Analytic continuation in QCD-like theories at finite density and finite isospin

Abstract

The method of analytic continuation is one of the most powerful tools to circumvent the sign problem in lattice QCD. The present study is part of a larger project which, based on the investigation of QCD-like theories which are free of the sign problem, is aimed at testing the validity of the method of analytic continuation and at improving its predictivity, in view of its application to real QCD. We have shown that a considerable improvement can be achieved if suitable functions are used to interpolate data with imaginary chemical potential. We present results obtained in a theory free of the sign problem such as two-color QCD at finite chemical potential. Comment: 8 pages, 6 figures, Contribution to The XXVII International Symposium on Lattice Field Theory, Beijing, July 25-31, 2009

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arXiv:1001.4390v1 [hep-lat] 25 Jan 2010
Analytic continuation in QCD-like theories at finite
density and finite isospin
P. Cea
Università di Bari & INFN - Bari, Italy
E-mail: paolo.cea@ba.infn.it
L. Cosmai∗
INFN - Bari, Italy
E-mail: leonardo.cosmai@ba.infn.it
M. D’Elia
Università di Genova & INFN - Genova, Italy
E-mail: massimo.delia@ge.infn.it
A. Papa
Università della Calabria & INFN - Cosenza, Italy
E-mail: papa@cs.infn.it
The method of analytic continuation is one of the most powerful tools to circumvent the sign
problem in lattice QCD. The present study is part of a larger project which, based on the investi-
gation of QCD-like theories which are free of the sign problem, is aimed at testing the validity of
the method of analytic continuation and at improving its predictivity, in view of its application to
real QCD. We have shown that a considerable improvement can be achieved if suitable functions
are used to interpolate data with imaginary chemical potential. We present results obtained in a
theory free of the sign problem such as two-color QCD at finite chemical potential.
The XXVII International Symposium on Lattice Field Theory - LAT2009
July 26-31 2009
Peking University, Beijing, China
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Analytic continuation in QCD-like theories
L. Cosmai
1. Introduction
The investigation the phase diagram of QCD in the temperature-chemical potential plane has
a deep relevance and implications on cosmology, astrophysics and in the phenomenology of heavy
ions collisions. The lattice formulation of QCD is the only tool to approach this important issue
starting from first principles. Unfortunately the study of QCD at non-zero baryonic density by nu-
merical simulations on a space-time lattice is plagued by the well-known sign problem: the fermion
determinant is complex and the Monte Carlo sampling becomes unfeasible. One of the possibilities
to circumvent this problem is to perform Monte Carlo numerical simulations for imaginary values
of the baryonic chemical potential, where the fermion determinant is real and the sign problem is
absent, and to infer the behavior at real chemical potential by analytic continuation. The method
of analytic continuation [1–16] is well-founded and works fine within the limitations posed by the
presence of non-analyticities and by the periodicity of the theory with imaginary chemical poten-
tial [17].
It is very important to answer the question about which is the optimal way to extract infor-
mation from data taken at imaginary values of the chemical potential. This is equivalent to an-
swer which is the best interpolating function for data at imaginary chemical potential that could
analytically continued in order to get physical predictions for real values of µ. The aim of our
investigations in the past two years has been to to study limitations and possible improvements of
the method of analytic continuation [9, 18, 19]. In order to study this problem we have considered
SU(2) (two-color QCD) and SU(3) at finite isospin. Indeed these theories are free of the sign prob-
lem and Monte Carlo numerical simulations at real values of the chemical potential or at real values
of isospin potential are feasible. Therefore it is possible to compare the analytic continuations with
the data from direct simulations allowing at the same time to discriminate between interpolating
functions an to test the range of the reliability of the analytic continuation. Here we briefly review
results obtained in studying analytic continuation of physical observables and of the critical line in
two-color QCD. Results obtained for SU(3) at finite isospin has been reviewed in ref. [20].
2. Analytic continuation of physical observables
As already shown long ago [17], the partition function of any SU(N) gauge theory with non-
zero temperature and imaginary chemical potential, µ = iµI, is periodic in θ ≡ µI/T with period
2π/N and that the free energy F is a regular function of θ for T < TE, while it is discontinuous at
θ =2π(k+1/2)/N, k =0,1,2,..., for T >TE, where TEis a characteristic temperature, depending
on the theory.
We have considered SU(2) in presence of nf = 8 degenerate staggered fermions of mass
am = 0.07. Figure 1 shows the the tentative phase diagram in the (µI,β) plane for this theory
(in correspondence of a fermion mass am = 0.07) with βE≃ 1.55 [21, 22] and βc≃ 1.41 [23]. We
have performed numerical simulations on a 163×4 lattice using the exact φ algorithm [21] with
dt = 0.01 (typical statistics: 20k trajectories). We have performed a careful test of the analytic
continuation of physical observables. A detailed discussion of the results obtained is reported in
ref. [9]. In order to show the importance of a careful choice of the interpoIating function of the
imaginary chemical potential data, in figure 2 we display results for two different observables in
2

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Analytic continuation in QCD-like theories
L. Cosmai
?
??
?
?
??
?
??
??
?
??????
?
???? ??
?
Figure 1: Phase diagramin the (β, ˆ µI)-plane; N is the numberof colors, Nτthe extensionof the lattice in the
temporal direction. The numerical values for βEand βcare valid for SU(2) in presence of nf= 8 degenerate
staggered fermions with mass am = 0.07
correspondence of two different values of β. We have shown that the use of ratio of polynomials as
interpolating function can lead to a dramatic improvement in the analytic continuation of physical
observables.
The aim of our subsequent investigations has been to understand if a careful choice of the
interpolating function can also improve the continuation of the critical line.
-0.4 -0.20.00.2 0.40.60.81.01.21.4 1.61.82.0
µ^I (left), µ^ (right)
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
Im(quark density) (left); quark density (right)
Aµ^I+Bµ^I
[Aµ^I+Bµ^I
3+Cµ^I
5
3]/[1+Cµ^I
2]
RW transition line
β=1.45
-0.2 0.00.2 0.4 0.60.81.0 1.2 1.41.6 1.82.0 2.22.4
µ^2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
chiral condensate
A+Bµ^I
[A+Bµ^I
2+Cµ^I
4
2]/[1+Cµ^I
2]
RW transition line
β=1.90
Figure 2: (Left) Negative side of the horizontal axis: imaginary part of the fermion number density vs. the
imaginary chemical potential at β = 1.45. Positive side of the horizontal axis: real part of the fermion num-
ber density vs. the real chemical potential at β =1.45. The green (blue) solid lines represent the polynomial
(ratio of polynomials) interpolating function; the dashed lines give the corresponding uncertainty, coming
from the errors in the parameters of the fit. (Right) Chiral condensate vs. µ2at β = 1.90. The green (blue)
solid lines represent the polynomial (ratio of polynomials) interpolating function; the dashed lines give the
corresponding uncertainty, coming from the errors in the parameters of the fit.
3. Analytic continuation of the critical line
The determination of the critical line in the (T,µ) plane is of overwhelming importance for the
study of strong interactions at finite temperature and baryon density. The analytic continuation of
the (pseudo-)critical line on the temperature-chemical potential plane is well-justified, but a careful
3

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Analytic continuation in QCD-like theories
L. Cosmai
1.51.55
β
1.6
0
0.5
1
chiral condensate susceptibility
βc=1.5440(16)
(Lorentzian fit)
βc=1.5421(14)
(reweighting)
Figure 3: Chiral susceptibility at (aµ)2= −0.1225 vs. β. Full red line is the Lorentzian fit. Dashed blue
line is the multihistogram reweighting within its bootstrap error (blue strip).
test in two-color QCD and three-color QCD with finite isospin chemical potential has cast some
doubts on its reliability [18–20].
(aµ)2
-0.1225
-0.09
-0.0625
-0.04
-0.0225
-0.01
?ψψ?
χ2/d.o.f.
1.34
0.65
0.88
0.50
1.20
1.86
0.07
2.91
1.34
1.09
?L?
χ2/d.o.f.
0.85
0.25
0.31
0.76
0.80
0.43
0.49
1.01
0.87
0.98
?P?
χ2/d.o.f.
0.93
1.06
0.65
1.21
0.80
1.45
0.07
1.16
1.28
0.60
1.5440(16)
1.5068(15)
1.4775(29)
1.4532(16)
1.4324(22)
1.4197(16)
1.4102(18)
1.3528(22)
1.3145(30)
1.2433(59)
1.5349(43)
1.5019(29)
1.4665(32)
1.4453(36)
1.4240(28)
1.4104(33)
1.3989(61)
1.3388(72)
1.2976(62)
1.2508(62)
1.5418(24)
1.5046(21)
1.4755(37)
1.4522(26)
1.4300(39)
1.4199(26)
1.4117(32)
1.3631(46)
1.3286(50)
1.2864(109)
0.
0.04
0.0625
0.09
Table 1: Summary of the values of βc(µ2) obtained by fitting the peaks of the susceptibilities of chiral
condensate ?ψψ?, Polyakov loop ?L? and plaquette ?P? in SU(2) on a 163×4 lattice with fermion mass
am=0.07. For each interpolation the χ2/d.o.f. is given.
In this section we present our results in the determination of the critical line in two-color QCD
using the method of analytic continuation. Contrary to the case of physical observables discussed
in the previous section, the theoretical basis is not straightforward since it relies on the assumption
that susceptibilities, whose peak signals the presence of the transition, be analytic functions of the
parameters on a finite volume [1, 2]. We have tested the method of analytic continuation in the case
of two-color QCD and in the case of QCD ad finite isospin density [18, 19]. As for usual QCD
simulations, we have determined the critical line for imaginary values of the chemical potential
(µ2< 0) and interpolate them by suitable functions to be continued to µ2> 0. In order to test the
reliability of the analytic continuation, the prediction obtained at real µ has been compared with
direct determinations of the transition line.
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Analytic continuation in QCD-like theories
L. Cosmai
-0.2 -0.15 -0.1-0.050 0.050.1 0.150.2
(aµ)2
1.2
1.3
1.4
1.5
1.6
βc
from
the chiral condensate
SU(2) 163 × 4
am=0.07
-0.2-0.15 -0.1-0.050 0.050.10.15 0.2
(aµ)2
1.5
1.55
1.6
1.65
1.7
1.75
1.8
βc
from
the chiral condensate
SU(2) 163 × 4
am=0.2
Figure 4: Critical couplings obtained from the susceptibility of chiral condensate in SU(2) on a 163×4
lattice with am=0.07 (left) and am=0.2 (right), together with a linear fit (dotted line) in (aµ)2to data with
µ2≤ 0. The solid lines around the best fit line delimit the 95% CL region.
-0.2-0.15-0.1 -0.050 0.050.1 0.150.2
(aµ)2
1.2
1.3
1.4
1.5
1.6
βc
from
the Polyakov loop
SU(2) 163 × 4
am=0.07
-0.2-0.15 -0.1-0.050 0.05 0.10.15 0.2
(aµ)2
1.5
1.55
1.6
1.65
1.7
1.75
1.8
βc
from
the Polyakov loop
SU(2) 163 × 4
am=0.2
Figure 5: Critical couplings obtained from the susceptibility of Polyakov loop in SU(2) on a 163×4 lattice
with am=0.07 (left) and am=0.2 (right), together with a linear fit (dotted line) in (aµ)2to data with µ2≤ 0.
The solid lines around the best fit line delimit the 95% CL region.
It should be remarked that on a finite volume there are no true nonanalyticities, therefore the
location of the critical line may depend on the observable chosen to probe the transition. Conse-
quently we have determined the (pseudo-)critical coupling βc(µ2) by looking at the peaks of the
susceptibilities of three different observables: the chiral condensate, the Polyakov loop, and the
plaquette. In figure 3 we show an example of this determination. We have fitted the peak according
to a Lorentzian function. The result obtained in this way for βc(µ2) agrees well with the result
obtained by means of the multihistogram reweighting.
In table 1 the values of βc(µ2) obtained in correspondence of the different "probe" observable
5