arXiv:1402.6042v2 [hep-lat] 4 Aug 2015
Simple QED- and QCD-like models at finite density
Jan M. Pawlowski,
1, 2
Ion-Olimpiu Stamatescu,
1, 3
and Christian Zielinski
1, 4,
∗
1
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
2
ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany
3
FEST, Schmeilweg 5, 69118 Heidelberg, Germany
4
Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371
(Dated: 30 July 2015)
In this paper we discuss one-dimensional models rep roducing some features of quantum electro-
dynamics and quantum chromodynamics at nonzero density and temperature. Since a severe sign
problem makes a numerical treatment of QED and QCD at high density difficult, such models help
to explore various effects peculiar to the full theory. Studying them gives insights into the large
density behavior of the Polyakov loop by taking b oth bosonic and fermionic degrees of freedom into
account, although in one dimension only the implementation of a global gauge symmetry is possible.
For t hese models we evaluate t he respective partition functions and discuss several observables as
well as the Silver Blaze phenomenon.
PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.-t
Keywords: Sign problem, finite density, quantum chromodynamics, quantum electrodynamics
I. Introduction
One of the open challenges of lattice gauge theory is
the ab initio treatment of full quantum chromodynam-
ics (QCD) at finite density and low temperature. The
fermion determinant is rendered complex and rapidly os-
cillating after the introduction of a finite chemical poten-
tial µ. The same holds for a wide spectrum of theories at
nonzero density. The resulting near-cancellations make
an evaluation of expectation values extremely challeng-
ing. Several metho ds for meeting the sign problem have
been advanced, but are either limited in their applicabil-
ity or are about to be tested for QCD, see e.g. Refs. [
1–
11].
In the past, studies of models of QCD have been proved
insightful [
12–19], including in the special case of heavy
quarks [20–22]. The inte rest in the present and many
other models discussed in the literature is that they pro-
vide a testing ground for new simulation algorithms to
be applied to the full theory. Examples can be found in
Refs. [
18, 19, 21, 23].
In this paper, we construct and study models that ex-
hibit certain characteristic properties of quantum elec-
trodynamics (QED) and q uantum chromodynamics at
nonzero µ and temperature T . These models are for-
mulated on a one-dimensional lattice using staggered
fermions [
24–27]. In comparison to the full theories these
models have several simplifications. In particular, as it
is not possible to define a plaquette variable in one di-
mension, there is no Yang-Mills action and the models
can only respect a g lobal gauge symmetry. Nonetheless,
the introduction of a bosonic field allows us to go beyond
models with only fermionic degrees of free dom. More-
over, by the introduction of a suitable bosonic action S
g
∗
Corresponding author.
zielinski@pmail.ntu.edu.sg
we can mimic some features of Yang- Mills theory, which
cannot be directly translated to one dimension.
We note that our models generalize the one-link mod-
els presented in Ref. [
21]. With an explicit incorporation
of bosonic degrees of freedom with a corresponding ac-
tion, a particular form of the fermion matrix and an in-
tegration over co njugacy classes we are able to construct
a novel low-dimensional QCD-like model, which extends
and cross-checks previous work.
As a result these models allow us to investigate some
unive rsal phenomena also found in other models from a
different perspective . In particular our findings presented
in Sec. III shine new light on the behav ior of the Polyakov
loop at large densities.
The resulting partition function can be fully integrated
for a U(1) gauge group. In the case of SU(3) we are left
with an integral expression, whose sign problem is man-
ageable and which c an be numerically evaluated. We
discuss some observables and investigate the Silver Blaze
phenomenon [
28] in these models. A theory exhibits
Silver Blaze behavior, if in the zero-temperature limit
T → 0 observables become independent of the chemical
potential for µ ≤ µ
crit
, where µ
crit
is some critical on-
set value of the chemical potential. In realistic models
µ
crit
= m
phys
q
corresponds to the physical fermion mass,
or, more generally, the mass of the lowest excitation with
nonvanishing quark number.
We organize the paper as follows: first we introduce a
QED-like model in Sec.
II. We derive a closed expression
for the partition function and discuss the dependence of
some observables on chemical potential and temperature.
In Sec.
III we deal with the case of QCD and derive an
integral ex pression, which can be numerically evaluated.
In Sec. IV we discuss and summarize our findings.
2
II.
A soluble, QED-like model at nonzero density
For the c onstruction of a QED-like model we formulate
an Abelian U(1) lattice gauge theory on a finite one-
dimensional lattice with staggered fermions and couple
them to a chemical potential, following a similar ansatz
as employed in previous works.
A. Partition function
The actio n we employ mimics the usual compact lattice
QED in one dimension. The lattice is assumed to have a
lattice spacing of a and an extension of N sites, where N
is a ssumed to be even. We set a = 1, i.e., we measure all
dimensionful quantities in appropriate powers of a. We
consider a single staggered fermion field and couple it
to a chemical potential µ. The temperature is identified
with the inverse of the lattice extensio n T = N
−1
. The
action S = S
f
+ S
g
consists of the fermionic part
S
f
=
N
X
t,τ =1
χ (t) K (t, τ ) χ (τ) , (1)
and the pure bosonic part
S
g
= β
N
X
t=1
1 −
1
2
U
t
+ U
†
t
. (2)
Here χ and χ denote the staggered fermion field, K (t, τ)
the fer mion matrix, β = 1/e
2
the inve rse coupling con-
stant and U
t
∈ U(1) the link varia bles. The fermion
matrix reads
K (t, τ ) =
1
2
U
t
e
µ
δ
t+1,τ
− U
†
τ
e
−µ
δ
t−1,τ
+ mδ
tτ
, (3)
with m denoting the mass of the fermion. The intro-
duction of the chemical po tential µ follows the prescrip-
tion by Hasenfratz and Karsch [
29]. Furthermore we im-
pose an antiperiodic boundary condition for the fermionic
field. After integrating out the fermionic degree s of free-
dom, the partition function reads
Z =
ˆ
N
Y
t=1
dU
t
det K e
−S
G
. (4)
In this case the fermion determinant can be evaluated
analytically using identity (1) derived in Ref. [
30]. We
find
2
N
det K = e
Nµ
Y
t
U
t
+ e
−Nµ
Y
τ
U
†
τ
+ 2ρ
+
, (5)
where we have introduced
ρ
±
= λ
+
± λ
−
, λ
±
=
1
2
m ±
p
1 + m
2
N
. (6)
Note that Eq. (
5), like full QED, satisfies the identity
det K (µ) = [det K (−µ
⋆
)]
⋆
, (7)
which shows that in general the fer mion determinant is
complex for µ > 0. We parametrize the link variables
as U
t
= exp (iφ
t
) in terms of algebra-valued fields φ
t
∈
[0, 2π). The corresponding U(1)-Haar measure reads
ˆ
dU
t
=
ˆ
2π
0
dφ
t
2π
. (8)
With this parametrization, the action in Eq. (
2) takes
the form
S
g
= β
N
X
t=1
(1 − cos φ
t
) . (9)
This allows us to integrate the partition function given in
Eq. (
4) by using the expression we derived for the fermion
determinant in Eq. (
5), to find
Z =
e
−βN
2
N−1
ρ
+
I
N
0
(β) + cosh (Nµ) I
N
1
(β)
. (10)
Here I
n
denotes the modified Bessel functions of the first
kind. As a cross-check we verified that Z reduces to
the previo usly derived partition function in Ref. [
21] for
N = 1 up to a normalization constant, which depends on
m and β.
B. Observables
Given the final expression for the partition function in
Eq. (
10), we can easily calculate any observable of inter-
est. The density follows from hni = N
−1
∂
µ
log Z, the
respective susceptibility is defined as hχ
n
i = ∂
µ
hni and
the fermion condensate is given by h
χχi = N
−1
∂
m
log Z.
For the density we then find,
hni =
sinh (N µ) I
N
1
(β)
ρ
+
I
N
0
(β) + cosh (N µ) I
N
1
(β)
. (11)
Again this expression reduces to the known res ult in
Ref. [
21] for N = 1. The fermion condensate follows
as,
h
χχi =
1 + m
2
−1/2
ρ
−
I
N
0
(β)
ρ
+
I
N
0
(β) + cosh (N µ) I
N
1
(β)
. (12)
By directly evaluating the respective path integral ex-
pression, we find for the Polyakov loop P =
Q
t
U
t
the
exp ectation value
hPi =
e
−βN
2
N
Z
2ρ
+
I
N
1
(β) + e
Nµ
I
N
2
(β) + e
−Nµ
I
N
0
(β)
.
(13)
The conjugate Polyakov loop P
†
=
Q
t
U
†
t
follows
from a simple symmetry argument as
P
†
µ
= hPi
−µ
,
cf. Ref. [
18].
In Fig. 1 we show the density given by Eq. (11), the
fermion condensate by Eq. (
12) and the Polyakov loop
by Eq. (13) as functions o f the chemical potential µ. We
3
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
Chemical potential µ
U(1) Model - β = 6, m = 1, N = 10
<n>
<χ
n
>/<χ
n
>
max
<χ
-
χ>
(a) Density, condensate and normalized susceptibility.
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2
Chemical potential µ
U(1) Model - β = 6, m = 1, N = 10
<P>
<P
+
>
(b) Polyakov lo op and conjugate Polyakov loop.
Figure 1: Observables in the U(1) model.
see that already for T = 1/N = 1/10 the observables
only s how a weak dependence on the chemical potential
below some critical value µ
crit
. This shows how the Sil-
ver Blaze behavior [
28] becomes apparent in this model,
which holds strictly in the limit T → 0.
Close to µ ≈ µ
crit
we also observe a fast incr ease or
decrease of the obse rvables before re aching the saturation
regime. Note that the limits µ → ∞ and β → 0 do not
commute. Density and condensate behave as one would
exp ect. The ex pectation values of P and P
†
approach
hPi →
I
2
(β)
I
1
(β)
N
,
P
†
→
I
0
(β)
I
1
(β)
N
, (14)
for µ → ∞. The Polyakov loop quickly drops to a typ-
ically small value with increa sing µ while the conjugate
Polyakov loop grows to a saturation value which diverges
when β → 0.
III. A QCD-like model at nonzero density
Now we extend the previous model to the non-Abelian
gauge group SU(3). By restricting the integra tion over
the full gauge group to the respective conjugacy classe s o f
SU(3), we will be able to reduce the partition function to
an integral expression with a manageable sign problem.
A. Partition function
Our starting point is again the path integral expression
for the partition function in Eq. (
4), where now the pure
bosonic part of the action reads
S
g
= β
N
X
t=1
1 −
1
6
Tr
c
U
t
+ U
†
t
. (15)
Here β = 6/g
2
denotes the inverse coupling, U
t
∈ SU (3)
the link variables and Tr
c
a trace in color space. Further-
more we replace the fermion matrix by
K (t, τ) =
1
2
σ
+
U
t
e
µ
δ
t+1,τ
− σ
−
U
†
τ
e
−µ
δ
t−1,τ
+ mδ
tτ
,
(16)
with σ
±
=
1
2
(
1
± σ
3
) and the third Pauli matrix σ
3
=
diag (1, −1). In the loop expansion this suppresses
back s teps, thus simulating a special feature of Wilson
fermions. This choice results in a factorization of the
fermion determinant of the form
det K = det
t,c
K
f
· det
t,c
K
b
, (17)
where we introduced
det
t,c
K
f
= det
t,c
mδ
tτ
+
1
2
U
t
e
µ
δ
t+1,τ
,
det
t,c
K
b
= det
t,c
mδ
tτ
−
1
2
U
†
τ
e
−µ
δ
t−1,τ
. (18)
Here U
N+1
= −U
1
and det
t,c
refers to a determinant in
position and color space.
In the following we restrict ourselves to observables
which only depend on the conjugacy class of the link
variables. We then replace the integration over the full
gauge group SU(3) with an integration over these con-
jugacy classes. This idea and the factorization given
in Eq. (
17) were also pre viously exploited in a one link
model in Ref. [21]. We thus parametrize the links by
U
t
= diag
e
iφ
t
, e
iϑ
t
, e
−i(φ
t
+ϑ
t
)
, (19)
with φ
t
, ϑ
t
∈ (−π, π]. Ignoring a normalization constant,
the Haar measure is given by dU
t
∝ J (φ
t
, ϑ
t
) dφ
t
dϑ
t
with
J (φ
t
, ϑ
t
) = sin
2
φ
t
− ϑ
t
2
× sin
2
φ
t
+ 2ϑ
t
2
sin
2
2φ
t
+ ϑ
t
2
, (20)
4
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2
Chemical potential µ
SU(3) Model - β = 6, m = 1.5, N = 6
<n>
<χ
-
χ>
(a) Density and condensate.
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.5 1 1.5 2
Chemical potential µ
SU(3) Model - β = 6, m = 1.5, N = 6
<P>
<P
+
>
(b) Polyakov lo op and conjugate Polyakov loop.
Figure 2: Observables in the SU(3) model.
while the bosonic part of the action takes the form
S
g
= β
N
X
t=1
1 −
1
3
(cos φ
t
+ cos ϑ
t
+ cos (φ
t
+ ϑ
t
))
.
(21)
The determinant in position space has a simple structure
and can be analytically e valuated, e.g. directly or by re-
summation of the loop expansion for Eq. (
16). For the
remaining determinant in color space we use the identity
det
c
(
1
+ αU
t
) = 1 + α Tr
c
U
t
+ α
2
Tr
c
U
−1
t
+ α
3
, (22)
valid for all A ∈ SL
3
(C), see Ref. [
21]. We can express
the result in terms of the (conjugate) Polyakov loop
det
t,c
K
f
= m
3N
det
c
1
+ ξ
f
Y
t
U
t
!
= m
3N
1 + ξ
f
P + ξ
2
f
P
†
+ ξ
3
f
, (23)
with ξ
f
= [κ exp (µ)]
N
and hopping parameter κ =
1/ (2m). The Polyakov loo p P and c onjugate Polyakov
loop P
†
are defined by
P = Tr
c
N
Y
t=1
U
t
, P
†
= Tr
c
N
Y
t=1
U
†
t
. (24)
Analogously, we find
det
t,c
K
b
= m
3N
1 + ξ
b
P
†
+ ξ
2
b
P + ξ
3
b
, (25)
with ξ
b
= [κ exp (−µ)]
N
. We o bserve that, as in the
QED-like model, the fermion determinant in Eq. (
17)
satisfies the relation in Eq. (7).
Putting all pieces together, we find that the partition
function of the model reads
Z = m
6N
π
ˆ
−π
"
Y
t
dφ
t
dϑ
t
J (φ
t
, ϑ
t
)
#
×
1 + ξ
f
P + ξ
2
f
P
†
+ ξ
3
f
×
1 + ξ
b
P
†
+ ξ
2
b
P + ξ
3
b
e
−S
g
, (26)
where an irrelevant numerical no rmalization constant has
been dropped. The measure term J (φ
t
, ϑ
t
) was given in
Eq. (
20), S
g
was introduced in Eq. (21).
B. Observables
Considering Eq. (
26) as a partition function for a model
of QCD, we c an derive integral expressions for the den-
sity, the susceptibility and the fermion condensate by tak-
ing corresponding derivatives of log Z. For the Polyakov
loop and the conjugate Polyakov loop we insert a
1
Z
P or
a
1
Z
P
†
term in Eq. (
26).
The resulting integral expressions are numerically e val-
uated. Typical exa mples of these o bservables can be
found in Fig.
2. The density and the condensate show
similar qualitative behavior to the corresponding observ-
ables in the U(1) model, where we now find hni → 3 for
µ → ∞.
The Polya kov loop and conjugate Polyakov loop show
some nontrivial behavior. Close to the critical onset µ
crit
,
we find peaks in hPi and
P
†
with the peak in the con-
jugate Polyakov loop appearing at smaller µ. Similar
behavior was previously observed in a simulation of a
gauge theory with ex c e ptio nal group G
2
[31], a strong
coupling limit in HQCD [
23], a three-dimensional effec-
tive theory of nuclear matter [22] and in recent studies of
one-dimensional QCD [
18, 19]. The drop of the Polyakov
loop at high density is easily understood as an effect of
saturation, while the displacement of the peaks has a dy-
namical basis, see, e.g. Ref. [
23].
5
Despite making use of a different approach, in general
we find good qualitative agreement with the res ults re-
ported in Refs. [
18, 19] after dropping S
g
, i.e. for β = 0.
IV. Conclusions
In this paper we have constructed one-dimensional lat-
tice models re sembling QED and QCD to investigate the
finite density and finite temperature regime. Despite the
drastic simplifications in these models, they capture some
essential physical properties e xpected from the full theory
and show an interesting behavior of the Polyakov loop.
We found that they—like their four-dimensional contin-
uous counterparts—exhibit the Silver Blaze property in
the zero temperature limit N → ∞. The µ-dependence
of the SU(3) (conjuga te) Polyakov loop P (P
†
) show s
the pec uliar µ-dependence also found in other approx-
imations of QCD. The models presented her e ca n also
serve as a starting point for the construction of mo re
elaborated models.
V. Acknowledgments
This work is supported by the Helmholtz Alliance
HA216/EMMI and by ERC-AdG-290623. J.M.P. thanks
the Yukawa Institute for Theoretical Physics, Kyoto Uni-
versity, where this work was completed during the YITP-
T-13-0 5 on ’New Frontiers in QCD’. I.-O.S. thanks the
Deutsche Forschungsgemeinschaft by STA 283/1 6-1 and
C.Z. thanks Nanya ng Technological University for sup-
port.
[1] P. d e Forcrand, PoS LAT2009, 010 (2009),
arXiv:1005.0539 [hep-lat].
[2] M. P. Lombardo, Mod.Phys.Lett. A22, 457 (2007),
arXiv:hep-lat/0509180 [hep-lat].
[3] G. Aarts, PoS LAT2009, 024 (2009), arXiv:0910.3772
[hep-lat]
.
[4] G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty,
and I.-O. Stamatescu,
Eur.Phys.J. A49, 89 (2013),
arXiv:1303.6425 [hep-lat].
[5] G. Aarts, PoS LATTICE2012, 017 (2012),
arXiv:1302.3028 [hep-lat].
[6] A. Schmidt, Y. D. Mercado, and C. Gattringer, PoS
LATTICE2012, 098 (2012),
arXiv:1211.1573 [hep-lat].
[7] Y. D. Mercado, C. Gattringer, and A. Sch midt, Com-
put.Phys.Commun. 184, 1535 (2013)
, arXiv:1211.3436
[hep-lat]
.
[8] A. Alexandru, M. Faber, I. Horvath, and K.-F. Liu,
Phys.Rev. D72, 114513 (2005), arXiv:hep-lat/0507020
[hep-lat]
.
[9] A. Alexandru and U. Wen ger, Phys.Rev. D83, 034502
(2011)
, arXiv:1009.2197 [hep-lat].
[10] K. Langfeld, B. Lucini, and A. Rago, Phys.Rev.Lett.
109, 111601 (2012)
, arXiv:1204.3243 [hep-lat].
[11] K. Langfeld and J. M. Pawlowski, Phys.Rev. D88,
071502 (2013)
, arXiv:1307.0455 [hep-lat].
[12] N. Bilic, H . Gausterer, and S. Sanielevici, Phys.Rev.
D37, 3684 (1988)
.
[13] N. Bilic and K. Demeterfi, Phys.Lett. B212, 83 (1988).
[14] K. Splittorff and J. J. M. Verbaarschot, Phys.Rev.Lett.
98, 031601 (2007)
, arXiv:hep-lat/0609076 [hep-lat].
[15] K. Splittorff and J. J. M. Verbaarschot, Phys.Rev. D75,
116003 (2007)
, arXiv:hep-lat/0702011 [HEP-LAT].
[16] K. Splittorff and J. J. M. Verbaarschot, Phys.Rev. D77,
014514 (2008)
, arXiv:0709.2218 [hep-lat].
[17] L. Ravagli and J. J. M. Verbaarschot, Phys.Rev. D76,
054506 (2007)
, arXiv:0704.1111 [hep-th].
[18] J. Bloch, F. Bruckmann, and T. Wettig, JHEP 1310,
140 (2013)
, arXiv:1307.1416 [hep-lat].
[19] J. Bloch, F. Bruckmann, and T. Wettig, (2013),
arXiv:1310.6645 [hep-lat].
[20] R . De Pietri, A. Feo, E. Seiler, and I.-O. St amatescu,
Phys.Rev. D76, 114501 (2007), arXiv:0705.3420 [hep-
-lat]
.
[21] G. Aarts and I.-O. Stamatescu, JHEP 0809, 018 (2008),
arXiv:0807.1597 [hep-lat].
[22] M. Fromm, J. Langelage, S. Lottini, M. Neuman,
and O. Philipsen,
Phys.Rev.Lett. 110, 122001 (2013),
arXiv:1207.3005 [hep-lat].
[23] E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys.Lett.
B723, 213 (2013)
, arXiv:1211.3709 [hep-lat].
[24] J. B. Kogut and L. Susskind, Phys.Rev. D11, 395 (1975).
[25] T. Banks, L. Susskind, and J. B. Kogut, Phys.Rev. D13,
1043 (1976)
.
[26] T. Banks, S. R aby, L. Susskind, J. Kogut, D. R . T. Jones,
P. N. Scharbach, and D. K. Sinclair (Cornell-Oxford-
Tel Aviv-Yeshiva Collaboration), Phys.Rev. D15, 1111
(1977)
.
[27] L. Susskind, Phys.Rev. D16, 3031 (1977).
[28] T. D. Cohen, Phys.Rev.Lett. 91, 222001 (2003),
arXiv:hep-ph/0307089 [hep-ph].
[29] P. Hasenfratz and F. Karsch, Phys.Lett. B125, 308
(1983)
.
[30] L. G. Molinari, Linear Algebra and its Applications 429,
2221 (2008)
.
[31] A. Maas, L. von Smekal, B. Wellegehausen, and A. Wipf,
Phys.Rev. D86, 111901 (2012), arXiv:1203.5653 [hep-
-lat]
.
