Complex Bosonic Many–body Models:
Overview of the Small Field Parabolic Flow
Tadeusz Balaban
Department of Mathematics
Rutgers, The State University of New Jersey
110 Frelinghuysen Rd
Piscataway, NJ 08854-8019
tbalaban@math.rutgers.edu
Joel Feldman∗
Department of Mathematics
University of British Columbia
Vancouver, B.C.
CANADA V6T 1Z2
feldman@math.ubc.ca
http://www.math.ubc.ca/∼feldman/
Horst Knörrer, Eugene Trubowitz
Mathematik
ETH-Zentrum
CH-8092 Zürich
SWITZERLAND
horst.knoerrer@math.ethz.ch, eugene.trubowitz@math.ethz.ch
http://www.math.ethz.ch/∼knoerrer/
May 5, 2015
∗
Research supported in part by the Natural Sciences and Engineering Research Council of Canada and
the Forschungsinstitut für Mathematik, ETH Zürich.
It is our long term goal to rigorously demonstrate symmetry breaking in a gas of
bosons hopping on a three dimensional lattice. Technically, to show that the correlation
functions decay at a non–integrable rate when the chemical potential is sufficiently positive,
the non–integrability reflecting the presence of a long range Goldstone boson mediating
the interaction between quasiparticles in the superfluid condensate. It is already known
[BFI, BFII] that the correlation functions are exponentially decreasing when the chemical
potential is sufficiently negative. See, for example, [C] and [W, §19] for an introduction
to symmetry breaking in general, and [AGD, Bog, FW, PS] as general references to BoseEinstein condensation. See [Ben, CG, LSSY, Sei] for other mathematically rigorous work
on the subject.
In [BFKT1, BFKT2] we expressed the positive temperature partition function and
thermodynamic correlation functions in a periodic box (a discrete three–dimensional torus)
as ‘temporal’ ultraviolet limits of four–dimensional (coherent state) lattice functional integrals (see also [NO]). By a lattice functional integral we mean an integral with one (in this
case complex) integration variable for each point of the lattice. By a ‘temporal’ ultraviolet
limit, we mean a limit in which the lattice spacing in the inverse temperature direction
(imaginary time direction) is sent to zero while the lattice spacing in the three spatial directions are held fixed. In [BFKT5], by a complete large field/small field renormalization
group analysis, we expressed the temporal ultraviolet limit for the partition function(1) , as
a four–dimensional lattice functional integral with the lattice spacing in all four directions
being of the order one, preparing the way for an infrared renormalization group analysis
of the thermodynamic limit.
In [BFKT7](2) , we initiate the infrared analysis by tracking the evolution of the effective interaction generated by the iteration of a renormalization group map that is characteristic of a parabolic covariance(3) : in each renormalization group step the spatial lattice
directions expand by a factor L > 1, the inverse temperature direction expands by a factor
L2 and the running chemical potential grows by a factor of L2 , while the running coupling constant decreases by a factor of L−1 . Consequently, the effective potential, initially
close to a paraboloid, develops into a Mexican hat with a moderately large radius and a
moderately deep circular well of minima. [BFKT7] ends after a finite number (depending
on the coupling constant) of steps once the chemical potential, which initially was of the
order of the coupling constant, has grown to a small ‘ǫ’ power of the coupling constant.
The renormalization group iterates have moved away from the trivial noninteracting fixed
point and it is no longer possible to solve the equations giving the critical point of the
(1)
A similar analysis will yield the corresponding representations for the correlation functions.
(2)
See also [BFKT6] for a more pedagogical introduction.
(3)
Morally, the 1 + 3 dimensional heat operator.
1
effective action simply by perturbing off of the linearized equations.
In the next stage of the construction, we plan to continue the parabolic evolution, but
expanding around fields concentrated at the bottom of the well rather about zero (much
as is done in the Bogoliubov Ansatz) and track it through an additional finite number
of steps until the running chemical potential is sufficiently larger than one, so that the
well is sufficiently deep. At that point we will turn to an elliptic renormalization group
map that expands both the temporal (inverse temperature) and spatial lattice directions
by the same factor L. It is expected that the elliptic evolution can be controlled through
infinitely many steps, all the way to the symmetry broken fixed point. The system is
superrenormalizable in the entire parabolic regime because the running coupling constant
is geometrically decreasing. However in the elliptic regime, the system is only strictly
renormalizable.
The technical implementation of the parabolic renormalization group in [BFKT7]
proceeds much as in [BFKT4, BFKT5]. We use 1+3 dimensional block spin averages, as in
[KAD, B, GK]. In [BFKT5], we had used decimation, which was suited to the effectively one
dimensional problem of evaluating the temporal ultraviolet limit. Otherwise, the stationary
phase calculation that controls oscillations is similar, but technically more elaborate. The
essential complication is that the critical fields and background fields are now solutions to
(weakly) nonlinear systems of parabolic equations. The Stokes’ argument that allows us to
shift a multi dimensional contour to the ‘reals’ and the evaluation of fluctuation integrals
is similar. However, there is an important new feature: the chemical potential has to be
renormalized.
To analyze the output of the block spin convolution (a single renormalization group
step), it is de rigueur for the small field/large field style of renormalization group implementations to introduce local small field conditions on the integrand and then decompose
the integral into the sum over all partitions of the discrete torus into small and large field
regions on which the conditions are satisfied and violated, respectively. Small field contributions are to be controlled by powers of the coupling constant v0 (a suitable norm
of the two body interaction) uniformly in the volume of the small field region. Large
ε
field contributions are to be controlled by a factor e−1/v0 , ε > 0 , raised to the volume
of the large field region. Morally, in small field regions, perturbation expansions in the
coupling constant converge and exhibit all physical phenomena. Large field regions give
multiplicative corrections that are smaller than any power of the coupling constant. So, in
the leading terms, every point is small field.
If the actions in our functional integrals were sums of positive terms (as in a Euclidean
O(n) model) it would be routine to extract an exponentially small factor per point of a
large field region. They are not. There are explicit purely imaginary terms. In [BFKT7]
2
we analyze the parabolic flow of the leading term, in which all points are small field, as
long as it is possible to expand around zero field. Nevertheless, we show (see, [BFKT7,
Appendix D]) that our actions do have positivity properties and consequently there is at
ε
least one factor e−1/v0 whenever there is a large field region. A stronger bound of a
factor per point of a large field region is reasonable and would be the main ingredient for
controlling the full parabolic renormalization group flow in this regime.
We now formally introduce the main objects of discussion and enough machinery
to allow technical statements of the main results of [BFKT7] and the methods used to
establish them.
One conclusion of our previous work in [BFKT5] is that the purely small field contribution to the partition function for a gas of bosons hopping on a three dimensional
discrete torus X = ZZ3 /Ls ZZ3 (where L is a fixed, sufficiently large odd natural number
and s ∈ IN is the spatial infrared regulator which will ultimately be sent to infinity) takes
the form
Z
dλ(ψ) eA0 (ψ
∗
, ψ)
S0
where
• X0 = ZZ/Lt ZZ × X , is a 1 + 3 dimensional discrete torus with points x = (x0 , x) .
1
⌋ ∈ IN is the inverse temperature infrared regulator, which will
Here, t ≈ ⌊logL kT
ultimately be sent to infinity.
• ψ ∈ CX0 is a complex valued field on X0 , ψ ∗ is the complex conjugate field and
Q dψ(x)∗ ∧dψ(x)
dλ(ψ) =
is the standard Lebesgue measure on the finite dimensional
2πı
x ∈ X0
complex vector space CX0 .
−1/ +ε
−1/ +ε
• S0 = ψ ∈ CX0 |ψ(x)| ≤ v0 3 , |∂ν ψ(x)| ≤ v0 3 , ν = 0, 1, 2, 3 , x ∈ X0 ,
where the small ‘coupling constant’ v0 is an exponentially, tree length weighted L1 L∞ -norm (see the discussion of norms at the end of this introduction or [BFKT7,
Definition I.9]) of an effective interaction V0 (see [BFKT7, Proposition II.1]).
• Let ψ∗ be another arbitrary element of CX0 . ( ψ∗ is not to be confused with the
complex conjugate ψ ∗ of ψ .)
• A0 (ψ∗ , ψ) = − A0 (ψ∗ , ψ) + p0 (ψ∗ , ψ, ∇ψ∗ , ∇ψ) . The action A0 (ψ ∗ , ψ) deter
mining the partition function is the restriction A0 (ψ ∗ , ψ) = A0 (ψ∗ , ψ) ψ =ψ ∗ of
∗
A0 (ψ∗ , ψ) to the ‘real’ subspace ψ∗ = ψ ∗ of CX0 × CX0 . Here, ∂ν , ν = 0, 1, 2, 3 , is
the forward difference operator in the xν direction and ∇ is the associated discrete
gradient operator.
• ‘Morally’, A0 (ψ∗ , ψ) = hψ∗ , (−∂0 + h)ψi0 + V0 (ψ∗ , ψ) − µ0 hψ∗ , ψi0 , where
P
◦ hf , gi0 =
f (x)g(x) is the natural real inner product on CX0
x ∈ X0
3
◦ h is a nonnegative, second order, elliptic (lattice) pseudodifferential operator acting on X
P
V0 (x1 , x2 , x3 , x4 ) ψ∗ (x1 )ψ(x2 )ψ∗ (x3 )ψ(x4 ) is a quartic mono◦ V0 (ψ∗ , ψ) = 21
X04
P
V0 (0, x2 , x3 , x4 ) > 0
mial whose kernel V0 is translation invariant with
X03
◦ µ0 is essentially the chemical potential.
• Let ψ∗ν , ψν , ν = 0, 1, 2, 3 , be the names of new arbitrary elements of CX0 . The per
turbative correction p0 ψ∗ , ψ, {ψ∗ν }3ν=0 , {ψν }3ν=0 , to the principal contribution A0
to A0 , is a power series in the ten variables ψ∗ , ψ, {ψ∗ν , ψν }3ν=0 with no ψ∗ (x)ψ(y)
terms, such that each nonzero term has as many factors with asterisks as factors
without asterisks. That is, the series conserves particle number. It converges on
n
ψ∗ , ψ, {ψ∗ν , ψν }3ν=0
∈C
10X0
−1/3+ε
, 0 ≤ ν ≤ 3 , x ∈ X0
|ψ(∗) (x)|, |ψ(∗)ν (x)| ≤ v0
where “(∗)” means “either with ∗ or without ∗”.
See [BFKT7, Proposition II.1] for more details.
For convenience, set
F0 (ψ ∗ , ψ) = eA0 (ψ
∗
, ψ)
With this notation the partition function becomes
R
χS0 (ψ)
ε dλ(ψ) F0 (ψ ∗ , ψ) + O e−1/v0 .
It is natural to study the partition function by means of a steepest descent/stationary
∗
phase analysis. The exponential ehψ , ∂0 ψi is purely oscillatory because the quadratic form
hψ ∗ , ∂0 ψi is pure imaginary. Fortunately, our partition function, Z , has the essential
feature that there is an analytic function A0 (ψ∗ , ψ) on a neighborhood of the origin
in CX0 × CX0 whose restriction to the real subspace is the ‘small field’ action. Our
renormalization group analysis of the oscillating integral defining Z is based on the critical
points of A0 (ψ∗ , ψ) = hψ∗ , (−∂0 + h)ψi + V0 (ψ∗ , ψ) − µ0 hψ∗ , ψi in CX0 × CX0 that
typically do not lie in the real subspace, and a multi dimensional Stokes’ contour shifting
construction that is only possible because p0 (ψ∗ , ψ) is analytic.
We now formally introduce the ‘block spin’ renormalization group transformations
that are used in this paper. Let X−1 be the subgroup L2 ZZ/Lt ZZ × LZZ3 /Ls ZZ3 of X0 .
Observe that the distance between points on the inverse temperature axis is now L2 and
on the spatial axes is now L , and that |X−1 | = L−5 |X0 | . Also, let Q(0) : CX0 → CX−1
be a linear ‘block spin averaging’ operator that commutes with complex conjugation. (We
will make a specific choice of Q(0) later.) Let θ , θ∗ be two arbitary elements of CX−1 . By
4
definition, the block spin transform of F0 (ψ∗ , ψ) associated to Q(0) with external fields
θ and θ∗ is
Z
(0) ∗
(0)
1
dλ(ψ) N1(0) e− L2 h θ∗ − Q ψ , θ − Q ψ i−1 F0 (ψ ∗ , ψ)
B1 (θ∗ , θ) =
CX0
where hf , gi−1 = L5
f (y)g(y) is the natural real inner product on CX−1 and the
P
y ∈ X−1
constant N (0) normalizes the Gaussian convolution kernel.
It can be awkward to compare functions defined on discrete tori with different lattice
(1)
spacings. So, we introduce the unit discrete torus X0 = ZZ/Lt−2 ZZ × ZZ3 /Ls−1 ZZ3 and
(1)
the ‘parabolic’ scaling map x ∈ X0 → (L2 x0 , Lx) ∈ X−1 which is an isomorphism
of Abelian groups. Abusing notation, we consciously use the symbol ψ(x) as the name
(1)
of a field on the unit torus X0 even though it was used before as the name of a field
on the unit torus X0 . By definition, the block spin renormalization group transform of
(1)
F0 (ψ ∗ , ψ) associated to Q(0) with external fields ψ and ψ∗ in CX0 is
F1 (ψ∗ , ψ) = B1 S−1 ψ∗ , S−1 ψ
where
3
S−1 ψ (y0 , y) = L− /2 ψ
y0 y
L2 , L
(1)
(1)
for any ψ ∈ CX0 . The ‘parabolic’ exponent −3/2 has been chosen so that(4)
hSθ ∗ , (∂0 + ∆)Sθi0 = hθ ∗ , (∂0 + ∆)θi−1 . We have, interchanging the order of integration,
L
(1)
−3|X0 |
Z
(1)
X
C 0
∗
dλ(ψ) F1 (ψ , ψ) =
Z
CX 0
dλ(ψ) F0 (ψ ∗ , ψ)
the original partition function.
Repeat the construction.
(2)
(1)
◦ Let X−1 be the subgroup L2 ZZ/Lt−2 ZZ × LZZ3 /Ls−1 ZZ3 of X0 and
(2)
(1)
◦ let Q(1) : CX0 → CX−1 be a linear ‘block averaging’ operator that commutes
with complex conjugation.
(2)
◦ Introduce the unit discrete torus X0 = ZZ/Lt−4 ZZ × ZZ3 /Ls−2 ZZ3 and
(2)
(2)
◦ the isomorphism x = (x0 , x) ∈ X0 → (L2 x0 , Lx) ∈ X−1 .
As before, integrate against the normalized Gaussian to obtain the block spin transform
of F1 associated to Q(1)
Z
(1) ∗
(1)
1
B2 (θ∗ , θ) =
dλ(ψ) N1(1) e− L2 h θ∗ − Q ψ , θ − Q ψ i−1 F1 (ψ ∗ , ψ)
C
(4)
X
(1)
0
In hθ ∗ , (∂0 + ∆)θi−1 , ∂0 is the forward difference operator on
Similarly, for spatial difference operators.
5
X−1
:
(∂0 f )(y) =
f (y0 +L2 , y) − f (y0 , y)
L2
.
and then rescale to obtain the block spin renormalization group transform
F2 (ψ∗ , ψ) = B2 S−1 ψ∗ , S−1 ψ
(2)
y
where S−1 ψ (y0 , y) = L−3/2 ψ( Ly02 , L
) for any ψ ∈ CX0 . Interchanging the order of
integration,
L
(2)
−3|X0 |
L
(1)
−3|X0 |
Z
∗
(2)
X
C 0
dλ(ψ) F2 (ψ , ψ) =
Z
CX 0
dλ(ψ) F0 (ψ ∗ , ψ)
We keep repeating the construction to generate a sequence Fn (ψ∗ , ψ) , n ≥ 1 , of functions
(n)
(n)
defined on spaces CX0 × CX0 . [BFKT7] concerns a sequence Fn(SF) (ψ∗ , ψ) of ‘small
field’ approximations to the Fn ’s. We expect, and provide some supporting motivation for,
ε
but do not prove, that Fn = Fn(SF) +O e−1/v0 . For the precise definition, see [BFKT7, §I.2
and, in particular, Definition I.6]. For the supporting motivation see [BFKT7, Appendix
D].
To make a specific choice for the, to this point arbitrary, sequence Q(0) , · · · , Q(n) , · · ·
of block averaging operators, let q(x) be a nonnegative, compactly supported, even function on ZZ × ZZ3 and Q the associated convolution operator(5)
(Qψ)(y) =
X
x∈ZZ×ZZ3
q(x) ψ y + [x]
(n)
, ψ ∈ CX0
(n)
where [x] is the point in the quotient X0
(n+1)
, y ∈ X−1
(n)
⊂ X0
= ZZ/Lt−2n ZZ × ZZ3 /Ls−n ZZ3 represented
(n+1)
by x ∈ ZZ × ZZ3 . By construction, Qψ ∈ CX−1 . We fix q(x) to be the convolution
2
2
3
of the indicator function of the (discrete) rectangle [− L 2−1 , L 2−1 ] × [− L−1
, L−1
] in
2
2
ZZ × ZZ3 convolved with itself four times and normalized so that its sum over ZZ × ZZ3 is
one. In [BFKT7] the basics objects are the ‘small field’ block spin renormalization iterates
Fn(SF) (ψ∗ , ψ) , where at each step Q is chosen to be convolution with the fixed kernel q .
If we had defined Q by convolving just with the indicator function of the rectangle
itself, properly normalized, then (Qψ)(y) would be the usual average of ψ(x) over the
(n)
rectangular box in X0 centered at y with sides L2 and L . We work with the smoothed
averaging kernel rather than the sharp one for technical reasons: commutators [∂ν , Q]
are routinely generated and are small enough when Q is smooth enough. For the rest of
this overview we will pretend that q is sharp and formulate our results as if this were the
case. We will also pretend that the operator h on X appearing in the integrand of the
(5)
By abuse of notation, we use the same symbol Q for the convolution operator acting on all of the
(n)
spaces CX0
.
6
partition function is (minus) the lattice Laplacian. Full, technically complete, statements
are in [BFKT7, §I.6].
Our main result is: If ε > 0 and v0 are small enough and L is large enough, there
5/
9/
log 1/
is a
µ∗ = O(v0 ) , such that for all(7) µ∗ + v0 4 < µ0 < v0 10 and all n < 25 log Lv0 ,
the ‘small field approximations’ Fn(SF) to the Fn ’s are
(6)
Fn(SF) (ψ∗ , ψ) =
1
Zn
exp − An ψ∗ , ψ, φ∗n (ψ∗ , ψ), φn (ψ∗ , ψ) + pn (ψ∗ , ψ, ∇ψ∗, ∇ψ)
An = an h(ψ∗ − Qn φ∗n ) , (ψ − Qn φn )i0 + hφ∗n , (−∂0 − ∆)φn in
− µn hφ∗n , φn in + Vn (φ∗n , φn )
on the domain
n
(n)
Sn = (ψ∗ , ψ) ∈ C2X0 |ψ(∗) (x)| ≤ κn , |∂ν ψ(∗) (x)| ≤ κ′n ,
0 ≤ ν ≤ 3 , x ∈ X0
and zero on its complement. Explicit expressions for the radii κn and κ′n are given in
3
− 1 +ε
and κ′n as
[BFKT7, Definition I.13.a]. You can think of κn as being roughly L 4 n v0 3
3
− 1 +ε
being roughly L 8 n v0 3 . Here,
• φ∗n (ψ∗ , ψ) and φn (ψ∗ , ψ) are (nonlinear) maps from an open neighborhood of the
(n)
(n)
origin in CX0 ×CX0 to CXn , where Xn is the discrete torus, isomorphic to X0 , but
with lattice spacing L−2n in the time direction and L−n in the spatial directions(8) .
We say more about them in the last of this sequence of bullets. For fixed ‘external
fields’ ψ∗ , ψ , the functions φ∗n (ψ∗ , ψ)(u) , φn (ψ∗ , ψ)(u) on Xn are referred to as
the “background fields” at scale n .
P
P
f (x)g(x) and hf , gin = L−5n
f (u)g(u) are the natural real
• hf , gi0 =
u ∈ Xn
(n)
x ∈ X0
inner products on C
(n)
X0
and CXn .
(n)
• Qn : CXn → CX0 is the linear map for which (Qn f )(x) is the average of f ∈ CXn
(n)
over the square box in Xn centered at x ∈ X0 with sides 1. (This box contains
3
L2n × (Ln ) points of Xn .)
• an =
1−L−2
1−L−2n
• −∂0 − ∆ is the natural heat operator on the ‘fine’ discrete torus Xn .
(6)
An explicit formula for µ∗ is given in [BFKT7, (I.19)].
(7)
We are weakening some of the statements, for pedagogical reasons. In particular, the sets of allowed
µ0 ’s and n’s are a bit larger than the sets specified here.
(8)
Xn = L−2n Z/Lt−2n Z × L−n Z3 /Ls−n Z3 and the map u ∈ Xn 7→ x = (L2n u0 , Ln u) ∈ X0 is an
isomorphism of Abelian groups.
7
• For each f∗ , f ∈ CXn
Vn (f∗ , f ) =
1
2
4
1
5n
L
X
Vn (u1 , u2 , u3 , u4 ) f∗ (u1 )f (u2 )f∗ (u3 )f (u4 )
uj ∈ Xn
j=1,2,3,4
where Vn (u1 , u2 , u3 , u4 ) is close to
Vn(u) (u1 , u2 , u3 , u4 ) =
3
1
(L5n )
Ln
V0 (U1 , U2 , U3 , U4 ) ,
Uj = (L2n uj0 , Ln uj )
• The perturbative correction pn ψ∗ , ψ, {ψ∗ν }3ν=0 , {ψν }3ν=0
(n)
X0
ψν }3ν=0
(2)
is a power series in the
, with no ψ∗ (x)ψ(y) or constant terms,
ten variables ψ∗ , ψ, {ψ∗ν ,
∈ C
such that each nonzero term has as many factors with asterisks as factors without
asterisks. It converges(9) when |ψ(∗) (x)| ≤ κn and |ψ(∗)ν (x)| ≤ κ′n for all 0 ≤ ν ≤ 3
(n)
and x ∈ X0 .
• Zn is a normalization constant(10) .
• µn is the ‘renormalized’ chemical potential(11) . It is close to L2n µ0 .
• For each pair in the polydisc
n
(n)
(ψ∗ , ψ) ∈ CX0
(n)
× CX0
o
(n)
|ψ(∗) (x)| ≤ κn for all x ∈ X0
the fields φ∗n (ψ∗ , ψ)(u) , φn (ψ∗ , ψ)(u) on Xn are critical points of the functional
(f∗ , f ) 7→ an h(ψ∗ − Qn f∗ ) , (ψ − Qn f )i0 − hf∗ , (∂0 + ∆ + µn )f )in + Vn (f∗ , f )
The maps φ∗n (ψ∗ , ψ) , φn (ψ∗ , ψ) are holomorphic on that polydisc.
In practical terms, what have we achieved? If ψ = z is a constant field on X0 , then
∗
∗
∗
A0 (ψ , ψ) = V0 (ψ , ψ) − µ0 hψ , ψi0 = |X0 |
v0 h 2 µ0 2
µ2 i
= |X0 |
|z| −
− 20
2
v0
v0
v
0
2
4
|z| − µ0 |z|
2
(9)
It is necessary to measure the size of pn by introducing an appropriate norm. See the last paragraphs
of this introduction.
(10)
When we take logarithms and ultimately differentiate with respect to an external field to obtain
correlation functions, it will disappear.
(11)
We will describe the inductive construction of µn later on in this introduction. The dependence of
pn on the derivatives of the fields arises because of the renormalization of the chemical potential.
8
where, v0 =
P
V0 (0, x2 , x3 , x4 ) . The graph of the real valued function
X03
µ20 i
v0 h 2 µ0 2
|z| −
− 2
2
v0
v0
over the complex plane z = x1 + ıx2 is a surface of revolution
q around the x3 –axis with
µ0
. Our hypothesis on µ0
the nondegenerate circular well of absolute minima |z| =
v0
implies that the radius and depth of the well are of order one and order v0 respectively.
After n renormalization group steps,
h
µn 2
µ2n i
(n) v0
2
An ψ , ψ, φ∗n (ψ , ψ), φn (ψ , ψ) ψ=z ≈ |X0 | n |z| −
−
v0/Ln
2L
(v0/Ln )2
∗
∗
∗
since, by [BFKT7, Remark G.1], φn (ψ ∗ , ψ) |ψ = z ≈ z and φ∗n (ψ ∗ , ψ) |ψ = z ≈ z ∗ . The
graph is again q
a surface of revolution with the nondegenerate circular well of absolute
3
µn
2 n and order L5n v
minima |z| =
0
v0/Ln , but now the radius and depth are of order L
respectively; the well is developing. We stop the flow when the well becomes so wide and so
deep that we can no longer construct background fields by expanding around ψ∗ , ψ = 0 .
This happens as µn approaches order one.
P
K(x, y) ψ∗ (x)ψ(y)
If the power series expansion of pn had a quadratic part
(n)
x,y∈X0
the discussion of the evolving well in the last paragraph would be misleading, because
the minimum of the total action An − pn would not be close enough to the minimum of
the dominant part An . The requirement that pn must not contain quadratic terms is
the renormalization condition for the chemical potential. (See, Step 9 below.) Under the
scaling map (1), the local monomials
hψ∗ , ψi0
hψ∗ , ∂ν ψi0 1 ≤ ν ≤ 3
are relevant and the local monomials
hψ∗ , ∂0 ψi0
h∂ν ψ∗ , ∂ν ′ ψi0 1 ≤ ν, ν ′ ≤ 3
are marginal. The local monomials hψ∗ , ∂ν ψi0 , 1 ≤ ν ≤ 3, do not appear, because of
reflection invariance. See [BFKT7, Definition A.1 and Lemma A.4]. So pn does not
contain any relevant monomials.
The parabolic renormalization group flow drives the system away from the trivial
(noninteracting) fixed point. To continue, we will have to construct background fields
by expanding about configurations supported near the bottom of the developing well,
analogously to the ‘Bogoliubov Ansatz’. At present, we expect to continue the parabolic
9
flow, but expanding about configurations supported near the bottom of the well, through
a transition regime (which overlaps with the regime of [BFKT7]) until µn becomes large
enough, and then switch to a new ‘elliptic’ renormalization group flow for the push to the
symmetry broken, superfluid fixed point.
The next part of this introduction is an outline, in nine steps, of the inductive
construction that uses a stationary phase calculation to build the desired form for
Fn+1 (ψ∗ , ψ) = Bn+1 S−1 ψ∗ , S−1 ψ , from that of Fn (ψ∗ , ψ) , n ≥ 1 , where
Z
∗
1
Bn+1 (θ∗ , θ) =
dλ(ψ) N 1(n) e− L2 hθ∗ − Qψ , θ − Qψi−1 Fn (ψ ∗ , ψ)
C
X
(n)
0
We are expecting that, by induction,
Z
∗
1
ε
Bn+1 (θ∗ , θ) =
dλ(ψ) N 1(n) e− L2 hθ∗ − Qψ , θ − Qψi−1 Fn(SF) (ψ ∗ , ψ) + O e−1/v0
Sn
Z
∗
∗
1
ε
=
dλ(ψ) N (n)1 Zn e− L2 hθ∗ − Qψ , θ − Qψi−1 − An (ψ ,ψ, φ∗n ,φn ) + pn + O e−1/v0
Sn
=
Dominant Part
+
Non Perturbative Correction
(3)
We emphasise that Steps 1 and 6, which control the difference between Fn+1 (ψ∗ , ψ) and
(SF)
its, dominant, ‘small field’, part Fn+1
(ψ∗ , ψ), have not been proven, though we do supply
some motivation.
(n+1)
is ‘large field’, that is Ψ ∈
/ Sn+1 ,
Step 1 (Large field generates small factors). If Ψ ∈ CX0
−1 ∗
−1
−1/vε0
, since the exponent appearing in the
then we expect Bn+1 (S Ψ , S Ψ) = O e
1
integrand of (3) is of order vε . See [BFKT7, Proposition D.1, “Corollary” D.2 and the
0
subsequent Steps 1 and 2].
(n+1)
is ‘small field’, that is
We assume for the rest of the discussion that Ψ ∈ CX0
Ψ ∈ Sn+1 , and therefore construct holomorphic functions of (Ψ∗ , Ψ) on the product
Sn+1 × Sn+1 . Let θ∗ = S−1 Ψ∗ and θ = S−1 Ψ.
Step 2 (Holomorphic form representation). For each fixed (θ∗ , θ) , the ‘action’
n
o
An (θ∗ , θ, ψ∗ , ψ) = − L12 hθ∗ − Qψ∗ , θ − Qψi−1 − An ψ∗ , ψ, φ∗n (ψ∗ , ψ), φn(ψ∗ , ψ)
+ pn (ψ∗ , ψ, ∇ψ∗, ∇ψ)
= −An,eff (θ∗ , θ, ψ∗ , ψ) + pn (ψ∗ , ψ, ∇ψ∗ , ∇ψ)
is a holomorphic function of (ψ∗ , ψ) on Sn × Sn . By design, the Dominant Part of
Bn+1 (θ∗ , θ) is expressed as the integral of the holomorphic form
^ dψ∗ (x) ∧ dψ(x)
eAn (θ∗ ,θ, ψ∗ ,ψ)
2πı
(n)
x ∈ X0
10
(n)
of degree 2|X0 | over the real subspace in Sn × Sn given by ψ∗ = ψ ∗ . This representation permits a stationary phase analysis in which the ‘real contour’ is shifted, by Stokes’
theorem(12) , to a non real contour that contains the critical point of the principal terms of
the action. The shift will be implemented in Step 6.
Step 3 (Critical Points). There is a unique pair of holomorphic maps(13) ψ∗cr (θ∗ , θ) ,
ψ∗
ψcr (θ∗ , θ) from (S−1 Sn+1 ) × (S−1 Sn+1 ) to Sn such that the gradient ∇
of
∇ψ
An,eff (θ∗ , θ, ψ∗ , ψ) vanishes when ψ∗ = ψ∗cr (θ∗ , θ) , ψ = ψcr (θ∗ , θ) . This pair of ‘critical
field maps’ can be constructed by solving the critical point equations, a nonlinear parabolic
system of partial difference equations, using the natural contraction mapping argument to
perturb off of the linearized equations(14) . The analysis of the linearized equations is
based on a careful examination of some linear operators given in [BFKT8]. Beware that,
∗
in general, ψ∗cr (θ ∗ , θ) 6= ψcr (θ ∗ , θ) .
To start the stationary phase calculation, we factor the integral of the holomorphic
form over the real subspace (ψ∗ , ψ) ∈ Sn ×Sn ψ∗ = ψ ∗ as the product of
eAn (θ∗ ,θ, ψ∗cr (θ∗ ,θ),ψcr (θ∗ ,θ) )
and the ‘fluctuation integral’
Z
eAn (θ∗ ,θ, ψ∗ ,ψ) − An (θ∗ ,θ, ψ∗cr (θ∗ ,θ),ψcr (θ∗ ,θ) )
^
(n)
x ∈ X0
real subspace of
Sn ×Sn
dψ∗ (x) ∧ dψ(x)
2πı
Step 4 (The Value of the Action at the Critical Point). Let the ‘checked’ field
φ̌(∗)n+1 (θ∗ , θ) = φ(∗)n ψ∗cr (θ∗ , θ) , ψcr (θ∗ , θ)
be the background field evaluated at the critical point, and set
Ǎn+1 (θ∗ , θ, f∗ , f ) =
an+1
L2
hθ∗ − QQn f∗ , θ − QQn f i−1 − hf∗ , (∂0 + ∆ + µn )f in
+ Vn (f∗ , f )
(12)
The argument is similar to the use of Cauchy’s theorem in stationary phase arguments for functions
of one variable.
(13)
In [BFKT7] these maps are called ψn∗ , ψn .
(14)
In [BFKT7]
constructed
background
functions of
we take another route to the critical field maps. The background fields φ(∗)n (ψ∗ , ψ) are
first, using the natural contraction mapping argument to perturb off of the linearized
field equations. See [BFKT7, Proposition G.3]. The critical fields can be expressed as
the background fields. See [BFKT7, Proposition IV.4].
11
One can show (see [BFKT7, Definition B.2.c and Proposition B.3.c]) that the ‘checked’
fields are critical for the map (f∗ , f ) → Ǎn+1 (θ∗ , θ, f∗ , f ) and
An,eff (θ∗ , θ, ψ∗ , ψ) ψ
∗ =ψ∗cr (θ∗ , θ) , ψ=ψcr (θ∗ , θ)
= Ǎn+1 θ∗ , θ, φ̌∗n+1 (θ∗ , θ), φ̌n+1 (θ∗ , θ)
(See [BFKT7, Proposition IV.4.c].) Consequently,
eAn (θ∗ ,θ, ψ∗cr (θ∗ , θ),ψcr (θ∗ , θ)) = e−Ǎn+1 (θ∗ , θ, φ̌∗n+1 (θ∗ , θ), φ̌n+1 (θ∗ , θ)) + pn (ψ∗cr ,ψcr ,∇ψ∗cr ,∇ψcr )
Remark. Bear in mind that the checked fields depend implicitly on µn . In the next steps,
we will evaluate the fluctuation integral, up to corrections that we expect to be nonperturbatively small, and construct its logarithm, which will be almost entirely absorbed in
pn+1 . Quadratic terms (in (θ∗ , θ)) are also generated. They will be subsumed in the new
‘renormalized’ chemical potential µn+1 appearing in An+1 . If, for the purposes of discussion, we ignored the effects of renormalization, then the new chemical potential would
be, by scaling, just L2 µn . In this hypothetical scenario, An+1 would be a rescaled Ǎn+1
(see [BFKT7, Definition III.3 and Lemma III.4.c]) and the new background field φ(∗)n+1
would be a rescaled φ̌(∗)n+1 (see [BFKT7, Definition IV.2 and Proposition IV.4.b]). So,
we are not far off.
Step 5 (Diagonalization of the Quadratic Form in the Fluctuation Integral). If we change
variables (θ∗ , θ, ψ∗ , ψ) → θ∗ , θ, δψ∗ = ψ∗ − ψ∗cr (θ∗ , θ) , δψ = ψ − ψcr (θ∗ , θ) , then the
kernel (Hessian) of the quadratic term(15) in the Taylor expansion with respect to δψ∗ , δψ
of the difference
An,eff (θ∗ , θ, ψ∗ , ψ) − An,eff θ∗ , θ, ψ∗cr (θ∗ , θ), ψcr(θ∗ , θ)
has a dominant part (see [BFKT7, Lemma V.1, (V.13) and Lemma VI.5]) that is independent of θ∗ , θ and that has all of its eigenvalues bounded away from the negative real
axis, uniformly in n .(16) So, the dominant part is invertible and its inverse has a square
root D(n) , all of whose eigenvalues have strictly positive real parts. See [BFKT8, Corollary IV.5]. Now, the Taylor expansion of the difference of the effective actions in the new
T
variables δψ∗ = D(n) ζ∗ , δψ = D(n) ζ becomes
hζ∗ , ζi0 + smaller terms of degree 2 in ζ∗ , ζ + terms of degree at least 3 in ζ∗ , ζ
(15)
By criticality, the linear term vanishes.
(16)
A major part of [BFKT8] is devoted to proving this vital technical statement.
12
and the fluctuation integral becomes
Z
e− hζ∗ ,ζi0 + qn (θ∗ , θ, ζ∗ , ζ) det(D(n) )2
^ dζ∗ (x) ∧ dζ(x)
2πı
(n)
x ∈ X0
Ωn (θ∗ , θ)
(n)
(n)
where the domain of integration Ωn (θ∗ , θ) is the set of all pairs (ζ∗ , ζ) ∈ CX0 × CX0
T
such that ψ∗cr (θ∗ , θ)+D(n) ζ∗ , ψcr (θ∗ , θ)+D(n) ζ is in the real subspace of Sn ×Sn , and
the term qn is holomorphic on the complex domain of all quadruples (θ∗ , θ, ζ∗ , ζ) with
T
(θ∗ , θ) ∈ (S−1 Sn+1 )×(S−1 Sn+1 ) and ψ∗cr (θ∗ , θ)+D(n) ζ∗ , ψcr (θ∗ , θ)+D(n) ζ ∈ Sn ×Sn .
See [BFKT7, (V.4) and Corollary V.3].
Step 6 (Stokes’ Theorem). For each pair (θ∗ , θ) ∈ (S−1 Sn+1 ) × (S−1 Sn+1 ) , we construct,
(n)
in [BFKT7, Appendix D following (D.17)], a 2|X0 |+1 (real) dimensional cycle inside the
(ζ∗ , ζ) domain of analyticity of qn whose boundary consists of Ωn (θ∗ , θ) , the polydisk
Bn =
(ζ∗ , ζ) ζ∗ = ζ ∗ , |ζ(x)| <
1 Ln+1
4
v0
ε/2
(n) for all x ∈ X0
ε
and components on which e− hζ∗ ,ζi0 + qn (θ∗ , θ, ζ∗ , ζ) is O(e−1/v0 ) . See [BFKT7, (D.18)].
The holomorphic differential form in Step 5 has maximal rank and is therefore closed.
It follows from Stokes’ theorem that the fluctuation integral is equal to the small field
contribution
Z
∗
∗
(n) 2
dλ(ζ) e− hζ ,ζi0 + qn (θ∗ , θ, ζ , ζ)
F̌n (θ∗ , θ) = det(D )
Bn
plus corrections that are expected to be nonperturbatively small.
Step 7 (The Logarithm of the Fluctuation Integral). In [BFKT3] we developed a simple
variant of the polymer expansion that can be directly applied to the result of Step 6 to
h F̌ (θ , θ) i
n ∗
as an analytic function on (S−1 Sn+1 ) × (S−1 Sn+1 ).
obtain the logarithm Log
F̌n (0, 0)
See [BFKT7, Proposition VI.6].
Step 8 (Rescaling). We have
1
L2
−1
S Ψ∗ , S−1 Ψ −1 = hΨ∗ , Ψi0
−1
S f∗ , S−1 f n = L2 hf∗ , f in+1
SQQn S−1 = Qn+1
−1
S f∗ , (∂0 + ∆)S−1 f n = hf, , (∂0 + ∆)f in+1
See [BFKT7, Remark III.2.c and Lemma III.4.a]. So, by Steps 4, 6 and 7, the dominant
part of Fn+1 is expected to be
′
′
′
′
∗
(SF)
Fn+1
(Ψ∗ , Ψ) = e−An+1 (Ψ∗ , Ψ , φ∗n+1 (Ψ∗ ,Ψ) , φn+1 (Ψ∗ ,Ψ)) + pn+1 (Ψ∗ ,Ψ, ∇Ψ
13
,∇Ψ)
on Sn+1 × Sn+1 , where
A′n+1 Ψ∗ , Ψ, f∗ , f = Ǎn+1 S−1 Ψ∗ , S−1 Ψ, S−1 f∗ , S−1 f
= an+1 hΨ∗ − Qn+1 f∗ , Ψ − Qn+1 f i0 − f∗ , (∂0 +∆+L2 µn )f n+1
′
+ Vn+1
(f∗ , f )
φ′(∗)n+1 (Ψ∗ , Ψ) = S φ̌(∗)n+1 (S−1 Ψ∗ , S−1 Ψ)
and
p′n+1 Ψ∗ , Ψ, {Ψ∗ν }3ν=0 , {Ψν }3ν=0 = pn ψ∗cr , ψcr , ∇ψ∗cr , ∇ψcr
h F̌ (S−1 Ψ , S−1 Ψ) i
n
∗
+ Log
F̌n (0, 0)
(u)
′
If the kernel Vn of Vn were exactly the Vn of (2), then the kernel of Vn+1
would be exactly
(u)
−1
Vn+1 . See [BFKT7, Remark III.2.h]. Above, ψcr(∗) is a function of S Ψ∗ , S−1 Ψ and
the variables Ψ(∗)ν appear implicitly in ∇ψcr(∗) through the chain rule. Also, by Step 4,
the ‘primed’ fields φ′(∗)n+1 are critical for the map (f∗ , f ) 7→ A′n+1 ψ∗ , ψ, f∗ , f .
Step 9 (Renormalization of the Chemical Potential). At this point, we are close to the end
of the induction step, but not there yet because the power series p′n+1 contains (renormalization group) relevant contributions, in particular a quadratic term hΨ∗ , KΨi0 , where
(n+1)
to itself. By
K is a translation and reflection invariant linear operator mapping CX0
the discrete fundamental theorem of calculus, for any translation invariant K,
hψ∗ , Kψi0 = K hψ∗ , ψi0 +
3
X
hψ∗ , K ν (∂ν ψ)i0
ν=0
(n+1)
where K ∈ C and K ν , ν = 0, 1, 2, 3 , are linear operators on CX0
. See [BFKT7,
P3
ν
Corollary F.2]. By reflection invariance, K is real and ν=1 hψ∗ , K (∂ν ψ)i0 can be rewritten as a sum of marginal and irrelevant monomials. See [BFKT7, Lemma F.3.c].
Now the chemical potential term in A′n+1 (ψ∗ , ψ , φ′∗n+1 (ψ∗ , ψ) , φ′n+1 (ψ∗ , ψ)) is expressed in terms of φ′(∗)n+1 (ψ∗ , ψ) rather than directly in terms of ψ(∗) . But because
φ′(∗)n+1 (ψ∗ , ψ) = B(∗) ψ(∗) plus terms of degree at least three in (ψ∗ , ψ) (see [BFKT7,
Proposition G.3.a]) and because the linear operators B(∗) have left inverses (see [BFKT8,
Lemma V.7] and the beginning of the proof of [BFKT7, Lemma VII.3]), one can
show that(17) K hψ∗ , ψi0 = K′ φ′∗n+1 (ψ∗ , ψ), φ′n+1 (ψ∗ , ψ) n+1 plus a power series in
(17)
For reasons that will be explained shortly, we do not actually use this fact expressed in this way.
14
ψ∗ , ψ, ∇ψ∗ , ∇ψ that converges on the desired domain of analyticity and does not contain
any relevant contributions. See [BFKT7, Lemma VII.3]. We have
p′n+1 = K′ φ′∗n+1 , φ′n+1 n+1 + p′′n+1
where p′′n+1 has no hψ∗ , ψi0 term. Moving K′ φ′∗n+1 , φ′n+1 n+1 from p′n+1 into A′n+1 ,
we obtain
−A′n+1 ψ ∗ , ψ, φ′∗n+1 , φ′n+1 + p′n+1 = −A′′n+1 ψ ∗ , ψ, φ′∗n+1 , φ′n+1 + p′′n+1
with
A′′n+1 ψ ∗ , ψ, f∗ , f = an+1 hψ∗ − Qn+1 f∗ , ψ − Qn+1 f i0
′
(f∗ , f )
− f∗ , (∂0 + ∆ + (L2 µn + K′ )f n+1 + Vn+1
But we are still not done. The prime fields φ′∗n+1 (ψ∗ , ψ), φ′n+1 (ψ∗ , ψ) are background
fields for A′n+1 , which has chemical potential L2 µn , and not for A′′n+1 , which has chemical
potential L2 µn + K′ . That is, the prime fields are critical for f∗ , f 7→ A′n+1 ψ∗ , ψ, f∗ , f
and not for f∗ , f 7→ A′′n+1 ψ∗ , ψ, f∗ , f , as they must be to have An+1 = A′′n+1 . One can
construct field maps φ′′∗n+1 (ψ∗ , ψ), φ′′n+1 (ψ∗ , ψ) that are critical for A′′n+1 and substitute
the double primed fields for the primed fields at the cost of another power series of correction terms that is combined with p′′n+1 . But the correction terms include a quadratic term.
The way out of this vicious circle is a fixed point argument that yields a self consistent
µn+1 ≈ L2 µn . See [BFKT7, Lemmas VII.2 and VII.6].
So far we have skirted the issue of bounding the perturbative correction pn in our
main result. To measure the size of pn , we introduce a norm whose finiteness implies that
all the kernels in its power series representation are small with v0 and decay exponentially
(n)
as their arguments separate in X0 . For this purpose, write
X
X
pn r (x) ψ̃ r (x)
pn (ψ̃) =
r ∈ IN10
0
r6=0
(n) |r|
x∈ (X0
)
with the notations, IN0 = IN ∪ {0}, ψ̃ = ψ∗ , ψ, {ψ∗ν }3ν=0 , {ψν }3ν=0
and
ψ̃ r (x) = ψ∗ (x1 ) · · · ψ∗ (xr1 ) ψ(xr1 +1 ) · · · ψ(xr1 +r2 ) · · · ψ3 (x|r|−r10 +1 ) · · · ψ3 (x|r| )
Each kernel pn r (x) is separately symmetric in each block xr1 +...+rj +1 , . . . , xr1 +...+rj+1
of arguments. The norm of pn is
X
|r|−r1 −r2
kpn r km κnr1 +r2 κ′n
kpn k(n) =
r ∈ IN10
r6=0
15
For a translation invariant kernel with four arguments, like the interaction kernel
V0 (x1 , x2 , x3 , x4 ) , kV0 km is the exponentially weighted tree length L1 –L∞ norm of V0 :
kV0 km = max
sup
j=1,2,3,4 xj ∈X0
X
|V0 (x1 , x2 , x3 , x4 )| emτ (x1 ,x2 ,x3 ,x4 )
xk ∈X0
k6=j
where τ (x1 , x2 , x3 , x4 ) is the minimal length of a tree graph in X0 that has x1 , x2 , x3 , x4
among its vertices and m > 0 is a fixed decay rate. (The small ‘coupling constant’
v0 = 2kV0 k2m .) The norm kwkm of a kernel w with an arbitrary number of arguments
is defined in much the same way. For details see [BFKT7, §I.4 and Definition I.11].
Ideally, kpn k(n) would be bounded (and in fact small) uniformly in n . Unfortunately,
such a bound is too naive to achieve the upper limit on n stated in our main result. The
reason is that, while the coefficient of an irrelevant monomial decreases as the scale n
increases, the maximum allowed size of fields in the domain Sn also increases, so the
monomial as a whole can be relatively large. So we have chosen
◦ to move all quartic ψ∗ ψ)2 monomials out of pn into An , i.e. to also renormalize the
interaction Vn , and
◦ to split pn into two parts,
◦ one, called En (ψ∗ , ψ), is an analytic function whose size is measured in terms of
a norm like k · k(n) and
◦ the other, called Rn , is a polynomial of fixed degree, the size of whose coefficient
kernels are measured in terms of a norm like k · km .
The details are stated in our main result, [BFKT7, Theorem I.19].
16
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18