arXiv:1302.2249v2 [hep-lat] 22 May 2013

Thirring model at ﬁnite density in 2 + 1 dimensions

with stochastic quantization

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: May 23, 2013)

We consider a generalization of the Thirring model in 2+1 dimensions at ﬁn ite density. We employ

stochastic quantization and check for the applicability in th e ﬁnite density case to circumvent the

sign problem. To this end we derive analytical results in the heavy dense limit and compare with

numerical ones obtained from a complex Langevin evolution. Furthermore we make use of indirect

indicators to check for incorrect convergence of the underlying complex Langevin evolution. The

method allows the numerical evaluation of observables at arbitrary values of the chemical potential.

We evaluate the results and compare to the (0 + 1)–dimensional case.

PACS numbers: 05.50.+q, 71.10.Fd

Keywords: Thirring model, ﬁnite density ﬁeld theory, complex Langevin evolution, stochastic quantization

I. INTRODUCTION

The sign problem remains one of the biggest challenges

of lattice ﬁeld theory until this day. It is c aused by

a highly os c illatory and complex path integral mea sure

after introducing a chemical potential µ. This renders

many theo ries, like four-dimensional quantum chromo-

dynamics (Q C D), inaccessible by commo n simulation al-

gorithms in wide regions of the phas e diagram. Many

solutions have been proposed, see e.g., [

1–8]. However,

reliable numerical calculations in theories with a severe

sign pr oblem remain extremely challenging.

One of the approaches to the sign problem is stochastic

quantization [

9], i.e., a complex Langevin evolution [10].

For a review of stochastic quantization see e.g., [

11], and

for its application to the ﬁnite density case refer to [

3].

It has been applied to many theories suﬀering from a se-

vere sign problem, often successfully [

12–18]. It was also

applied outside the context of ﬁnite density calculations,

e.g., quantum ﬁelds in nonequilibirum [

19] and in real

time [20, 21]. However, there are cases known where the

complex Langevin evolution converges towards unphysi-

cal ﬁxed points [

22, 23]. Starting from early studies of

complex Langevin evolutions [

24–26] until this day, the

convergence properties of Langevin equations ge neralized

to the complex case are not well understood. In this work

we focus on the stochastic quantization o f the Thirring

model at ﬁnite density in 2 + 1 dimensions, cf. [

27]. This

serves as a toy model for the matter sector of QCD. Fur-

thermore, the (2 + 1)–dimensional model appear s in ef-

fective theo ries of high temperature superconductors and

graphene, see e.g., references given in [

28].

Here we extend the (0 + 1)–dimensiona l studies car-

ried out in [

29]. In 2 + 1 dimensions we lose the analytic

benchmarks which facilitated the interpretation of the

∗

Permanent address: Division of Mathematical Sciences, Nanyang

Technological University, Singap ore 637371

numerical results with stochastic quantization in 0 + 1

dimensions. Still, we can exploit the similarities to the

lower-dimensional case. Further benchmarks of our nu-

merical results are given by those in the heavy dense

limit, as introduced in [

30]. This limit describes the

regime of large fermion masses and lar ge chemical poten-

tials. In addition, we evaluate indirect indicators, namely

the consistency requir e ments presented in [

31–33] and the

analyticity of the fermion condensate at µ

2

= 0, cf. [

13].

The paper is organized as follows: In Sec. II we intro-

duce a generalized Thirring model in the continuum and

on the lattice. We also formulate the associated Langevin

equation. In Sec.

III we employ the heavy dense limit

to derive numerous analytical results. At the end of the

section we introduce indirect indicators of correct conver-

gence, namely analyticity of observables at µ

2

= 0 and

a se t of co nsistency conditions. In Sec.

IV we discuss

numerical results and use the analytical results among

other indicators as a benchmark to evaluate the co mplex

Langevin evolution. Finally in Sec.

V we discuss and

summarize our ﬁndings.

II. THE GENERALIZED THIRRING MODEL

A. Continuum formulation

We begin with a short reca pitulatio n of the mo del,

which we introduced in [

29]. It is a generalization of

the historical 1 + 1 dimensional Thirring model [34], but

formulated in d dimensions with N

f

fermion ﬂavors at

ﬁnite density. The Euclidean Lagrangian 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)

2

Here i = 1 , . . . , N

f

is a ﬂavor index, m

i

and µ

i

are the

bare mass and bare chemical potential of the respective

ﬂavor a nd g

2

is the bare c oupling strength. The γ matri-

ces satisfy {γ

µ

, γ

ν

} = 2δ

µν

1

.

In d = 2 + 1 dimensions,

Ψ and Ψ denote four-

component spinors. While it is possible to use an irre-

ducible two-dimensional representation of the Dir ac alge-

bra, this representation does not allow for a chira l sym-

metry in the massless case. See [

28] for details and a

discussion of symmetries. It is noted that the model als o

shows breaking of chiral symmetry at vanishing chemical

potential [

35].

We reformulate the generalized Thirring model using

an auxillar y ﬁeld and integrate out the fermionic degrees

of free dom. We ﬁnd for the partition function

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

(2)

with temperature T , fermionic term K

i

=

/

∂ + i

/

A + m

i

+

µ

i

γ

0

and S

eﬀ

= S

A

−

P

i

Tr log K

i

. The fer mion deter-

minant obeys

det K

i

(µ) = [det K

i

(−µ

⋆

)]

⋆

, (3)

yielding in general a complex action. The sign prob-

lem can be avoided by taking the absolute value of the

fermion determinant. This corresponds to an isospin

chemical potential and is also referred to as the phase-

quenched case. We als o note that observables shall be-

come independent of the chemical potential µ up to some

threshold µ

c

in the zero- tempera tur e limit. This goes un-

der the name of the Silver Blaze problem [

36].

B. Lattice formulation

We consider the generalized Thirring model in 2 + 1

dimensions on a s pace-time lattice with N

t

time slices and

N

s

slices in spatial direction. Furthermor e , we require

that N

t

be even [29]. The spatial volume and the space-

time volume are denoted by

V = N

2

s

, Ω = N

t

N

2

s

. (4)

We use staggered fermions [

37–40], where the number of

staggered fermion ﬁelds—denoted also as lattice ﬂavors—

is given by N. The introduction of a chemical po tential

follows the prescription by Hasenfra tz and Karsch [

41].

In the following, all dimensionful quantities are measured

in appropriate powers of a, so that we only deal with

dimensionless parameters.

In three dimensions, N staggered fer mion ﬂavors cor-

respond to N

f

= 2N continuum ﬂavors [

27] as each stag-

gered fermion ﬁeld encodes two tastes. In principle the

partition function for a single continuum ﬂavor can be

obtained by taking the square root of the fermion deter-

minant, but it is still under debate if this ro oting pre-

scription is consistent [

42].

In d = 2 + 1 dimensions (where generalizations to ar-

bitrary d are evident), the lattice action we employ reads

S =

X

x,y,i

χ

i

(x) K

i

(x, y) χ

i

(y) +

Nβ

2

X

x,ν

A

2

ν

(x) . (5)

Here

χ

i

and χ

i

denote staggered fermion ﬁelds with ﬂavor

index i = 1, . . . , N, and the sums extend over x, y =

1, . . . , Ω and ν = 0, . . . , d − 1. The fermion matrix reads

K

i

(x, y) =

1

2

d−1

X

ν=0

ε

ν

(x)

(1 + iA

ν

(x)) e

µ

i

δ

ν0

δ

x+ˆν,y

−(1 − iA

ν

(y)) e

−µ

i

δ

ν0

δ

x−ˆν,y

+ m

i

δ

xy

(6)

with staggered phase factor

ε

ν

(x) = (−1)

P

ν−1

i=0

x

i

(7)

and ˆν denoting a unit vector in ν direction, cf. [

27, 43].

We impose periodic boundary conditions in spatial and

antiperiodic boundary conditions in temporal direction.

The lattice partition function is, like in the continuum in

(

2), given by

Z =

∞

ˆ

−∞

Y

x,v

dA

v

(x)

Y

i

det K

i

!

e

−S

A

, (8)

where S

A

=

1

2

Nβ

P

x,ν

A

2

ν

(x). The central observables

in our analysis are the fermion density, the fermion con-

densate, the energy density and the phase fa c tor of the

fermion determinant. In the following, sums over ﬂavor

indices are not implied. The fermion density of ﬂavor i

is given by

hn

i

i =

1

Ω

∂ log Z

∂µ

i

V,T

=

1

Ω

Tr

∂K

i

∂µ

i

K

−1

i

. (9)

The fermion condensate follows from

h

χ

i

χ

i

i =

1

Ω

∂ log Z

∂m

i

V,T,µ

i

=

1

Ω

Tr K

−1

i

, (10)

and the energy density reads

hε

i

i = −

∂ log Z

∂N

t

V,µ

i

+ µ

i

hn

i

i. (11)

Usually we normalize the latter one to hε

i

i(µ = 0) = 0.

The phase factor of the fermion determinant is deﬁned

by exp (iφ) = det K/ |det K| with phase φ. For N de-

generated staggered fermion ﬂavors we set K

i

= K. The

expectation value of exp (iNφ) [

44, 45] follows in the N

ﬂavor phase-quenched theory from

e

iN φ

pq

N

=

Z

N

Z

pq

N

∈ [0, 1] , (12)

3

where Z

N

is given by (8) and Z

pq

N

denotes the phase-

quenched partition function. The expectation value of

the fermion phase factor is a measure of the sign problem,

where smaller values indicate a more severe problem.

C. Complex Langevin equation

To deal with the associated sign problem we apply

stochastic quantization [

9], namely a co mplex Lange vin

evolution [

10], to the Thirring model at ﬁnite density.

To this end we determine the stationary solution of the

corresponding L angevin equation, which rea ds

∂

∂Θ

A

ν

(x, Θ) = −

δS

eﬀ

[A]

δA

ν

(x, Θ)

+

√

2 η

ν

(x, Θ) . (13)

Here Θ denotes ﬁctitious time, and η

ν

(x, Θ) is a Gaussian

noise with

hη

ν

(x, Θ)i = 0,

hη

ν

(x, Θ) η

σ

(x

′

, Θ

′

)i = δ

νσ

δ (x − x

′

) δ (Θ − Θ

′

) . (14)

We use an adaptive ﬁrst-order stepsize integration

scheme [

27, 46, 47], w here the stepsize ǫ

L

is adjusted

with respec t to the modulus of the drift term.

For the numerical treatment of the Langevin equation

we use a ﬁrst-order integration scheme. Our discretiza-

tion of (

13) reads

A

ν

(x, Θ + ǫ

L

) =

A

ν

(x, Θ) + ǫ

L

D

ν

(x, Θ) +

√

2ǫ

L

η

ν

(x, Θ) , (15)

where D

ν

(x, Θ) = −dS

eﬀ

/dA

ν

(x, Θ) is the drift term

and ǫ

L

the (adaptive) integration stepsize. The drift term

takes the ex plicit form

D

ν

(x, Θ) = −NβA

ν

(x, Θ)

+

i

2

ε

ν

(x)

X

i

K

−1

i

(x + ˆν, x) e

µ

i

δ

v0

+

i

2

ε

ν

(x + ˆν)

X

i

K

−1

i

(x, x + ˆν) e

−µ

i

δ

v0

, (16)

cf. [

27]. The stepsize is upda ted after each integration

step according to

ǫ

L

≡ ǫ

L

(Θ) =

δ

max

x,ν

|D

ν

(x, Θ)|

. (17)

Typically we use δ = 10

−2

, see also [

29].

For a correctly converging complex Langevin evolution

we can generalize the real noise to an imaginary noise

in (

13) while keeping expectation values unchanged [27].

We parametrize the complex noise using the repla c e ment

η

ν

(x, Θ) →

√

I + 1 Re η

ν

(x, Θ) + i

√

I Im η

ν

(x, Θ) (18)

with I > 0. Furthermore, we require that

hRe η

ν

(x, Θ) Re η

σ

(x

′

, Θ

′

)i

= hIm η

ν

(x, Θ) Im η

σ

(x

′

, Θ

′

)i

= δ

νσ

δ (x − x

′

) δ (Θ − Θ

′

) (19)

and

hRe η

ν

(x, Θ) Im η

σ

(x

′

, Θ

′

)i = 0. (20)

While ma intaining numerical stability, we check if ob-

servables turn out to be indepe ndent of I.

III. ANALYTICAL RESULTS

In the following we derive approximate expressions for

some observables in the lattice Thirring model in d di-

mensions. We begin with a hopping parameter expa nsion

and then take the so-called heavy dense limit in Sec.

III B.

This renders the model eﬀectively one-dimensional and

allows us to obtain a simple expression for the fermion

determinant. Later we discuss an extended version in

Sec.

III F.

A. Hopping parameter expansion

Applying a hopping parameter expansion to the

fermion determinant in (

8) yields

det K

m

Ω

=

Y

ℓ

Y

{C

ℓ

}

1 − κ

ℓ

γ

C

ℓ

P

C

ℓ

. (21)

Here κ = 1/ (2m) is the hopping parameter, {C

ℓ

} are

closed contours of perimeter ℓ, n is the number of times

a given contour is traced out and P

C

ℓ

is the product of

hopping terms M (x, y) at µ = 0 along the given contour

C

ℓ

. The matrix elements M (x, y) read

M (x, y ) = −

d−1

X

ν=0

ε

ν

(x) [(1 + iA

ν

(x)) δ

x+ˆν,y

−(1 − iA

ν

(y)) δ

x−ˆν,y

] , (22)

compare with the fermion matrix in (

6). Furthermore,

we represent the dependence on the chemical potential

explicitly via

γ

C

ℓ

= [−exp (µN

t

)]

!C

ℓ

, (23)

where

!C

ℓ

∈ Z (24)

denotes the temp oral winding number of the path C

ℓ

counted in positive direction. The minus sign in (

23)

stems from antiperiodic temporal boundary conditions.

4

B. Heavy dense limit

The heavy dense limit projects out the leading contri-

butions of the hopping parameter expans ion in the limit

of a large mass m and large chemical potential µ. This

limit was introduced in [30] and employe d e.g., in [15, 48–

53]. It is deﬁned by

κ → 0, µ → ∞, κe

µ

ﬁxed. (25)

In this regime the fermion determinant is dominated by

contributions from Polyakov loops in positive time di-

rection with !C = 1, s e e (

24). Due to the absence of

spatial paths, the model is eﬀectively one dimensional.

The fermionic contribution det K in this limit reads

det K

m

Ω

=

Y

C∈ P

(1 + ξP

C

) , (26)

where P denotes the set of Polyakov loops and

ξ ≡ ζ

N

t

, ζ ≡ κe

µ

. (27)

Let P

x

denote a Polyakov loop

P

x

=

N

t

Y

t=1

(1 + iA

0

(t, x)) (28)

starting and ending in the space point x. We can then

express the fermion determinant in the regime of m and

µ being la rge by

det K = m

Ω

Y

x

(1 + ξP

x

) , (29)

where Ω is the space-time volume deﬁned in (

4). Note

that in this limit the relation in (

3) is violated. Due

to this approximation we will ﬁnd hni

µ=0

→ 0 only for

m → ∞. In Sec.

III F we intro duce a modiﬁed version of

this limit, which preserves this relation.

C. The case of one ﬂavor

By replacing the full fermion determinants in (

8) with

the simpler expression in (29), we can derive analytical

expressions for several observables of interest. We be-

gin with one ﬂavor and later generalize to more ﬂavors.

In the heavy dense limit the pa rtition function can be

integrated exactly, and we ﬁnd

Z

1

=

∞

ˆ

−∞

Y

x,v

dA

v

(x) det Ke

−S

A

=

(1 + ξ)

V

(2κ)

Ω

2π

β

Ωd

2

.

(30)

Using (

9) and (10) the fermion density and condensate

read

hni =

1

1 +

1

ξ

, hχχi =

2κ

1 + ξ

. (31)

(a) Plot of the fermion density hni.

(b) Plot of the fermion condensate hχχi.

Figure 1: The phase structure in the heavy dense limit

for one ﬂavor.

We point out that the exact (0 + 1)–dimensional results

derived in [29] reproduce the above results in the cor-

responding limit. Applying (

11), the normalized energy

density reads

hεi =

ξ − κ

1/T

log κ

−1

1 + κ

1/T

(1 + ξ)

, (32)

5

where we identify the temperature with T = N

−1

t

. Fur-

thermore, we ﬁnd

c

V

=

1

V T

2

∂

2

log Z

1

∂N

2

t

V,µ

=

ξ log

2

ζ

T

2

(1 + ξ)

2

> 0 (33)

for the heat capacity. The cor responding mechanical

equation of state P V = T log Z

1

takes the form

P = log

"

(1 + ξ)

1/T

2κ

2π

β

d/2

#

(34)

with P denoting pressure.

Figure

1 shows the phase structure in the heavy dense

limit. Note that β dropped out in most considered ob-

servables. For large N

t

we ﬁnd two well-separated phases,

where the system is in a condensed phase for large µ.

In the limit of vanishing temperature we ﬁnd

hni

T =0

= Θ (µ − µ

c

) ,

hχχi

T =0

= 2κ Θ (µ

c

− µ) , (35)

hεi

T =0

= µ

c

Θ (µ − µ

c

) ,

where Θ is the Heaviside step function and the critica l

chemical p otential onset is found to be µ

c

= log (2m).

Note that µ

c

> 0 in the heavy dense regime. The model

clearly exhibits Silver Blaze behavior a s mentioned in

Sec. I.

D. The case of two ﬂavors

We continue with determining the partition function

for two ﬂavors in the heavy dense limit. For nondegen-

erated ﬂavor s we label the parameters κ, µ and ξ with a

ﬂavor index i ∈ {1, 2} and ﬁnd after an e xact integration

Z

2

=

∞

ˆ

−∞

Y

x,v

dA

v

(x) det K

1

det K

2

e

−S

A

=

(1 + ξ

1

+ ξ

2

+ ξ

1

ξ

2

∆)

V

(4κ

1

κ

2

)

Ω

π

β

Ωd

2

, (36)

where we introduce

∆ =

1 −

1

2β

N

t

(37)

for brevity. For simplicity we quote the observa bles only

in the cas e of degenerated ﬂavors and a common chemical

potential, i.e., κ

i

= κ, µ

i

= µ and ξ

i

= ξ. The genera l-

ization to nondegenerated ﬂavor s is straightfor ward. T he

total fermion density is given by

hni =

2ξ (1 + ξ∆)

1 + 2ξ + ξ

2

∆

(38)

with hni → 2 for µ → ∞ (if β 6= 1/2), while the fermion

condensate reads

h

χχi =

4κ (1 + ξ)

1 + 2ξ + ξ

2

∆

. (39)

The mechanical equation of state takes the form

P = log

"

1 + 2ξ + ξ

2

∆

1/T

4κ

2

π

β

d/2

#

. (40)

Like in 0 + 1 dimensions [

29] we ﬁnd a plateau in the

density, co ndens ate and energy density for N = 2. They

bec ome visible for couplings of the order β ≈ 1/2, see

Fig.

7. For the special case of β = 1/2 (i.e. ∆ = 0), the

density never goes into full saturation for µ → ∞, but is

stuck on the plateau. These structur e s ca n be understood

in the (0 + 1)–dimensional continuum case [

29], see also

[

54].

Phase-quenched case. We also consider the partition

function with a phase-quenched fermion determinant as

mentioned in Sec.

II B. Then in the case of two degener-

ated ﬂavors, the partition function reads

Z

pq

2

=

∞

ˆ

−∞

Y

x,v

dA

v

(x) |det K|

2

e

−S

A

=

1

(2κ)

2Ω

"

1 + 2ξ + ξ

2

∆

pq

π

β

N

t

d/2

#

V

(41)

with

∆

pq

=

1 +

1

2β

N

t

. (42)

The observables ar e of the same form as in the full theory

with ∆ replaced by ∆

pq

.

Phase factor. The previous results allow us to derive

an analytical expression for the ex pectation value of the

phase factor of the fermion determinant in the two-ﬂavor

theory. It serves as a measure for the severity of the sign

problem. We ﬁnd

e

2iφ

pq

N =2

=

Z

2

Z

pq

2

= e

−V/V

0

, (43)

where we deﬁne

V

0

= log

1 + 2ξ + ξ

2

∆

pq

1 + 2ξ + ξ

2

∆

. (44)

As expected, the sign problem gets more severe for larger

lattices. In the limit of large chemical potentials the ex-

pec tation value approaches

lim

µ→∞

e

2iφ

pq

N =2

=

2β − 1

2β + 1

Ω

. (45)

We see that the choice of β has a signiﬁcant impact on

the severity of the sign problem. In the heavy dense limit

we ﬁnd that it is most severe for β = 1/2.

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fermion density <n>

Chemical potential µ

Heavy dense limits - N

t

= 10, m = 0.5

Heavy dense

Symmetric, β = 10

Symmetric, β = 3

Figure 2: The density in the strict and in the

symmetric heavy dense limit.

E. The case of N ﬂavors

Finally, we generalize (

30) to an arbitrary number of

N degenerated ﬂavors by raising the fermion determinant

to the power of N in the partition function. We obtain

Z

N

=

1

(2κ)

N Ω

2π

Nβ

Ωd

2

×

"

N

X

k=0

N

k

Λ

k

ξ

k

U

N

t

1 − k

2

,

3

2

,

Nβ

2

#

V

(46)

in the heavy dense limit, where we introduce

Λ

k

=

2

Nβ

N

t

(k−1)

2

. (47)

Here Γ (z) denotes Euler’s gamma function, and

U (a, b, z) is the conﬂuent hypergeometric function of the

second k ind [

55], also known as Kummer’s function. We

do not want to quote the obser vables here explicitly, but

as required they reproduce previous results for N = 1, 2.

F