arXiv:1302.1622v2 [hep-lat] 22 May 2013
Thirring model at finite density in 0 + 1 dimensions
with stochastic quantization: Crosscheck with an exact solution
Jan M. Pawlowski
1, 2
and Chr istian Zielinski
1, ∗
1
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
2
ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany
(Dated: July 8, 2018)
We consider a generalized Thirring model in 0 + 1 dimensions at finite density. In order to
deal with the resulting sign problem we employ stochastic quantization, i.e., a complex Langevin
evolution. We investigate the convergence properties of this approach and check in which parameter
regions complex Langevin evolutions are applicable in this setting. To this end we derive numerous
analytical results and compare directly with numerical results. In addition we employ indirect
indicators to check for correctness. Finally we interpret and discuss our findings.
PACS numbers: 05.50.+q, 71.10.Fd
Keywords: Thirring model, finite density field theory, complex Langevin evolution, stochastic quantization
I. INTRODUCTION
Despite all effor ts, one of the outstanding problems of
lattice field theory until this day is the sign problem. The
introduction of a finite chemical potential µ > 0 renders
the path integral measure complex and rapidly oscillating
in many theories of interest, like quantum chromodynam-
ics (QCD) in 3 + 1 dimensions. The oscillatory behavior
significantly increases the numerical costs, in particular
in the continuum limit. A hard sign pr oblem ex ists for
theories where the costs grow more than polynomially
with the volume. This obstacle hinders numerical ab ini-
tio studies of str ongly interacting matter under extreme
conditions and the understanding of the phas e diagram of
QCD. There is no satisfactory solution known to the sign
problem, despite the large number of propose d solutions.
Among the proposed solutions we can find reweighting
techniques, Taylor expansions about µ = 0, ex trapola-
tions from imaginary chemical potential, the introduction
of dual variables and a canonical ensemble approach. For
recent reviews see e.g. [
1, 2]. However, due to the over-
lap problem reweighting techniques are computationally
expensive and can only be used for small µ, while the
numerical determination of Taylor coefficients is no isy
and the expa nsion converges slowly [
3]. Also the contin-
uation from imaginary chemical potential is a nontrivial
task [
4]. The applica tion of dual variables and the canon-
ical ensemble approach is still under active research, see
e.g. [
5, 6] for dua l observables and [7, 8 ] for simulations
with canonical ensembles.
In this paper we employ a different a pproach. Parisi
proposed already in 1983 that stochastic quantization
[
9]—for a review see e.g. [10]—could circumvent the sign
problem in terms of a complex Langevin evolution [1 1].
However, it is well known that the Langevin evolution
may converg e towards unphysical fixed points. It has
∗
Permanent address: Division of Mathematical Sciences, Nanyang
Technological University, Singap ore 637371
bee n successfully applied to the SU(3) spin model [12, 13],
to an effective theo ry of QCD in the strong- c oupling limit
[
14], simple models of quantum chromodynamics [15, 16]
and to the relativistic Bose gas [17, 18] at finite den-
sity. Furthermore it has been applied to quantum fields
in Minkowski time [
19, 20], also in nonequilibrium [21].
Counterexamples are given by the three-dimensional XY
model at finite chemical potential for small β [
22] and in
cases of gauge theories with static charges [
23]. Early in-
vestigations of complex Langevin evolutions can be found
in [
24–26], while for reviews see e.g. [27, 28]. Rec e ntly,
a set of consistency conditions indicating correct conver-
gence could be derived [29–31]. When truncating this
infinite towe r of identities one obtains necessary condi-
tions for correctness.
In this work we apply a complex Langevin evolution
to a generalized Thirring model at finite density. Here it
serves us as a model theor y to check for the applicabil-
ity of this method. Our results extend the studies ca r-
ried out in [
32], which led to ambiguous results. In this
paper we restrict our selves to the case of 0 + 1 dimen-
sions and dea l with the ques tion of whether a complex
Langevin evolution can enable finite density c alculations
in this setting. Further investigations of this approach
in the 2 + 1-dimensional generalized Thirring model are
presented in [
33]. The 2 + 1-dimensional model appears
for example in effective theories of high temperature su-
perconductors and graphene, see e.g. [34] and references
given therein. It is also worth mentioning that in the
case of the three-dimensional massless Thirring model, a
fermion bag approach was successfully applied in [
35].
We organized the paper as follows: In Sec .
II we intro-
duce a generalized Thirring model and its formulation on
the lattice. We discuss the Langev in equation and its nu-
merical implementation. In Sec.
III we present a closed
expression for the partition function of the lattice theory
and derive some obser vables of interest. We also discuss
additional indicators to evaluate the convergence prop-
erties of the complex Langevin e volution. In Sec .
IV we
discuss the results of the numerical part o f this wor k and
aim to answer the question in which parameter regime
2
results are reliable. We end this paper with concluding
remarks in Sec. V.
II. THE GENERALIZED THIRRING MODEL
A. Continuum formulation
We consider a generalization o f the Thirring mo del.
The historical model was intro duced in 1958 by Walter
E. Thirring and is one of the rare examples of an exactly
solvable quantum field theory [
36]. While the original
model describ es self-interaction fermions in 1 + 1 dimen-
sions, we c onsider N
f
fermion flavors at finite density.
We begin with a ge neralization to d dimensions and
then later specialize to the case of 0 + 1 dimensions. The
Euclidean Lagr angian in the continuum reads
L
Ψ
=
N
f
X
i=1
Ψ
i
/
∂ + m
i
+ µ
i
γ
0
Ψ
i
+
g
2
2N
f
N
f
X
i=1
Ψ
i
γ
ν
Ψ
i
2
. (1)
The index i = 1, . . . , N
f
enumerates fermion flavors, m
i
and µ
i
denote the bare mass and bare fer mion chemi-
cal potential of the r e spective flavor a nd g
2
is the bare
coupling s trength. The γ matrices satisfy the Clifford
algebra {γ
µ
, γ
ν
} = 2δ
µν
1
.
The model shows breaking of chiral symmetry at µ = 0
in 2 + 1 dimensions [
37]. For the 1 + 1-dimensional
Thirring model, the e quivalence to the sine-Gordon
model can be shown [
38, 39].
The four-po int intera ction can be resolved w ith the
introduction of an auxiliary field A
ν
. This formulation
reads
L =
X
i
Ψ
i
/
∂ + i
/
A + m
i
+ µ
i
γ
0
Ψ
i
+ N
f
βA
2
ν
. (2)
Here we introduced the inverse coupling β = 1/
2g
2
.
When integrating A
ν
out, we recover (
1). Although the
auxiliary field A
ν
is not a gauge field, the model can be
interpreted as a more ge neral gauge theory after gauge
fixing, see e.g. [
40]. After integrating out the fermionic
degrees of freedom we find
Z =
ˆ
DA
Y
i
det K
i
!
e
−S
A
=
ˆ
DA e
−S
eff
,
S
A
= N
f
β
1/T
ˆ
0
dt
ˆ
d
d−1
x A
2
ν
. (3)
Here we introduced the temperature T and K
i
=
/
∂ +
i
/
A+ m
i
+ µ
i
γ
0
. Including the fermion determinant in the
exponential term yields
S
eff
= S
A
−
X
i
Tr log K
i
. (4)
For the fermion determinant the relation
det K
i
(µ) = [det K
i
(−µ
⋆
)]
⋆
(5)
holds, thus rendering the path integr al measure complex
for µ > 0. At vanishing or purely imaginary chemical
potential, the determinant is real and the theory is free
of a sign problem. If the fermion determina nt is replaced
by its modulus, we refer to this as the phase-quenched
case. Physically this corr e sponds to the introduction of
an isospin chemical potential.
Like quantum chromodynamics, the Thirring model
exhibits Silver Bla z e behavior [
41, 42]. It implies that
at vanishing temperature there is a thre shold µ
c
, so that
observables are independent of the chemical potential µ
for µ < µ
c
. While in the full theory the onset is given by
the physical fermion mass m
phys
, in the phase-quenched
theory we have µ
c
= m
π
/2, where m
π
is the physical
pion mass.
B. Lattice formulation
We consider the case of 0 + 1 dimensions—
corresponding to a quantum mechanical system—with
lattice spacing a and N
t
lattice po ints. We employ stag-
gered fermions [
43–46] and denote the number of lattice
flavors, i.e., the number of staggere d fermion fields, by N.
Furthermore we assume that N
t
is even, as otherwise the
formulation of stagge red fermions is conceptually prob-
lematic and (
5) is violated. In order to introduce a finite
chemical potential µ, we use the prescription by Hasen-
fratz and Ka rsch [
47]. Fo r notational ease we refer to
the one-component auxiliary field as A
t
= A
0
(x = t)
and all dimensionful quantities are scaled dimensionless
by appropriate powers of a. Furthermore we introduce
the hopping parameter κ = 1/ (2m). The tempera tur e
corresponds to the inverse temporal extension T = N
−1
t
.
Using this fo rmulation the lattice partition function
reads
Z =
∞
ˆ
−∞
N
t
Y
t=1
dA
t
Y
i
det K
i
!
e
−S
A
(6)
with S
A
=
1
2
Nβ
P
t
A
2
t
and flavor index i = 1, . . . , N.
The fermion matrix takes the form
K
i
(t, τ) =
1
2
(1 + iA
t
) e
µ
i
δ
t+1,τ
−
1
2
(1 − iA
τ
) e
−µ
i
δ
t−1,τ
+ m
i
δ
tτ
, (7)
where we impose antiperiodic boundary conditions,
cf. [
32, 48]. In our analysis we focus on a few observ-
ables, namely the fermion density and condensate, the
3
energy density and the phase factor o f the fermion de-
terminant. In the following, sums over the flavor index i
are not implied. The fermion density of a given flavor is
given by
hn
i
i =
1
N
t
∂ log Z
∂µ
i
T
=
1
N
t
Tr
∂K
i
∂µ
i
K
−1
i
. (8)
The fermion condensate follows from
h
χ
i
χ
i
i =
1
N
t
∂ log Z
∂m
i
T,µ
i
=
1
N
t
Tr K
−1
i
(9)
and the energy density reads
hε
i
i = −
∂ log Z
∂N
t
µ
i
+ µ
i
hn
i
i, (10)
which we normalize to hε
i
i(µ = 0) = 0.
The phase factor of the determinant is defined by
exp (iφ) = det K/ |det K|. It can be expres sed in terms
of the partition function
Z
N
=
∞
ˆ
−∞
Y
t
dA
t
(det K)
N
e
−S
A
, (11)
for N degenerated flavors and Z
pq
N
for the phase-
quenched case, where the fermion determinant in (
11)
is r e placed by its modulus. The expectation va lue of
exp (iNφ) follows in the N flavor phase-q uenched theory
[
49, 50] as
e
iN φ
pq
N
=
Z
N
Z
pq
N
∈ [0, 1] . (12)
A value close to zero indicates a rapidly oscillating path
integral measure with a severe s ign problem.
C. Complex Langevin evolution
The idea of stochastic quantization is that observables
in a Euclidean quantum field theory can be obtained as
the equilibrium values of a statistical system coupled to
a heat bath [
10]. The problem of quantizing a field the-
ory is then reduced to finding the static solutions of an
associated Langevin equation. If the action is real and
bounded from below, c orrectness of this approach can be
ensured. We can also formally generalize to the case of a
complex action [
11]. This situation naturally arises when
considering field theor ie s at finite density. Until this day
there is a lack of rigor mathematical understa nding re-
garding the validity of this procedure. However, in cases
where it is converging correctly one has a very elegant
solution for the sign problem at hand.
We aim to check for the applicability of complex
Langevin evolutions to the Thirring model. To this end
we have to find the static solution of the Langevin equa-
tion
∂
∂Θ
A
t
(Θ) = −
δS
eff
[A]
δA
t
(Θ)
+
√
2 η
t
(Θ) , (13)
where Θ denotes a fictitious time. The noise term η
t
(Θ)
follows a Gaussian distribution with
hη
t
(Θ)i = 0,
hη
t
(Θ) η
t
′
(Θ
′
)i = δ (t − t
′
) δ (Θ −Θ
′
) . (14)
A simple approach to solve the Langevin equation nu-
merically is a first order integration scheme with fixed
stepsize ǫ
L
. Higher order integration schemes of O(ǫ
3/2
L
)
have been employed in the literature too [
13, 51]. How-
ever, in some models fixe d stepsize integration schemes
fail due to the occurrence of r un-away trajectories, which
can be avoided by the use of an ada ptive stepsize [
52, 53].
Although a constant stepsize proved here to be sufficient
[
32], we employ an adaptive stepsize algorithm due to
better convergence properties. For N deg enerated fla -
vors our discretization of (
13) reads
A
t
(Θ + ǫ
L
) = A
t
(Θ) + ǫ
L
D
t
(Θ) +
√
2ǫ
L
η
t
(Θ) (15)
with drift term
D
t
(Θ) = −NβA
t
(Θ)
+
Ni
2
K
−1
(t + 1, t) e
µ
+ K
−1
(t, t + 1) e
−µ
. (16)
After each integration step the stepsize ǫ
L
will be up-
dated according to
ǫ
L
≡ ǫ
L
(Θ) =
δ
max
t
|D
t
(Θ)|
(17)
with stepsize parameter δ = 10
−3
(compare to [
32]).
It is possible to generalize the real noise term in (15)
to an imagina ry one [
32] via the replacement
η
t
(Θ) →
√
I + 1 Re η
t
(Θ) + i
√
I Im η
t
(Θ) (18)
with I ≥ 0. The noise correlators then read
hRe η
t
(Θ) Re η
t
′
(Θ
′
)i = hIm η
t
(Θ) Im η
t
′
(Θ
′
)i
= δ (t − t
′
) δ (Θ −Θ
′
) (19)
and hRe η
t
(Θ) Im η
t
′
(Θ
′
)i = 0. Assuming correctness of
the complex Langevin evolution and numerical stability,
we expect ex pectation values to be independent of I.
III. ANALYTICAL RESULTS
A. Exact partition function
We begin with the partition function for one staggered
fermion field, i.e., N = 1. We incorpo rate antiperiodic
boundary conditions and for brevity we introduce
B
±
=
1
2 (2κ)
N
t
1 ±
p
B
c
/β
N
t
,
4
(a) Plot of the fermion density hni.
(b) Plot of the fermion condensate hχχi.
Figure 1: In the phase structur e in d = 0 + 1 we find a
condensed phase for large µ.
B
c
= β + 4 (β + 1) κ
2
. (20)
Then the partition function (
6) reads
Z
1
= 2
π
2β
N
t
/2
[B
+
+ B
−
+ cosh (N
t
µ)] . (21)
This can be shown for example by systematic saturation
of the Grassmann integral or the help of the determinant
identities in [
54]. For the fermion density we find
hni =
sinh (µ/T )
B
+
+ B
−
+ cosh (µ/T )
, (22)
while the fermion condensate is given by
h
χχi =
2κ
p
β/B
c
(B
+
− B
−
)
B
+
+ B
−
+ cosh (µ/T )
. (23)
The expression for the energy density is rather lengthy
and we will