values in Eq. (5.15) using Wick’s theorem again, we get
expectation values of a connected part Qij and a disconnected part Qi ,
i
1 X h
Qij (x, t + τ, y, τ ) − Qi (x, t + τ ) Qj (y, τ ) , (5.18)
Cij (t) =
L2 L21 τ,x,y
with the two parts
Qij (x, tx , y, ty ) = M −1 (x, tx |x, tx ) Ωi M −1 (y, ty |y, ty ) Ωj
− M −1 (x, tx |y, ty ) Ωi M −1 (y, ty |x, tx ) Ωj ,
Qi (x, t) = − M −1 (x, t|x, t) Ωi .
(5.19)
(5.20)
5.4. Numerical aspects
33
The Dirac indices do not appear explicitly in these expressions, but they are
summed as usual. Thus we have to compute in our MC simulation
N
1 X 1 X
Cij (t) =
det(k) Qij (x, t + τ, y, τ )k
L2 L21 τ,x,y N k=1
N
N
X
1 X
− 2
det(k) Qi (x, t + τ )k ×
det(k) Qj (y, τ )k . (5.21)
N
k=1
k=1
The sums run over all configurations k. The argument/index k means the determinant/observable has to be evaluated for the configuration k. The normalization
factor N is defined as the sum over all determinants, more precisely
N =
5.4
N
X
det(k) .
(5.22)
k=1
Numerical aspects
When we want to evaluate the above discussed observables, we need to invert the
Dirac operator on the one hand, and to compute the determinant on the other.
For this purpose we use Fortran90 routines from the LAPACK package [29]. But
let us go through this procedure step by step.
The first routine we use is called ZGETRF. It provides a LU factorization
of an arbitrary matrix A, that we pass to this routine. This factorization is
needed later for inverting the matrix. The routine uses partial pivoting with row
interchanges. More precisely, it writes a complex matrix A as A = P LU, where
P is a permutation matrix and L, U are lower and upper triangular matrices. L
has only unit diagonal elements. The matrix A is overwritten by the factors L
and U, but the diagonal elements of L are not stored (they are equal to 1). The
Pivot indices are stored in an array called IPIV.
Now we are able to compute the determinant of the given matrix. We simply
need to multiply the determinant of P , which is either +1 or −1, with the diagonal
elements of U, since det(L) = 1. This results from the fact that the determinant
of a (lower or upper) triangular matrix is the product of the diagonal elements.
Thus we only have to determine det(P ). The implementation is explained in
Fig. 5.1.
34
Chapter 5. Observables
!
!
!
!
!
!
!
IPIV(i) gives the exchange of row IPIV(i) with row i
e.g.:
2 2 3
implies: row 1 has been exchanged with 2,
then no more changes (2 with 2, 3 with 3)
for every change a factor of -1
pivfac = 1
do i=1, 2*VOL
if( IPIV(i) .ne.
end do
i ) pivfac = - pivfac
det = cmplx(1.0,0.0,kind=dp) * cmplx(real(pivfac),0.0,kind=dp)
do i=1, 2*VOL
det = det * A(i,i)
end do
Figure 5.1: The Fortran90 code to compute the determinant of the pivoting
matrix P and ultimately det(A). IPIV is the array with the Pivot indices, VOL
is the total number of lattice sites.
The next step is to invert the matrix A with a routine called ZGETRI. Therefore we have to pass the LU factorized matrix A, together with IPIV, to the
subroutine ZGETRI. The inverse of A is computed by forming U −1 and then
solving the equation XP L = U −1 for X. In the end A is overwritten by its
inverse A−1 .
Chapter 6
Tests for the free case
It is important to test the computer code in analytically known special cases.
For the free case (g = 0) we have a powerful tool to do such tests, the Fourier
transformation (FT), see App. B. We use it to test the correct implementation of
M, its inversion and the evaluation of the determinant. The observables presented
in Chap. 5 also have to pass the tests we apply.
6.1
Testing the entries of the Dirac matrix and
its inverse
The Dirac operator in the free case is
M(n, m) = [2 + m]δn,m −
±2
X
Γµ δn+µ̂,m .
(6.1)
µ=±1
For the sake of simplicity, we introduce the following abbreviations,
N(p) = 2 + m −
D(p) = N 2 (p) +
2
X
2
X
µ=1
35
cos apµ ,
(6.2)
sin2 apµ .
(6.3)
µ=1
36
Chapter 6. Tests for the free case
For notational convenience we suppress the factor a, that would occur together
with each momentum, in all expressions from now on. When applying FT, we
get
1 X −ipn
M̃(p, q) =
e
M(n, m)eiqm
V n,m
√
V δ(p − q) M̂(p) ,
(6.4)
=
2
X
M̂(p) = N(p) + i
γµ sin pµ ,
(6.5)
µ=1
Because Eq. (6.4) is diagonal in momentum space, it is simple to invert M. First
we invert M̃ :
1
M̃ −1 (p, q) = √ δ(p − q) M̂ −1 (p) ,
(6.6)
V
"
#
2
X
1
M̂ −1 (p) =
N(p) − i
γµ sin pµ .
(6.7)
D(p)
µ=1
When we apply FT back to real space (cf. (B.9)), we obtain
1 X ip(n−m) −1
M −1 (n, m) =
e
M̂ (p) .
V p
(6.8)
This expression was used to check the implementation of M (using M ·M −1 = 1),
and for direct comparison to the result of M −1 computed with the LAPACK
routines (see Sec. 5.4).
6.2
Tests for the derivative of the partition
function
With Eq. (6.8) we have the means to calculate the three observables χ, Cχ and ρ,
defined in Sec. 5.2, analytically. Combining (6.8) with (5.10), (5.12) and (5.14)
we obtain
2 X N(p)
χ(m) = −
,
(6.9)
V p D(p)
Cχ (m) =
2
i
X
2 X 1 h 2
2
N
(p)
−
sin
p
µ ,
V p D 2 (p)
µ=1
(6.10)
6.3. The correlator
37
2
4 X N(p)
ρ(m) =
V 2 p D(p)
P2
2 X N(p)N(q) − µ=1 sin pµ sin qµ
− 2
.
V p,q
D(p)D(q)
(6.11)
The bulk observables are functions of m and we can compare the analytic results
with those from the general program, running at g = 0. This was done for two
different volumes and the results of this test are documented in Fig. 6.1. It is
obvious that the data points fall on the curve for all values of m and the program
passes the test for the free case.
6.3
The correlator
For the free case it is also possible to calculate the correlator exactly. The general
relations for the connected and disconnected parts of our correlator are given in
Eqs. (5.19) and (5.20). In the free case the scalar fields in the Dirac matrix M
are set to zero. Thus no average over the gauge fields is necessary and so for each
value of m only one run of the program is needed to compute the final result.
Here the first piece of the connected part and the product of the disconnected
parts are canceling each other out. What remains is
i
1 X h −1
(free)
−1
M
(x,
t
+
τ
|y,
τ
)
Ω
M
(y,
τ
|x,
t
+
τ
)
Ω
(6.12)
Cij (t) = −
i
j .
L2 L21 τ,x,y
To evaluate the expression above we use again Eq. (6.8).
As for the bulk observables we tested the 2–point functions against the exact
results in the free case and found agreement up to machine precision, see Fig. 6.2.
In that figure we also plot the corresponding effective masses, which we can
extract from the correlators. Such a fitting procedure will be discussed later in
Sec. 8.2. In the limit L → ∞ the effective mass plateaus appear at a value of 2m.
On the finite lattice we observe corrections of O(1/L) which also depend on the
mass parameter m.
38
Chapter 6. Tests for the free case
-0.65
-0.65
free case
computed, g=0.0
free case
computed, g=0.0
-0.7
χ(m)
χ(m)
-0.7
-0.75
-0.75
-0.8
-0.8
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
-0.3
(a) χ(m), 162
-0.1
0
m
0.1
0.2
0.3
0.2
0.3
(b) χ(m), 322
0.2
0.2
free case
computed, g=0.0
free case
computed, g=0.0
0
Cχ(m)
0
Cχ(m)
-0.2
-0.2
-0.4
-0.2
-0.4
-0.6
-0.6
-0.8
-0.3
-0.2
-0.1
0
m
0.1
(c) Cχ (m), 16
0.2
0.3
-0.3
2
-0.2
-0.1
0
m
0.1
(d) Cχ (m), 32
0.3
ρ(m)
ρ(m)
0.3
2
0.25
0.25
free case
computed, g=0.0
0.2
-0.3
-0.2
-0.1
0
m
(e) ρ(m), 16
0.1
2
0.2
0.3
free case
computed, g=0.0
0.2
-0.3
-0.2
-0.1
0
m
(f) ρ(m), 32
0.1
0.2
0.3
2
Figure 6.1: In these figures a comparison between the exact results from FT
and the general program (running for g = 0) are shown. The results for 162 –
lattices are on the left–hand side, on the right–hand side one finds the results for
322 –lattices. The observables are χ, Cχ and ρ (from top to bottom).
6.3. The correlator
39
-2
10
-3
0.5
-4
0.4
10
10
meff(t)
C11(t)
10
-5
-6
m = 0.05
m = 0.20
-7
0.1
-8
10 0
0.3
0.2
10
10
m = 0.05
m = 0.20
fitted value
fitted value
10
20
30
t
40
50
0
0
60
10
20
30
t
40
50
60
50
60
(b) meff for C11
(a) C11 (t)
-2
10
-3
0.5
-4
0.4
10
10
meff(t)
C44(t)
10
-5
-6
m = 0.05
m = 0.20
-7
-8
10 0
0.3
0.2
10
10
m = 0.05
m = 0.20
fitted value
fitted value
10
20
30
t
(c) C44 (t)
40
50
60
0.1
0
0
10
20
30
t
40
(d) meff for C44
Figure 6.2: On the l.h.s. of this figure the correlators in the free case for the
combination γ1 γ1 and γ5 γ5 are shown. The solid lines are exact results from the
calculations with FT, the crosses mark the results, computed with the general
program for g = 0. On the r.h.s. the corresponding effective masses are plotted.
The solid lines in these plots mark the values for meff extracted from a fit of the
correlators. The ordinates are cut such that the boundary points can not be seen
here. The two different curves on both sides correspond to the two different bare
masses m = 0.05 and m = 0.20. These calculations were done on a 24×64 lattice.
Chapter 7
Analysis of the Dirac matrix M
In this chapter we want to analyze the Dirac operator M itself. First we discuss
the properties of the determinant of M and study the distribution of the effective
action. Then the eigenvalues of the Dirac matrix are calculated and analyzed. In
the last part we discuss properties of the phase diagram extracted from the Dirac
spectrum.
7.1
The distribution of det(M )
As already mentioned in Chap. 4, one interesting property of the GN–model is
the distribution of det(M). But it is not as easy as it seems to observe that
distribution because of the numerical instability for this calculations. When we
use the following representation of M,
M(n, m) = [2 + m +
√
g φ(n)] δn,m −
±2
X
Γµ δn+µ̂,m ,
(7.1)
µ=±1
we are getting into trouble. Why? Because the determinant is easily reaching
orders of magnitude of 10500 , the fluctuations of the spectra are also dramatically
increasing.
41
42
Chapter 7. Analysis of the Dirac matrix M
One easy way out is to rescale M and write it as
M(n, m) = κ Mκ (n, m) ,
(7.2)
κ = 2+m ,
√
±2
X
g
1
Mκ (n, m) = 1 +
Γµ δn+µ̂,m .
φ(n) δn,m −
2+m
2 + m µ=±1
(7.3)
where
(7.4)
Then one finds:
det(M) = κ2V det(Mκ ) ,
(7.5)
and the trivial factor κ2V can be removed from the calculation of the determinant.
When we calculate observables with Eq. (4.8), this factor also cancels out.
Another remarkable property of the determinant is that it is always real.
This is not obvious since the matrix M is complex–valued. When we use the
γ5 –hermiticity of the Dirac operator (see App. C),
γ5 M γ5 = M † ,
(7.6)
this fact can be shown in the following way:
det(M) = det(γ5 ) det(M) det(γ5 ) = det(γ5 M γ5 )
| {z }
| {z }
=1
=1
†
= det M = det (M ∗ )T = det(M)∗ ,
(7.7)
where the asterisk denotes complex conjugation. It is clear now that det(M) ∈
R. In Fig. 7.1 we present the histograms for Seff , not calculated as defined in
Eq. (4.12), but using Mκ instead of M. This amounts to a trivial shift of the
distribution, since Seff can be written in the following way,
Seff = − ln [det(M)] = − ln κ2V det(Mκ )
= − ln [det(Mκ )] − 2V ln [2 + m] .
(7.8)
We then neglect the constant term in the calculation of Seff , because it is only a
shift for the distribution of the effective action. In Fig. 7.1 one can clearly see
that the smaller the coupling g is, the sharper is the peak of the distribution.
7.1. The distribution of det(M)
43
0.14
1
0.12
0.8
0.1
0.6
0.08
0.06
0.4
0.04
0.2
0.02
0
-52
-51.8
-51.6
Seff
-51.4
-51.2
-51
0
-60
-40
(a) g = 0.0
0.14
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
-60
-40
-20
0
Seff
20
40
60
0
-60
-40
(c) g = 0.06
0.14
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
-60
-40
-20
0
Seff
20
(e) g = 0.3
20
40
60
-20
0
Seff
20
40
60
40
60
(d) g = 0.1
0.14
0
0
Seff
(b) g = 0.02
0.14
0
-20
40
60
0
-60
-40
-20
0
Seff
20
(f) g = 0.7
Figure 7.1: In these figures spectra for Seff = − ln[det(Mκ )] are shown. 2000
configurations are calculated for m = 0.0 on a 20 × 40 lattice. On the upper left
figure the free case (g = 0) is depicted. For the other figure we used the same
axis scaling to emphasize the evolution of the distribution.
44
Chapter 7. Analysis of the Dirac matrix M
This implies that our reweighting strategy is most efficient for small values of g,
where the scatter of the effective action around its mean value is small.
We also calculated the expectation values for the effective action for different
couplings g. Therefore we define the first moment, given by
Seff =
PNbin
i=1
(7.9)
PNbin
(7.10)
H(i) Seff (i)
,
PNbin
H(i)
i=1
and the second moment
2
Seff
=
H(i) Seff(i)2
.
PNbin
i=1 H(i)
i=1
Here H(i) is the discrete distribution function for the effective action in bin
number i and the sums run over all the bins of the histogram. Now we can define
the standard deviation σS2 by
2
2
− Seff .
σS2 = Seff
(7.11)
Thus the effective action is distributed around its mean value Seff with a width
of σS . In Fig. 7.2 we plot the dependence of the mean value Seff and the width σ
as a function of g for m = 0.
7.2
Eigenvalue spectrum of M
A characteristic property of the Dirac operator is the spectrum of its eigenvalues
and their distribution.
An important fact for this discussion is that the eigenvalues always come in
complex conjugate pairs. Let λ be an eigenvalue of M and vλ the corresponding
eigenvector, i.e., M vλ = λ vλ . Then the characteristic polynomial P (λ) has to
vanish. More precisely, we find
P (λ) = det(M − λ1) = det(γ5 ) det(M − λ1) det(γ5 )
= det(γ5 Mγ5 − λ1) = det(M † − λ1)
= det( (M − λ∗ 1)∗ ) = det(M − λ∗ 1)∗
= P (λ∗ )∗ = 0 .
(7.12)
7.2. Eigenvalue spectrum of M
45
80
60
< Seff >
40
20
0
-20
-40
-60
0
0.2
0.4
g
0.6
0.8
1
Figure 7.2: The expectation value of Seff for a 20 × 40 lattice for 2000 configurations. The mean value Seff (circles) and the width σS of its distribution (bars)
as a function of g for m = 0.
In the first line we used det(γ5 ) = −1, in the second the γ5 –hermiticity of M
(cf. Eq. (7.6)) and γ52 = 1. Complex conjugation on both sides of the last line of
Eq. (7.12) leads to
P (λ∗ ) = 0 .
(7.13)
That means if λ is an eigenvalue of M and thus a zero of the characteristic polynomial, the complex conjugate eigenvalue λ∗ also is a zero and thus an eigenvalue
of M, as well.
From the γ5 –hermiticity follows another consequence for the γ5 matrix elements of the eigenvectors vλ† γ5 vλ :
1 †
1
vλ γ5 M vλ = vλ† M † γ5 vλ
λ
λ
1
λ∗ †
=
(M vλ )† γ5 vλ =
v γ5 vλ .
λ
λ λ
vλ† γ5 vλ =
(7.14)
46
Chapter 7. Analysis of the Dirac matrix M
When we multiply (7.14) with λ we find
(λ − λ∗ ) · vλ† γ5 vλ = 0
⇐⇒
Im(λ) · vλ† γ5 vλ = 0 .
(7.15)
Consequently for complex eigenvalues with Im(λ) 6= 0 the γ5 –sandwich of the
eigenvectors must vanish. For the other case, Im(λ) = 0, it is possible that
v † γ5 v 6= 0.
In Fig. 7.3 we show a scatter plot of eigenvalues near the origin for 100 configurations of a 20 × 40 lattice for various couplings g. What one can clearly see in
this plots is that the edges of the spectra (dotted red lines) are shifted to negative
values as the coupling g increases. The theory becomes critical when the Dirac
operator has small eigenvalues, since then its inverse, the propagator, diverges.
In order to achieve such small eigenvalues, the edge of the spectrum must be
shifted to the origin by adding a mass term m. The value necessary to shift the
edge for various values of g will be an estimate for the critical mass mcrit . This
critical mass is studied in more detail in the next section.
We also observe that the scatter of the eigenvalues is increasing with increasing
g (at a fixed volume). That again stresses the fact that calculations are sharper
at small couplings g.
7.3
A rough estimate for the critical mass
In this section we want to analyze the distribution of the eigenvalues in a bit more
detail, i.e., we want to figure out where the theory becomes critical. As already
seen before (Fig. 7.3), the edge of the distribution of the eigenvalues is shifted
in negative Re(λ)–direction. This shift is now evaluated to find which value of
m has to be added to the Dirac operator to find criticality. This is done in the
following way: We first calculate all the eigenvalues for 1000 configurations of
the Dirac matrix for different couplings g and for the same m = 0.0. For every
configuration we search for the eigenvalue with the smallest real part and then
define the gap r as the distance of its real part to the origin (see Fig. 7.4). The
next step is to set up a histogram H(r) for the distances r and then appoint
(heuristically!) the distance dpeak of the peak of these histograms to zero as the
negative critical mass,
dpeak ≡ −mcrit (g) .
(7.16)
0.4
0.4
0.2
0.2
Im(λ)
Im(λ)
7.3. A rough estimate for the critical mass
0
0
-0.2
-0.2
-0.4
-0.4
-0.4
-0.2
0
Re(λ)
0.2
0.4
-0.4
0.4
0.4
0.2
0.2
0
-0.2
-0.4
-0.4
0
Re(λ)
(c) g = 0.2
0
Re(λ)
0.2
0.4
0.2
0.4
0
-0.2
-0.2
-0.2
(b) g = 0.05
Im(λ)
Im(λ)
(a) g = 0.0
-0.4
47
0.2
0.4
-0.4
-0.2
0
Re(λ)
(d) g = 0.7
Figure 7.3: Here eigenvalue spectra of the Dirac operator near the origin are
plotted for four different values of the coupling. In the free case g = 0 (Fig. 7.3(a))
we only used one configuration, whereas 100 different configurations were taken
into account where g 6= 0. These spectra have been calculated on a 20 × 40
lattice with m = 0. To have a better overview we inserted the imaginary–axis
(solid black line), as well as a dotted red line, which indicates the edge of the
spectrum in each plot.
One has to add these critical masses to the Dirac operator as the bare mass
parameter m, on the one hand to compensate for this unwanted shift, on the
other hand to be able to do calculations with “massless fermions”.
One example for such a histogram is depicted in Fig. 7.5. There it can be
easily seen that the statistical spread of the eigenvalues around their mean value
48
Chapter 7. Analysis of the Dirac matrix M
Im(λ)
r
Re(λ)
Figure 7.4: In this plot the gray dots represent the eigenvalues of our Dirac operator.
The distance of the eigenvalue with the smallest real part to the origin is defined as r.
is increasing with decreasing lattice volumes and thus calculations are getting
more expensive with smaller lattice volumes. We estimate the statistical spread
in the same way as it was done in Sec. 7.1 for the effective action. We define the
first moment r as
Prmax
H(r) r
min
,
(7.17)
r = Pr=r
rmax
r=rmin H(r)
and the second moment r 2 as
Prmax
2
r=rmin H(r) r
2
P
.
r =
rmax
r=rmin H(r)
(7.18)
Then the deviation σr around the fixed mean value −mcrit (g) is given as
p
(7.19)
σr = r 2 − r 2 .
These calculations are done for g running from 0 to 0.1 in steps of 0.02.
In Fig. 7.6 these mean values r are plotted, together with the corresponding
standard deviations σr . The critical lines in the g–m–plane correspond to those
7.3. A rough estimate for the critical mass
49
L=8
L = 16
L = 32
0.2
H(r)
0.15
0.1
0.05
0
-0.02
0
0.02
r
0.04
0.06
Figure 7.5: Histograms H(r) for the distribution of the smallest eigenvalue.
Three different lattice extensions L are taken into account. The corresponding
lattice volume is V = L2 . The coupling is g = 0.06 and the bare mass parameter
m was set to zero.
values where the model shows massless excitations. It is obvious from this picture
that the spread is getting less with smaller couplings g. This confirms again the
fact that our strategy of doing the calculations with the reweighting procedure,
is more efficient for small couplings.
50
Chapter 7. Analysis of the Dirac matrix M
0.02
L=8
L = 16
L = 32
0.01
mcrit
0
-0.01
-0.02
-0.03
-0.04
0
0.02
0.04
g
0.06
0.08
0.1
Figure 7.6: The critical mass mcrit as a function of the coupling g for various
lattice extensions L. The mean values of the histograms (squares) and the corresponding deviations (bars) are plotted (the symbols are connected by a thin line
to guide the eye). We sketched the endings of the bars as horizontal lines to see
their length, because of the overlap of several bars.
Chapter 8
Correlation functions and the
mass spectrum
In this chapter we investigate the correlators that were introduced in Chap. 5.
As discussed earlier, correlators are a powerful tool for spectroscopy, i.e., for
extracting masses.
After a short overview of the operators used in the correlators, we present
the basic analysis–techniques to extract the masses for ground states. Then
a description of the variational method is given, which allows to compute also
excited state energies. In the last section we discuss the connection between the
mass spectrum and the eigenvalues of the Dirac matrix.
8.1
Short review of correlation functions
To set the ground we shortly review the basic formula that is needed for the
discussion of correlation functions, Eq. (5.15) from Sec. 5.3:
†
i
1 X h
†
q
(x,
t
+
τ
)q
(y,
τ
)
−
q
(x,
t
+
τ
)
q
(y,
τ
)
.
(8.1)
Cij (t) =
i
i
j
j
L2 L21 τ,x,y
In that definition we used
qi (x, t) = ψ α (x, t) (Ωi )αβ ψβ (x, t) .
51
(8.2)
52
Chapter 8. Correlation functions and the mass spectrum
In Fig. 6.2 we only plotted the correlators where either γ1 or γ5 is sandwiched
between the fermion fields. The reason is that only these two correlation functions
show an exponential decay as discussed in the second chapter, Sec. 2.3. The other
operators Ωi = 1 or γ2 give rise to condensates. So from now on we concentrate
on the correlators with the matrices γ1 or γ5 between the fermion fields.
In the next section we present techniques to extract masses from correlation
functions. For the variational method we need a set of operators, which have all
the same quantum numbers but are linearly independent. They should span a
sub–manifold in the space of interpolators. To set up such a basis we generalize
the qi in the following way:
qi (x, t) −→ qi′ (x, t) = ψ α (x + n, t) (Ωi )αβ ψβ (x − n, t) ,
(8.3)
where Ωi now is either γ1 or γ5 and n = 0, 1, . . . , Nneib − 1. For further notational
convenience we suppress the prime and the Dirac indices. Altogether a total
amount of 2Nneib operators now form the basis.
An important characteristic of these qi , or combinations of them, is their
behavior under time evolution. To point out clearly if either the connected or
disconnected part of the correlator (cf. Eqs. (5.19), (5.20)) shows an exponential
decay, we present some tables in App. D where we collect properties of these
operators.
8.2
How to extract masses from correlation
functions
For computing the mass spectrum of the model we need to fit the correlators
according to the exponential decay (compare Eq. (2.38) from Sec. 2.3)
C(t) = c1 e−taE1 1 + O e−ta∆E ,
(8.4)
where c1 is a constant, E1 the ground–state energy and ∆E the energy difference
to the first excited state. The lattice spacing a, shown explicitly here, will be
neglected again from now on. All energies and masses are measured in units of
the inverse lattice spacing then.
By having a closer look at the expression for the correlator given in Eq. (8.1),
one notices two sums over the spatial lattice components x, y. These sums, to-
8.2. How to extract masses from correlation functions
53
gether with the factor 1/L21 , are a projection to momentum zero. We have to
apply this Fourier transformation (only for spatial components!) to filter out the
masses. From the dispersion relation
p
E = E(p) = m2 + p2 ,
(8.5)
it follows that for p = 0 the energy reduces to the rest mass (because in natural
units the speed of light, as well as Planck’s constant, are equal to one).
An important tool for determining the fit ranges in (8.4) are effective mass
plots, discussed next.
Effective mass plots
The general case for the form of a correlator, including the excited states, is given
by
C(t) = c1 e−tE1 + c2 e−tE2 + c3 e−tE3 + . . . ,
(8.6)
where E1 < E2 < E3 < . . . As a consequence only for large t the ground state
dominates, while for small t contributions from excited states are also mixing in
and veiling the relevant information we are interested in – the lowest energy. So
for a start let us ignore the excited states and only incorporate the term with E1
to extract the ground state mass.
As one can see in Figs. 8.1(a), 8.1(c), the correlator has not just a decreasing
part, but again increases from L2 /2 on. The reason is that the states propagate in
positive, as well as in negative time–direction. Thus the correlator is characterized
by a cosh behavior:
C(t) = c1 e−tm1 + c1 e−(L2 −t)m1
L2
−L2 m1 /2
− t m1 .
= 2 c1 e
cosh
2
(8.7)
In order to decide which range of t is suitable to fit C(t), we calculate effective
masses. The “quick–and–dirty” way is defining them as
C(t)
1
≡ ln
.
(8.8)
meff t +
2
C(t + 1)
As soon as the correlator is governed by the ground state, the effective mass forms
a plateau at meff = m1 . If we want to take into account the time–periodicity, we
54
Chapter 8. Correlation functions and the mass spectrum
have to solve the following equation for m1 at each t,
cosh L22 − t m1
C(t)
,
=
C(t + 1)
cosh L22 − t − 1 m1
(8.9)
and thus get meff (t + 1/2). This more efficient procedure was applied to the
correlators C11 and C44 and the results can be seen in Figs. 8.1(b) and 8.1(d).
One can clearly see the formation of effective mass plateaus here. We must now
decide in what region they are “flat enough”, such that we can fit the correlator.
If, for example, one decides that meff (t = 8.5) is already on the plateau, then the
lower boundary for the fitting–region is t = 8, because meff (t = 8.5) is built from
C(t = 8) and C(t = 9) and one wants to have the largest possible interval for t.
Another important criteria is to choose a symmetric fit range relative to L2 /2.
For the correlators in Fig. 8.1 we choose 9 < t < 55 for the γ1 correlator
C11 and for C44 we take the interval 13 < t < 51 for t. We now have a first
approximation for our ground state mass m1 , namely the plateau height. In
addition to that we also gained the more important result, the region of t where
we can apply a fit to the correlator.
The two–parameter fit of the correlator
We are still interested only in the ground state energy, that means we fit the C(t)
to a cosh,
L2
− t m1 .
(8.10)
C(t) = A1 cosh
2
Therefore we minimize a χ2 –functional and thus find out the best parameters
(opt)
(opt)
A1
and m1 . This functional is defined as
2
tmax X
L2
(opt)
(opt)
(opt)
(opt)
2
− t m1
. (8.11)
C(t) − A1
cosh
=
χ A1 , m1
2
t=t
min
For tmin and tmax in (8.11) we use the intervals previously obtained from the
effective mass plateaus. Minimizing means the two following equations have to
be fulfilled:
∂ χ2
∂
(opt)
A1
2
∂χ
∂
(opt)
m1
!
(8.12)
!
(8.13)
= 0,
= 0.
8.2. How to extract masses from correlation functions
10
-3
0.8
m = -0.05
m = 0.05
m = 0.10
m = 0.20
-4
0.6
10
meff(t)
C11(t)
10
-5
55
0.4
-6
10
10
m = -0.05
m = 0.05
m = 0.10
m = 0.20
-7
0.2
-8
10 0
10
20
30
t
40
50
0
0
60
10
(a) C11
10
0.8
-3
-4
40
50
60
50
60
m = -0.05
m = 0.05
m = 0.10
m = 0.20
0.6
meff(t)
C44(t)
30
t
(b) meff for C11
10
10
20
-5
0.4
-6
10
10
m = -0.05
m = 0.05
m = 0.10
m = 0.20
-7
-8
10 0
10
20
30
t
40
50
60
0.2
0
0
10
(c) C44
20
30
t
40
(d) meff for C44
Figure 8.1: The two exponential decreasing correlators C11 and C44 for different
masses m (l.h.s. of the plot). The corresponding effective masses are shown on
the r.h.s. The results are from a MC simulation on a 24 × 64 lattice with coupling
g = 0.1.
(opt)
Eq. (8.12) is easy to solve analytically for the amplitude A1
(opt)
A1
(opt)
m1
=
Ptmax
t=tmin C(t) cosh
h
Ptmax
2
t=tmin cosh
h
L2
2
and one obtains
(opt) i
− t m1
(opt) i .
− t m1
L2
2
(8.14)
This expression can be inserted in (8.13), which is then solved numerically for
(opt)
m1 .
56
8.3
Chapter 8. Correlation functions and the mass spectrum
Excited state masses via the variational
method
Theoretical framework
With the method from the last section we can only extract masses for the lowest
energy state, the ground state. But what about higher excited state masses? For
that enterprise we use the variational technique [30, 31]. The procedure is as
follows:
A matrix K(t) is defined, the so–called cross correlation matrix, as:
with
1 X
Qi (t + τ ) Q†j (τ ) ,
Kij (t) ≡ Qij (t) −
L2 τ
1 X
qi (x, t + τ ) qj† (y, τ ) ,
L21 L2 x,y,τ
1 X
Qi (t) =
qi (x, t) .
L1 x
Qij (t) =
(8.15)
(8.16)
(8.17)
The indices i, j = 1, 2, . . . , 2Nneib are now labeling both, the n steps of the relative
shift of ψ and ψ and the index µ, determining which γµ is sandwiched between the
fermion fields (for the qi see Eq. (8.3)). In other words i and j are multi–indices,
i = (n, µ) ,
(8.18)
j = (m, ν) .
(8.19)
The next step is to diagonalize this matrix K(t) thus extracting its eigenvalues.
It has been shown, that diagonalization of the cross correlation matrix permits us
to untangle the physical states to some amount. When one solves the generalized
eigenvalue problem
K(t) vi = λi (t, t0 ) · K(t0 ) vi ,
(8.20)
the t–behavior of the eigenvalues is given by
λi (t, t0 ) = c e−(t−t0 )mi 1 + O e−(t−t0 )∆Ei ,
i = 1, . . . , 2Nneib .
(8.21)
c is some constant, t0 6 t a fixed timeslice and ∆Ei the energy difference to
the nearest energy level. So for the calculations that means we have to order
8.3. Excited state masses via the variational method
57
the eigenvalues in every timeslice t from big to small. The eigenvalues can then
be fitted according to (8.21). The ground state mass is now given through the
time–dependence of the largest eigenvalue, the first excited mass through the
time–dependence of the second largest, etc.
In our calculations we set Nneib = 8, t0 = 1 and thus included 16 different
operators. However, since it is an a priori choice of operators, it is not clear
whether the resulting eigenvalues are reasonable or not. One first has to analyze
the different entries of the matrix in detail. Also the symmetry properties of K,
in particular its hermiticity within error bars, has to be checked. Our results pass
these checks, indicating that the correlation matrix was implemented correctly.
The set of operators
An important assumption is that the operators couple differently to the excited
states. In other words, the operators used, should be as orthogonal to each other
as possible. Unfortunately it turned out that the relative shift n of ψ and ψ
does not give rise to operators which differ strongly in their overlap with the
excited states. This is demonstrated in Fig. 8.2, where we show the quenched
correlators for different values of n for the γ1 case (l.h.s.) and the γ5 case (r.h.s.).
It is obvious that the correlators behave almost identically in the small–t–range,
where the excited states dominate.
However, comparing γ1 and γ5 shows that changing the γ–matrix indeed
changes the coupling to the excitations. Altogether we are unfortunately reduced
to a 2 × 2 sub matrix with entries constructed from the two following operators:
q1 (x, t) = ψ(x, t) γ1 ψ(x, t) ,
(8.22)
q2 (x, t) = ψ(x, t) γ5 ψ(x, t) .
(8.23)
A possible other class of operators could be obtained by additionally inserting
a discretized version of γ1 ∂1 into (8.22) and (8.23), respectively, so–called p–
waves. This may give rise to operators which are more orthogonal as the simple
bilinears.
Combination of cosh– and sinh–type correlators
A second problem appears due to the different decay properties of the diagonal
and the off–diagonal correlators. On the diagonal we find a cosh behavior, while
58
10
-2
10
n=0
n=3
n=6
γ1
10
10
Chapter 8. Correlation functions and the mass spectrum
-3
-2
10
-4
10
0
5
10
20
15
t
-3
-4
30
25
n=0
n=3
n=6
γ5
0
5
10
15
t
20
25
30
Figure 8.2: Correlators with three different shifts n. On the l.h.s. the correlators
with γ1 , on the r.h.s. these with γ5 sandwiched between the fermion fields, are
plotted. We included 9000 quenched configurations on a 32×64 lattice for g = 0.1
and m = 0.1.
10
-5
6.1×10
10
0
1st EV, GEVP
2nd EV, GEVP
1st EV, SEVP
2nd EV, SEVP
-5
4.7×10
-3
-5
3.6×10
10
-1
-5
2.8×10
-4
10
10
-5
18
19
t
20
21
-2
10
K11(t)
K12(t), K21(t)*
K22(t)
-3
0
5
10
15
t
20
(a) Matrix elements
25
30
10 0
5
10
15
t
20
25
30
(b) Eigenvalues
Figure 8.3: On the l.h.s. the three different matrix elements of the correlation
matrix are plotted, while on the r.h.s. two sets of eigenvalues are shown. On
the one hand the eigenvalues for the generalized eigenvalue problem (GEVP) are
plotted in black, on the other hand one sees the eigenvalues for the standard
eigenvalue problem (SEVP) as red lines/symbols. These results are extracted
from 9000 configurations on a 32 × 64 lattice for g = 0.1 and m = 0.1.
8.3. Excited state masses via the variational method
59
for the two off–diagonal entries a sinh behavior is seen. We illustrate this in
Fig. 8.3. Although we do not use quenched data, the precision is very high and
thus allows us to see the subtle effect from the different symmetries of cosh and
sinh, which in usual QCD calculations [32, 33] is hidden under statistical noise.
In the left–hand side plot of Fig. 8.3 we show the two diagonal elements and
the off–diagonal element γ1 γ5 . Near t = L2 /2 the diagonal entries bend up again,
while the sinh–type off–diagonal entries go through zero. Consequently the curves
have to cross. This crossing takes place around tcross = 19.5 in the plot (see the
insert in the upper right corner of Fig. 8.3(a)).
When analyzing the eigenvalues of the generalized eigenvalue problem (see
r.h.s. plot of Fig. 8.3), we find that slightly below this critical value tcross = 19.5
the eigenvalues start to show irregular behavior. We conclude that for an analysis
only values t < tcross should be taken into account, where the different behavior
of sinh– and cosh–type correlators can be neglected. This restriction is also
strengthened by a comparison of the eigenvalues for the generalized eigenvalue
problem (GEVP) with those for the standard eigenvalue problem (SEVP), which
can be found in Fig. 8.3(b). The different sets of eigenvalues match again for
t = L2 /2 because for that point in time the off–diagonal sinh–type correlator
goes through zero, that means it vanishes.
Reweighting compared to pruned and quenched results
When studying which operators are best for the calculation of the eigenvalues we
also experimented with the weight factor det(M). Two alternative approaches are
studied qualitatively. On the one hand we omit the contributions with the largest
determinants (pruning) to suppress the strong fluctuations of the observables. On
the other hand we present a quenched calculation, i.e., we set det(M) = 1 for
every configuration. In Fig. 8.4 we show the largest eigenvalue (corresponding
to the ground state) for these different processes. It can be clearly seen that by
pruning, as well as quenching, the curves of the eigenvalues are shifted upwards,
i.e., the mass goes down (in comparison to the full theory). The factor by which
the mass decreases is not studied here, this observation should just be seen as a
qualitative aspect of the used calculation techniques.
60
Chapter 8. Correlation functions and the mass spectrum
0
10
full theory
1% omitted
2% omitted
5% omitted
10% omitted
quenched
-1
10
-2
10 0
5
10
15
t
20
25
30
Figure 8.4: The first eigenvalue (corresponding to the ground state mass) for the
full theory, omitting x% of the largest determinants and a quenched calculation.
9000 configurations on a 32 × 64 lattice are used, the coupling is g = 0.1 and the
bare mass parameter is m = 0.1.
Eigenvalues and the mass spectrum
Analytic calculations concerning the mass spectrum of the GN–model were done
in [34, 35] in the large Nf limit. When one expands the given formula in a power
series in 1/Nf the result is
mj = j m1 1 + O 1/Nf2 , j = 2, 3, . . .
(8.24)
In other words, in leading order the mass of the excited states is a multiple of
the ground state mass m1 . For our case, however, this statement has to be taken
with a grain of salt because we analyze the GN–model only for Nf = 1!
The eigenvalues of some 2 × 2 matrices are plotted in Fig. 8.5 for different
couplings g and masses m. Therefrom we extracted the ground state masses and
the masses for the first excited states. These results can be found in Fig. 8.6. As
one can see from these plots, Eq. (8.24) can not be confirmed. Of course, these
calculations are “a shot in the dark” due to the included number of flavors.
8.3. Excited state masses via the variational method
10
10
0
10
ground state
1st excited
-1
10
-2
ground state
1st excited
-1
10
-3
10
-4
10 0
0
-2
10
10
-3
-4
5
10
15
t
20
25
30
10 0
5
(a) g = 0.02, m = 0.1
10
10
15
t
20
10
ground state
1st excited
-2
-2
10
10
-4
-4
10
10
-6
-6
5
10
15
t
20
25
30
10 0
5
(c) g = 0.02, m = 0.2
10
ground state
1st excited
-2
10
10
15
t
20
25
30
(d) g = 0.1, m = 0.2
0
0
ground state
1st excited
-2
10
-4
-4
10
10
-6
-6
10
10
-8
10 0
30
0
ground state
1st excited
10
25
(b) g = 0.1, m = 0.1
0
10 0
61
-8
5
10
15
t
20
(e) g = 0.02, m = 0.3
25
30
10 0
5
10
15
t
20
25
30
(f) g = 0.1, m = 0.3
Figure 8.5: Eigenvalues of different 2 × 2 correlation matrices. From top to
bottom the mass rises from m = 0.1 to m = 0.3, where the coupling is g = 0.02
on the l.h.s. plots and g = 0.1 on the r.h.s. plots. We used 9000 configurations
on a 32 × 64 lattice for each of the parameter sets.
62
Chapter 8. Correlation functions and the mass spectrum
0.8
0.8
ground state
1st excited
ground state
1st excited
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.1
0.2
m
0.3
0.0
0.1
(a) g = 0.02
0.2
m
0.3
(b) g = 0.1
Figure 8.6: Masses from a fit of the eigenvalues from Fig. 8.5. On the l.h.s. the
results for g = 0.02 for the bare mass parameters m = 0.1, 0.2, 0.3 can be found.
On the r.h.s. we show results for the same set of masses m, but for coupling
g = 0.1.
However, one could most likely improve this method by including the mentioned p–wave sources in the used set of operators, and take into account several
flavors, since the only change is that det(M) is raised to the power Nf in the
calculations. Thus this model can serve as a “toy–model”, which maybe useful to
obtain deeper insight into 4–dimensional QCD calculations.
8.4
Lines of constant physics
In this section we want to pick up the topic from 7.3 again. There we found
lines in the g–m–plane, which correspond to a “massless theory”. Now these
calculations are expanded to a bigger range of the coupling g and the mass m.
We systematically compute the correlators C11 and C44 in the g–m–plane.
Afterwards the fitting procedure (as explained before in Sec. 8.2) is applied. Thus
for several points (g, m) we get a physical mass mphys (g, m), which is plotted as
a surface in Fig. 8.7 for C11 . The masses for C44 are so similar to those from
C11 that another plot of the type of Fig. 8.7 would not differ enough to see any
difference.
Now we are able to find the “lines of constant physics”, i.e., that combination
of the parameters g and m where the physical mass stays the same within errors.
8.4. Lines of constant physics
63
To see things clearly, we also plotted the contour of the surface in Fig. 8.8. We
can compare the evolution of these lines with the coupling g with those from
Fig. 7.6. Although the lattice extension is not the same in both calculations,
all the lines behave similar with increasing coupling g. That confirms again the
consistency of our calculations, because we obtained these (similar) results with
completely different methods.
Chapter 8. Correlation functions and the mass spectrum
m
-0.2
-0.3
0
0.2
0.4
0.6
0.8
1
mphys (g, m)
-0.1
0
0.1
0.2
0.3
0.25
0.2
0.15
0.1
0.05
g
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
64
Figure 8.7: The physical mass of the correlator C11 for a wide range of the coupling g and mass m. For each point in the g–m–plane we used 1000 configurations
on a 24 × 64 lattice.
0.2
0.1
-0.2
-0.3
0.15
0.1
0.05
0
65
g
0.2
0.25
8.4. Lines of constant physics
-0.1
m
0
Figure 8.8: The contour plot to Fig. 8.7. The lines indicate where the value of
the physical mass is constant.
0.3
0.8
0.5
0.3
0.2
0.1
0.05
0.01
Chapter 9
A comparison between the
standard and the loop approach
In this chapter we systematically compare the results and techniques for the
two representations of the GN–model, the standard formalism as presented in
Chaps. 2–4 on the one hand, and the model in terms of loop variables, discussed
in Sec. 3.5 and App. A of this work and also in more detail in [3], on the other
hand.
Even though the free case (g = 0) is trivially solved in the standard approach
by Fourier transformation, it has the same significance for this comparison as
g 6= 0, because in the loop representation one has to run a complete Monte–Carlo
simulation also for g = 0.
First we want to give a review of the formulas needed. The expressions in
loop variables are presented only and will not be derived. Then results will be
compared and discussed.
9.1
The basic formulas
Let us first collect the basic formulas for the observables that we will compare.
The partition function in terms of loop variables (cf. Sec. 3.5) for the one–flavor
67
68
Chapter 9. A comparison between the standard and the loop approach
model (the two sets of loops are characterized by the colors red (r) and blue (b))
reads
X 1 c n1 n0
√
ZL =
2+m
(2 + m)2 + g
.
(9.1)
2
r,b
The sum is over the species r, b of self avoiding loops. c denotes the total number
of corners and n0 , n1 are the numbers of empty and singly occupied sites. When
we define the factors f1 and f2 as
f1 = f1 (g, m) ≡
2+m
,
(2 + m)2 + g
f2 = f2 (g, m) ≡
1
,
(2 + m)2 + g
the partition function can be written as
V X c
1
1
√
ZL =
f1n1 f2n2 ,
f2
2
r,b
(9.2)
(9.3)
where n2 now is the number of double occupied sites.
The observables we address are mainly bulk observables, they are obtained
from derivatives of the logarithm of the partition function with respect to one of
the bare parameters m or g.
The chiral condensate
The chiral condensate is defined as the derivative of the partition function with
respect to the mass parameter m. The expression, suitable for a MC simulation
in the standard formalism, is given in Eq. (5.10). For the loop approach it is
1 f2
1 ∂
(L)
ln ZL = −
hn1 i + 2f2 hn0 i .
(9.4)
χ =−
V ∂m
V f1
The chiral susceptibility
The chiral susceptibility is obtained by another derivative of the chiral condensate
with respect to m. For the standard approach see Eq. (5.12), in loop variables
the susceptibility is
Cχ(L)
1 ∂2
ln ZL
= −
V ∂m2
h
1
=
[4f12 − 2f2 ] (n0 − hn0 i)2 + (f2 /f1 )2 − 2f2 (n1 − hn1 i)2
V
i
+ 2f2 (n0 + n1 − hn0 + n1 i)2 − [4f12 − 2f2 ]hn0 i − (f2 /f1 )2 hn1 i . (9.5)
9.2. Results for bulk observables in the free case
69
The interaction density
We consider another observable, the interaction density, which is given by the
derivative of the partition function with respect to the coupling g. Eq. (5.14) is
the formula for the standard approach, while for the loop approach one has
ρ(L) =
2 ∂
2f2
ln ZL =
hn0 i .
V ∂g
V
(9.6)
Having collected all the formulas we compare the results from the two formalisms now, we start the comparison with the free case and then look at the
case with g 6= 0.
9.2
Results for bulk observables in the free case
In this section we present results for both approaches to the GN–model for the
special case of a vanishing coupling constant g = 0. The way to calculate results
for this case is completely different in the two representations, concerning time
and effort, as well as the way of the implementation.
The standard approach at g = 0 can be solved analytically using Fourier
transformation. We can compute the inverse of the Dirac matrix in a few lines of
algebra (compare Eqs. (6.7), (6.8)) and thus obtain formulas for the bulk observables as analytical exact expressions. Although already mentioned in Sec. 6.3, we
want to point out again the following fact: In the free case g = 0 the scalar fields,
φ(n), do not couple to the fermion fields ψ(n) and ψ(n). As a consequence for
each value of m only one run of our program is needed to get the final result and
no statistical errors appear. Furthermore we can do calculations for much bigger
lattices and less computer time is needed.
One encounters a completely different situation in the simulation with loops.
Although the factors f1 , f2 from (9.2) simplify to
f1 =
1
,
2+m
f2 =
1
,
(2 + m)2
(9.7)
a complete Monte–Carlo simulation has to be run in the loop representation at
g = 0. This implies that also in the free case the results from the loop calculations
have statistical errors.
70
Chapter 9. A comparison between the standard and the loop approach
In Figs. 9.1, 9.2 and 9.3 we plot the three bulk observables discussed in Sec. 9.1.
For these plots four different lattices sizes were always taken into account, from
162 up to 1282 . In the plots for the chiral condensate (Fig. 9.1) and the interaction
density (Fig. 9.3) one sees that the different data sets agree very well for the
lattice sizes 322 and larger. The biggest gap between our data from the standard
approach (solid lines), and the data sets from loop calculations (symbols), is found
for the smallest lattice volume of 162 . For the chiral susceptibility the discrepancy
is becoming acceptable from a lattice size of 642 on, but one has to bear in mind
the big difference in the available data points for this region of m. The reason for
this discrepancy in the two approaches are finite size effects from different types
of boundary conditions, which we briefly analyze now.
The finite volume effects of the loop approach are discussed in detail in [3],
based on mean field arguments. This approach has (for quadratic lattices) three
distinct sectors, corresponding to even/odd numbers of windings for red and blue
loops. It was shown that these sectors differ by corrections proportional to 1/L.
For the loop representation of the Ising model, it is expected (see [36]) that
the standard approach with periodic boundary conditions, can be written as a
linear combination of the three sectors in the loop approach. A similar picture
is expected to hold for the loop approach in the Gross–Neveu model. Since we
here compare to a single fixed sector (only even windings), we expect to find
O(1/L)–effects when comparing standard and loop approach. Such a finite size
effect is evident in Figs. 9.1 to 9.3.
In order to test whether the effect is of order O(1/L), we compare the chiral
condensate in the free case g = 0 for m = 0 and study this as a function of
L. In Fig. 9.4 we plot the difference of the loop and the standard results as a
function of ln(L). It can be clearly seen in this plot, that the data points fall
nicely on a straight line for all three sectors of the loop approach. We fit these
lines using the data points for lattice extensions of L = 128 and below. For the
“empty” sector in the loop representation data for L = 256 and L = 512 are also
available. But these two data points do not fall on the straight line. The reason
is most likely that for such big lattices the local update algorithm used in [3]
would have to run longer to compensate for critical slowing down. A way out
would be the implementation of cluster algorithms to update the system in the
loop representation.
9.2. Results for bulk observables in the free case
71
2
-0.65
standard, 16
2
loop, 16
2
standard, 32
2
loop, 32
χ(m)
-0.7
-0.75
-0.8
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(a) χ for lattice sizes 162 and 322
2
-0.65
standard, 64
2
loop, 64
2
standard, 128
2
loop, 128
χ(m)
-0.7
-0.75
-0.8
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(b) χ for lattice sizes 642 and 1282
Figure 9.1: The chiral condensate χ for the free case g = 0 as a function of
m. At top we show the lattice sizes 162 and 322 , below 642 and 1282 . The solid
lines correspond to results from the standard approach and the symbols represent
points, calculated with the loop approach, respectively.
72
Chapter 9. A comparison between the standard and the loop approach
0.2
0
Cχ(m)
-0.2
-0.4
-0.6
2
standard, 16
2
loop, 16
2
standard, 32
2
loop, 32
-0.8
-1
-1.2
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(a) Cχ for lattice sizes 162 and 322
0.2
0
Cχ(m)
-0.2
-0.4
-0.6
2
standard, 64
2
loop, 64
2
standard, 128
2
loop,128
-0.8
-1
-1.2
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(b) Cχ for lattice sizes 642 and 1282
Figure 9.2: Same as Fig. 9.1, now for the chiral susceptibility Cχ .
9.2. Results for bulk observables in the free case
0.35
ρ(m)
0.3
2
standard, 16
2
loop, 16
2
standard, 32
2
loop, 32
0.25
0.2
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(a) ρ for lattice sizes 162 and 322
0.35
ρ(m)
0.3
2
standard, 64
2
loop, 64
2
standard, 128
2
loop, 128
0.25
0.2
-0.3
-0.2
-0.1
0
m
0.1
0.2
0.3
(b) ρ for lattice sizes 642 and 1282
Figure 9.3: Same as Fig. 9.1, now for the interaction density ρ.
73
74
Chapter 9. A comparison between the standard and the loop approach
-1
10
10
empty
single
double
-2
~ 0.80717 / L
~ 0.30358 / L
~ 0.25125 / L
-3
10
10
100
L
Figure 9.4: The modulus of the difference between results for χ from the standard and the loop approach, as a function of the linear lattice extension L, which
varies from 8 up to 512. These numbers are calculated for g = 0.0 and m = 0.0.
The curves correspond to the three distinct sectors for quadratic lattices in the
loop approach, whereas for the calculations with standard methods mixed boundary conditions are always used. The solid lines are fits to those data points where
L 6 128.
The investigated comparison for the free case is quite meaningful because for
the standard method exact analytic results are known. In that case large lattice
volumes can be analyzed, what has as a consequence that finite size effects can
be studied in great detail. Since for the free case the presented results from the
two different approaches match up to finite size effects, it is expected that in
the coupled case g 6= 0 the mapping from [2] is also correct. This case will be
analyzed and discussed now.
9.3. Results for bulk observables at nonvanishing coupling
9.3
75
Results for bulk observables at nonvanishing coupling
In this section we compare the same observables as before, but now for the situation where the coupling has a finite value of g = 0.1. Due to large computer costs
for the standard approach in that case, we are unfortunately reduced to lattices
322 and smaller.
In Fig. 9.5 the chiral condensate is shown for the two different lattices 162
and 322 . The chiral susceptibility Cχ is plotted in Fig. 9.6 for the same lattices
and the interaction density can be found in Fig. 9.7. Comparing these data to
that from the free case the results do not match so well and the finite size effect
has a larger amplitude. The physical mechanism for this observation can be
understood as follows: Here, we only compare data from the empty sector of the
loop representation with that from the standard approach. As afore mentioned, it
is expected that the standard approach with antiperiodic boundary conditions can
be written as a linear combination of the three sectors in the loop approach. That
means we simply neglect contributions from other sectors than the empty one in
the comparison. Since the weight factors f1 , f2 decrease with increasing coupling
g, the occupation density of loops goes down and finite size effects dominate more
because of the missing sectors. However, for a more detailed analysis using mean
field techniques and numerical data see [3].
We conclude with a couple of comments. The overall behavior of the observables is qualitatively the same for both approaches: The curves for χ become
steeper and the peak in Cχ gets sharper and is shifted towards zero mass for
bigger lattices. Also the slope of ρ increases with larger lattice volumes. Although a comparison for large lattices (say > 642 ) is not possible here, there is
no doubt that results from the fermion loop representation of the Gross–Neveu
model match those from the standard approach for infinite volumes.
9.4
n–point correlators in both formalisms
Not only bulk quantities but also n–point correlators have to match in the two
approaches. Thus, again data for the correlators from the free case and for non–
vanishing coupling are compared.
76
Chapter 9. A comparison between the standard and the loop approach
2
-0.65
standard, 16
2
loop, 16
χ(m)
-0.7
-0.75
-0.8
-0.3
-0.2
-0.1
0
0.1
m
(a) χ for lattice size 162
0.2
0.3
2
-0.65
standard, 32
2
loop, 32
χ(m)
-0.7
-0.75
-0.8
-0.3
-0.2
-0.1
0
0.1
m
(b) χ for lattice size 322
0.2
0.3
Figure 9.5: The chiral condensate χ for g = 0.1 as a function of m. On the top
plot we show the lattice size 162 , below 322 . The solid lines correspond to results
from the standard approach and the symbols represent points, calculated with
the loop approach, respectively.
9.4. n–point correlators in both formalisms
0.2
2
standard, 16
2
loop, 16
Cχ(m)
0
-0.2
-0.4
-0.6
-0.8
-0.3
0.2
-0.1
0
0.1
m
(a) Cχ for lattice size 162
0.2
0.3
0.2
0.3
2
standard, 32
2
loop, 32
0
Cχ(m)
-0.2
-0.2
-0.4
-0.6
-0.8
-0.3
-0.2
-0.1
0
0.1
m
(b) Cχ for lattice size 322
Figure 9.6: Same as Fig. 9.5, but now for the chiral susceptibility Cχ .
77
78
Chapter 9. A comparison between the standard and the loop approach
0.35
ρ(m)
0.3
2
standard, 16
2
loop, 16
0.25
0.2
-0.3
-0.2
-0.1
0
0.1
m
(a) ρ for lattice size 162
0.2
0.3
0.35
ρ(m)
0.3
2
standard, 32
2
loop, 32
0.25
0.2
-0.3
-0.2
-0.1
0
0.1
m
(b) ρ for lattice size 322
0.2
0.3
Figure 9.7: Same as Fig. 9.5, but now for the interaction density ρ.
9.4. n–point correlators in both formalisms
79
-2
10
12x64
16x64
20x64
24x64
-3
10
-4
10
-5
10
0
10
20
30
t
40
50
60
Figure 9.8: The scalar correlator (i.e., the unit matrix is sandwiched between
the fermion fields) for the free case g = 0 and m = 0.1 as a function of t. Four
different lattice sizes are included. The solid lines are results from the standard
approach, the triangles represent data points, calculated with the loop approach
(no error bars are depicted for that points!).
In Fig. 9.8 2–point correlators are shown for different lattice volumes. There
it is obvious that by going to larger lattices the corresponding curves approximate
each other more and more. In addition to that, two other qualitative arguments
hold, namely the bottom values decline in the same way for both curves and the
plateaus are shrinking in their extension.
The coupled case with g = 0.1 is found in Fig. 9.9. In those plots we are no
longer able to use a logarithmic scale on the y–axis, because the fluctuations of
the data from the standard approach is so high that negative values appear in the
correlator. Hence, no error bars are inserted so that the picture is not additionally
blurred. However, we stress that the errors are at the one percent level. Anyway,
the agreement (within the precision of the measurement) of the data sets and
thus also the correctness of the mapping is seen in that plots. The remaining
discrepancy is again, of course, concerned with different boundary conditions of
the approaches and the finite site effect.
80
Chapter 9. A comparison between the standard and the loop approach
-4
2×10
-4
1×10
0
-4
-1×10 0
10
20
30
t
40
50
60
50
60
(a) 12 × 64 lattice
-4
2×10
-4
1×10
0
-4
-1×10 0
10
20
30
t
40
(b) 16 × 64 lattice
Figure 9.9: Here the scalar correlator is plotted for the coupling g = 0.1 and
the mass m = 0.1. In these four plots the lattice extension L1 increases from 12
up to 24, the lattice extension in time–direction always is L2 = 64. The black
squares
9.4. n–point correlators in both formalisms
81
-4
2×10
-4
1×10
0
-4
-1×10 0
10
20
30
40
t
50
60
(c) 20 × 64 lattice
-4
2×10
-4
1×10
0
-4
-1×10 0
10
20
30
t
40
50
60
(d) 24 × 64 lattice
are results from the standard approach, the red triangles are from the loop approach (the symbols are connected to guide the eye). No error bars are plotted
here to emphasize the shape of the curves.
82
Chapter 9. A comparison between the standard and the loop approach
The mapping presented in [2] is exact for infinite volume. We have convinced
ourselves that the numerical approach, based on the two representations, gives
the same results up to corrections of order O(1/L). What remains is a discussion
of advantages and disadvantages for working with the different representations.
This is done in the next section.
9.5
Standard techniques vs.
loop approach –
pros and cons
In the previous sections we could convince ourselves by a comparison of results,
that the numerical treatment of the GN–model in the two representations produces the same results up to boundary terms of order O(1/L). However, a final
important question is: Is it lucrative to use this loop approach? This issue will
be discussed now.
The main advantage of the standard approach is that it is “standard”. This
means that the techniques are well known, the used algorithm and the implementation is straight forward (routines like LAPACK can be used, etc.). Also the
formalism itself is less complicated than the loop approach, where a lot of work is
built into finding the representation and expressions for observables. Once having
such a mapping one has to develop algorithms to update the system. Depending
on the chosen algorithm the ergodicity has to be checked, whereas it is obvious
in the standard approach. Actually the Gross–Neveu model is a model with Nf
flavors. To do a simulation with Nf > 1, the determinant just has to be raised
to the power Nf in the standard approach. This enterprise is much harder for
the loop representation, since the model is a 72Nf vertex model. In other words,
already for Nf = 2 the system would consist of 2401 vertices, instead of 49 in the
one flavor case. A simulation would have to update 4 sets of self–avoiding loops.
Of course the standard approach does not win every competition one can
think of. Two advantages of the loop model carry really much weight for lattice
calculations. These are on the one hand drastically reduced numerical costs
and on the other hand the manageable lattice volumes. Observables in the loop
approach only consist of simple quantities like occupation numbers of loops or
number of corners. Since these numbers are even integers, one can imagine that
9.5. Standard techniques vs. loop approach – pros and cons
83
hardware requirements are very low and thus the analysis can also be done with
a desktop PC. The other great advantage is that really huge lattice volumes from
the order of O(106 ) lattice points are feasible with the loop approach. Due to
the enormous computer costs only about O(103 ) lattice points are realistic with
standard methods (on a small PC cluster).
To summarize the facts above one can say that loop models in lattice field
theories maybe of advantage if one accepts the (mostly) hard preparative work.
But always the standard approach (especially analytical known cases) have to be
used to check the mappings.
Chapter 10
Summary and outlook
We shortly summarize the calculations and present conclusions. To finish, an
outlook is given regarding further analysis of this model, and questions of interest
to be addressed in future studies of loop representations for fermionic models.
Summary
In this diploma thesis we analyzed the 2–dimensional Gross–Neveu model on
the lattice, using Monte–Carlo methods. The aim of this work was on the one
hand to obtain deeper insight into the GN–model and on the other to check the
applicability of a fermion loop representation through a comparison of results.
The loop representation avoids the awkward fermion sign problem one faces in
the standard approach.
Our simulation used a partial reweighting strategy for the importance sampling in the Monte–Carlo simulation. Observables were computed for configurations of the scalar field, generated according to a Gaussian distribution, and the
explicitly calculated fermion determinant was multiplied as weight factor. In the
end the averaged observables were normalized with the sum of all determinants.
The results are divided into two main parts, namely the computation of observables with the standard approach, and their comparison to the data from the
loop representation.
85
86
Chapter 10. Summary and outlook
For an assessment of the reweighting strategy we studied the Dirac matrix M
in great detail. First the distribution of the values of det(M) was analyzed. We
found that the distribution of the determinant is peaked sharply for small couplings and that the peak is getting less pronounced as the coupling increases and
fluctuations become more important. Subsequently the eigenvalues of the Dirac
operator were studied. These results showed that the scatter of the eigenvalues
is less for smaller couplings, confirming the picture obtained from the analysis of
the distribution of det(M). The edge of the eigenvalue distribution was used to
compute the critical mass parameter of the system.
The conclusion from this analysis is that results from reweighting are much
more precise for small couplings and thus this strategy is more reliable in the
weak coupling regime.
An important quantity, also studied with standard methods, were 2–point correlators. Such correlation functions are the key ingredient in mass spectroscopy,
and using them we presented results for ground state masses. To extract the
masses of excitations we performed a first exploratory study with the variational
method. It turned out that only two operators of our initial set were suitable for
calculations, and thus just one excited mass could be computed. We believe that
the use of p–wave sources could improve this simulation.
However, already with this small set of basis operators interesting properties of the variational method could be seen from the toy–model. Since it is
a 2–dimensional model which we are investigating, the statistics are much better than in a 4–dimensional model, due to reduced computational costs. We
found effects from different correlator symmetries that are normally hidden under statistical noise in 4–dimensional QCD calculations. In particular it was the
difference between the sinh– and cosh–type elements in the correlation matrix.
The extracted eigenvalues started to show an irregular behavior, as soon as the
two different parts began to cross each other. Such an unwanted effect can be
avoided by using larger lattice extensions in time direction, to shift this crossing
to higher t. We argue that there the numerical effort is a better investment than
in a naive enhancement of the statistics.
With these correlation functions we also computed the physical mass in a wide
range of the couplings, confirming the results for the critical mass parameter of
the system. That in the end demonstrates the consistency of our calculations.
87
A third important part is the comparison of results for the two different representations of the Gross–Neveu model, the standard approach and the loop representation.
First bulk observables were compared to each other in the free and in the
coupled case. In the free case analytically exact results are available for the
standard approach from Fourier transformation, while in the loop formulation
a complete simulation is necessary. Thus the comparison is rather meaningful
even in the free case. Indeed results for the condensates were found to match
already for small lattice volumes, whereas the susceptibility was found to be
more influenced by finite size effects. In the coupled case we were reduced to
smaller lattices due to the large computational costs in the standard approach.
Nevertheless, results for the two representations were found to match also here,
but a comparison on larger lattices would be helpful. The discrepancies can be
explained with finite size effects, which are more important in the coupled case.
In the second part of the comparison we investigated correlators. Again the
two cases, free and coupled, were taken into account. We found agreement of
the two approaches, but since the fluctuations are higher in these quantities, a
comparison is harder than for the bulk observables.
To summarize this comparison, one can say that results for the two approaches
do match up to finite size effects. Since each representation has its advantages, it
depends on the problem under consideration, which formulation should be used.
Outlook
Of course a lot more work can be done on the 2–dimensional Gross–Neveu model.
But since it is a 2d model, one should ask what can really be learned from that.
However, we want to mention two interesting questions, which could be important
for further QCD calculations.
One of these tasks is a simulations with more than only one flavor. Especially
for the standard approach this is rather easy, because one only has to rise the
fermion determinant to the power of the number of flavors. In the loop approach
one would have to update 4 sets of self avoiding loops.
Another property of the model should be investigated more, namely its spectrum of excitations. These excitations are known exactly in the large Nf limit.
Reproducing those in a dedicated lattice simulation would certainly improve our
techniques for computing excitations with the variational method.
Appendix A
The loop representation of the
Gross–Neveu model
Here we want to summarize the loop representation for the Gross–Neveu model.
It can be shown [2], that the fermion determinant for Wilson fermions, coupled to
an external field, is equivalent to self–avoiding loops interacting with this external
field. More precisely, we write the fermion determinant as an exponential of a
sum over closed loops. The auxiliary fields along such loops are collected as
factors for this loop. In principle the loops can have arbitrary length and may
iterate themselves several times. Thus we would expect any desired power of
the fields, but on a finite lattice we have a finite polynomial in the fields for the
determinant. This means an infinitely number of contributions cancel each other
out in the expansion of the exponential.
Now we want to perform this expansion with the so–called hopping expansion.
The first step is writing the Dirac matrix as
M(n, m) = h(n) [1 − H(n, m)] ,
(A.1)
with the hopping matrix
H(n, m) =
±2
X
1
Γµ δn+µ̂,m ,
h(n)
µ=±1
(A.2)
and the (locally varying) hopping parameter
h(n) = 2 + m +
89
√
g φ(n) .
(A.3)
90
Appendix A. The loop representation of the Gross–Neveu model
When we use the well–known trace–logarithm formula,
det(1 − H) = exp Tr [ln(1 − H)] ,
we obtain the following form for the determinant,
Y
det(M) =
h2 (n) det(1 − H)
n
=
Y
n
∞
X
1
2
h (n) exp −
Tr [H n ]
n
n=1
(A.4)
!
.
(A.5)
In the second step we wrote the logarithm as a power series. Because of the
Kronecker delta in Eq. (A.2) we only get contributions to Tr [H n ] for even n = 2k.
More precisely, we only have to take into account closed loops, and thus loops of
even length. For even n we get
Y
X X Y
1
Tr
Γµ .
(A.6)
Tr H 2k =
h(m)
(2k)
n
µ∈L
L∈Ln
m∈P (L)
(2k)
Here Ln is the set of loops of length 2k and starting point n, and P (L) the
amount of visited lattice points. The last product is a ordered product of the Γµ
as they appear along the loop L. We also want to remark that
Γ±µ Γ∓µ ∝ (1 ∓ γµ )(1 ± γµ ) = 0 .
(A.7)
As a consequence back–tracking loops are excluded, that are loops which turn by
180 degrees on a lattice point.
One more step is needed, namely computing the trace of the product of the
matrices Γµ . The result is [37, 38, 39]
c(L)
Y
1
s(L)
√
,
(A.8)
Tr
Γµ = −(−1)
2
µ∈L
with s(L), c(L) as the number of self intersections of the loop and the number of
corners, respectively. The final result for the determinant, using (A.6) and (A.8),
is


c(L) Y
∞
Y
X
X
X
1
1
1 
det(M) =
h2 (n) exp 
(−1)s(L) √
.
2k m
h(l)
2
(2k)
n
k=1
L∈Lm
l∈P (L)
(A.9)
91
Now we are prepared to compare this result to the hopping expansion of a
generalized 8–vertex model [20]. In such a model one has a local varying field ϕ(n)
coupled to every vertex. The eight vertices are represented by eight quadratic
tiles with different weight factors. To build up allowed configurations, one has to
arrange the tiles in a way such that only closed loops are created. The partition
function then is a sum over all possible tilings and the Boltzmann weight is given
by a product of the weights wi for all the tiles used to get the specific tiling. In
addition to that we acquire a product of the external fields for the tiling. The
result is
8
XY
Y
n (T )
Z8V =
wi i
ϕ(n) .
(A.10)
T ∈T i=1
n∈P (T )
Here P (T ) is the set of all tiles occupied by the special tiling T and ni (T ) is the
number how often tile number i is present.
Now we can again apply the hopping expansion to Eq. (A.10). What we get
is


∞
8
X
X
X
Y
Y
1
1
1
n (L)
(−1)s(L) |L|
Z8V = (−1)V w1V exp 
wi i
ϕ(n) .
2
2k m L∈L
w1 i=3
k=1
n∈P (L)
m
(A.11)
The sum runs over all closed loops of arbitrary length, which is denoted by |L|,
but one has to count only one of the possible directions. The expression above
does not contain w2 because it is related to the other weights via the free fermion
condition [40]
w1 w2 + w3 w4 = w5 w6 + w7 w8 .
(A.12)
When we compare Eqs. (A.9) and (A.11), we find that the sums in the exponent
become identical when we set
w1 = w3 = w4 = 1,
1
w5 = w6 = w7 = w8 = √ ,
2
ϕ(n) =
1
.
h(n)
(A.13)
w2 vanishes because of the free fermion condition. An important observation is
that the arguments of the exponents still differ by a factor of 2. This comes from
the fact that the action for the Wilson fermions is a bilinear form, and thus gives
rise to a determinant. The Grassmann action for (A.10) on the other hand is
quadratic and gives a Pfaffian when integrating out the Grassmann variables.
92
Appendix A. The loop representation of the Gross–Neveu model
The final form for the determinant, including all the facts mentioned above,
is

2
c(L) Y
Y
X
1
1
√
2
 .
√
det(M) =
2+m+ g φ(n) 
√
2
+
m
+
g
φ(m)
2
n
L∈Lsa
m∈P (L)
(A.14)
Here Lsa is the set of self avoiding loops.
Now we can bring the GN–model into play. We integrate [det(M)]N (for N
flavors) over the auxiliary field using a Gaussian weight factor. Then we obtain
the following partition function,
X 1 c(L1 )+...+c(L2N ) Y
√
G[2N − On (L1 , . . . , L2N )] .
(A.15)
ZL =
2
n
L1 ,...,L2N
The new function On (L1 , . . . , L2N ) gives us the number how many of the 2N
independent loops occupy the lattice point n. The function G is defined as
Z
x
dφ(n) −φ2 (n)/2 √
√
e
2 + m + g φ(n) , x = 1, . . . , 2N .
(A.16)
G[x] =
2π
To summarize the calculations above, we see that the Gross–Neveu model is
equivalent to a model of 2N independent and self avoiding loops. Thus we created
a mapping from the GN–model to a 72N vertex model (because w2 = 0) with 2N
different line colors.
For only one flavor, the case studied with the loop representation in [3], the
integrals from Eq. (A.16) are resulting in
G[1] = 2 + m ,
G[2] = (2 + m)2 + g .
(A.17)
In the sum in (A.15) two loop species then have to be taken into account. For
graphic purposes let us give the self avoiding loops two colors, namely red (r)
and blue (b). The partition function for the one–flavor Gross–Neveu model in the
fermion loop representation is given by
X 1 c(r)+c(b) n1 n0
√
2+m
(2 + m)2 + g
,
(A.18)
ZL =
2
r,b
where n1 and n0 denote the number of single occupied or empty lattice sites,
respectively.
Appendix B
Fourier transformation on the
lattice
In this appendix, we collect the main formulas for Fourier transformation on the
lattice. We consider a function f (n), defined on the lattice
Λ = {n = (n1 , n2 ) | nµ = 0, . . . , Lµ − 1} ,
(B.1)
with the volume V = L1 L2 .
For our lattice we use toroidal boundary conditions in both directions µ,
f (n + µ̂Lµ ) = e2πiϑµ f (n) .
(B.2)
µ̂ is the unit-vector in µ-direction and ϑµ = 0 for periodic boundary conditions,
or ϑµ = 1/2 for anti–periodic boundary conditions.
e corresponding to Λ with the boundary conditions
The momentum space Λ,
(3.2) is
e = {p = (p1 , p2 ) | pµ = 2π (kµ + ϑµ ), kµ = 0, . . . , Lµ − 1} .
Λ
aLµ
(B.3)
The key formula for Fourier transformation on the lattice is (l integer, 0 ≤ l ≤
N − 1)
N −1
1 X
2π
(B.4)
exp i l · k = δl,k .
N
N
k=0
The proof of this identity is trivial for l = 0 and otherwise follows from the
93
94
Appendix B. Fourier transformation on the lattice
geometric sum
N
−1
X
qk =
k=0
1 − qN
,
1−q
for q = e2πiN/l .
(B.5)
From (B.4) we derive the following expressions:
1 X
exp ip[n − m]a = δ(n − m) = δn1 ,m1 δn2 ,m2 ,
V
(B.6)
e
p∈Λ
1 X
exp i[p − q]na = δ(p − q) .
V n∈Λ
We define the Fourier transform fe(p) of f (n) as
1 X
f (n) exp(−ipna) .
fe(p) = a2 √
V n∈Λ
(B.7)
(B.8)
The inverse transformation then is
f (n) =
1
√
a2 V
X
e
p∈Λ
fe(p) exp(ipna) .
The last equation follows from inserting (B.8) in (B.9) and using (B.6).
(B.9)
Appendix C
γ5–hermiticity of the lattice Dirac
operator
Our Dirac operator on the lattice is given by
±2
X
√
M(n, m) = 2 + m + g φ(n) δn,m −
Γµ δn+µ̂,m .
(C.1)
µ=±1
When we expand the sum and use that Γ±µ = (1 ∓ γµ )/2, we end up with the
following equation,
2
X 1 − γµ
√
δn+µ̂,m
M(n, m) = 2 + m + g φ(n) δn,m −
2
µ=1
−
2
X
1 + γµ
µ=1
2
δn−µ̂,m .
(C.2)
To be γ5 –hermitian, the matrix M has to obey the identity
γ5 M γ5 = M † .
95
(C.3)
96
Appendix C. γ5 –hermiticity of the lattice Dirac operator
To proof that the Dirac operator is in fact γ5 –hermitian, we calculate both sides
of Eq. (C.3). For the l.h.s. we get
2
X
1 − γ5 γµ γ5
√
γ5 M(n, m) γ5 = γ5 2 + m + g φ(n) γ5 δn,m −
δn+µ̂,m
2
µ=1
−
= 2+m+
√
2
X
1 + γ5 γµ γ5
2
µ=1
g φ(n) δn,m −
−
2
X
1 + γµ
µ=1
2
2
X
1 − γµ
µ=1
2
δn−µ̂,m
(C.4)
δn+µ̂,m
δn−µ̂,m , (C.5)
where we used that γ52 = 1 and the anti–commutation relation for the γµ – matrices: γ5 γµ = −γµ γ5 . Now we compute the hermitian conjugate of M,
M(n, m)
†
2
X
√
1 − γµ
= 2 + m + g φ(m) δm,n −
δm+µ̂,n
2
µ=1
−
2
X
1 + γµ
µ=1
2
δm−µ̂,n
(C.6)
2
X
1 − γµ
√
δn−µ̂,m
= 2 + m + g φ(n) δn,m −
2
µ=1
−
2
X
1 + γµ
µ=1
2
δn+µ̂,m .
(C.7)
When one compares Eq. (C.5) with (C.7), it is obvious that our Dirac operator
fulfills Eq. (C.3) and thus is γ5 –hermitian.
Appendix D
Discussion of the correlation
matrix
In Tab. D.1 we summarize the functional form of the disconnected pieces in the
correlation matrix, i.e., the expectation value Qi (t) . In Tab. D.2 we summarize
the functional form of the connected piece in the correlation matrix, i.e., the
expectation value Qij (t) . In Tab. D.3 we summarize the functional form of the
complete correlation matrix Kij (t). The indices i, j are used according to the
definitions from Eqs. (8.18) and (8.19).
i
(0, 1)
(0, 5)
(1, 1)
(1, 5)
(2, 1)
(2, 5)
(3, 1)
(3, 5)
Qi (t)
i
0
0
∈R
∈ iR
∈ R
∈ iR
∈ R
∈ iR
(4, 1)
(4, 5)
(5, 1)
(5, 5)
(6, 1)
(6, 5)
(7, 1)
(7, 5)
Qi (t)
∈ R
∈ iR
∈ R
∈ iR
∈ R
∈ iR
∈ R
∈ iR
Table D.1: Functional form of the disconnected part Qi (t) of the correlation
matrix.
97
98
i|j
(0, 1)
(0, 5)
(1, 1)
(1, 5)
(2, 1)
(2, 5)
(3, 1)
(3, 5)
(4, 1)
(4, 5)
Appendix D. Discussion of the correlation matrix
(0, 1) (0, 5) (1, 1) (1, 5) (2, 1) (2, 5) (3, 1) (3, 5) (4, 1) (4, 5)
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
Table D.2: Functional form of the disconnected part Qij (t) of the correlation
matrix. The symbol r denotes a real number ∈ R, which is proportional to a
hyperbolic cosine. c stands for a imaginary number ∈ iR, proportional to a
hyperbolic sine.
i|j
(0, 1)
(0, 5)
(1, 1)
(1, 5)
(2, 1)
(2, 5)
(3, 1)
(3, 5)
(4, 1)
(4, 5)
(0, 1) (0, 5) (1, 1) (1, 5) (2, 1) (2, 5) (3, 1) (3, 5) (4, 1) (4, 5)
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
r
c
r
c
r
c
r
c
r
c
c
r
c
r
c
r
c
r
c
r
Table D.3: Same as Tab. D.2, but now for the complete correlation matrix
Kij (t).
References
[1] D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically
free field theories, Phys. Rev. D10, 3235–3253 (1974).
[2] C. Gattringer, Loop representation for 2D Wilson lattice fermions in a
scalar background field, Nucl. Phys. B543, 533–542 (1999).
[3] V. Hermann, Monte Carlo simulation of the Gross–Neveu model in a
fermion loop representation. Master thesis, 2006.
[4] J. E. Hirsch, Discrete Hubbard–Stratonovich transformation for fermion
lattice models, Phys. Rev. B28, 4059–4061 (1983).
[5] J. E. Hirsch, Erratum: Discrete Hubbard–Stratonovich transformation for
fermion lattice models, Phys. Rev. B29, 4159 (1984).
[6] Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles
based on an analogy with superconductivity. I, Phys. Rev. 122, 345–358
(1961).
[7] Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles
based on an analogy with superconductivity. II, Phys. Rev. 124, 246–254
(1961).
[8] W. E. Thirring, A soluble relativistic field theory, Annals Phys. 3, 91–112
(1958).
[9] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics. John
Wiley and Sons, New York London Sidney Toronto, 1977.
[10] A. Messiah, Quantum Mechanics, vol. 1 and 2. Dover Publications,
Mineola New York, 1958.
99
100
References
[11] K. G. Wilson, Confinement of quarks, Phys. Rev. D10, 2445–2459 (1974).
[12] M. Creutz, Quarks, gluons and lattices. Cambridge, Uk: Univ. Pr. (1983)
169 p. (Cambridge Monographs on Mathematical Physics).
[13] I. Montvay and G. Münster, Quantum fields on a lattice. Cambridge, UK:
Univ. Pr. (1994) 491 p. (Cambridge Monographs on Mathematical Physics).
[14] J. Smit, Introduction to quantum fields on a lattice: A robust mate,
Cambridge Lect. Notes Phys. 15, 1–271 (2002).
[15] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena. Oxford
University Press, New York, 2002.
[16] C. Gattringer and C. B. Lang, QCD on the lattice – an introduction for
beginners. Unpublished notes.
[17] S. Chandrasekharan and U.-J. Wiese, Meron–cluster solution of a fermion
sign problem, Phys. Rev. Lett. 83, 3116–3119 (1999).
[18] M. Karowski, R. Schrader, and H. J. Thun, Monte Carlo Simulations for
Quantum Field Theories Involving Fermions, Commun. Math. Phys. 97,
5–29 (1985).
[19] K. Scharnhorst, The exact equivalence of the one–flavour lattice Thirring
model with Wilson fermions to a two–colour loop model, Nucl. Phys. B503,
479–504 (1997).
[20] C. Gattringer, A formula for the hopping expansion of 8–vertex models
coupled to an external field, Int. J. Mod. Phys. A14, 4853–4863 (1999).
[21] C. Gattringer, The lattice Schwinger model as a discrete sum of filled
Wilson loops, Nucl. Phys. B559, 539–562 (1999).
[22] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and
E. Teller, Equation of state calculations by fast computing machines, J.
Chem. Phys. 21, 1087–1092 (1953).
References
101
[23] C. R. Gattringer, I. Hip, and C. B. Lang, Topological charge and the
spectrum of the fermion matrix in lattice QED(2), Nucl. Phys. B508,
329–352 (1997).
[24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
Numerical Recipes in C: The Art of Scientific Computing. Cambridge
University Press, New York, 1992.
[25] M. Lüscher, A portable high–quality random number generator for lattice
field theory simulations, Computer Phys. Comm. 79, 100–110 (1994).
[26] F. James, RANLUX: A Fortran implementation of the high–quality
pseudorandom number generator of Lüscher, Computer Phys. Comm. 79,
111–114 (1994).
[27] L. Lyons, A practical guide to data analysis for physical science students.
Cambridge University Press, 1991.
[28] G. C. Wick, The Evaluation of the Collision Matrix, Phys. Rev. 80,
268–272 (1950).
[29] http://www.netlib.org/lapack/
.
[30] M. Lüscher and U. Wolff, How to calculate the elastic scattering matrix in
two–dimensional quantum field theories by numerical simulation, Nucl.
Phys. B339, 222–252 (1990).
[31] C. Michael, Adjoint sources in lattice gauge theory, Nucl. Phys. B259,
58–76 (1985).
[32] T. Burch et al., Excited hadrons on the lattice: Mesons, Phys. Rev. D73,
094505 (2006).
[33] T. Burch et al., Excited hadrons on the lattice: Baryons, Phys. Rev. D74,
014504 (2006).
[34] R. F. Dashen, B. Hasslacher, and A. Neveu, Semiclassical bound states in
an asymptotically free theory, Phys. Rev. D12, 2443–2458 (1975).
102
References
[35] A. B. Zamolodchikov and A. B. Zamolodchikov, Excact S matrix of the
Gross–Neveu “elementary” fermions, Phys. Lett. B72, 481–483 (1978).
[36] F. Wegener, Grassmann–Variable. Online scriptum:
www.tphys.uni–heidelberg.de/˜wegner/grassmann.ps
.
[37] I. O. Stamatescu, Note on the lattice fermionic determinant, Phys. Rev.
D25, 1130 (1982).
[38] C. R. Gattringer, S. Jaimungal, and G. W. Semenoff, Loops, Surfaces and
Grassmann Representation in Two– and Three–Dimensional Ising Models,
Int. J. Mod. Phys. A14, 4549–4574 (1999).
[39] C. Gattringer, Hopping expansion as a tool for handling dual variables in
lattice models, Nucl. Phys. Proc. Suppl. 73, 772–774 (1999).
[40] C. Fan and F. Y. Wu, Ising model with second–neighbor interaction. I.
Some exact results and an approximate solution, Phys. Rev. 179, 560–570
(1969).
Danksagung
An dieser Stelle ist es mir eine große Freude allen Menschen danken zu können,
die zum Entstehen dieser Arbeit beigetragen haben, direkt oder auch indirekt.
Als erstes möchte ich mich bei meinem Betreuer Prof. Dr. Christof Gattringer
bedanken. Er bot mir dieses interessante Thema, hatte immer genug Zeit für
ausführliche Diskussionen und gab mir die Motivation zurück wann immer ich an
meinen Vorhaben zweifelte.
Genauso gebührt mein Dank Prof. Dr. Andreas Schäfer. Er ermöglichte
mir diese Diplomarbeit an seinem Lehrstuhl zu schreiben und an Konferenzen
teilzunehmen.
Bei Prof. Dr. Christian Lang und der ganzen Gittergruppe der Karl–Franzens–
Universität Graz möchte ich mich für die wertvollen Diskussionen und Vorträge
bedanken.
Für die zahlreichen Hilfestellungen und das angenehme Arbeitsklima möchte
ich allen Kollegen in Regensburg und Graz danken, besonders Erek Bilgici, Rafael
Frigori, Markus Kloker, Dominik Nickel, Philipp Huber, Dr. Bernd–Jochen Schaefer und Martin Volk. Speziell bei Verena Hermann bedanke ich mich für alle
Diskussionen jenseits“ der Physik.
”
Allen Korrekturleserinnen und –lesern sei an dieser Stelle für die hilfreichen
Verbesserungsvorschläge gedankt.
Der meiste Dank jedoch gebührt meinen Eltern, dabei ganz besonders meiner
Mutter. Ohne Sie wäre diese Arbeit und mein ganzes Studium nicht möglich
gewesen. Sie waren mir eine konstante Stütze in jeder Situation, und nicht zuletzt
große Vorbilder.
Ich möchte mit einem Zitat meines Betreuers Christof Gattringer schließen.
Es beschreibt kurz und prägnant das Verhätnis von Programmierer zu Computer.
Der Depp sitzt immer davor.“
”
Christof Gattringer
Graz, im Dezember 2006.
Erklärung
Hiermit erkläre ich, dass ich die Diplomarbeit selbständig angefertigt und keine
Hilfsmittel außer den in der Arbeit angegebenen benutzt habe.
Regensburg, 6. Dezember 2006
....................................
(Markus Limmer)