ECT*
Marco Cristoforetti
ECT*
Abhishek Mukherjee
ECT*
Luigi Scorzato
INFN/TIFPA
Francesco Di Renzo
University of Parma
PRD86, 074506 (2012)
PRD Rapid 88, 051501 (2013)
PRD Rapid 88, 051502 (2013)
Centre for Information Technology - CIT_irst
Centre for Materials and Microsystems - CMM
Centre for Italian-German Historical Studies Centre for Religious Sciences - ISR
Lattice QCD at finite
temperature and density
KEK - January 21 - 2014
Quantum field theory
on the Lefschetz thimble
FBK Research Centres
European Centre for Theoretical Studies in Nu
Introduction: dense systems
Nuclear physics
High energy physics
Condensed matter
Introduction: dense systems
!
Studied using Monte Carlo simulations
Introduction: dense systems
Condensed matter
High Tc superconductivity
Hubbard model
h
(HK +HV +Hµ )
Z = Tr e
HK =
Hµ =
t
X
i
(c†i cj + c†j ci )
hi,ji,
µ
HV = U
X
(ni" + ni# )
i
X✓
i
ni"
◆✓
1
ni#
2
1
2
◆
.
Introduction: dense systems
Nuclear physics
stellar nucleosynthesis
Shell model
h
(HM F +HVres )
Z = Tr e
HM F =
X
✏↵ a†↵ a↵
i
↵
HVres
1 X
=
h↵ |V |
2
↵
ia†↵ a† a a
Introduction: dense systems
High energy physics
Heavy ion collisions
QCD on the lattice
Z=
Z
DU det[M (U )]Nf e
SG [U ] =
3
XX
n2⇤ µ<⌫
Re tr[1
SG [U ]
Uµ⌫ (n)]
Introduction: sign problem
Hubbard Model
Shell Model
QCD
!
What do they have in common?
SIGN PROBLEM
Introduction: dense systems
Z=
DA det[D
/ +m
The sign problem
at finite chemical potential
the fermionic determinant is complex:
standard Monte Carlo methods fails
det M(μ) = ¦det M(μ)¦ eiθ
µ 0 /2]eSY M
[det M(μ)]* = det M(-μ*)
Unfortunately we cannot simply neglect the
phase of the determinant. Phase quenched
theory can be very different from the real
world
An example of that difference that we will treat later is
the Silver Blaze phenomenon
Introduction: Lefschetz thimble
We want to overcome sign problem for Lattice QCD
We must be extremely careful not destroying physics
[ Silver Blaze phenomenon ]
whatever machinery we use to solve the theory
Integration on a Lefschetz thimble
Before applying the idea to full QCD we choose to
start from something more manageable: we consider
here integration on Lefschetz thimbles for the case
of a simple 0-dim field theory and
the 4-dim scalar field with a quartic interaction
Lefschetz thimble on a lattice
Saddle point integration
the Airy function
1
Ai(x) =
2⇡
Z
1
1
e
i
⇣
t3
3
⌘
+x t dt

Re e
1.0
tI
i
⇣
t3
3
+x t
⌘
0.5
0.0
-0.5
tR
-1.0
-10
-5
0
5
10
Lefschetz thimble on a lattice
Saddle point integration
the Airy function
1
Ai(x) =
2⇡
Z
complexify
1
e
i
1
1
2⇡
⇣
t3
3
Z
⌘
+x t dt
e
i
⇣
z3
3
+xz
⌘
dz
Complexify the variable t ! tR + i tI
Stationary point
Steepest descent for the real part of the
exponent starting at the stationary point
Imaginary part of the exponent is constant
integrate on SD
1 i
e
2⇡
Z
e
h ⇣ 3
⌘i
z
R i 3 +xz
dz
tI
SR
steepest descent
tR
Lefschetz thimble on a lattice
Saddle point integration
the Airy function
1
Ai(x) =
2⇡
Z
1
1
e
i
⇣
t3
3
Re e
⌘
+x t dt
1.0
i
⇣
t3
3
+x t
⌘
0.5
Complexify the variable t ! tR + i tI
Stationary point
Steepest descent for the real part of the
exponent starting at the stationary point
Imaginary part of the exponent is constant
tI
steepest descent

0.0
-0.5
-1.0
-10
SR
-5
0
5
10
0.5
0.4
0.3
0.2
0.1
tR
0.0
-4
-2
0
2
4
From saddle point to Lefschetz thimble
Saddle point integration
!
Works extremely well for low dimensional oscillating integrals.
!
Usually combined with an asymptotic expansion around the stationary
point (sort of perturbative expansion).
!
The phase is stationary +
important contributions localized =
good for sign problem
tI
SR
What about a Monte Carlo integral
along the curves of steepest descent
tR
Lefschetz thimble on a lattice
Path integral and Morse theory
E. Witten arXiv:1009.6032 (2010)
Complexify
the degrees of freedom
Deform appropriately the
original integration path
(Morse theory)
L
C=
( )
R
z = x + iy
( )
( )
( )
C
( )
( )
=
C
( )
( )
L
for each stationary point pσ the Lσ (thimble) is the union
of the paths of steepest descent that fall in pσ at
L
the thimbles provide a basis of the relevant
homology group, with integer coefficients
Generalization of the
one dimensional SD to
n-dim problems is
called Lefschetz thimble
Lefschetz thimble on a lattice
Path integral and Morse theory
E. Witten arXiv:1009.6032 (2010)
Complexify
the degrees of freedom
( )
R
( )
C
( )
=
( )
( )
C
Deform appropriately the
original integration path
(Morse theory)
( )
z = x + iy
steepest descent
tI
( )
( )
L
tR
# of intersections between steepest
ascent and original integration domain
steepest ascent
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
Can we use the thimble basis to compute
the path integral for a QFT ?
R Q
d
CR x
Q
hOi =
C
xe
xd
S[ ]
xe
O[ ]
S[ ]
hOi =
P
n
P
R
n
J
R
Q
J
In principle yes but we have to discuss the details
xd
Q
xe
xd
S[ ]
xe
O[ ]
S[ ]
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
R Q
d
CR x
Q
hOi =
C
S[ ]
xe
x d xe
O[ ]
S[ ]
hOi =
P
n
P
R
n
J
R
Q
J
xd
Q
xe
xd
S[ ]
xe
Computing the contribution from all the thimbles
is probably not feasible
On a Lefschetz thimble the imaginary part of the action is constant
but the measure term does introduce a new residual phase, due to the
curvature of the thimble
O[ ]
S[ ]
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
the residual phase
hOi =
R
J0
R
Q
J0
xd
Q
xe
xd
S[ ]
xe
O[ ]
S[ ]
Additional phase coming from the Jacobian of
the transformation between the canonical
complex basis and the tangent space to the
thimble
Does it lead to a sign problem?
No formal proof but ...
dΦ=1 at leading order and <dΦ> 1 are strongly
suppressed by e-S
there is strong correlation between phase and weight
(precisely the lack of such correlation is the origin of the
sign problem)
In fact this residual phase is completely neglected in the
saddle point method
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
the residual phase
hOi =
R
J0
R
Q
J0
xd
Q
xe
xd
S[ ]
xe
O[ ]
S[ ]
Additional phase coming from the Jacobian of
the transformation between the canonical
complex basis and the tangent space to the
thimble
Does it lead to a sign problem?
Hopefully not but nevertheless the calculation of that phase
cannot be avoid!
highly demanding in terms of computation power
H. Fujii et al JHEP 1310 (2013) 147
Next talk: Y Kikukawa
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
R Q
d
CR x
Q
hOi =
C
S[ ]
xe
x d xe
O[ ]
S[ ]
hOi =
P
n
P
R
n
J
R
Q
J
xd
Q
Computing the contribution from all the thimbles
is probably not feasible
Is it necessary in order to have the correct physics?
I will try to convince you that in many cases we can
consider only one thimble
xe
xd
S[ ]
xe
O[ ]
S[ ]
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
choosing the stationary point
The study of the stationary points of the complexified theory is mandatory
and has to be done on a case-by-case
The system has a single global minimum?
There are degenerate global minima, that
are however connected by symmetries?
These cases
are good!
There is a large number of stationary points
that accumulate near the global minimum
giving a finite contribution?
This case
can be bad!
There are degenerate global minima, with
vanishing probability of tunneling?
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
choosing the stationary point
hOi =
P
n
P
R
n
J
R
Q
J
d
Q
x
xe
xd
S[ ]
xe
O[ ]
S[ ]
This is an exact reformulation of the original
path integral.
For a QFT reproducing the original integral
is both impractical and unnecessary
Consider the stationary point with the lower value
of the real part of the action and with nσ 0
Define a QFT on the thimble attached to this point. If
The degrees of freedom are the same
The symmetries are the same
The same perturbative expansion
The same continuum limit
By universality this is a legitimate regularisation of the original QFT
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
choosing the stationary point
hOi =
P
n
P
R
n
J
R
Q
J
d
Q
x
xe
xd
S[ ]
xe
O[ ]
S[ ]
This is an exact reformulation of the original
path integral.
For a QFT reproducing the original integral
is both impractical and unnecessary
Consider the stationary point with the lower value
of the real part of the action and with nσ 0
Define a QFT on the thimble attached to this point. If
The degrees of freedom are the same
The symmetries are the same
The same perturbative expansion
Universality is not a
theorem BUT it is an
assumed property
studying QFTs on a
lattice
The same continuum limit
By universality this is a legitimate regularisation of the original QFT
Lefschetz thimble on a lattice
Lefschetz thimbles and QFT
choosing the stationary point
hOi =
P
n
P
R
n
J
R
Q
J
xd
Q
xe
xd
S[ ]
xe
O[ ]
S[ ]
hOi =
Consider the stationary point with the lower value
of the real part of the action and with nσ 0
Define a QFT on the thimble attached to this point. If
The degrees of freedom are the same
The symmetries are the same
The same perturbative expansion
R
J0
R
Q
J0
xd
Q
xe
xd
S[ ]
xe
O[ ]
S[ ]
Universality is not a
theorem BUT it is an
assumed property
studying QFTs on a
lattice
The same continuum limit
By universality this is a legitimate regularisation of the original QFT
Lefschetz thimble: algorithm
Integration on a Lefschetz thimble
M. C., F. Di Renzo and L. Scorzato
PRD86, 074506 (2012)
Is it numerically applicable to QFT s on a Lattice?
Before applying the idea to full QCD we choose to
start from something more manageable: we consider
here integration on the Lefschetz thimbles for the
case of
a 0 dimensional field theory with U(1)
symmetry
the four dimensional scalar field with a
quartic interaction
Lefschetz thimble: algorithm
Langevin
PRD Rapid 88, 051501 (2013)
d
R
i (⌧ )
d⌧
d
I
i (⌧ )
d⌧
=
S R ( (⌧ ))
R
+
⌘
i (⌧ )
R (⌧ )
i
=
S R ( (⌧ ))
I
+
⌘
(⌧ )
i
I (⌧ )
i
d⌘i (⌧ ) X
=
⌘(⌧ )k @k @j SR
d⌧
k
projection of the noise
on the tangent space
Metropolis
PRD Rapid 88, 051502 (2013)
d i (r)
1
=
dr
r
S
i (r)
i (n
+ 1) =
i (n)
+ r
S
i
Other methods (example HMC)
H. Fujii et al JHEP 1310 (2013) 147
Next talk: Y Kikukawa
Lefschetz thimble: algorithm
Langevin
PRD Rapid 88, 051501 (2013)
We want to compute this:
1
hOi =
e
Z0
iSI
Z
boundend from below on J0
Y
d
xe
SR [ ]
J0 x
constant on J0
O[ ]
preserve J0
by
construction
We can use a Langevin
algorithm but how can
we stay on the thimble?
d
R
i (⌧ )
d⌧
d
I
i (⌧ )
d⌧
=
S R ( (⌧ ))
R
+
⌘
(⌧ )
i
R (⌧ )
i
=
S R ( (⌧ ))
I
+
⌘
i (⌧ )
I (⌧ )
i
Need to be
projected on
the tangent
space to J0
Lefschetz thimble: algorithm
Langevin
PRD Rapid 88, 051501 (2013)
preserve J0
by
construction
We can use a Langevin
algorithm but how can
we stay on the thimble?
d
R
i (⌧ )
d⌧
d
I
i (⌧ )
d⌧
=
S R ( (⌧ ))
R
+
⌘
(⌧ )
i
R (⌧ )
i
=
S R ( (⌧ ))
I
+
⌘
i (⌧ )
I (⌧ )
i
Need to be
projected on
the tangent
space to J0
The tangent space at the stationary point is easy to compute
!
We can get tangent vectors at any point if we can transport the noise along the
gradient flow so that it remains tangent to the thimble
L@SR (⌘) = 0
[@SR , ⌘] = 0
d⌘i (⌧ ) X
=
⌘(⌧ )k @k @j SR
d⌧
k
projection of the noise
on the tangent space
Lefschetz thimble: algorithm
Langevin
PRD Rapid 88, 051501 (2013)
d
R
i (⌧ )
d⌧
d
I
i (⌧ )
d⌧
=
S R ( (⌧ ))
R
+
⌘
i (⌧ )
R (⌧ )
i
=
S R ( (⌧ ))
I
+
⌘
(⌧ )
i
I (⌧ )
i
1
hOi =
e
Z0
iSI
d⌘i (⌧ ) X
=
⌘(⌧ )k @k @j SR
d⌧
Z
Y
J0 x
d
xe
SR [ ]
O[ ]
k
Start from the global minimum of the
real part of the action, generate a noise
vector projected on the thimble and
follow the steepest ascent
Perform a Langevin step using the noise
evolved along the steepest descent and
compute the observables
Go back along the steepest descent
until you are in a region where
quadratic approx. is valid and then
project the configuration on the thimble
Generate a new noise and go
back along the steepest ascent
* in the plot you have -SR that is the exponent in the
integrand of the partition function : where I wrote ascent
(descent) you see the opposite in the figure
Lefschetz thimble: algorithm
Langevin on the Lefschetz thimble
vs Complex Langevin
Lefschetz Langevin
d
R
i (⌧ )
d⌧
d
I
i (⌧ )
d⌧
Complex Langevin
=
S R ( (⌧ ))
R
+
⌘
i (⌧ )
R (⌧ )
i
d
=
S R ( (⌧ ))
I
+
⌘
(⌧ )
i
I (⌧ )
i
d
d⌧
I
i (⌧ )
d⌧
3
3
2
2
0
-1
-2
-2
-3
-3
0
1
2
fR
3
4
5
=
S R ( (⌧ ))
I
+
⌘
i (⌧ )
I (⌧ )
i
0
-1
-1
=
S R ( (⌧ ))
R
+
⌘
(⌧ )
i
R (⌧ )
i
1
fI
The relation between
the two approaches
should be studied
carefully!
1
fI
R
i (⌧ )
-1
0
1
2
fR
3
4
5
Lefschetz thimble: algorithm
Metropolis
PRD Rapid 88, 051502 (2013)
In the neighbourhood
of a critical point
3
i
S[ ] = S[ 0 ] + SG [⌘] + O(|⌘| )
1X
2
SG =
k ⌘k
2
Hwk =
0
i
+
wki ⌘k
k
G0 is the flat thimble
associated to the gaussian
action SG
k
The λ and w are
solutions of
=
X
where H is the Hessian
k w̄k
ηreal are the direction of steepest descent of SR and the equations of
steepest descent of η for the Gaussian action can be explicitly solved in term
of a new parameter r=e-τ
¯G
d⌘k
1 @S
1
=
=
dr
r @⌘k
r
but for r=ε infinitesimal
the Lefschetz and
Gaussian thimbles coincide
k ⌘k
i (✏)
=
0
i
+
¯
d i
1 @S
=
dr
r@ i
X
⌘k / r
k
✏ k wki ⌘k
k
r 2 [✏, 1]
Start with a random
real η vector, compute
Φ(ε) and evolve using
steepest descent
Lefschetz thimble: algorithm
Metropolis
PRD Rapid 88, 051502 (2013)
In the neighbourhood
of a critical point
3
i
S[ ] = S[ 0 ] + SG [⌘] + O(|⌘| )
1X
2
SG =
k ⌘k
2
η
Hwk =
k w̄k
¦η¦/δr
+
wki ⌘k
G0 is the flat thimble
associated to the gaussian
action SG
where H is the Hessian
n-dim random vector living on the
manifold defined by the
eigenvectors of the Hessian
computed at the critical point with
positive eigenvalues
¦η¦
0
i
k
k
The λ and w are
solutions of
=
X
distance along the thimble
number of steps along the steepest
descent
d i (r)
1
=
dr
r
S
i (r)
i (n
+ 1) =
i (n)
+ r
S
i
Lefschetz thimble: algorithm
Gaussian thimble
Gaussian manifold:
flat manifold defined by the directions of
steepest descent at the critical point
r!0
Lefschetz thimble
number of steps along the steepest
descent
Decreasing δr your manifold get closer and closer to the Lefschetz thimble
¦η¦/δr = N
If the action decreases fast away from the stationary point
integrating on the Gaussian thimble can be sufficient
U(1) one plaquette model
PRD Rapid 88, 051502 (2013)
Can be seen as the limiting case of the more
interesting three-dimensional XY model
One dimensional problem: the integration on the Lefschetz
thimble can be plotted
ACTION
S=
OBSERVABLE
i
2
U +U 1 =
i cos
J1 ( )
he i = i
J0 ( )
i
On the thimble
P
hO( )i = P
m
m
R
RJ
J
d O( )e
d ( )e
S( )
S( )
SR =
sin
SI =
cos
R
sinh
I
R cosh I
constant on the thimble
U(1) one plaquette model
PRD Rapid 88, 051502 (2013)
The stationary points are in (0,0) and (π,0) and the
thimble can be computed also analytically
Exact thimbles: have to pass from the critical point
and the imaginary part of the action has to be
constant
SI (⌧ ) =
cos
R (⌧ ) cosh
CRITICAL POINTS
THIMBLES
I (⌧ )
=
cp
SI
U(1) one plaquette model
PRD Rapid 88, 051502 (2013)
The stationary points are in (0,0) and (π,0) and the
thimble can be computed also analytically
In order to perform the integration on the thimble we use a
Metropolis algorithm
Gaussian manifold
Increasing the accuracy in the
integration of the steepest descent
we move closer to the exact
thimble
U(1) one plaquette model
PRD Rapid 88, 051502 (2013)
OBSERVABLE
J1 ( )
he i = i
J0 ( )
i
There are parameter regions where
integration on the Gaussian
manifold is sufficiently accurate
U(1) one plaquette model
PRD Rapid 88, 051502 (2013)
0.1
Residual phase is well under control and is not a source
0.5
1
of additional sign problem (at least in0 this case)
hO( )i = P
m
m
R
RJ
J
d O( )e
d ( )e
This phase should be essentially
constant over the portion of
phase space which dominates the
integral.
Residual phase
2
β
There is an additional phase coming from the
Jacobian of the transformation
between the
i
FIG. 2. Expectation
value of
e and
as athe
function
ofspace
.
canonical complex
basis
tangent
S( )
to the thimble
S( )
1
0.8
cos (arg {Jφη e-S })
P
1.5
0.6
0.4
0.2
0
0.0001
Probability measure
Gaussian
Nτ=6
Nτ=20
Nτ=200
0.001
Log scale
0.01
⏐Jφη e-S⏐
0.1
1
λΦ4 theory on the lattice
PRD Rapid 88, 051501 (2013)
[ ,
]=
(|
| +(
µ )| | + | | + µ(
)
Continuum action
T
Silver Blaze problem
when T=0 and μ<μc physics is
independent from the chemical potential
<n>=0
<n>≠0
μc
We will study the system at zero temperature
μ
λΦ4 theory on the lattice
PRD Rapid 88, 051501 (2013)
[ ,
[ ,
]=
(
]=
[(
µ
+
,
| +(
(|
)
+ˆ
+ (
+
+ˆ
µ
µ )| | + | | + µ(
Continuum action
Lattice action:
chemical potential introduced as
an imaginary constant vector
potential in the temporal direction
)
))
,
)
=
in term of real fields
[ ]=
( = , )
(
=
+
(
)
,
+
+
)
(
,
)
,
=
cosh µ
,
, +ˆ
+ sinh µ
,
, +ˆ
, +ˆ
λΦ4 theory on the lattice
PRD Rapid 88, 051501 (2013)
PHASE QUENCHED
[ ,
O
]=
full
=
(|
| +(
µ )| | + | | + µ(
D |e S |ei O
ei O pq
=
S
i
ei pq
D |e |e
Let us try ignoring the phase
O
pq
=
D |e S |O
D |e S |
)
0
0
λΦ4 theory on the lattice
PRD Rapid 88, 051501 (2013)
PHASE QUENCHED
[ ,
0.2
]=
| +(
(|
µ )| | + | | + µ(
1
L=4
L=6
L=10
L=4
L=6
L=10
0.8
Re < eiφ >
0.15
<n>
)
0.1
0.05
0.6
0.4
0.2
0
0
0
0.2
0.4
1
n =
V
0.6
μ
ln Z
µ
0.8
1
1.2
0
0.2
0.4
0.6
μ
0.8
1
λΦ4 on a Lefschetz thimble
PRD Rapid 88, 051501 (2013)
On the Lefschetz thimble
M. C., F. Di Renzo, A. Mukherjee and L. Scorzato
arXiv:1303.7204 (2013)
Fields are complexified
a
!
R
a
+i
I
a
The integration on the thimble performed with a Langevin
algorithm
In this case calculations in Gaussian approximation are sufficient
to obtain the exact result
λΦ4 on a Lefschetz thimble
PRD Rapid 88, 051501 (2013)
On the Lefschetz thimble
0.2
0.8
pq L=6
pq L=10
AA 64
AA 84
0.6
φ
i
Re[<e >]
0.15
<n>
1
pq L=6
pq L=10
AA 64
AA 84
0.1
0.05
0.4
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
0.5
µ
Silver Blaze
!
solving sign problem we have the correct physics
0.6
0.7
0.8
0.9
µ
1
1.1
1.2
λΦ4 on a Lefschetz thimble
On the Lefschetz thimble
0.7
0.6
AA 84
WA 84
Comparison with
Worm Algorithm
0.5
(courtesy of C. Gattringer
and T. Kloiber)
<n>
0.4
!
0.3
1
n =
V
0.2
0.1
0
-0.1
0.95
1
1.05
1.1
µ
1.15
1.2
1.25
ln Z
µ
What about QCD?
Consider the stationary point with the lower value
of the real part of the action and with nσ 0
Define a QFT on the thimble attached to this point.
In QCD this is the trivial vacuum
Complexification:
a,R
Aa⌫ (x) ! Aa,R
(x)
+
iA
⌫
⌫ (x)
a = 1...Nc
1
SU (3)4V ! SL(3, C)4V
Covariant derivative:
@
rx,⌫,a F [U ] :=
F [ei↵Ta U⌫ (x)]|↵=0
@↵
rx,⌫,a = rR
x,⌫,a
irIx,⌫,a
I
rx,⌫,a = rR
+
ir
x,⌫,a
x,⌫,a
Equation of steepest descent:
d
U⌫ (x; ⌧ ) = ( iTa rx,⌫,a S[U ])U⌫ (x; ⌧ )
d⌧
Defining the thimble for gauge theories is possible: substitute the concept of nondegenerate critical point with that of non-degenerate critical manifold
All the ingredients are there
What about QCD?
The symmetries are the same? Yes
It can be proved starting from the invariance of
the SD equation:
d
U⌫ (x; ⌧ ) = ( iTa rx,⌫,a S[U ])U⌫ (x; ⌧ )
d⌧
Under gauge transformations it changes as:
(Ta rx,⌫,a S[U ]) ! (⇤(x)
1 †
) (Ta rx,⌫,a S[U ])⇤(x)†
U⌫ (x) ! ⇤(x)U⌫ (x)⇤(x + ⌫ˆ)
1
The full SD equation is invariant only under the SU(3) subgroup of SL(3,C)
this is very interesting: the gauge links are not in SU(3) but the gauge
invariance is exactly the same!
The perturbative expansion is the same? Yes
It is an expansion around the trivial vacuum where the integrand in the
partition function has the form of a gaussian times polynomials
(let me skip the details)
Something else on a Lefschetz thimble
Next steps
- Theoretical questions (single thimble,
residual phase, reflection positivity ... )
!
- 0-dim Φ4 (finished)
!
- Chiral random matrix theory
!
- Thirring model
!
- Hubbard model (repulsive almost done)
!
- SU(3) pure gauge with theta term
!
- QCD in 0+1 dimension
!
-...
!
- QCD
thank you