
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