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