High cumulants from the 3-dimensional $O(1, 2, 4)$ spin models

0.24 MB·English

by  Xue Pan

High cumulants from the 3-dimensional O(1,2,4) spin models
Xue Pan,1 Lizhu Chen,1 X.S. Chen,2 and Yuanfang Wu1
1Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China
2Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Considering different universality classes of the QCD phase transitions, we perform the Monte
Carlo simulations of the 3-dimensional O(1,2,4) models at vanishing and non-vanishing external
field,respectively. Interestinghigh cumulantsof the order parameter and energy from O(1) (Ising)
spin model, and the cumulants of the energy from O(2) and O(4) spin models are presented. The
3 generic features of the cumulants are discussed. They are instructive to the high cumulants of the
1 net baryon numberin the QCD phase transitions.
0
2 PACSnumbers: 25.75.Nq,21.65.Qr,75.10.HK

n
a I. INTRODUCTION temperaturefromthe3-dimensionalIsingmodel,anden-
J
ergyfromthe 3-dimensionalO(2) andO(4) spinmodels,
3
respectively. The simulation is performed in a finite-size
A thermodynamic system may undergo a phase tran-
system. The obtained results should be instructive to a
] sition when temperature and/or pressure vary. For the
h finite system formed in ultra-relativistic heavy-ion colli-
strong interacting QCD system, it is known that the
t
- hadronic matter will change to partonic phase at very sions.
l The paper is organized as follows, firstly, the cumu-
c high temperature and density [1]. The data from ultra-
u lants of the order parameter and energy from the O(N)
relativistic nuclear-nuclearcollisions atRHIC has shown
n spin models are derived in section II. Then, their rela-
that the partonic phase of QCD matter–quark-gluon
[ tions to the net baryonnumber fluctuations in the QCD
plasma (QGP) has been formed [2]. One of the main
2 goals of current ultra-relativistic heavy-ion collision ex- phase transitions are discussed. In section III, the high
v periments is to map the QCD phase diagram [3]. cumulants from the 3-dimensional Ising, O(2) and O(4)
2 spin models are presented and discussed. Finally, the
The critical point at the QCD phase boundary is par-
4 generic features of the cumulants from the three models
5 ticular interested, as the large fluctuations are expected. are summarized in section IV.
2 The sensitive probes of QCD critical point are the high
. cumulantsoftheconservedcharges,i.e.,netbaryonnum-
1
1 ber, net electric charge, and net strangeness [4–9]. II. CUMULANTS IN THE O(N) SPIN MODELS
2 Thenetbaryonnumberfluctuationshavebeenstudied
1 using lattice QCD simulations and QCD effective mod-
The O(N)-invariant nonlinear σ-models (O(N) spin
: els [10–13]. However,due to the difficulties of the lattice
v models) are defined as,
i calculations and model estimations, the high cumulants
X
of the net baryonnumber are still not final [14, 15]. The
r universalityofthe criticalfluctuationsallowsustostudy βH=−J XS~i·S~j −H~ ·XS~i, (1)
a
therelevantcumulantsinasimplesystem,wherethehigh hi,ji i
cumulants can be precisely obtained.
where H is the Hamiltonian, J is an interaction energy
It is known that the QCD critical point, which termi- between nearest-neighbor spins hi,ji, and H~ is the ex-
nates the first order phase transition line, is in the same
ternal magnetic field. J and H~ are both reduced quan-
universality class as the 3-dimensional Ising model [16–
tities which already contain a factor β = 1/T. S~ is a
19]. Sothehighcumulantsoforderparameterandenergy i
unit vector of N-components at site i of a d-dimensional
in the vicinity of the critical point of the 3-dimensional
hyper-cubic lattice. It is usually decomposed into the
Ising model are called for. They should be a good refer-
longitudinal (parallel to the magnetic field H~) and the
ence for various relevant calculations.
transverse component
In the chiral limit, the chiral phase transition for 2-
flavorQCD belongs to the same universality class as the
S~ =Sk~e +S~⊥, (2)
3-dimensionalO(4)spinmodel[19]. Thesingularbehav- i i H i
ior of the net baryon number fluctuations is expected to
where ~e = H~/H. For the 3-dimensional Ising, O(2),
be governedby the universal O(4) symmetry group [20]. H
and O(4) spin models, d = 3, and N = 1, 2 and 4, re-
Owing to the staggered fermions in lattice calculations,
spectively.
the (2+1)-flavorchiralphase transition may fall into the
The (reduced) free energy per unit volume is
3-dimensional O(2) universality class [21–23].
In this paper, we present the high cumulants of the 1
orderparameterandenergyinthe vicinity ofthe critical f(T,H)=− lnZ, (3)
V
2
whereZ = dNS δ(S~2−1)exp(−βE+HVSk)isthe we set J = β and choose the approximate critical tem-
R Qi i i
partitionfunction. E =− S~ ·S~ is the energyofa peratures, TIsing = 4.51 [25], TO(2) = 2.202 [26], and
Phi,ji i j c c
spinconfiguration,Sk = 1 Sk isthelatticeaverageof TcO(4) =1.068 [26] for the 3-dimensional Ising, O(2) and
V Pi i
thelongitudinalspincomponents,V =L3 isthevolume, O(4) spin models, respectively.
In order to map the result of the 3-dimensional Ising
and L is the number of lattice points of each direction.
model to that of the QCD, the following linear ansatz is
As we known, the cumulants of the order parameter
suggested [27–29],
are the derivatives of the free energy density (Eq. (3))
with respect to H. We can get the cumulants of the
t≈T −T +a(µ−µ ), h≈µ−µ +b(T −T ). (9)
order parameter from the generating function [24], cp cp cp cp
T and µ are the temperature and chemical poten-
dn cp cp
κSn = dxn lnhexSki(cid:12)(cid:12)(cid:12)(cid:12)x=0. (4) ttwiaol aptartahmeeQteCrsDdcertietrimcailnpedoibnyt,QreCsDpe.cTtihveelby.arayoann-dbabryaorne
correlationlength diverges with the exponent y and ex-
So the first, second, third, fourth and sixth order cumu- t
ponenty whenthecriticalpointisapproachedalongthe
lants of the order parameter are as follows, h
t-direction and h-direction, respectively [11].
The cumulants of the net baryon number are the
κS =hSki, κS =hδSk2i, κS =hδSk3i,
1 2 3 derivatives of the QCD free energy density with respect
κS =hδSk4i−3hδSk2i2, to µ. In the vicinity of the critical point, it is the com-
4 (5) bination of the derivatives with respect to t and h in the
κS =hδSk6i−10hδSk3i2+30hδSk2i3 3-dimensionalIsingmodel. Sincey islargerthany [30],
6 h t
−15hδSk4ihδSk2i, the critical behavior of the net baryon number fluctua-
tionsis mainly controlledbythe derivativeswithrespect
to h, i.e., the fluctuations of the order parameter in the
where δSk = Sk − hSki, and κS is the magnetization
1 3-dimensional Ising model.
(order parameter) of the system. At vanishing external
The singular part of the free energy density for the
magnetic field, due to the spatial rotation symmetry of
chiral phase transition is suggested as [9]
the O(N) groups, such defined order parameter is zero.
In the case, an approximated order parameter definition f (T,µ ,h)
is suggested as, M =h|V1 PiS~i|i [25]. s T4q =Ah(1+1/δ)ff(z), z =t/hβδ, (10)
On the other hand, the cumulants of the energy are
where β and δ are the universalcriticalexponents of the
the derivatives of the free energy density with respect to
3-dimensionalO(4)spinmodel. f (z)isthescalingfunc-
the temperature T. The generating function is f
tion. The reducedtemperature t andexternalfieldh are
dn expressed as follows
κEn = dxn lnhexEi(cid:12)(cid:12)(cid:12)(cid:12)x=0. (6) t≡ 1(T −Tc +κ (µq)2), h≡ 1 mq. (11)
µ
t T T h T
So the first, second, third, fourth and sixth order cumu- 0 c 0 c
lants of the energy are as follows,
HereT isthe criticaltemperatureinthechirallimit. κ
c µ
isa parameterdeterminedbyQCD[23]. The netbaryon
κE =hEi, κE =hδE2i, κE =hδE3i,
1 2 3 numbersusceptibilityisthederivativeoffreeenergyden-
κE =hδE4i−3hδE2i2, sity with respective to the chemical potential µ . From
4 q
(7)
κE =hδE6i−10hδE3i2+30hδE2i3 Eqs.(10) and (11), we can get the derivatives of the free
6 energy density with respect to T, to chemical potential
−15hδE4ihδE2i, µˆ =µ /T at µˆ =0, and to µˆ 6=0 respectively,
q q q q
where δE =E−hEi. ∂f/T4 A
− =− h(2−α−n)/βδf(n)(z), (12)
In the vicinity of the critical point, the free energy ∂Tn (t T )n f
0 c
density (Eq.(3)) can be decomposed into two parts, the
regular and singular parts. The critical related fluctu-
ations are determined by the singular part. It has the ∂f/T4
scaling form − ∂µˆnq (cid:12)(cid:12)(cid:12)(cid:12)µq=0 =−A(2κq)n/2h(2−α−n/2)/βδff(n/2)(z),
f (t,h)=l−df (lytt,lyhh). (8) (13)
s s
∂f/T4 µ
Hpeerraetutr=e a(nTd−mTacg)n/eTt0icafinedldh, T=0Han/dHH0 0araereretdhuecnedortmemal-- − ∂µˆnq (cid:12)(cid:12)(cid:12)(cid:12)µq6=0 =−A(2κq)n(Tq)nh(2−α−n)/βδff(n)(z),
ized parameters. T is the critical temperature. y and (14)
c t
y are universal critical exponents. In our simulation, where n is even in Eq. (13).
h
3
Comparing Eqs. (13) and (14) with Eq. (12), we can thatthebasicfeaturesofthecumulants,i.e.,thepatterns
see thatthe scalingformofderivativesofthe free energy of the fluctuations, are not influenced by the magnitude
density with respect to T and µ are equivalent. So the ofthe externalfield. Withthe appearingorincreasingof
q
netbaryonnumberfluctuationsarerelatedtothederiva- the external field, the whole critical region is amplified
tives of the free energy density with respect to T in the and shifted toward the high temperature side.
3-dimensionalO(4)spinmodel. Then-thordercumulant In the vicinity of the critical temperature, κE has
2
of the energy from the 3-dimensional O(4) spin model is a maximum peak. κE oscillates and its sign changes
3
relevanttothe2n-th(orn-th)ordercumulantofthenet frompositivetonegativewhentemperatureincreasesand
baryonnumberatµq =0(orµq 6=0)inthechiralphase passesthecriticalone. κE4 hastwopositivemaximumslo-
transition. catingatthetwosidesofT ,respectively. The minimum
c
The cumulants of the order parameter from the 3- between them is negative. In contrast to the two pos-
dimensional O(2) and O(4) spin models are the deriva- itive maximums of κE, κE has two negative minimums
4 6
tivesofthe free energydensity withrespecttothe exter- and one positive maximum at the critical temperature
nal field. Their critical behavior has been presented and region. The sign change for the cumulants of the energy
discussed in Ref. [31]. starts at the third cumulant. It happens twice in the
fourth and sixth order cumulants.
Thecumulantsoftheorderparameteratnon-vanishing
III. CRITICAL BEHAVIOR OF THE HIGH
external fields, i.e., κS, κS, κS and κS, from the 3-
CUMULANTS 2 3 4 6
dimensional Ising model are presented in Fig. 2(a) to
2(d), respectively. We can see that the influences of the
The Monte Carlo simulations of the 3-dimensional external field are similar to those as discussed in Fig. 1.
Ising, O(2), and O(4) spin models are performed by the InthevicinityofT ,κS showsthesamepeakstructure
c 2
Wolff algorithm with helical boundary conditions [32]. as that for the energy. κS has a negative valley and no
3
We start simulation without magnetic field, and then signchangeinthecriticalregion. κS showsaobviouspos-
4
modify the algorithm to include a magnetic field H = itivemaximumandaverysmallnegativeminimumwhen
0.05 and H =0.1 [33]. thetemperatureincreasesandpassesthecriticalone. κS
6
Asweknow,thevalidregionofsystemsizeofthefinite- oscillatesfrompositivetonegative,andthe negativeval-
size scaling varies with magnetic field and observable. leyismoreobviousthanthatinκS. Herethesignchange
4
The typical size of an observable at a given magnetic starts at the fourth order cumulant in Fig. 2(c).
field is determined by the saturation of size dependence,
The high cumulants of the order parameter from the
as shown in Ref. [34]. The typical system sizes for each
3-dimensionalIsing model at vanishing externalfield are
kind of cumulants at a given magnetic field and model shown in Fig. 3(a) to 3(b), respectively. κS in Fig. 3(a)
2
are listed in Table I.
shows a narrow and sharp peak at the critical temper-
ature region. From Fig. 3(b), we can see that, both κS
3
TABLE I: The typical system size for each kind of cumulant
at a given external field and model
XfieXld(XH)XXobsXerXvaXbXle κSn(O(1)) κEn(O(1)) κEn(O(2)) κEn(O(4)) 1 O( 1H)= 0.1 (a) 1 (b)
H=0.05
0 24 20 20 20 0.8 H=0 0
0.05 12 10 10 8
E 20.6 E 3
0.1 8 8 8 8 k k -1
0.4
-2
0.2
In order to compare the basic structure of the cumu-
lants with and without external fields, each cumulant of 0
0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3
theenergyatvanishingandnon-vanishingexternalfields T/Tc T/Tc
isplottedinanidenticalsub-figure,andrescaledtounity
by its maximum or minimum (except for the first order 1 (c) 1 (d)
cumulantoftheenergyfromO(2)andO(4)spinmodels),
0 0.5
asshowninFig.1,Fig.4,andFig.5. FortheIsingmodel,
the cumulants of the order parameter at vanishing and E 4 E 6
k -1 k 0
non-vanishing external fields are quite different, which
may be caused by the definitions as discussed in section
-2 -0.5
II below Eq. (5). So their cumulants at non-vanishing
and vanishing external fields are presented in Fig. 2 and 0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3
T/T T/T
c c
Fig. 3, respectively.
The cumulants of the energy, i.e., κE, κE, κE and κE,
from the 3-dimensional Ising model a2t H3= 04.1,0.056,0 FIG.1: (Color online). κE2 (a),κE3 (b),κE4 (c) and κE6 (d) at
are shown in Fig. 1(a) to 1(d), respectively. We can see H =0.1,0.05,0 from the3-dimensional Ising model.
4
1 O(1) 0 -0.5 O(4) 1
H=0.1 H=0.1
0.8 H=0.05 -0.2 HH==00.05 0.8
-1
0.6 -0.4
S 2 S 3 E 1 E 20.6
k 0.4 k-0.6 k k
-1.5
0.4
0.2 -0.8
0 (a) -1 (b) -2 (a) 0.2 (b)
0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6
T/T T/T T/T T/T
c c c c
1 (c) 1 (d) 1 1
0.8 0.8
0
0
0.6 0.6
S 4 S 60.4 E 3 -1 E 4
k 0.4 k k k -1
0.2
0.2 -2
0
-2
0 -0.2 -3 (c) (d)
0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6
T/T T/T T/T T/T
c c c c
FIG. 2: (Color online). κS2 (a), κS3 (b), κS4 (c), κS6 (d) at FIG.4: (Color online). κE1 (a),κE2 (b),κE3 (c) and κE4 (d) at
H =0.1,0.05 from the3-dimensional Ising model. H =0.1,0.05,0 from the3-dimensional O(4) spin model.
andκS oscillate,buttheformerchangesfromnegativeto
4
positive, while the latter changes from positive to nega-
tive with the increasing temperature. The generic struc- σ-model [8].
ture of κS is similar to that of κE, having two positive
6 4
maximum locating at the two sides of Tc and a negative The cumulants of the energy, i.e., κE1, κE2, κE3 and
minimum between them. So the generic structure of the κE, from the 3-dimensional O(4) spin model at H =
4
high cumulants at non-vanishing external field is quite 0.1,0.05,0arepresentedinFig.4(a)to4(d),respectively.
different from that at vanishing external field. The sign Again, the external field shows the similar influences as
change in the former case appears in the fourth order discussed above. κE increases with the temperature. κE
1 2
cumulant, while the third one in the later case. has a peak and is positive in the whole critical temper-
Comparing the energy fluctuations in Fig. 1 with the ature region. κE oscillates and changes from positive to
3
orderparameterfluctuations inFig.2andFig.3,wecan negative with the increasing temperature. κE has two
4
seethatthegenericstructureofthesameordercumulant maximumsandaminimumbetweenthem. Thebehavior
of the energy is different from that of the order parame- of κE, κE and κE is similar to that of the energy fluctu-
2 3 4
ter,exceptforthe secondorderone. The wayofthe sign ations from the 3-dimensional Ising model, cf. Fig. 1(a)
changeofκS atH =0asshowninFig.3(b)isconsistent to 1(c).
3
with the expectation from the effective model [7]. The
generic structure of κS at non-vanishing external field As we discussed in section II, at vanishing baryon
4
as shown in Fig. 2(c) is very similar to that from the chemical potential, χB, χB, χB and χB in the chiral
2 4 6 8
phase transition are corresponding to κE, κE, κE and
1 2 3
κE fromthe3-dimensionalO(4)spinmodel. Thepositive
4
peak of κE is consistent with χB from the calculations
1 O(1) H=0 (a) 1 (b) 2 4
k S oflattice QCD[10]andthe estimationsofthe PNJL[12]
0.8 2 0 and PQM models [13]. The sign change of κE3 is also
0.6 observed in χB6 from the PQM model [9, 13].
-1
0.4
The cumulants of the energy, i.e., κE, κE, κE and
k S 1 2 3
0.2 -2 kk 3S4S6 0κ.E41,,0f.r0o5m,0tahreep3r-edsiemnetnedsioinnaFligO.5(2(a))stpoin5(dm)o,dreeslpaetctHivel=y.
0 -3
0.8 0.9 1 1.1 1.2 1.3 0.9 0.95 1 1.05 1.1 Wecanseethateachsub-figureinFig.5issimilartothat
T/T T/T
c c in Fig. 4. Each order cumulant of the energy from the
3-dimensionalO(2) andO(4) spinmodels showsqualita-
FIG. 3: (Color online). κS2 (a), κS3, κS4 and κS6 (b) at H =0 tively similar temperature dependence in the vicinity of
from the 3-dimensional Ising model. the critical temperature.
5
critical fluctuations.
-0.5
O(2) 1
H=0.1 For the 3-dimensional Ising model, the generic struc-
H=0.05
H=0 0.8 tureofthehighcumulantsoftheenergyaredifferentfrom
-1
E 1 E 20.6 thatoftheorderparameter. Forthe energyfluctuations,
k k the sign change starts at the third order cumulant no
-1.5
0.4
matter with or without external field. So does the order
0.2 parameter at vanishing external field. At non-vanishing
-2 (a) (b)
external field, the first sign change of the order param-
0.8 1 1.2 0.8 1 1.2
T/T T/T eter fluctuations appears at the fourth order cumulant.
c c
The common feature is that the higher the order of the
1 1
cumulant, the more complicated the fluctuation pattern
0 is.
0
The critical behavior of the third order cumulant of
E 3 -1 E 4
k k -1 the order parameter at vanishing external field, and the
-2 fourthordercumulantofthe orderparameterareconsis-
-2 tentwiththe expectationsofthe effective modelandthe
-3 (c) (d) σ-model, respectively.
0.8 1 1.2 0.8 1 1.2
T/T T/T For the 3-dimensional O(2) and O(4) spin models, the
c c
behavior of the second to fourth order cumulants of the
FIG.5: (Color online). κE1 (a), κE2 (b),κE3 (c) and κE4 (d)at energy is similar to that from the 3-dimensional Ising
model. The sign change also starts at the third order
H =0.1,0.05,0 from the 3-dimensional O(2) spin model.
cumulant.
ThenetbaryonnumberfluctuationsbasedontheO(4)
IV. SUMMARY spin model are qualitatively consistent with the calcula-
tions from the lattice QCD, and expectations from the
PNJL and the PQM models. Our results also show that
In this paper we perform the simulations of the 3-
atvanishingchemicalpotential,thesixthordercumulant
dimensional Ising, O(2) and O(4) spin models at a
ofthenetbaryonnumberisnecessaryinordertoobserve
given system size at three different external fields H =
a sign change in the chiral phase transition.
0.1,0.05,0. The critical behavior of the high cumulants
of the order parameter and energy in the 3-dimensional The authors are grateful for valuable comments and
Ising model, and the cumulants of the energy in the 3- suggestionsfromProf. F.Karsch,Dr. S. Mukherjee,Dr.
dimensional O(2) and O(4) spin models is presented, re- V. Skokov,and Dr. H. T. Ding.
spectively. We findthatthe externalfielddoesnotinflu- This work was supported in part by the National
ence the generic structure of the cumulants, except the Natural Science Foundation of China under Grant No.
cumulantsoftheorderparameterfromthe3-dimensional 10835005and MOE of China under Grant No. IRT0624
Isingmodel. Butitwidensthetemperatureregionofthe and B08033.
[1] K. Fukushima and T. Hatsuda, Rep. Prog. Phys. 74, [13] S. Chatterjee and K.A. Mohan, arXiv:1201.3352.
014001 (2011). [14] F. Karsch and K. Redlich, Phys. Rev. D 84, 051504
[2] M.Gyulassy,L.McLerran,Nucl.Phys.A750,30(2005); (2011).
J. Adams et al. (STAR Collaboration), Nucl. Phys. [15] S.Borsa´nyi(fortheWuppertal-BudapestCollaboration),
A757, 102 (2005). arXiv:1210.6901.
[3] H.Caines-for theSTARCollaboration, arXiv:0906.0305. [16] P. de Forcrand and O. Philipsen, Phys. Rev. Lett. 105,
[4] Y. Hatta and M.A. Stephanov, Phys. Rev. Lett. 91, 152001 (2010).
102003 (2003). [17] M.Stephanov,K.Rajagopal,andE.Shuryak,Phys.Rev.
[5] V.Koch, arXiv:0810.2520. Lett. 81, 4816 (1998).
[6] M.A. Stephanov,Phys.Rev.Lett. 102, 032301 (2009). [18] M. Asakawa, J. Phys. G: Nucl. Part. Phys. 36, 064042
[7] M.Asakawa,S.Ejiri,andM.Kitazawa,Phys.Rev.Lett. (2009).
103, 262301 (2009). [19] R.D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338
[8] M.A. Stephanov,Phys.Rev.Lett. 107, 052301 (2011). (1984).
[9] B.Friman,F.Karsch,K.Redlich,V.Skokov,Eur.Phys. [20] Y.HattaandT.Ikeda,Phys.Rev.D67,014028 (2003).
J. C 71, 1694 (2011). [21] C. Bernard et al., Phys. Rev.D 61, 054503 (2000).
[10] M. Cheng et al., Phys.Rev.D 79, 074505 (2009). [22] S. Ejiri et al., Phys.Rev.D 80, 094505 (2009).
[11] R.V.Gavai, S. Gupta, Phys.Lett. B 696, 459 (2011). [23] O. Kaczmarek et al., Phys.Rev. D 83, 014504 (2011).
[12] W.J.Fu,Y.X.LiuandY.L.Wu,phys.Rev.D81,014028 [24] SeeforinstanceS.L.Lauritzen,ThielePioneerinStatis-
(2010). tics, Oxford University Press, 2002.
6
[25] A.L.Talapov and H.W.J. Bl¨ote, J. Phys.A:Math. Gen. [30] J. Garc´ıa, J.A. Gonzalo, Phys. A 326, 464 (2003).
29, 5727 (1996). [31] Pan Xue’s talk at CPOD11,
[26] H.G. Ballesteros, L.A. Ferna´ndez, V. Mar´ın-Mayor, A. http://conf.ccnu.edu.cn/∼cpod2011/.
Mun˜oz Sudupe,Phys. Lett.B 387, 125 (1996). [32] U. Wolff, Phys.Rev. Lett.62, 361 (1989).
[27] J.J.RehrandN.D.Mermin,Phys.Rev.A8,472(1973). [33] I. Dimitrovi´c, P. Hasenfratz, J. Nager and F. Nieder-
[28] N.B Wilding, J. Phys.: Condens. Matter 9, 585 (1997). mayer, Nucl. Phys.B350, 893(1991).
[29] C. Nonaka and M. Asakawa, Phys. Rev. C 71, 044904 [34] M.A. York,G.D. Moore, arXiv:1106.2535.
(2005).