Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential

0.21 MB·English

by  J. C. Osborn

Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential
J.C. Osborn,1 K. Splittorff,2 and J.J.M. Verbaarschot3
1Physics Department, Boston University, Boston, MA 02215, USA
2Nordita, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
3Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794, USA
(Dated: February 1, 2008)
TherelationbetweenthespectraldensityoftheQCDDiracoperatoratnonzerobaryonchemical
potential and the chiral condensate is investigated. We use the analytical result for the eigenvalue
density in the microscopic regime which shows oscillations with a period that scales as 1/V and
4
an amplitude that diverges exponentially with the volume V =L . We find that the discontinuity
of the chiral condensate is due to the whole oscillating region rather than to an accumulation of
eigenvalues at the origin. These results also extend beyond the microscopic regime to chemical
potentials µ∼1/L.
5
0
0 Introduction. One of the salient features of QCD at low where the eigenvalues of the Dirac operatorare givenby
2
energy is the spontaneous breaking of chiral symmetry z , and the brackets denote the average over the Yang-
k
n characterizedbyadiscontinuityofthechiralcondensate. Millsaction. TheeigenvaluedensityoffullQCDincludes
a
Morethantwodecadesagoitwasrealizedby Banksand the fermion determinant in the average
J
Casher [1] that the discontinuity of the chiral conden-
6 δ2(x+iy z )detNf(D+m)
2 sdaetnesiattyzoefrothqeuQarCkDmDasisraiscporpoepraotrotiro.nTalhtisortehleateiiognenevsatalube- ρNf(x,y,m;µ)= hPk detN−f(Dk +m) i.(4)
h i
1 lishesthattheeigenvaluespectrumoftheanti-Hermitian
Strictly speaking, since ρ is in general complex, this
v Dirac operatorbecomes dense atthe originofthe imagi- Nf
0 is not a density. Due to chiral symmetry the eigen-
naryaxisinthethermodynamiclimit. TheBanks-Casher
1 relationisofsubstantialpracticalvaluefornonperturba- values occur in pairs ±zk so that ρNf(x,y,m;µ) =
2 ρ ( x, y,m;µ). A second reflection symmetry which
1 tive numerical studies of QCD. It allows one to extract hNolfds−onl−y after averagingoverthe gaugefieldconfigura-
0 the chiral condensate directly from the spectral density.
tions is that ρ∗ (x,y,m;µ)=ρ (x, y,m;µ). Because
5 Atnonzerobaryonchemicalpotential,µ,theEuclidean Nf Nf −
the fermion determinant vanishes for z = m we ex-
0 k
Dirac operator, D D γ +µγ , is non-Hermitian so ±
h/ that the support of≡its sηpeηctrum0is a 2 dimensional do- pect that ρNf(x = ±m,y = 0,m;µ) = 0. The chiral
condensate in the chiral limit can be expressed as
t maininthecomplexplane. Inthiscasewewillshowthat
-
ep the discontinuityofthe chiralcondensateatzeromass is Σ= lim lim 1 dxdy ρNf(x,y,m;µ). (5)
h not due to the accumulation of eigenvalues near zero. m→0V→∞V Z x+iy+m
The QCD partition function at zero temperature does
:
v At zero chemical potential the above quantity is known
not depend on the baryon chemical potential,
i to be proportional to the density of the imaginary
X
Z (m;µ)=Z (m;µ=0), for µ<µ , (1) eigenvalues at zero [1]. This happens because then
r Nf Nf c
a whereµcisthemassperunitquarknumberofthelightest ρuNesf(sxp,ryea,dmi;nµto=th0e)c∝omδp(xle)x. pWlanheenanµd6=thi0stahreguemigeenntvanlo-
excitation with nonzero quark number. Here and below longerholds. Inthiscasewewillshowthatthereisanex-
we only consider the case with Nf quark flavors with tended region of the eigenvalue density that contributes
equalmass m. Therefore,the chiralcondensate givenby to the chiral condensate.
1 For simplicity, here we will show how the condensate
ΣNf(m)= N V ∂mlogZNf(m;µ), (2) arises from the microscopic limit of the spectral density
f
which is believed to be universal. We have also verified
remains equal to its value at µ=0 for µ<µ . Our aim [2] that a similar mechanism occurs even for larger µ
c
is to understand this behavior from the spectrum of the providedthatthecontributingeigenvaluesarestillinthe
Dirac operator. universal region. The microscopic limit of the spectral
We will consider two types of gauge field averages, density is defined as [3]
quenched averages where the determinant of the Dirac
ρˆ (xˆ,yˆ,mˆ;µˆ) (6)
operator is not included in the average and unquenched Nf
averages which include the fermion determinant. The 1 xˆ yˆ mˆ µˆ
= lim ρ ( , , ; )
quenched spectral density can be expressed as V→∞(ΣV)2 Nf ΣV ΣV ΣV F √V
π
ρ (x,y;µ)= δ2(x+iy z ) , (3) In this limit, xˆ = xΣV, yˆ = yΣV, mˆ = mΣV and µˆ =
Q k
hXk − i µFπ√V are kept fixed with Σ given by (5) and Fπ the
2
pion decay constant. The expression for the condensate 100
(5) in this limit becomes
ρˆ (xˆ,yˆ,mˆ;µˆ)
Σ= lim Σ dxˆdyˆ Nf . (7)
mˆ,µˆ→∞ Z xˆ+iyˆ+mˆ 50
The microscopic limit of the spectral density at nonzero
chemical potential was recently calculated both for the yˆ
0
quenched case [4] and the unquenched case [5]. For
a nonzero number of flavors it was found [6] that the
eigenvalue density for mΣ < 2µ2F2 is a strongly oscil-
π
lating complex function. The oscillations cover a region -50
of the complex eigenvalue plane and, as we will see be-
-0.002
low, the entire region contributes to the integral in (7). -0.0001
0.001
This constitutes a new mechanism where a discontinu-
0.002
-100
ity of the chiral condensate in the complex mass plane 50 100 150
is obtained from an oscillating eigenvalue density in the Re[ρˆNf=1(xˆ,yˆ,mˆ;µˆ)] xˆ
complexplane. Thismechanismdoesnotrelyonthespe- FIG. 1: Top view of the real part of the eigenvalue density
cific formofthe eigenvaluedensity as is demonstratedin for one flavor with mˆ =60 and µˆ =8 in half of the complex
eigenvalueplane. Theoscillations arecut off toillustrate the
the simple example below.
2
eigenvaluerepulsionatxˆ=yˆ=0andthedropoffforxˆ>2µˆ .
The lack of Hermiticity properties of the Dirac op-
erator at nonzero chemical potential is a direct conse-
and has the property that it vanishes at the point where
quence ofthe imbalancebetweenquarksandanti-quarks
the fermion determinant is zero. The integral in (7) can
imposed in order to induce a nonzero baryon density.
be evaluatedanalyticallyby meansofacomplexcontour
Because of the phase of the fermion determinant, proba-
integral in yˆresulting in
bilistic methods are no longer effective in the analysis of
the partition function. This is known as the sign prob- Σ
lem. Although progresshas been made in some areaswe ΣEx =sg(mˆ)Σ+ e−|mˆ|(e−|mˆ| 1), (9)
mˆ −
believe that because of its physical origin, a paradigm
shift will be necessary to develop viable probabilistic al- which, for large mˆ, approaches sg(mˆ)Σ. What we have
gorithmsforthisproblem. Becauseofthis,itisouropin- learned from this example is that a discontinuity in the
ion that it is particularly important to improve our an- chiral condensate can be obtained from an oscillating
alytical understanding of chiral symmetry breaking for spectral density rather than from eigenvalues localized
QCD at nonzero baryon density. on the imaginary axis. We will show next that the same
EuclideanQCDatfinitebaryondensityisnottheonly mechanism is at work for QCD at µ=0.
6
system without Hermiticity properties that has received
The microscopic spectral density. The input for our
much attention recently. We mention the distribution of
calculation of the chiral condensate is the microscopic
the poles of S matrices which are given by the eigen-
spectral density derived for any number of flavors in
−
values of a non-Hermitian operator [7, 8], the Hatano-
[5]. Without loss of generality we will consider the case
Nelson model [9] (a random potential together with a
N =1andtopologicalchargeequaltozeroforwhichthe
f
nonzeroimaginaryvectorpotential), andthe description
equations are less extensive. The microscopic eigenvalue
of Laplacian growth in terms of the spectrum of non-
density can be decomposed as [5, 6]
Hermitianrandommatrices[10]. Theessentialdifference
from QCD is that in these problems the determinant of ρˆ (xˆ,yˆ,mˆ;µˆ)=ρˆ (xˆ,yˆ;µˆ) ρˆ (xˆ,yˆ,mˆ;µˆ), (10)
Nf=1 Q − U
theoperatoronlyentersinthegeneratingfunctionofthe
resolvent. We willseethatthe additionaldeterminantin with (zˆ=xˆ+iyˆ)
QCD completely changes the character of the theory.
ρˆ (xˆ,yˆ,mˆ;µˆ)= |zˆ|2 e−(zˆ2+zˆ∗2)/(8µˆ2) (11)
Example. As an example to illustrate our point, let us U 2πµˆ2
consider the eigenvalue density (sg is the sign function)
K (|zˆ|2)I0(zˆ) 1dtte−2µˆ2t2I (zˆ∗t)I (mˆt).
θ(2µˆ2 xˆ) × 0 4µˆ2 I0(mˆ)Z0 0 0
ρˆ (xˆ,yˆ,mˆ;µˆ)= −| | (8)
Ex 4πµˆ2
The firstterm in (10) is the quenched eigenvaluedensity
1 eisg(xˆ)(|xˆ|+2µˆ2)yˆ/4µˆ2+(|xˆ|−|mˆ|)(|xˆ|+2µˆ2)/4µˆ2 . [4]givenbyρˆ (xˆ,yˆ;µˆ)=ρˆ (xˆ,yˆ,xˆ+iyˆ;µˆ). Asexpected,
Q U
×h − i the microscopic spectral density vanishes at zˆ= mˆ. A
±
This eigenvalue density has the same reflection symme- plotof the realpartofthe eigenvaluedensity for mˆ =60
triesastheeigenvaluedensityoftheQCDDiracoperator andµˆ =8isshowninfigure1. Noticethattheoscillatory
3
region extends from the mass pole at zˆ=mˆ and toward Wenowderivethechiralcondensatefrom(7)usingthe
the boundary of the support of the spectrum. asymptotic form of the microscopic eigenvalue density.
The oscillations appear as the microscopic variables We first consider the integral over yˆ. The contribution
become large, i.e. as the thermodynamic limit is ap- from the quenched part of the spectral density (14) is
proached. In this region an asymptotic formula for the given by
eigenvalue density is accurate and will be used in order
1 ∞ 1 1
toanalyzetheroleoftheoscillationsforchiralsymmetry
dyˆ =sg(xˆ+mˆ) . (16)
breaking. We first derive the asymptotic formula for the 4πµˆ2 Z−∞ xˆ+iyˆ+mˆ 4µˆ2
eigenvalue density and then evaluate the chiral conden-
sate from (7). For µˆ2 1 and (xˆ+mˆ)/(4µˆ2) < 1, the The contributionfrom ρˆU in (7) is evaluatedby a saddle
≫ pointapproximation. Thecontourinthecomplexyˆplane
integral in (11) over t is very well approximated by [11]
isdeformedintoacontourfrom to overthesaddle
−∞ ∞
1 point at yˆ= isg(xˆ)(4µˆ2 xˆ mˆ ) and, if the contour
Z dtte−2µˆ2t2I0(zˆ∗t)I0(mˆt) (12) has crossed the pole, an−int|eg|r−al|ar|ound the pole at yˆ=
0
i(xˆ+mˆ). The saddle point contributionis exponentially
1 zˆ∗2+mˆ2 mˆzˆ∗
exp( )I ( ). suppressedfor xˆ <2µˆ2 leavingonlytheintegralaround
≈ 4µˆ2 8µˆ2 0 4µˆ2 | |
the pole. For mˆ > 0 (mˆ < 0) the pole contribution for
xˆ>0 (xˆ<0) is exponentially suppressed. We obtain
Furthermore, we are interested in the approach to
tzˆhe2/t(h4eµˆr2m)ody1naanmdicmˆlziˆm/i(t4µˆw2h)ere1|.mˆT|h≫isju1s,tifi|zeˆs| t≫her1e-, 1 ∞dyˆe−[yˆ+isg(xˆ)(|xˆ|+|mˆ|−4µˆ2)]2/8µˆ2−(|xˆ|−2µˆ2)2/2µˆ2
| | ≫ | | ≫ 4πµˆ2 Z
placement of the Bessel functions by their leading order −∞
1
asymptotic expansion including the Stokes terms. We (17)
obtain the following asymptotic result for the difference ×xˆ+iyˆ+mˆ
between the quenched and the unquenched eigenvalue 1
[θ(mˆ)θ( xˆ mˆ) θ( mˆ)θ(xˆ+mˆ)],
density ≃−2µˆ2 − − − −
ρˆ (xˆ,yˆ,mˆ;µˆ) (13) where we have used that the exponent vanishes at the
U
pole. For xˆ > 2µˆ2 the eigenvalue density is zero so it is
1 e−[yˆ+isg(xˆ)(|xˆ|+|mˆ|−4µˆ2)]2/(8µˆ2)−(|xˆ|−2µˆ2)2/(2µˆ2). now trivial to do the integration over xˆ to get
∼ 4πµˆ2
This expression has the reflection symmetries discussed Σ 2µˆ2 1
Σ = dxˆ sg(xˆ+mˆ) (18)
below (4). The asymptotic expansion of the quenched Nf=1 2µˆ2 Z−2µˆ2 (cid:2)2
part of the spectral density is simply given by
+θ(mˆ)θ( xˆ mˆ) θ( mˆ)θ(xˆ+mˆ)
− − − −
1 = sg(mˆ)Σ. (cid:3)
ρˆ (xˆ,yˆ;µˆ) θ(2µˆ2 xˆ). (14)
Q ∼ 4πµˆ2 −| |
Thisresultagreeswith(15)for mˆ 1wheretheasymp-
| |≫
For the argument presented below it is important that totic expansion of Σ (m) is valid. Using the exact
Nf=1
alsotheasymptoticexpansionofthespectraldensityvan- microscopicspectraldensitywewouldhaverecoveredthe
ishes at xˆ+iyˆ= mˆ. mass dependence of (15).
±
We also emphasize that a finite result for the chiral
The chiral condensate. As explainedinthe introduction,
condensate is not obtained due to a cancellation of the
the chiral condensate does not depend on the baryon
pole and a zero of the fermion determinant. The pole
chemical potential. Hence, in the microscopic limit it
termgivesafinite contributionfor eachofthe twoterms
is known that [12, 13] (momentarily we use the original
in (10) which do not vanish at zˆ=mˆ. We have checked
variables to emphasize the volume dependence)
numericallythatthesamemechanismresultsinadiscon-
I (mVΣ) tinuity ofthe chiralcondensate formorethan oneflavor.
1
Σ (m)=Σ , (15)
Nf=1 I (mVΣ) The contribution from the unquenched part of the
0
eigenvalue density to the chiralcondensate is dominated
whichisdiscontinuous,Σ (m)=sg(m)Σ,inthether- by the pole term because the exponential in (17) sup-
Nf=1
modynamic limit at fixed quark mass. The question we presses the integrand at the saddle point in the complex
wish to answer is how the oscillatory spectral density yˆ-plane. The simple result (17) which implies a nonzero
conspires into a µ independent condensate. We stress chiral condensate in the chiral limit is thus directly re-
thatthisisnotjustachallengingmathematicalproblem; lated to the complex phase of the spectral density. The
understandingwhichpartsoftheeigenvaluedensitycon- oscillatingexponentialhastosuppresstermsthatdiverge
tributes to the chiral condensate will give direct insight exponentiallywiththevolumewhichisachievedbyoscil-
in the physical consequences of the sign problem. lationsintheyˆ-directionwithaperiodthatscalesasthe
4
inverse of the volume. In contrast, for quenched QCD, from the smallest eigenvalues. On the contrary, we have
wheretheeigenvaluedensityisrealandpositive,thechi- found that the contributions from strips parallel to the
ral condensate vanishes in the chiral limit provided that imaginaryeigenvalueaxisdonotdependontherealpart
µ = 0 as follows immediately from (16). Instead a di- ofthe eigenvalueaslongthe eigenvalueisinside the sup-
6
quarkcondensateformsandbreakschiralsymmetryjust port of the spectrum. This resultarises from integrating
likeinphase quenchedtheories(see e.g. [14, 15]). Chiral aspectraldensitythatoscillateswithaperiodof1/(ΣV)
symmetry breaking in phase quenched and full QCD oc- and an amplitude that diverges exponentially with the
cur by meansoftwo differentmechanisms,andthe oscil- volume. Although we have shown only results using the
lations of the eigenvalue density due to the sign problem microscopic eigenvalue density, we have checked [2] that
in the unquenched case distinguish between the two. ourargumentsapplyuptoµ 1/L(forV =L4). Incon-
∼
clusion, we have uncovered a novel mechanism of chiral
The resolvent. One can easily convince oneself that for
symmetry breaking at nonzero chemical potential where
zˆ=mˆ the exponent in the integrand of the resolvent
6 an oscillatory spectral density results in a discontinuity
ρˆ (Re(uˆ),Im(uˆ),mˆ;µˆ) of the chiral condensate in the complex mass plane.
Gˆ (zˆ,zˆ∗,mˆ;µˆ)=Σ d2uˆ Nf (19)
Nf Z uˆ+zˆ
Acknowledgments: We wish to thank Gernot Ake-
resultsinexpressionsthatdivergeinthe thermodynamic mann, Ove Splittorff and Andrew Jackson for discus-
limit. Indeed, this was concluded from earlier Random sions. J.C.O. was supported in part by U.S. DOE grant
Matrix calculations of the resolvent [16]. Using the mi- DE-FC02-01ER41180, and J.J.M.V. was supported in
croscopic spectral density (11) one finds good numerical part by U.S. DOE grant DE-FG-88ER40388.
agreement with the results obtained in [16].
Phase transitions in generating functions. Our results
canbeunderstoodintermsofphasetransitionsforgener-
atingfunctionsforthespectraldensity. Usingthereplica [1] T. BanksandA.Casher, Nucl.Phys. B169, 103(1980).
trick [6, 14, 17] [2] J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, in
preparation.
1
ρNf(x,y,m;µ)=nli→m0πn∂z∗∂zlogZNf,n(m,z,z∗;µ) [3] EA.5V6.0,S3h0u6ry(a1k993a)n;dJ.JJ..MJ.M.V.erVbearabrasachrsocth,oPt,hyNs.uRcel.v.PLheytst..
72, 2531 (1994).
we are naturally lead to the generating functionals
[4] K.SplittorffandJ.J.M.Verbaarschot,Nucl.Phys.B683,
(m,z,z∗;µ)= det(D+m)Nf det(D+z)2n 467 (2004).
ZNf,n h | | i [5] J. C. Osborn, Phys. Rev.Lett. 93, 222001 (2004).
[6] G.Akemann,J.C.Osborn,K.SplittorffandJ.J.M.Ver-
for the eigenvalue density. The presence of conjugate
baarschot, arXiv:hep-th/0411030.
quarks in the generating function induces a coupling to
[7] C. Mahaux and H.A. Weidenmuller, Shell Model Ap-
the chemical potential in the low-energy effective theory
proach in Nuclear Reactions, North-Holland, Amster-
which is completely fixed by the pattern of chiral sym- dam, 1969.
metry breaking and determines the eigenvalue density [8] Y.V. Fyodorov and H.-J. Sommers, J. Math. Phys. 38,
uniquely in the microscopic regime. In order to calcu- 1928 (1997).
latethe microscopicspectraldensitythis way,onehasto [9] N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77, 570
(1996).
employ powerful integrability relations that exist for the
[10] R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin
effective partition function [18, 19].
and P. Wiegmann, Nucl. Phys.B704, 407 (2005).
Let us compare the phases of the generating function [11] G. Akemann,J. Phys.A 36, 3363 (2003).
for n = 1 to the regions in the complex plane charac- [12] J. Gasser and H. Leutwyler, Phys. Lett. B 188, 477
terized by a different behavior of the eigenvalue density (1987).
for N = 1. For real zˆ the phase structure follows from [13] H. Leutwyler and A. Smilga, Phys. Rev. D 46, 5607
f
[20]. The extension for complex zˆcan be obtained from (1992).
[14] M. Stephanov,Phys.Rev.Lett. 76, 4472 (1996).
the asymptotics of the microscopic generating functions
[15] D.Toublan and J.J.M. Verbaarschot,Int.J. Mod.Phys.
given in [6]. For Re(zˆ) >2µˆ2 the generating functional
| | B 15, 1404 (2001).
is in the normal state while for Re(zˆ) < 2µˆ2 two Bose
| | [16] M.A. Halasz, A. D. Jackson and J.J.M. Verbaarschot,
condensates are separated by a first order phase tran- Phys. Lett. B 395, 293 (1997).
sition. This phase transition occurs exactly where the [17] V.L.Girko,Theoryofrandomdeterminants,KluwerAca-
oscillating region of the eigenvalue density begins, while demic Publishers, Dordrecht,1990.
the transition to the normal phase corresponds to the [18] E. Kanzieper, Phys. Rev.Lett. 89, 250201 (2002).
drop off of the eigenvalue density for Re(zˆ) >2µˆ2. [19] K. Splittorff and J.J.M. Verbaarschot, Phys. Rev. Lett.
| | 90, 041601 (2003).
Conclusions. ForQCDatnonzerochemicalpotentialthe [20] J.B. Kogut and D. Toublan, Phys. Rev. D 64, 034007
chiral condensate is not dominated by the contribution (2001).