Phonons, Pions and Quasi-Long-Range Orderin Spatially Modulated Chiral Condensates

Kazuhiko Kamikado (RIKEN)

Y. Hidaka, K. Kamikado, T. Kanazawa and T. Noumi, <Phys Rev D.92.034003 >QCD phase diagram
Rep. Prog. Phys. 74 (2011) 014001⇥⇤¯/ + mqL=iD

K Fukushima

1+ trFµ⌫ F µ⌫
4

Figure 1. Conjectured QCD phase diagram with boundaries that define various states of QCD matter based on SχB pa

Fukushima-Hatsuda (2011)

and sets a natural scale for the critical temperature of light flavours it is appropriate to address more ochiral restoration. In the chiral perturbation theory (χPT) phase transition. Recent analyses on the basis of t
the chiral condensate for two massless quark flavours at fermion and Wilson fermion indicate a crossov
phase to the quark–gluon plasma for r
low¯temperature is known to behaveh ¯ias5⟨~ψ̄ψ⟩⌧ iT /⟨ψ̄ψ⟩ = hadronic
h
2i
2
4
4
1 − T /(8fπ ) − T /(384fπ ) − · · · with the pion decay and s quark masses [38, 39]. The pseudo-criticalconstant fπ ≃ 93 MeV [29]. Although the validity of Tpc , which characterizes the crossover location, is
χPT is limited to low temperature, this is clear evidence within the range 150–200 MeV as summarized in
Even for the temperature above Tpc the sysof the melting of chiral condensate at a finite temperature.
At low baryon density, likewise, the chiral condensate strongly correlated and show non-perturbative
decreases as ⟨ψ̄ψ⟩nB /⟨ψ̄ψ⟩ = 1 − σπ N nB /(fπ2 m2π ) − · · · such as the existence of hadronic modes or[30–32] where σπ N ∼ 40 MeV is the π–N sigma term. hadrons in the quark–gluon plasma at µB =as well as at µB ̸= 0 [41–43]. Similar phe(For higher order corrections, see [33, 34].)¯ ichiralh The= transition(x) is a notionh ¯iindependent⌧ i =of~⇡the(x) be seen in other strong-coupling systems such5~
deconfinement transition. In section 3.2 we classify the temperature superconductivity and in the BEC• Chiral phase transition is described by the chiral order parameters• Some model analyses suggest that spatially modulated chiralcondensate can appear near the QCD critical point.

Inhomogeneous chiral condensation
15

LNJL = ¯@/ + G ( ¯ )2 + ( ¯i 5~⌧ )2100

Gap-equation:100[ , ⇡]=(x)[ , ⇡]
=0
⇡(x)
!

50

⇤
@/100
+ (x) +200
i 5~⌧ · ~⇡ (x)300
Μq !MeV"80!

Critical point → Lifshitz point

60
40
6 0
20 h ¯ i =We need to solve Dirac equation⇥

T !MeV"

T !MeV"15000h¯ i = 0

D. Nickel 2009

= 400
0

0

260 280 300 320 340 360 380 400
Μq !MeV"•

someofathegapofequation.
FIG. 1: We
Left: know
Structureof the (one-dimensional)
NJL phase diagram in thesolutions
chiral limit asfunction
temperature T and
RealKink crystalline indicates the second order
Dual chiraldensityquark chemical
potentialµq wavefor M(DCDW)(short-dashed)q = 300MeV. The black
p first order phase transition
phase transition fromchirally
brokentorestoredphase,
thered(solid)linethe¯
h i = q ⌫sn(qz; ⌫)h ¯ i = M cos(qz)and the bullet the critical point. The spinodal region is enclosed by the blue (long-dashed) lines. Right:h ¯i 5the~⌧ energeticallyi=0Same plot ash ¯onthe orange (shaded) domain where
preferred groundi 5the⌧3 left
i =including
M sin(qz)state is inhomogeneous.
Nakano, Tatsumi (2004)

D. Nickel (2009)

the chiral limit and instead of adjusting ⟨ψ̄ψ⟩, we choose a value for Mq . These choices in turn fix
Typically
the periodthroughof condensationorder
fm. side of Fig. 1 we present the phase diagram
the model
parameters
Eq.(40).is On
theof left

M −in6Mmagnitude+ 6M ,and k as the same +6η
order
is consistent at
.
(24)
(a)
"$
(32)
(b)
2
originalELequation
in
3D
is
terms
are
shown
to
vanish
[10].
Then
the
functional
is
I)
this
transition.
At
the
onset
of
the
condensate,
however,
where the operator ∆ = ∂ + ∂ + ∂ is the three1.4
!
"
2
(FF)
2
2
2
dimensional
Laplacian.
is a 1D
fourth-ordernonlincast(x)in− thechirallimit,
using
theThisthe amplitude
M0 vanishes
while
k remainsfinite sothat
1
0 = ∆ M (x) + 3∆M
10 Mfollowing(∇M ) + Mform∆M in theear
partialdiﬀerential
equation.
The
solutionspacefor
2D than
(LO)
1.2
the
derivative
terms
are
more
importantthebulk
convention as Eq. (4):the partial diﬀerential equation is much wider than that
endence +6η
on 2ηM2 − 6Msame3+ 6M 5 ,(LO) of thehomogeneousterms.
In contrast,
the 3D
onset
do0.8
for the ordinary
diﬀerentialat
equation.
Therefore
it is not1
divergence of(32) 4 main wall
trivial thatthe
Jacobi’selliptic function
stays
as theamost
fa1D
(LO)
derivative
terms
play
minor
η2
1
2
′ 2 formation,
H.
Abuki,
D.
Ishibashi
andK.
Suzuki(2012)
2
2
2|m| + sgn(α4 )(|m| role.
+ |m |0.6) vorable structure when the restriction
to 1D modulations
Free
energy: ω̃ =theSN (solitonic)
0.8 operator ∆ = ∂x2 + ∂y + ∂z4 is the threen thewhere
divergentis taken away. Here we"do not search for the formal solu! a fourth-orderdimensional Laplacian. This1 isnonlin1 try′′some
6
2
′ 2
′ 2
∗tion,
2 but only
2 specificHomogeneous
crystals having the simple
+
|m|
+
4|m|
|m
|
+
Re(m
)
(m
)
+
|m
|
.
0.6
square or 2
cubic symmetry. In order to demonstrate how
ear partial diﬀerential equation.
6 The solution space for
0.4
2

GL expansion

2
x

2
y

2
z

mave

0.8

0.6

0.4

q

mave

0.2

0

ω [x10]ω [x10]the dimensionality of a crystal structure aﬀects the free
the partial
much
wider
than
that
B.
Most
generalcondensate
with higher harmonics
0.4 diﬀerential equation is LO(sinoidal)energy, we concentrateon the LO-type phase which has(26)
for the ordinary diﬀerential equation.
Therefore it is nota simpler form than the elliptic function. We set multi∗ 0.2 Higher
SN (solitonic)
dimensional
modulation
Pion
condensation
0.2
When
the
condensate
is
real
[m(z)
=
m(z)],
thisfuncdimensional
LO-typereal condensatesastrivialthat
Jacobi’selliptic
function
stays
as
the
most
faWe here try to see if Jacobi’s
elliptic
functionis thePATTERNS
Homogeneous√
vorable structurewhentherestriction
to
1D
modulations
0
MLO;1D(x) among= 2M
sin(kz),the favorable
restriction
to
the
1D
tional reduces to Eq. (4) with most01D0solution
modulation
patterns.
ONAL
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.05
0.1
0.15
0.2
0.25for0.3
0.35
0.4soluFF:
M(z)
=do
Δ not
exp(ipz)
is taken away.
Here
wesearchtheformalM(x)=M(sin(kx)+sin(ky)),We first note that
Jacobi’s 0elliptic
function just gives
modulation,
M (x) → M (z), substituted.
AsLO;2Dbefore,
we
η2
η2
#
LO:
M(z)
=
Δsin(pz)tion, but only try some specific crystals having the simple
2 solutions to the EL equaaIn
oneFig.
parameter
subgroup
ofamM=(b)LO;3D (x)the(c)
3 M0 (sin(kx) + sin(ky) + sin(kz)).
concentrate
on
the
case
α
<
0.
4
we
show
4
square or cubic symmetry. In order to demonstrate howtion. It should be stressed that the original EL equation(33)
the dimensionality ofa crystalofstructure
the freeplitudemassesaﬀects(a), themagnitudeofwave
vectors
(b),
M
(x)
is
equivalent
tothe“egg-carton”
Ansatzinfourth-order diﬀerential equation while(9) is a nonlinear
LO;2D
of whetheror
0 concentrate on the LO-type phase which has[34]. Each condensate is characterized by two real pa0
energy, we
Eq. (11)is basicallytheandsumof threediﬀerential
equaand corresponding thermodynamic
potentials
(c)
for
the
rameters
Mq
whose
valuesaretobedetermined0nsate,
witheia simpler form than the elliptic function. We set multitions
each ofofwhich
canbe obtained
theandsecondvia
minimizationof Ω. In from
Fig. 6(a)(b) we order
show
FF
and
LO
condensates,asafunction
η
.
The
wave
2
dimensional
LO-typerealcondensatesaseﬀective which
order parametersand freeenergies.obeys.
One canIt
ns, are possidiﬀerential equation
the elliptic
function
see
that
the
energyisanincreasing
functionofthe di√vector q is just k (kII ) for the FFis (LO)state,
while
forreally
thecovers all the solutions
-0.1
-0.1
thus
not
obviousthatit
we retain
up = 2M0 sin(kz),mensionality. Also it is notable thatthe critical points
MLO;1D (x)1D (FF)solitonic state it is given by 2π/ℓ
From atEL
thefigures,
wefrom theto theequation.
Keeping
thisin mind,try
which
the transitions
Wignerto the wecrystalP . original
previoussec3D(LO)phasestake place arefor
commonall threestates. To
MLO;2D (x) = Msee+thesin(ky)),0 (sin(kx)the most
generalassumption
the among
spatially
modulated
that
real
LO
phase
is
more
favorable
than
the
FFFF
(spiralwave)
see this analytically, we expand Ω 2D
in powersof M0 as(LO)
ynamics -0.2of the # 2
chiral condensate.
We
set
the
condensate
intheform of
%
$
-0.2
η
3
6
2
4
6
LO
(sinoidal
wave)
like
chiralspiral
over
the
whole
range
of
η
,
while
they
are
MLO;3D (x) =M0 (LO)+ O(M0 ),ΩLO;1D = 2 − 16 M0 + 24 1D
2
3 M0 (sin(kx) + sin(ky) + sin(kz)).
a harmonic expansion
seriesFF-likechiral$ η as3 % 2
9SN (solitonicwave) the solitonic phase, whichΩLO;2D= 2often− 16 M0 + 24
M04 +O(M06 ),
SN
(34)
less favorable
than
will
(33)
!
%
$
ese Mtwo statesη3
Homogeneous state2 H 106+ 24
M04 + O(MH
Homogeneous
MHH (z;hereafter.
{ℓH , MΩnH}) ==TheM
sin((2π/ℓ)nz),
to the“egg-carton”
Ansatz in(SN) state
LO;3D
0 n
0 ).
LO;2D (x) is equivalent
2 −FF
16 M
be
denotedby
thesolitonic-0.3-0.3
n, then=1,3,5,···[34].analysis
Each0 condensate
characterizedby two0.05
0.1 is
0.15
0.2
0.25
0.3
0.35 real
0.4 pa00.05at0.15
0.2 0.25 take
0.3exactly
0.35 the0.4
Since
the0.1
quadratic
coeﬃcients
same
and
LO
(and
SN)
statesbecomedegenerate
the
onset
η
3
1
II
(30)
η
M0 and q whose values
are to be II
determined
form, 2 − 16 ≡ 2 (η2 −η2η2 ), the critical point is shared.
hen rameters
we explore
2
On thefrom the coeﬃcientsof quarticofΩ.condensateat η(b)= show
3/8. Neartheit hand,is easy
where {MandℓHotherarevariationalparameters.
Weterms,note
via
minimization
of
In
Fig.
6(a)
and
we
n } point2
II A, we try a
we see that the free energies are on the order of the ditheevencomponents
areabsentif orderweformulas
restricttheFIG.
6. (color
online).
The
eﬀectiveparameters
eﬀective
freetheenergies.
One canto perform
expansionin Mthatresultfor (a)theLOmensionalityofmodulation.
The abovefor enFIG.
4. order(color parameters
online).
(a)and
Eﬀectiveorderparameters,
0 . The., thecondenfor the 1D-LO(dashed,
red),
2D-LO
(dotted,density
can beeasily period
generalized
towith=to ergya halfantiperiodic
Mgreen),(zthe+ ℓcase/2)
see wavethat
the energyis(c)anpotentials
increasing
ofofthediH3D-LO(b)
vectors,
andphase
as function
a function
η2 foris alreadygiven
byEq. condensate
(6),
and
that
for
the
FF
(dot-dot-dashed,magenta),and thefor
solitonic
state(solid)as−M(z). This is quite
reasonable
the caseof thechiralmensionality.
it is(dotted,
notableblue)
thatand
theSNcriticalpointsomparison,
we Alsothe
LO (dashed,red), FF(solid) states.
a functionofFor[10]comparison,
the quantity
theeven
FF
phase
isthe
obtainedatthe
thesame orderMηwe
02 . aslimit.
Ininfact
havechecked numericallythatfortheThequantities
correspondingtohomogeneouscondensate
at
which
the
transitions
from
theWignerto
crystaltonic
state.
In
phase is also are
depictedby thethinifsolidline (blue).
(b) Theareshownby place
dot-dashedlines. The
solidalllines
arestates.
the samecomponents
vanishingeven
included.
Hereafter,
wephasestakeare common
among
three
To
thermodynamic potentials as a function of η2 . as the higher
analogs
the in Fig.we2.expand Ω
asdepictedseethosethisofanalytically,Ω in=powersof FF
M0(x))⟩
as WSabbreviate the state characterized by MHH⟨ω(Mharmonic (HH) state. We took%!
"(27)into account up to theuch higher di- $ω [x10]FF (spiral)

0

-0.1

-0.2

-0.3

FIG. 6. (cfor the 1D(dot-dot-da
a function
phase is althermodyn22

the LO coby22

condensation tends to be energetically unfavored near the• PionLifshitz point.
• Higher-dimensional condensations are energetically unfavored.

MLO;dD (with x = (In this casremarks. Some technicalities are summarized in appenmension, the phonon mode (u) appears as the Nambu-dices. rotation by ✓ about the y axis, the gradientsthgiven byGoldstone (NG) mode of translational symmetry breakthiting, in addition to pions (⇡). In the real kink crysII. SYMMETRY@z u = CONSIDERATION
1 + cos ✓ , @x u = sin ✓ , cotal phase discussed later, the vectorial isospin symmeontry SU(2)V is unbroken and there is no mixing between
in
2
2
When
a chiralcondensate
is modulated
along onedi- the
hence
(1+@
u)
+(r
u)
is
invariantunderz?
phonons and pions. Therefore, the symmetry breakingmension, the phonon mode (u) appears as the Namburo
1
Then
the
eﬀective
theoryofuinvariantunderpattern reads
Goldstone (NG) mode of translational symmetry break- 2 gi¯¯
as a In
function(1+@h ⇤i ing,⌧ inican= be
0 constructedh ⇤ i⇥ = M (z)z u) +(raddition
to pions (⇡).
the realofkink
crys5~⇥
2
discussed
isospin derivative
symme2@z u+ (ru)later,+ 1 theandvectorial
higher-orderte
R3 o SO(3) ! R2 o SO(2) ⇥ discrete symmetry tal phase
unbroken and there is no mixing between
V 2isu.
as r(1)try SU(2)?
hephonons and pions. Therefore, the symmetry breaking
A
total
derivative
term
@
u
would
be
allowedz
Chiral symmetry
Translationalandrotational symmeties
1Th
pattern reads
in addition to the chiral symmetry breaking pattern,
fective theory
if we only
require the stationary
cac
⇥
⇤
⇥
⇤
d
2@
SU(2)R ⇥ SU(2)L ! SU(2)V . Here R and SO(d) denote R3 ofor
SO(3)! R2aovariation
SO(2) ⇥ discrete
symmetry
F underof withfixed boundaasthe d dimensional translational and rotational symmetions. However, global minimization of (1)F exclu00i⇡(x)
try groups, respectively.
The
discretesymmetry
includes
M
=asM
u(x))
term,
which
can
be
seen
follows:
consider fea
Phonon:0 (z +pattern,
Pion: M = M0 (z)ein addition to the chiral symmetry
breaking
a discrete translational symmetry along the modulatedSU(2)Rtionu =L "zover Vthe
groundstate(z). It cor⇥ SU(2)! SU(2). HereRd andSO(d)0denotefo
direction of the condensate as a subgroup, which is athe d todimensionaltranslationalrotationala dilatationz ! (1 and+ ")z.
In thesymmepresencetioo
The discrete symmetry includes
teremnant of the translation R in this direction. Other el-try groups,/ A@respectively.
z u in F, this u will generate an energy shi
a discrete translational symmetry along the modulatedtio
We expects
lowshapeenergyexcitations.
ements of the discrete groupdepend fouron theof thedirection of the condensate as a subgroup, which
is a
to
2
F
=
"A
+
O("
)
,
kink crystal2 .
remnant of the translation R in this direction. Other el/
In this symmetry breaking pattern, two rotational andements of the discrete group depend on the shape of the2
indicatingthat there exists a lower-energy direct.
one translational symmetries are spontaneously broken.kink crystal
symmetrybreaking
pattern,
and theA 6=
0. In otherwords,
we twohaverotational
A = 0 when
We note that there appears no gapless mode associated In thisinone
symmetries
are spontaneously broken.
T-G.translational
Lee,
E.
Nakano,et.al,state.
(2015)
DCDW
special.
sate
is
the
ground¯
withisthebroken rotational
symmetry.
Ingeneral,theh i = M cos(qz)A
We note that there appears no gapless mode associatedThisobservation
imposesaconstraint
that
number of NG modes does not coincide with that of bro-with the broken rotational symmetry. In general, thesaF
2depend
on
2@
u
+
(ru)
atwithleastz
ken global spacetime
symmetries[40–42].
number
of
NG
modes
does
not
coincidethatquadraticallof broh ¯i 5 ⌧3 i = M sin(qz)dewe conclude
the freeenergy density for the
spacetime thatsymmetries[40–42].
In the remainder of this section, we present a generalken global
weremainderof this section, we present a generalassumesa formdiscussion on the eﬀective low-energy theory of phonons In theasdiscussion on the eﬀective low-energy theory of phonons
We
can
not
distinguishthechiralrotationandspacialtranslation(see also [25, 31, 32, 43, 44]). LetR us consider a the-(see also [25, 31, 32, 43,44]). Let us consider
2
a theR 3 2
3
1
2
2
ory with the free energy F [ ] = d x F( , @ ) andory with the freeF energyFz[ u] += (ru)
d x F ( ,+
@ C(r
) and
=B @
u)
,
?
2 solution minimizassume that h i = 0 (x) is a static solution minimiz-assume that h i = 0 (x) is a staticThree
low energy
excitations
appearing F [ ]. In thefollowing
we assumethat thefreeing F [ ]. In the following we assume that the freeenergyatdensity
F respects
and translational
the leading
orderrotationalof the derivative
expansion,atw

Symmetry breaking• Rotational and translational symmetries are spontaneously broken.

Landau’s discussion
Deformation of condensation:

M 0 = M (z + u(x))

Energy unchanged
Translation (u = cons)z

Rotation (u =θx or u = θy)LLs of Hu in (21) at (T, µ) = (70, 286.0)
top) and for kz = 0 (bottom). The 0.5 is the first Brillouin zone. kz and
Q and k? is normalized bythe2 UV cutoﬀu(x)zFIG. 5. B and C at T = 70 MeV (top) and T = 10 MeV(bottom). B and C are normalized by the UV cutoﬀ parameter ⇤ with appropriate dimensions. The dotted vertical lines
mark the boundaries of the modulated
phase.2(r u(x))?

pertainsof long-wavelength phonons.
Energy changed
Stretch(u = az)to the free energyzmposed into a plane wave and a periBy adopting a variational approach, one can rigorously
show that E0 behaves for kz ⇠ k? ⇠ 0 as
L’ = L/(1+a)
4
E0 ⇠ Bkz2 + Ck?
,(23)= eik? ·x? eikz z (z) ,(22)

where B and C are functions of T and µ. The absence
mentum in transverse directions, kz
2 2 ) term in (23) is guaranteed by the property
of
the
O(k
al momentum, and (z) is a periodic (@z u(x)) ?
(17). The proof of (23) is somewhat technical and isL) = (z). Substituting (22) into
relegated to Appendix E. Equation (23) shows that thevalue equation for , which we have
Operations which change
the system
elasticthe
freeenergy
energy of of
low-energy
phononsappear
becomes on theby way ofp a Fourier decomposition
inQz free energy.
Z/ L with nmax = 20. To
ne
⇥
⇤
1
3
u
2
2
2
d x B(@z u) + C(r? u) .
(24)
Fel =
have increased nmax up to 30 and
2esults are unchanged.

•

A. Phase diagram
expansion are given as functions of temperature(T ) and A.
chemicalpotential (µ):
Phase diagram5

GL
expansion
of
NJL
, ↵ =

0 GL expansion
Accordingto Nickel’s[4], (3the+GLexpansion of the NJL(3 + 1)-dimensional
NJLsixthmodel,order,
to the reads
sixth orderAccordingofPhasethe1)-dimensionalmodel, to the
A. work
diagram↵20 to Nickel’s↵20 work [4],↵the
2

↵2 =

2

, ↵4 =

A. Phase2 diagram46⌦[M] =2↵2 M + ↵4 M6 + (rM )22 + ↵6 2M+ 10M 22(rM )2 + ( M )2 .
GL
4
2

6

4 [M ] = ↵2 M
122 + ↵4 M + (rM )⌦GLZ+ ↵6 2M + 10M (rM ) + ( M )

.

(22)

3expansion
XrdingEffective
to Nickel’s
work
[4],
the
GL
ofexpansionthe(3+1)-dimensional1)-dimensional
NJL model,
the (23)sixthorder,
The
coeﬃcientsof thethe GLgiven
as functionsof temperature(T ) sixth
andtochemicalpotential(µ):
Accordingto
Nickel’s
work
[4],
the
GL expansionofthe point
(3are
+
NJL
model,
to theorder,
readsdp1
potential
of
NJL
model
near
Lifshitz
2,n
0
n/2

↵n =
( 1) of4N
+ (T
,
The
coeﬃcients
theGLgiven as4 functionsof temperature
potential (µ):
FNc Texpansion are
0 2 ) and2 chemical
2 3
2 0
2
n/20 6↵2+ ↵↵2 +6 10M
↵2 (rM2) + ( M
2= ↵2(2⇡)
4 M +
2=
2)
2
2G
⌦
[M
]
M
+
↵
+
(rM
)
2M
. M )2 . (22)
GL
4
6
↵
,
↵
=
,
↵
=
[(!
iµ)
+
p
]
2
4
6
⌦GL [M ] = ↵2 M
+
↵
M
+
(rM
)
+
↵
2M
+
10M
(rM
)
+
(
n
4
6 4
n
2
12
Z
↵20
↵20
↵20
3
X
p chemical
1 potential (µ):
2,nThe coeﬃcients of theof↵2GL= expansion, ↵4 =are given, ↵6 as= functions
0
n/2temperature (T )d and
↵
=
(
1)
4N
N
T
+
,
F
c
n
3 ) and chemical
n/2
2 are
4 as
12 the of22
eﬃcients
GLNJL
expansion
temperature(Tpotential
(µ):
(2⇡)
2G
couplingofinthe
themodel,
N0cgivenandN0 ffunctionsarenumberofcolor
andflavorrespectively.
[(!n + iµ) + p ]
Z 0 3
n
(23)
X
↵2
↵2
↵2d p
1
2,n
0
n/2
↵
,4N
↵the
=regularized, ↵6 =couplingve the UV divergenceandshouldberegularization
scheme. The GL
4F
↵
(2 =
1) G2 isfour
in thesome
NJL model,
Nn/2
Coefficients:
c4T Fermi
0N
012 withc and+Nf are, the number of color and flavor resp
n =0 where
3↵2The integrals↵in↵2the2 + p2 ]2G with some regularization
Z(2⇡)
2↵the =
[(!
+
iµ)
↵
have
UV
divergenceandshould be regularizedn(23)scheme.
nn3Xar the QCD critical
point
at
which
↵
=
0.
↵2 = expansion,0 ↵4 =isn/2,
2↵6 = 4
d
p
1
2,n
only
the QCD critical point at which +↵2 = ↵,4 = 0.
↵n = ( 1) 4
4Nvaried2
12
F Nc T near
3
n/2
2
2
ect
groundof thecoupling
GL potential,
we
have
solve
GL
equation:
(2⇡)
2G
Z Nto
In orderto findthemodel,correct
state
the
we
to solve
GL equation:
[(!
+are
iµ)GL
+potential,
pnumber]
where
G is thestate
four Fermi
in theNJL
Nnfofthetheofhavecolor
andtheflavorrespectively.
n groundc 3and
X

d p

1

2,n
Z
0
n/2
The integrals in the ↵n↵havetheUV
divergenceandshouldberegularized
with
some
regularization
The GL
Z
=
(
1)
4N
N
T
+
, scheme.
3 of color and flavorn Fermi coupling inFthec NJL model, Nc andwhere G is the four
N
are
the
numberrespectively.
3
n/2
⌦
d
x
=
0
.
f
2 +GLp2 ](2⇡) ↵[(!
2G
expansionis
onlyvariednear
theQCD
critical
point
at
which
=
↵
=
0.
M
(y)
+
iµ)
3
2
4
n
n
The integrals in the ↵n have the ⌦UVdivergenceandshouldberegularized
with
some
regularizationGL= GL
0 . potential, we have to solve the GL equation:scheme. The(24)
GL dofxthe
In order
to
find
the
correct
groundstate
expansion is only varied
near
theassume
QCD criticalpoint at which↵2 = ↵in
0.
M (y)
4 =
If we
an one-dimensional
modulationz direction,
the GL equation, reduces to that of the GN2
Zto potential,
G is the In
four
Fermi
in
the state
NJL
thebythenumberof color and flavor respeorder
to findcoupling
the correct
groundofmodel,
the
GL
have
to solve
GL equation:
whose
generalsolution
(up
aN
shift
in z)weisNgiven[13–15]
c and
f are
3
pwith some regularization scheme.
Z⌦GL dbe
x regularized=reduces
0 .M (z) = qto(24) Tegrals inmodulationthe ↵n haveGap-equation:thez UV
divergenceandshouldnsionalin
direction,theMGLequation,
that
of
the
GN
model,2⌫sn(qz;⌫),0(y)⌦GL d3 x = 0 .
(24)on
the[13–15]
QCD criticalpoint
at which ↵2 = ↵4 = 0.
M(y)to isa only
shift variedin z) isnear
givenbywhere q is a function of ⌫ through the relationderfind thecorrect groundmodulation
state of thepotential,
we haveto solve
thetoGLf wetoassumean one-dimensionalin zGL
direction,
the GLequation,
reduces
thatequation:of the GN2 model,pIf
we
assumean
one-dimensional
modulationinz
direction,theGLequation,
reducestothatof
the GN2 model,
⌫+1
↵4 2
1
↵2
4
whose generalsolution
(up
to
a
shift
in
z)
is
given
[13–15]
by
q + 2
q + 2
= 0.
Z
M0 (z)=aqshift⌫ insn(qz;⌫) , [13–15](25)
Solution:
with
whose
general solution(up toz) is givenby⌫ + 4⌫ + 1 ↵6
⌫ + 4⌫ + 1 ↵6p⌦p d3 x = 0 .
GL
M
(z)
=
⌫)
0that
One can easily check
M0qsatisfiesby(25)
M=withq⌫ sn(qz;⌫(26)
sn(qz;⌫),, the GL equation (24). The period of M0 is given
(25)
0 (z)
M
(y)
hrough the relation

where q whereis a functionof ⌫ throughthe relation
q is a functionof ⌫ throughthe relation

4K[⌫],
ssume an one-dimensional modulation in z direction, the GL equation,
reduces to that of the GN2qL=•⌫ free+ 1 energy↵4 2 of the1+ 1NJL↵model2
Expandand momentumof4⌫↵
1 in amplitude
↵22
⌫
+
1
↵
1
↵
4
general solution
(up
to
a
shift
in
z)
is
given
[13–15]
by
4
2
4
q + 2
q
+
=
0
.
(26)
4
2
where
complete ellipticof [14,15], w
q is+the+ 2 of the first kind,=2+
q K[⌫]+q ↵q+integral= 00in.. this article following the convention(26)
(26)
2 +4⌫
R
⌫
+
4⌫
+
1
↵
⌫
+
1
2
2
⌫
4⌫
+
1
↵
⌫
+
4⌫
+
1
↵
6 ⌘ ⌫ (1+ 4⌫
6 6 ⌫ + 4⌫ + 1 ↵6
6
+ t)
1 ↵6 dt.
condensation. K(⌫)⌫ sin
⌫ 2 [0, 1] controls the shape of the solution; sn(z; 0)
pThe elliptic parameter
⇡/2
0

2

1/2

and withsn(z; (26)1) = tanhWe
canidentify
the solution1) as thebecause thermodyna
Mz.0 (z)=GLq equation⌫ sn(qz;⌫) The,sn(z;period
One can easily check that M
satisfies
the(24).
of homogeneous
M0 is given solution,by0Onethat Mthe
satisfies totheequation(24).
of M0byis given bytanh(26)z equation
is equivalent
theGLspatially
constant
configuration.
M0 canwitheasily(26)checksatisfies
GL
(24).
The
periodofTheM0period
is given
0 withOnce we get a set of temperatureand chemical potential, the values of ↵2 , ↵4 and ↵6 are determined. U
4K[⌫]
is a function of ⌫ through values,
the relation
we have to find the configurationL=,which minimizes the thermodynamic potential;
(27)
4K[⌫]
q
4K[⌫]
L=
,
(27)
I
I

• Solution of the GL equation is known.
L=

, 1
⌫+

↵

q

2

14

↵ 2

6

2

2

(27) 2

A.

Phase diagram

Phase diagram

ording to Nickel’s work [4], the GL expansion of the (3 + 1)-dimensional NJL model, to the sixth order⌦GL [M ] = ↵2 M 2 + ↵4 M 4 + (rM )2 + ↵6 2M 6 + 10M 2 (rM )2 + ( M )2 .

oeﬃcients of the GL expansion are given as functions of temperature (T ) and chemical potential (µ):

�����������

���

00Order parameter (T=10)
↵20
↵
↵
2
2
���������
↵2 =, ↵4 =, ↵6 =
����2
4
12������������
��������
Z
�
3������������������ X
d
p
1
2,n
0
n/2
↵
=
(
1)
4N
N
T
+
,
F c
����
n �����������
�����
3
n/2
(2⇡) [(!n + iµ)2 + p2 ]
2G
n
����

�����

����

G is the ���four Fermi coupling in the NJL model, Nc and Nf are the number of color and flavor resp
����
�����
tegrals in the ↵n have the UV divergenceand should be regularizedwith some regularization scheme.
���sion is only varied near the QCD critical point at which ↵2 = ↵4 =����0.
����
rder to findthecorrect
ground
state
of
the
GL
potential,
wehave
to solve the GL equation:
�����Z���� ���� ���� ���� ���� ���� �����������
��⌦GL d3 x = 0 .
M (y) ����
����
����
����
����
2⇡
Q=
�������
assume an one-dimensional modulation in z direction, the GL equation, reduces L
to that of the GN2
generalsolution(up to of
a the
shift
in modelz) is ingiven
[13–15]
FIG. 1.
Phase diagram
NJL
the chirallimitbywith 6th order GL expansion. The Lifshitz point is locatedpM0 (z) = q ⌫ sn(qz; ⌫) ,at (T, µ) = (81.4, 276.4) [MeV].

•⌫+1↵calculation.
1• Consistent with non-expanded
q +
q +

����

Modulatedchiral thecondensate
is energetically favored than uniform one.
q is a functionof ⌫ throughrelationthat the real kink crystal phase in Fig. 1 appears to bebroader at low temperature. The
4 our
4 phase boundary in2
results are consistent with Nickel’s observation⌫ 2 + 4⌫ + 1[14]↵ that↵2= 0.
2⌫ + 4⌫ + 1 ↵⌘ M0 (z) + M (r)rive a free energy of the phonon which is associated with the translation symmetry in the real
)bstituting
and expanding in series of u, we obtain

L [M0 +

M (r) = M0 z + u(r)Phonon fluctuation
(32)

M]

1
= M0 (z) + M00 (z)u(r) + M000 (z)u(r)2 + · · ·2 4
2(M0 + M ) ⌘+4 + (M
0 + M ) + (r(M0 +
M0↵
(z)
M (r)
GL
0
GL
0
6
2
↵22)
2(M
+ of
10(M
+ M ) (r(M0 +expandingin )seriesu, we0obtain
6 and
0+ M

⌦

(34)

[M (z + u(x))] = ⌦[M ] +M ))2⌦GL + O[u3 ]

(33)
(34)

M ))2 + ( (M0 + M ))2(35)

FIG. 3. Order parameter M0 and the coeﬃcient functions f1 and g1 at (T, µ) = (70, 286.0)[MeV] (left) and (T, µ)⌦GL [M0 + Mf] 1 (70,(z)
f2 (z)
g2 parameter(z)
286.5)[MeV]All
functionsaregnormalized
by thecut oﬀ
�����
1 (z)2 (right).
22
2
2 ⇤ with
2 appropriate dimensions.
f
(z)
f
(z)
1
2
2
2
2
����
[M
]
+
(@
u)
+
(@
u)
+
(r
u)
+
(r
u)
(36)
4 +
2
GL
z
?
����
z
?
(@
u)
(@
u)
�
�
↵2 (M00+⌦GLM )2 ⌘
+
↵
(M
+
M
)
+
(r(M
+
M
))
z
z
0
���� ���
2 4
2 0
2
2

2
2
�����
���������
22
2↵6 2(M0 + M ) +210(M0 + M ) (r(M20 + M ))2 + ( (M0 +3 M ))���� (35)Theeivenlalueequation
leads
to
++ h
(z)(@
u)(r
u)
+
h
(z)(r
u)(@
u)
+
O[u
]
(37)
g
(z)
g
(z)
1
z
2
? 1
? 2 z
2
2
2
�����
f1 (z)(z)(z)+ f2 (z)(r?2u)(r g2u)h+ g1+i⌦GL [M0 ] +(@z u)2 + 2 (@z2 u)(r2? u)22+ ? 2 (r2? u)22
(36)
����
2
2
@z 2(f1 @z ) + @z (f2 @2z ) g1 r? + g2 r4? h01 r2? + r2? h2 , @z2 + u = Eu(42h2+arefunctionsofzdefined
as
�����
2
2
2
2
2
2
3
h1 (z)(@z u)(r? u)++hh21(z)(r(37)(z)(@u) +] h2 (z)(r? u)(@z u)? u)(@z u) +z u)(r?O[u��
2the anti-commutation
0 2
00 2 and h2 ,0@z2 +000u�����
where{}relation= h2 @z2 u + @z2 (h2 u). Because the operator actin+ denotesofzdefinedas
f1functions
= 2(↵
+
10↵
M
)(M
)
+
4↵
(M
)
2M
M
1 , h2 are
4
6 0
6
0
0
0 0
6

on u is a hermitian operator, the eigenvalue E is real.
����
2
0
2
00
2
0
000
0M2the10↵6all)(Mcoeﬃcient
(M0 )
2M
M0
Since
are 0periodic
functions sharing �����
the same period, according to the Bloch’s theorem
4+0 ) + 4↵6 functions
ff21 ==2(↵
2↵
6 (M0 ) 0
0 2 decompose u into the plane wave and a periodic function: ����(38)
f2 = 2↵6we
(Mcan
0)
��
����
����
����
����
��
2
0 2
0
000
(38)
gg11 ==2(↵
2(↵2 6M
0 2 )(M0 )0000 4↵6 M0 M0
4 + 10↵���
~
4 + 10↵6 M0 )(M0 ) 0 4↵6 M
0 M0
u(x) = eik? ·~x? eikz z (z)(43
0 2 0 2
0
00
0 2
0
00
0
2
) , )h1 ,= 4↵h
M04↵
, 6hM
(M ) .
6 (M6M
2 = 2↵
gg22 ==2↵2↵
6 0(M
1 0=
0
0 M60 ,0 h2 = 2↵6 (M0 ) .
where k? and kz are momentums for each direction and (z) isFIG.
a periodic
periodL. Substituting
3. Order function
parameter with
M0 andthe coeﬃcient
functions f1 (43aner parameter and
some
of we
the obtaincoeﬃcient(T, µ) =for
(70,the286.0)[MeV]functionand286.5)[MeV]
(T,of
µ) =
(70,
All functions
normalized
the cut oﬀ
into
(42),
the functions
eigenvalueatequation
periodic. (right).
In commonwith areelectrons
inbymetals,thparameter
and
some
of
the
coeﬃcientfunctions
at
(T,
µ)
=
(70,
286.0)
[MeV]
and
(T,
µ)
=
. 3. Obviously thefunctionsshare theperiodwiththe order
parameter.
As proventhephononfluctuaion
onsamethe realkink
crystalcondensation
fomrs
a band instructure.
1220
equence
of
the
fact
that
M
realizes
theglobalminimumofthethermodynamic
potential(28),
0
Obviously the functions share the same period with theTheorderparameter.
eivenlalueequation leads toAs proven in the
one period vanishes;huence of the fact thatI M0 realizes the global minimum of the thermodynamic
potential
(28),
2
2
2
4
@
(f
@
)
+
@
(f
@
)
g
r
z
1
z
2
1
z
z
? + g 2 r?
0
g1 (z) = 0 .
(39)period vanishes;• f , f ,…, h are periodic functions sharing the same period with M .
• Since M realises the global minimum of the GL potential,

where {}+ denotes the anti-commutation relation and h2 , @z2 +I the followingZueigenvalue
is a hermitian operator, the eigenvalue E is real.
phonon fluctuaion on the real kink crystal condensation is determined by
1 L onSince
all the coeﬃcient functions are periodic functions sharin⌘dz
1
L 0 we can decompose u into the plane wave and a periodic functionIg (z) = 0 .

(39)

H
~
(40)
u(x) = eik? ·~x? eikzu
non fluctuaion on the real kink
crystal condensation is determined by the following eigenvalue
Eu =

IG.
(left) and (h 3. Order parameter M0 and the coeﬃcient functions f1 and g1 at (T, µ) =i (70,B.286.0)[MeV]
Phonon mode
2 All functions
2
2 normalized
4 by the0 cut
2 oﬀ parameter
20, 286.5)[MeV] (right).
are⇤2 with appropriate dimensions.

@z (f1 @z ) + @z (f2 @z )Band structure
h r + r h ,@g 1 r ? + g 2 r?

2 z + u = Eu
1 ?
?
We would like to derive a free energy of the phonon which is associated wit
Ikink crystal phase. Substituting
2
he eivenlalueequationtoheanti-commutationand h2 , @z2 + u = h2 @z2 u + @M(hu).
Becausethe operator a⌦GLleads=relation
Eu(x)
2=
z
(r)
M
z
+
u(r)
0
u(x)
h
i
perator, the eigenvalue E is real.
1 00
2
2
2
4
0
2
2
2
0
=
M
(z)
+
M
(z)u(r)+M0 (z)u(r)2@z (f1 @z ) + @z (f2 @z ) g1 r? + g2 r? h1 r? + r? h2 ,0@z + u0 = Eu
2cient functions are periodic functions sharing the same period, according to the Bloch’stheo⌘ M0 (z) + M (r)into the plane wave and a periodic function:•

here {}+ denotes
the theorem:
anti-commutation
relationand h2 , @z2 + u =into
h2 @z2 u + @z2 (h2 u). Because the operator
Bloch’s
u canbedecomposed
into the GL potential (22) and expanding in series of u, we obtainn u is a hermitian operator, the eigenvalue
E xis real.
i~k? ·~? ikz z(z)(z +accordingL)⌦GLthe
[M0 +
M ]=
u(x)
=
e
e
(z)
Since all the coeﬃcient functions are periodic functions
sharingsame
period,to the Bloch’s th= ↵2 (M0 + M )2 + ↵4 (M0 + M )4 + (r(M0 + M ))2
e can decompose u into the plane wave and a periodic function:coeﬃcientbands
f1 and g1 at (T, µ) = (70, 286.0)[MeV] (left) and (T, µ)6 =0 and the Energy
phonon2
momentums
forfunctions
eachof directionand (z) is a periodic function with period L.2 Substituting+ ↵6 2(M0 + M ) + 10(M0 + M ) (r(M0 + M )) +unctions are normalized by the cut oﬀ parameter ⇤ with appropriate
dimensions.
~u(x)= eik? ·~x?ofeikz z. (z)f1 (z)
g1in(z) metals
the eigenvalue equation for the periodic
function
In common
withf2 (z)electrons= ⌦GL [M0 ] +(@z u)2 +(@z2 u)2 +(r? u)2
2 kz = 022k? = 0n the real kink crystal
condensation
fomrs
a
band
structure.
22
2 L. Substituti
3
hereare��momentums for each direction and (z)�����
is a��+periodic
functionwith period
h1 (z)(@z u)(rs to k? and kz�����? u) + h2 (z)(r? u)(@z u) + O[u ]to (42), we obtain
the eigenvalue equation for the periodic
functionof . In common with electrons in met
i, g , g �������
�����������
where
f
,
f
,
h
,
h
are
functions of z defined as
12 2
1
2
1
2
2
2
2
4kink 0crystal
2
2
honon
fluctuaion
on
the
real
condensation
fomrs
a
band
structure.
@
)
+
@
(f
@
)
g
r
+
g
r
h
r
+
r
h
,
@
u
=
Eu
(42)
1 z
1 ?
2 ?
2 z +
z 2 z
1 ?
?
�������

������

�������

f1 = 2(↵4 + 10↵6 M02 )(M00 )2 + 4↵6 (M000 )2

2M������

0 2
2
2
2
��
f2 = 2↵acting
commutation relation
operator
6 (M0 )
��������� and h2 , @z + u = h2 @z u + @z (h2 u). Because�������the, the eigenvalue E is ��real.
g1 = 2(↵4 + 10↵6 M02 )(M00 )2 4↵6 M00 M0000
��
�����
�����
nctions are periodic
functions sharing the same period, according to
the Bloch’s
theorem,
g2 = 2↵6 (M00 )2 , h1 = 4↵6 M00 M000 , h2 = 2↵6��e plane wave and�����
a periodic
function:4�������
2~������ ���� ���� ����E⇠k
E ⇠ kThe
shapes of the order parameter and some of the?coeﬃcient functions at (z

�� Obviously the functions share the same period with
(70, 286.5) [MeV] in fig. 3.
(43)�� ���� ����
����
����
����
����
�� M0����
����the global minimum
Appendix. A, as a consequence of the fact thatrealizes����������
the averageof gperiod
period vanishes;
1 over one
tums for each direction and (z) is a��periodic
functionwith
L. Substituting(43)
I
nvalue equation for the periodic function of . In common with2⇡
electrons in metals, theg1 (z) = 0 .
Q=quadratic
quartic
al kink crystal condensation fomrs
a band structure.

u(x) = eik? ·~x? eikz z (z)L

The property of the phonon fluctuaion on the real kink crystal condensation iswhere B and C are functions of T and µ. The absek? is the momentum in transverse directions, kz
2of the O(k?
) term in (23) is guaranteed by the prope
so-called crystal momentum, and (z) is a periodic(17). The proof of (23) is somewhat technical andon, viz. (z + L) = (z). Substituting (22) into
relegated to Appendix E. Equation (23) shows thatyields an eigenvalue equation for , which we haveelastic free energy of low-energy phonons becomes◆d numericallyby way✓ofdecomposition
Ip a Fourier
Pnmax
inQz
Z
u
2 nmax =4 20. To=withn e2 + / L⇥⇤
n=E0nmax
⇠ Bk
g
k
+
Ck
1
1
3
z
?
?
u
2
2
2
d x B(@z u) + C(r? u) .
(Fel =onvergence, we have increased nmax up to 30 and224unchanged.
med that theresultsare= Bkz + Ck?
Fig. 4, we show the eigenvalue E numerically comOne
H may suspect that the coeﬃcient B would be gi
for varying kz and k? . A marked diﬀerence
fromof freeby
f1at. THowever,as shown in Appendix E, this na
Coefficientsenergy= 10
genvalue of particles in a free spaceguess is incorrect: the coupling between 0 and n6=������is that there���� even in the perturbation series in kz .
finitely many levels for given momenta, in analogy
not negligible�������
����
ctrons in metals which develop a band structure.
In order to extract the values of B and C from eig������
s the lowest eigenvalue E0 (red curves
in Fig. 4) thatvalues, we have numerically fitted the curve of E0 w

Free energy of phonon�������
������
�������
������
�������������

in (21) at (T, µ) = (70, 286.0)
d for kz = 0 (bottom). The
the first Brillouin zone. kz and? is normalized by the UV cutoﬀ

����

����

���� ����
�������

����

����

FIG. 5. B and C at T = 70 MeV (top) and T = 10 MeV(bottom). B and C are normalized by the UV cutoﬀ parameter ⇤ with appropriate dimensions. The dotted vertical lines
mark the boundaries of the modulated phase.

• We can obtain the free energy of the phonon fluctuation from thecurvatures of the lowest energy band.

(right).
n,m

L

2

Quasi long range order"#Zikxbehavior. For |x|
X Mn M
Under thermodynamic limit, the systemis mconsidered
to showorder
L,m2long+ n2range+ 2mne! 2 thed3 kquasi
i!nz= parameter ebehaves
explike [22],(49)
2 + Kk 4
the correlation function of the order
3
L
2
(2⇡)
Bk
?
k
n,m
"
#
X
M
M
m i!nzX MnnMmd3 k
1 eikxhM (x)M (0)i ==e(47)
i!nz hexp [i! (nu(x)2 +2 mu(0))]i
e
n !
(50)
n, m expL
2
4
3 Bk + Kk
L
(2⇡)
n,m
?
k
n,m

X Mn M2 m⇢↵! 2n2⌦⌘c
X
i!nz
2
|M
|
(|x?+
| =mu(0))0)
n=e 2 cos(!nz)zexp(nu(x)
(48)
⇠
(51)
2
L
2
2n ⌘c
n,m
L
|x
|
(z = 0)
?
n 1
"
#
Z
2
2
ikx
X M n Mmm + n + 2mne p!2
d3 k P
i!nz
=
e
exp
(49)
2ein!(z+u(x))
42⇡3where we have made a Taylor expansion
the order
parameter
M(2⇡)
(x) =
M
/
L
and
took
averaged
Q2ofTL
n
2
Bk
+
Kk
n
?
k
n,mp
Q
=
⌘
=
c
over the thermal fluctuation of u(x). The exponent ⌘c is defined
as
"
#
L
8⇡
BK
3
ikx
X Mn Mm
1 e
i!nz
2 2 d k
2
=
e
n !
(50)
n, m
! exp
T
3 Bk 2 + Kk 4
L
(2⇡)
p
⌘c =
.
(52)
?
k
n,m
8⇡parameter
BK 2
Correlation function
of
the
order
shows a pawer law.
⇢2n ⌘c
X |Mn |
2 cos(!nz)z(|x? | = 0)⇠
2 and T = 10 [MeV]. Since B and K become zero near (51)
In fig. 6, we show the critical exponent ⌘c at
T
=
70
[MeV]
the
2n
⌘c
L
|x
|
(z
=
0)
?
1
Exponent
ηc isnotuniversal
and depends on temperature andphase boundaries,⌘c rapidly
goes
to ninfinity.
If the size of the system is finite, hu2 i remains finite. This is true on the mattersuch
as insideP
7 pthe neutron star. We
in!(z+u(x))
parameter.
where
we have
madefroma Taylorof theorder becomesparameterlarge.
M (x)If =Mn e/ L withand took
averagedcanevaluatea scalewhichexpansion
the thermal
fluctaion
hu2 i nbecomescomparable
the square
of
T
=
10
MeV
2
4
overperiod
the trial
thermaloffluctuation
exponent⌘c������
is large.
definedas the expression (46), we can define the crossover
functions
Eu(x).
0 = CkThez and? , respectively,thee.g.
hu2fluctuationi E⇠0 =L2Bk
, the
can
be regaredUsing��with B and C treated as fitting parameters. Figure 5
length (⇠shows
)
at
which
the
fluctuation
washedout
theone
dimensional
modulatedcondensation,?
B and C at T = 70 MeV and ⌘T = 10 MeV obc!2 T
hMway.
(z)M(0)i⇠Czin Fig.
tained thisNotably,B and
5 are positive ⌘ =p 2 p.
(52)pc
throughout the real kink crystal phase, which proves local
4⇡L
BK/T
8⇡ e BK�����
(53)⇠? ⇠ K/B
stability of this condensate in agreement with numerical••results in [17]. Because the phonon mode exists only in

In fig. 6,the
werealshow
the criticalexponent
T = 70 [MeV] and T = 10 [MeV]. Since B and K become zero near thec atcoeﬃkink crystal
phase, it isnatural that⌘both
cients tend to zeroat the phaseboundaries,
although thephase boundaries,⌘c rapidly
goes
to infinity.
����eigenvalues E were too small near the leftfor2
2 boundary
If
the
size
of
the
systemisfinite,huiremains
finite.
This
is true on the matter such as inside the neutron star. We
hM
(z)M
(0)i
⇠
M
cf) hadron
T = 70phase:
MeV to perform a reliable fitting.
2
can evaluatea scalethat
fromthermalfluctaion
becomeslarge.
Ifhu
i becomes comparable with the square of
We mention
the whichspectrumtheof gaplessexcitationsz(0)ie worked
overphase:the same(8) fluctuationhas
also⇠beenout regared large. Using the expression (46), we can define the crossovercf)period
QGP
the
e.g.
hu2 ibackground⇠ hM
L2 ,(z)Mthecan be���in [45] with entirely diﬀerent methods. However a direct
length (⇠comparison
theowing
fluctuation
washedoutin the one dimensional
modulated����������
���������� condensation,����������? ) at which
is diﬃcult
to the fact thatthe model[45] is non-relativistic and in one space dimension, while
our model is relativistic and in three space dimensions.

p

2

p

�������

eikz z (z) ,

(22)

(23)

E0 ⇠ Bkz + Ck? ,crystal. Taking the pFourier decomposition M (x) =P
inQ(z+u(x))
where B andC
arefunctions of/ T L,
andµ. TheuabsenceMetreating
in the Gaussian apn
n2of the O(kguaranteed
the fluctuation,
property
proximation
andisignoring
theby
pion
we obtain? ) term in (23)Impact of phonon fluctuationtransverse directions, kz
um, and (z) is a periodic(17). The proof of (23) is somewhat technical and is). Substituting (22) intoX
1
relegatedtoAppendixE.
Equation(23)
shows that theinQ(z+u(x))
0tion for , which we have
p
hM
(x)i
=
M
he
i
M
=
M
(z
+
u(x))
n
elastic free energy of low-energy
L n phonons becomes
p a Fourier decomposition
Z(25)
L with nmax = 20. To
h
i
X
⇥
⇤
1
1
1
3
2
2
2
inQz
2 2 2
d
x
B(@
u)
+
C(r
u)
(24)
Felu =
eased nmax up to 30 and
z
p
=
M
e
n
Q hu i? exp.
n22
L n
unchanged.
value E numerically comOne
H may suspect that the coeﬃcient B would be given
A marked diﬀerence fromby f1 . with
However, as shown in Appendix E, this naive
Z ⇤ between Z 0⇤and n6=0 is
a free space is that there
guess is incorrect: the coupling2⇡ perturbation series in k .
T2iven momenta,
in analogy value
notofnegligible
even
in
the
Expectationu:hu i =
dk? k?
dkzz 24
3velop a band structure.
In order to extract(2⇡)
the values
eigen`?1 of B and C⇤from Bk
z + Ck?
(26)(red curves in Fig. 4) thatvalues, we have numericallyfitted
the
curve
of
E
with
0
T
`

⇠

•
•

p

4⇡ BC

log p

?

C/B

,

where we have only incorporated the lowest Matsubara
mode since it is dominant in the infrared. The momen<u2> has a logarithmic
IRintegral
divergence.
tum
is IR divergent and is regularized by a cutoﬀ`? . One can regard `? as the transverse diameter of the
One-dimensional modulation
is violatedby thermal
fluctuation
inquark matterin a compact
star. In thethermodynamic
limit ->`? ∞).
! 1,
the condensate (25)drops to zero withthermodynamic limit (L⊥(Landau-Peierls
theorem)negative powers of `? , implying that the one-dimensional
modulation is wiped out by thermal fluctuations at any
low T > 0, a phenomenon known as the Landau-Peierls
instability. In fact, if the average amplitude of displacement fluctuation exceeds the interval of layers, it does not

FI
of
ph
do
ph

pa

h

Possibilities
We can evacuate from the large thermal phonon fluctuation if

1) Strictly zero temperature
2) Higer dimensional modulations
3) External magnetic field
4)Finite volume system
divergence is at most logarithmic to volume.
We roughly estimated that, at T = 10 MeV, one-dimensionalcondensation is stable up to L ~10 Km.

1
? and
d and the constantandterm
isomittedforsimplicity.
We
canincludeP
1
2
Fk is a positive constant. crystal
To derive(63),
we usedphase
is givenby the general soulution in Ap. We derived it in Happendix(C3).fluctuationZ
g1 6= 0, Pion
there appears
the quadratic term of k? in (63). The pion contribution
h to th1⇡
3
2
crystal
phase
is
given
by
F
=
d
x
F
(@z ⇡
el
k
edure that we have utilised to the phonon. Substituting
2

Z

h

i

1
⇡
3 the 2real kink2 crystal
2 condensation
2
PionFfluctuation
on
=
d
x
F
(@
⇡
)
+
F
(r
⇡
)
. is ani
z
0
?
0
el
?
k

i⇡0

M (r) = M0 e
2 thermal fluctuation asosciated with the pion d
Therefor the�����the long range order.
=
M
(z)
+
iM
⇡
+
·
·
·
0 fluctuation on
0 the
0 real kink crystal condensation is anisotripic but the dipspersion i
Pion
�����
T = 70 MeV
Thereforthethermalfluctuation
asosciatedwiththe pion does not cause any infrared d������������
⌘M+ M(r)the0 (z)long rangeorder.
�� ���
�����������

���������

��f ⇡0 , we obtain�����������

�������

�������
���������

�������
��

4

����

����

M | + |r(M0 + M )|2�������

2��������

2

����

����

⇥

�������
�������

2+ M | |r(M0 + M )| + 2 Re (r(M0 + �������M ) (M0 + M )�������
�������

2⇤+ | (M0 + M�������

f2⇡ (z) 2 2 g1⇡ (z)g1⇡ (z) ��2222(@z ⇡0 ) +(r? ⇡0 ) +(r������
⇡
)
+
h
(z)(@
⇡
)(r
z 0������ ������
? 0������ ����1⇡������ ������
? ⇡0 )
22
2�������
3⇡0 )�������
�������
�������
�������

������ ���� ���� ������ ���� ���� ���� ����������

FIG. 9. F?2 and Fk2 at T = 70 MeV (left) and T = 70 MeV
dimensions.

• We can repeat the same analyses to the pion fluctuation.
• Pion has an anisotorpy in the inhomogeneous phase but dispersion

In fig 9, we show F?2 and Fk2 at T = 70 [MeV]. We
2
E⇡,0 = F?2 k?
and E⇡,0 = Fk2 kz2 , respectively.
In the homo22FIG. 9. F? and Fk at T = 70 MeV (left) and T = 70 MeV (right). F?2 and Fk2 are nor
the kink crystal phase, the system is asymmetric and the codimensions.
crystal phase, Fk2 > F?2 is held at any temperature and che
is still quadratic.

Summary• We evaluate properties of “phonon” and “pion” fluctuations on
inhomogeneous chiral condensate.

• Pion and phono have strong anisotropic features.
• One-dimensional modulation is violated by the thermal fluctuation ofphonon (Landau-Peierls theorem).

• Inhomogeneous phase becomes the quasi-long range order phase.
We evaluate the critical exponent in this phase.