Solitonic modulation and Lifshitz point in an external magnetic field within Nambu--Jona-Lasinio model

0.12 MB·

by  Gaoqing Cao

Solitonicmodulation andLifshitzpoint inanexternal magneticfield withinNambu–Jona-Lasinio
model
Gaoqing Cao and Anping Huang
DepartmentofPhysics,TsinghuaUniversityandCollaborativeInnovationCenterofQuantumMatter,Beijing100084,China
(Dated:April25,2016)
WestudytheinhomogeneoussolitonicmodulationofachiralcondensatewithintheeffectiveNambu–Jona-
Lasiniomodelwhenaconstantexternalmagneticfieldispresent.Theself-consistentPauli-Villarsregularization
scheme isadopted tomanipulate the ultravioletdivergence encountered in thethermodynamic quantities. In
ordertodeterminethechiralrestorationlinesefficiently,anewkindofGinzburg-Landauexpansionapproachis
proposedhere.Atzerotemperature,wefindthatboththeupperandlowerboundariesofthesolitonicmodulation
oscillatewiththemagneticfieldintheµ–BphasediagramwhichisactuallythedeHass-vanAlphan(dHvA)
6 oscillation. ItisveryinterestingtofindouthowthetricriticalLifshitzpoint(T ,µ )evolveswiththemagnetic
L L
1 field: There are also dHvA oscillations in the T –B and µ –B curves, though the tricritical temperature T
L L L
0 increasesmonotonicallywiththemagneticfield.
2

r PACSnumbers:11.30.Qc,05.30.Fk,11.30.Hv,12.20.Ds
p
A
I. INTRODUCTION tion”introducedinRef.[21],therealsituationisstillunclear.
2
In this work, we explore the features of chiral symmetry
2
breakingandrestorationinaconstantexternalmagneticfield
The inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrell
bytakingthesolitonicmodulationchiralcondensateintoac-
] (LOFF)statehasattractedalotofinterestssinceitsproposi-
h count. The main advantageis that this situation can be han-
tion in 1960s [1]. Thoughno direct evidenceof LOFF state
-t hasbeendiscoveredinexperimentsyet,recentdevelopments dledself-consistentlybyadoptingthePauli-Villars(PV)reg-
l ularizationscheme.Besides,inrecentyears,inversemagnetic
c ofultracoldatomicphysicsmightprovideagoodopportunity
u to pin down the exotic state because of its strong control- catalysis (IMC) effect was found in QCD system [25] and
n inspired a lot of discussions relevant to the effects of mag-
lability [2–4]. In condensed matter physics, an external
[ neticfield [26–37]. Expertsmainlyattributedthisanomalous
magnetic field will split the single-particle dispersion of
2 electronswithspin-upandspin-downthroughZeemaneffect phenomenontothe asymptoticfreedomorearlierdeconfine-
v and a large mismatch of the Fermi surfaces usually favors ment transition [26, 32, 33, 37]. Nevertheless, at very large
3 the pairing with finite momentum [1]. While in quantum magneticfield,itispossiblethatthedeconfinementtransition
9 might happen earlier than chiral symmetry restoration with
chromodynamic (QCD) systems, baryon chemical potential
4 increasing temperature[37, 38]. In the case of finite chemi-
directly playsa mismatch between quarkandanti-quarkand
3
calpotential,itisusuallythoughtthatdeconfinementandchi-
0 isospin chemical potential plays a mismatch between u and
ralsymmetryrestorationtransition separateswith each other
. d quarks; the sources of LOFF state are very rich in quark
1 which then gives rise to the quarkyonicmatter [39]; thus, if
matterandnuclearmatter[5–19].
0 theIMC effectholdsin thiscase isstill unknown,especially
Specially, in the study of chiral symmetry breaking and
6
neartheLifshitzpoint. Forsimplicity, westill adoptthe ini-
1 restoration, it was found that the original first-order transi-
tial Nambu–Jona-Lasinio model and introduce the magnetic
: tionlineintheT–µphasediagram[20]wouldbecoveredby
v fieldthroughcovariantderivative.We’llgivesomecomments
theinhomogeneousphasewiththesolitonicmodulation(SM)
Xi or dualchiraldensity wave (DCDW) modulationchiralcon- whenevertheIMCeffectshouldbeconsidered.
r densate[7,9]. Butdifferentmodelsgivedifferentpredictions The paper is organizedas following: In Sec.II, we give a
a generaltheoreticalframeworkforthestudyofchiralsymme-
aboutwhichinhomogeneousphaseismuchmorefavored. A
trybreakingandrestorationwithSec.IIA presentingthefor-
recent study in the renormalizable quark-meson model sug-
mulism for solitonic modulation in NJL model and Sec.IIB
gestedthatthesolitonicmodulationusuallyhaslowerfreeen-
discussing a new kind of Ginzburg-Landau (GL) expansion
ergyandthephaseboundariesarebothofsecondorderwhich
approach. ThenumericalresultswillbeshowninSec.IIIand
indicates the existence of a tricritical Lifshitz point [9]. On
wefinallysummarizeinSec.IV.
the other hand, at the presence of a constant magnetic field,
thefeaturesofchiralsymmetrybreakingandrestorationwith
LOFF state alter a lot [21, 22]. Because of the asymmetry
II. THEORETICALFRAMEWORK
between the particle and antiparticle dispersions at the low-
est Landau level (LLL), the DCDW modulation was found
to be much more favored at non-vanishing chemical poten- A. Nambu–Jona-Lasiniomodelandsolitonicmodulation
tial [21] and the transitionpointwas foundto be a tricritical
point at vanishing chemical potential [22]. However, due to The Lagrangian of Nambu–Jona-Lasinio model in a con-
thesensitivityoftheinhomogeneousstatetothechoiceofreg- stantexternalmagneticfieldis
ularizationsinthechiraleffectiveNambu–Jona-Lasinio(NJL)
model [21, 23] and the ambiguous”intermediate regulariza- =ψ¯(iD/ m µγ0)ψ+G (ψ¯ψ)2+(ψ¯iγ τψ)2 , (1)
0 5
L − −
h i
2
whereψ(x) = (u(x),d(x))T denotesthetwo-flavorquarkfield thermodynamicpotentialforthechosensettingcanbeevalu-
withcolordegreesoffreedomN = 3andD = ∂ +iQA is atedas
c µ µ µ
the covariant derivative with the subscripts µ = 1,2,3 cor-
T
responding to the coordinates x,y,z and the charge matrix Ω = N Trln iD/ m(x) iγ τ3m(x) µγ0
c r 5 i
Q = diag(q ,q ) = diag(2e/3, e/3) in flavor space. With- V − − − −
u d (cid:2) (cid:0) (cid:1)
− 1
out loss of generality, we set the magnetic field to be along + d4x m2(x)+m2(x)
z-directionandthevectorpotentialtobeexpressedwithLan- 4G Z r i
(cid:0) (cid:1)(cid:3)
daugauge,thatis, A = (0,0, Bx,0) (B > 0) . Here, m is T 1+γ5τ3 1 γ5τ3
the currentmass of uµand d qu−arkswhich we set to zero0for = V −NcTrln iD/− 2 m(x)− −2 m∗(x)−µγ0
simplicity as it will notaffect the qualitative results [7], µ is (cid:2)1 (cid:0) (cid:1)
quarkchemicalpotentialandτ = (τ1,τ2,τ3) arePaulimatri- + d4xm(x)2 , (5)
4G Z | |
cesinflavorspace. Fortheconvenienceoflaterdiscussions, (cid:3)
we’d like to choose the explicitformsof γ-matricesin Weyl where the mass gap m(x) = m(x)+im(x), and we work in
r i
representation[40]: Euclideanspacewiththeintegraloftheimaginarytime x =
4
ix intheregion[0,1/T].
0
γ0 = 0 I , γi = 0 σi , (2) InordertoobtainanexplicitexpressionforΩ,it’sessential
I 0 ! σi 0 ! toevaluatethecontributionofquarks. Let’ssupposethatthe
−
inhomogeneous mass gap is one-dimensional and along the
where σ1,σ2, and σ3 are Pauli matrices and I is the 2 2 magneticfield, thatis, m(x) = m(z), asthe Taylorexpansion
×
identitymatrixinDiracspinorspace.Aswellknown,inchiral analysisindicatedthattheinhomogeneityalongotherdimen-
limitm0 =0,theLagrangianhasSUL(2) SUR(2)chiralsym- sionsisdisfavored[21].Then,themostimportantmissionleft
×
metry with the left-handedand right-handedtransformations istoevaluatetheeigenvaluesofthefollowingHamiltonian:
ψ(x) exp(i(1 γ )τ·θ/2)ψ(x),respectively. Iftheexpec-
5
tation→valueofψ¯∓ψorψ¯iγ τψisnonvanishing,itiseasytosee 1+γ5 1 γ5
that the symmetry of the5 Lagrangian will be spontaneously Hf = −γ0 iD/f − 2 m(z)− −2 m∗(z)
broken. Thus, ψ¯ψ and ψ¯iγ5τψ are both orderparametersfor = γ0(cid:2)iγ1∂x+iγ2(∂y iqfBx)+iγ3∂z (cid:3)
chiralsymmetrybreakingandrestoration. − −
1+(cid:2) γ5 1 γ5
Inordertostudytheexpectationvaluesoftheorderparam- m(z) − m∗(z) , (6)
eters, weintroducefourauxiliaryfieldsσ(x) andπ(x). Then − 2 − 2
(cid:3)
the original partition function = [ ψ¯][ ψ]ei can be
L wherethesubscriptfstandsfortheflavoruord. Ifweexpand
Z D D
changedtothefollowingform: R the spinors with Ritus’s method by separating variables, the
firsttwotermsinthesquarebracketgiverisetoaLandaulevel
1
= [ ψ¯][ ψ][ σ][ π]exp i d4x (σ2+π2) termi 2nq Bγ2[43,44].Thelefttermsactuallycorrespond
Z Z D D D D Z − 4G | f |
n h toaonpe-dimensionalNJLmodelorGross-Neveu(GN)model
+ψ¯(iD/ σ iγ π·τ µγ0)ψ , (3) withthefollowingHamiltonian:
5
− − −
io
which is actually the Hubbard-Stratonovich transforma- 1+γ5 1 γ5
tion[41]. Then,bycompletingthefunctionalintegrationover HGN = −γ0 iγ3∂z− 2 m(z)− −2 m∗(z)
the quark degrees of freedom, the model can be bosonized (cid:2) i∂ 0 m(z) 0 (cid:3)
z
withthepartitionfunctiongivenas 0 i∂ 0 m(z)
=  − z . (7)
Z = Z [Dσ][Dπ]expn−ih41G Z d4x(σ2+π2) m∗0(z) m∗0(z) −0i∂z i∂0z 
+iN Trln(iD/ σ iγ π τ µγ0) , (4) TheHamiltoniancanbebroughttoablockdiagonalformby
c 5
− − · − takingasimilitudetransformation,thatis,
io
wherethetraceistakenoverthecoordinate,spinorandflavor
H m(z) 0
spaces. Inmeanfieldapproximation,let’sset σ(x) =m(x), U 1H U = z , (8)
π3(x) = m(x), and π1(x) = π2(x) = 0.hThisisettinrg is − GN (cid:0)0 (cid:1) Hz m∗(z) !
i
ghenerailwhenthemagnheticfiieldihsabseintbecauseoftherota- (cid:0) (cid:1)
wheretheinvolvedmatricesarerespectively:
tional symmetryin the flavor space. However, when a finite
magnetic field is present, other cases with π1(x) or π2(x) 1 0 0 0
nonvanishingarequitedifferentfromthis. Ihnprinicipalh,thesei i∂ m (z) 0 0 0 1
H m(z) = z ∗ ,U = . (9)
cc(LahsaQerCsgeDasr)eccaharrlacireuddlatbtoiyoenπv1a[4lau2na]dteaπnp2dr.etcAhiescecGloyrindbzienbcguarutgos-eLthaoenfdlatahtuteicaeenlaeQlcyCtrsDiics z(cid:0) (cid:1) m(z) −i∂z ! 00 10 01 00
ofourpreviouswork[36],π1(π2)condensationisjustlikesu- For a given inhomogeneousstate, the explicit form of the
perconductorandthemagneticfielddisfavorsthepionsuper- thermodynamic potential usually can be evaluated with the
fluidity,sothesecasescanbesafelyneglectedhere.Then,the helpofthedensityofstates. However,oneshouldbecautious
3
whenm(z)isnotreal.Inthiscase,thespinorsareactuallyhalf whenmagneticfieldispresent[36].Thentheconvergentform
validattheLLLandtaketheformsu(x)= u (x),0,u (x),0 T ofthethermodynamicpotentialis
1 3
and d(x) = 0,d2(x),0,d4(x) T, respective(cid:0)ly [44]. Then th(cid:1)e M2(m,ν) qB
DspCecDtrWa {mε}odo(cid:0)uflaHtiGoNna(mre(zn)o=t s(cid:1)myme2mikze)troicf uwiqtuha{r−kεf}o.r eTxaakmeptlhee, Ω(T,µ,B;m,ν) = 4G −q=Xqu,qdNc|2π|Xn=0αn
sthigenspaesyctmramaeretri{c±sppepc2ztr+amm2a+kek}t.hAesrehgauslbareieznatmioennvtieornyeddi,ffithceuslet Z ∞dερ(ε;m,ν)fPV(T,µ,n,q,B,ε),(15)
0
at finit chemical potentialdue to the non-renormalizablena-
where f (T,µ,n,q,B,ε) = 3 c f(T,µ,n,q,B,√ε2+iΛ2)
ture of NJL model [21, 23]. For the solitonic modulation, PV i=0 i
withc = c =1,c = c = 3. TheadvantagesofthePV
m(z)isrealanditcanbecheckedthattheLLLspectraaresign 0 3 1 2P
− − −
regularizationschemearereadytosee: Inthelimitν 0or
symmetric.Thus,thiscasecanbetreatedself-consistently.
→
m 0,wecanreproducethePVregularizedthermodynamic
Forthesolitonicmodulation,themassgaptakesthefollow- →
potentialforthechiralsymmetryrestorationphase(χSR)asit
ingform[45]:
shouldbe. Inthelimitν 1,thePVregularizedthermody-
→
namicpotentialforthehomogeneouschiralsymmetrybreak-
M(m,ν,z) = m νsn(K(ν)ν)sn(mzν)sn(mz+K(ν)ν)
| | | ingphase(χSB)canalsobereproduced. Thus,thePVregu-
(cid:16)
cn(K(ν)ν)dn(K(ν)ν) larizationschemeguaranteestheself-consistencyofsolitonic
+ | | , (10)
sn(K(ν)ν) modulationinseverallimits. Finally,thegroundstateshould
| (cid:17) bedeterminedbyminimizingΩ(T,µ,B;m,ν)with respectto
where sn,cn and dn are elliptic Jocobi functions with ellip- m and ν for given parametersT,µ and B. And phase transi-
tic modulus √ν. And the thermodynamic potential can be tionhappenswiththechangeofparameters:Whenνchanges
derivedstraightforwardlyfromthe case with vanishingmag- from1 to asmaller one, thesystem transitsfromχSBphase
neticfield[7]becausethetransversaldegreesoffreedomare toSMphase;whenitchangesfrom0<ν<1to0,thesystem
irrelevanttothelongitudinalone. Byreplacingthetransverse transitsfromSMphasetoχSRphase.
momentawiththeLandauLevels,thethermodynamicpoten-
tialcanbeexpressedexplicitlyas
B. Ginzburg-Landauexpansionwithsmallν
1 L qB
Ω(T,µ,B;m,ν) = M2(m,ν,z)dz N | | α
4GLZ − c 2π n Inordertoevaluatethechiralsymmetryrestorationtransi-
0 q=Xqu,qd Xn=0 tion and the Lifshitz point more efficiently, we’d like to in-
∞dερ(ε;m,ν)f(T,µ,n,q,B,ε), (11) troduceadifferentGinzburg-Landauexpansionschemecom-
Z0 pared to that of Ref. [7], that is, expand the thermodynamic
potentialΩ(T,µ,B;m,ν)with respect to the elliptic modulus
whereL = 2K(ν)/mistheperiodof M(m,ν,z),αn = 2 δn,0 ν. The Taylor expansions of M2(m,ν) and ρ(ε;m,ν) around
−
stands for the degeneracy of the n-th Landau level, and the
smallνarerespectively
integrandis
7ν
M2(m,ν) = m2 √ν +o(ν3/2) , (16)
f(T,µ,n,q,B,ε) = ǫ(n,q,B,ε)+Tln(1+e (ǫ(n,q,B,ε) µ)/T) − 8
− −
(cid:0) (cid:1)
+Tln(1+e−(ǫ(n,q,B,ε)+µ)/T), (12) ρ(ε;m,ν) = 1 + m2√ν 7m2ν
π 2π(ε2 m2) − 16π(ε2 m2)
withtheexcitationenergyǫ(n,q,B,ε)= 2n|qB|+ε2 1/2. The mνδ(ε m−)+o(ν3/2), − (17)
corresponding density of states for soli(cid:0)tonic modu(cid:1)lation is −2π −
givenby[46] whereo(ν3/2)isthePeanoformoftheremainderandthethird
terminEq.(17)isfromthestepfunctions. Thus,thethermo-
1 ε2 m2E(ν)/K(ν)
ρ(ε;m,ν) = − dynamicpotentialhasthefollowingform
π (ε2 m2) ε2 (1 ν)m2
− − − Ω(T,µ,B;m,ν) = Ω(T,µ,B;m,0)+m2β(T,µ,B;m)√ν
θ(pε2 m2) (cid:0)θ ε2+(1 ν(cid:1))m2 , (13) 7
− − − − + m2β(T,µ,B;m)+γ(T,µ,B;m) ν
(cid:2) (cid:0) (cid:1)(cid:3) − 8
whereK(ν)isthequarterperiod,E(ν)istheincompleteellip- (cid:16) (cid:17)
+o(ν3/2), (18)
ticintegralandθ(x)isthestepfunction. Fortheconvenience
ofnumericalcalculations,theintegralinthefirstpartofΩcan
1 qB
beworkedoutpresicelytogive[45] β(T,µ,B;m)= N | | α ∞dε
4G − c 2π nZ
q=Xqu,qd Xn=0 0
1 2E(ν)
M2(m,ν)=m2 +1 ν . (14) 1
(cid:16)sn2(K(ν)|ν) − K(ν) − (cid:17) 2π(ε2 m2)fPV(T,µ,n,q,B,ε), (19)
−
qB
ThesecondpartisdivergentandwerefertoPVregularization γ(T,µ,B;m)=m N | | α f (T,µ,n,q,B,m).(20)
schemeasitismuchsofterthanothersandcanavoidartifacts q=Xqu,qd c(2π)2 Xn=0 n PV
4
NotethatΩ(T,µ,B;m,0)isjustthethermodynamicpoten- inthevacuumwhichisalittle smallerthantheLQCD result
tial for χSR phase which doesn’t depend on m. However, u¯u = d¯d = (0.25GeV)3.
h i h i −
m2β(T,µ,B;m)is massdependentandthevalueofm should BoththeNJLmodelandQMmodelpredictedthatthetran-
be determined by minimizing this coefficient for given pa- sitions from χSB phase to SM phase and from SM phase to
rameters which gives the lowest free energy around ν 0. χSR phase are both of second order in the absence of mag-
∼
The integral over ε should be understood as Cauchy Princi- netic field [7, 9]. We carefully check the case with a con-
pal value integration and then β(T,µ,B;m) is real and con- stantmagneticfieldandfinditremainsthesame: Asthereis
vergent. The minimum of β(T,µ,B;m) directly determines onlyoneminimumofΩ(T,µ,B;m,ν)withrespecttoνwhich
whichphaseisthesystemin: Iftheminimumisnegative,the alone determines the transition points for given parameters
χSBphaseorSMphaseismorefavored;ifpositive,theχSR andnonzeromandtheconventionalfirst-ordertransitionfrom
phaseismorefavored;andmin β(T,µ,B;m) = 0justdeter- χSB phase to χSR phase lies between these two, the transi-
minesthechiralsymmetryresto(cid:0)rationpoint. A(cid:1)tthetransition tionsmustbothbeofsecondorder.
point,thenext-ordercoefficientreducestoγ(T,µ,B;m)which In order to explore the effect of magnetic field, we’d like
issemi-positivedefinite(onlyequalstozerowhenm=0)and tostartfirstlyatzerotemperature. Inthiscase,theintegrand
this means that the phase transition is always of second or- becomes
der. Itshouldbeclarifiedthatanonzeroexpectationvalueof
1
m aroundν 0 doesn’tnecessarily meanthe transitioncan- f(0,µ,n,q,B,ε)= ǫ(n,q,B,ε) sµ, (24)
∼
notbesecondorder,becausewhatreallymattersis M(m,ν,z) 2 | − |
Xs=
whichofcoursevanishesatν=0. ±
Furthermore,thiskindofGLexpansionapproachisalsoca- andthenumericalresultsfortheupperandlowerboundaries
pabletofindtheLifshitzpointwherethesolutionwithν=0is ofSM phaseare shownin Fig.1. Itisinterestingtofind that
consistentwiththesolutionν=1.Thiscanhappenonlywhen the critical chemicalpotential µ oscillates with the magnetic
theexpectationvalueofmiszeroandtheLifshitzpointisin field B at both boundaries which is actually an illumination
fact the critical pointbetween m = 0 and m , 0. Neverthe- of the dHvA effect [48]. This is consistent with that found
less,thereisasmalldefectwiththisapproach:Thederivative in the study of homogeneouschiral condensate [35, 36, 49].
of the coefficientβ(T,µ,B;m)with respect to m is divergent Because theoscillationsare notstrictly coincidentwith each
aroundtheintegralregionε mascanbeseenfromthefol- other,thesizeoftheexistingregionfortheSMphasealsoos-
lowing ∼ cillates with B and is the smallest around √eB = 0.33 GeV
which is accidently the same as the mass in vacuum. For
∂β(T,µ,B;m) = N |qB| α ∞dεfPV(T,µ,n,q,B,ε). larger B, the region for the SM phase increases with B due
m∂m − c 2π nZ π(ε2 m2)2 tothecatalysiseffect.
q=Xqu,qd Xn=0 0 −
(21)
0.34
Therefore,theminimumcanonlybeevaluatedbydirectscan-
ning of β(T,µ,B;m) over m instead of a new kind of ”gap ΧSR
0.32
equation”.
V
e 0.30
G
(cid:144)Μ ΧSB
III. THEPHASEDIAGRAMSANDLIFSHITZPOINT 0.28 SM
0.26
Ashadalreadybeenilluminatedinthequark-meson(QM)
model [7], the expectation value of m in vacuum affects the
0.24
qualitativeresultsquitemuchabouttheexistenceofthesoli- 0.0 0.2 0.4 0.6 0.8
tonic modulation. In order to show the results explicitly, we
eB (cid:144)GeV
choosem=330MeVasamoderatechoiceandkeepthepion
decay constant f to the experimentalvalue 93 MeV. Then,
π FIG.1: Theµ–Bphasediagramatzerotemperature. Theupperand
accordingtothefollowingrelations[47]:
lowerboundariescorrespondtothetransitionfromsolitonicmodula-
tionphasetochiralsymmetryrestorationphaseandfromchiralsym-
m 6m 3 m2+iΛ2 metrybreakingphasetosolitonicmodulationphase,respectively.
ψ¯ψ = c(m2+iΛ2)ln ,(22)
h i ≡ −2G −4π2 i m2
Xi=0
Then,we’dliketoturnonthetemperatureandcheckhow
f2 = Ncm2 3 c lnm2+iΛ2, (23) would the magnetic field affect the T–µ phase diagram. To
π 4π2 i m2 showtheeffectobviously,wechooseastrongmagneticfield
Xi=0
√eB = 0.6GeV whichis nearthe minimaofthe boundaries
theparametersofthePVregularizedNJLmodelcanbefixed andcompareitwiththecaseintheabsenceofmagneticfield.
as Λ = 0.786 GeV and GΛ2 = 6.24. This corresponds to The results are illuminated in Fig.2. As can be seen, the re-
a chiral condensate u¯u = d¯d = ψ¯ψ /2 = (0.20 GeV)3 sult for B = 0 is qualitatively consistent with that obtained
h i h i h i −
5
0.25
0.14
0.20 0.6 GeV
0.12
V
V 0.15 IMC e
e eB = 0 G 0.10
G
(cid:144)
(cid:144) L
T 0.10 T
0.08
0.05
0.06
0.30
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Μ(cid:144)GeV 0.28
V
FIG.2: (coloronline) Thephasetransitionlinesforthecaseswith 0.26
e
B = 0(blackdashedlines)and √eB = 0.6GeV(reddottedlines). G
Thesolitonicmodulationphaseisinthesmallclosedregionsatlarge (cid:144)L 0.24
Μ
chemicalpotentialandthebulletsaretheLifshitzpoints.Theinverse
magnetic catalysis effect to the case with √eB = 0.6 GeV is also
0.22
illuminated(bluedot-dashedline).
0.20
0.0 0.2 0.4 0.6 0.8
in Ref. [7]. However,one shouldnotice here thata mostre-
eB (cid:144)GeV
cent study of a more general form of inhomogeneous order
parameterbyusingfinite-modeapproachgivesa verydiffer-
entresult: thehomogeneous-inhomogeneousphasetransition
becomes of first order and the inhomogeneousregion is en-
FIG.3: TheevolutionsofthetemperatureT andthechemicalpo-
larged a lot even for a moderate constitute quark mass [24]. L
tentialµ attheLifshitzpointwiththemagneticfieldB.
DuetothedHvAeffectatzerotemperature,thesecond-order L
transition linesfrom χSBphase to χSR phase intersect with
eachother. ItmightnotbethecaseinrealQCDsystem: Be-
cause of the IMC effect [25], it is much more probable that magneticfieldduetothesignsymmetryofthequarkspectra.
Andinordertoevaluatethechiralsymmetryrestorationtran-
the transition line would be substituted by another one for
sition more efficiently, we develop a new kind of Ginzburg-
√eB = 0.6GeV(seethebluedot-dashedline)andthetransi-
Landauexpansionapproacharoundsmallν.
tionlinesarecoveredbythoseof B = 0. However,forlarger
ThetransitionsfromχSBphasetoSMphaseandfromSM
magnetic field (√eB 0.8 GeV), the transition lines would
startto intersectwith≥thoseof B = 0 dueto the dHvAoscil- phasetoχSRphasearebothofsecondorderinthepresenceof
magneticfield.Atzerotemperature,boththeupperandlower
lation at low temperature. There are two tricritical Lifshitz
boundariesoftheSM phaseoscillate withthemagneticfield
pointswherethreesecond-ordertransitionlinesintersectwith
andsodoesthesizeoftheSMregionwhichareilluminations
eachotherintheplotandthepointisfoundtoshifttoupper
of the dHvA effect. At finite temperature, the tricritical Lif-
leftbythemagneticfield.
shitz pointis also found to oscillate with the magnetic field.
Finally, the evolution of the Lifshitz point with the mag-
Generally, only one minimum is found for each µ–B phase
netic field is evaluated as illuminated in Fig.3. The dHvA
diagramwhichisconsistentwithpreviousworks[35,36,49].
oscillation showsup in both curves: The oscillation is much
Ourrecentstudyisverypreliminary. How’sthefateofthe
moreobviousintheµ –Bcurvewiththeminimumataround
L
√eB = 0.7 GeV. On the other hand, the temperatureT in- SM phase when the fluctuations of collective modes are in-
L cludedeitherthroughGaussianexpansion(thoughverydiffi-
creases monotonicallywith the magnetic field B and the flat
cult)orquark-mesonmodel[7,9,50]isanimportantquestion.
oscillationintheregion √eB [0.2GeV,0.5GeV]ismainly
∈ Asitisimpossibletocheckwhichoneismuchmorefavored
duetothefastdecreasingofµ .
L
betweentheDCDWmodulationandthesolitonicmodulation
phasesbythefirstprincipalLQCDcalculationatfinitechem-
ical potential[51], it is very importantto find a method that
IV. SUMMARY canconsistentlytreatbothcasesatfinitemagneticfield. The
Dyson-Schwinger equation and the functional renormaliza-
In this work, we explore the magnetic field effect to the tion groupapproachmay serve as possible candidates. Still,
phase transitions among the homogeneous chiral symme- when taking a more general form of inhomogeneous order
trybreaking,inhomogeneoussolitonicmodulationandchiral parameter into account [24], how would the phase diagram
symmetry restoration phases. The thermodynamic potential changeatfinitemagneticfielddeservesfurtherstudy.Finally,
canbeobtaineddirectlybygeneralizingfromthecasewithout thoughthedHvAoscillationsurelyexistsatlowtemperature,
6
how would the IMC effect affect the T–µ phase diagram at geneousstates and Lifshitz point. The work is supportedby
finitemagneticfieldstillneedfurthercheck. the NSFC under Grant No. 11335005and the MOST under
Acknowledgments: G.C. thanks Bernd-Jochen Schaefer GrantsNo. 2013CB922000andNo. 2014CB845400.
forprovidingtherelevantpapersaboutthe studyofinhomo-
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