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