Ward identity and Homes’ law in a holographic superconductor with momentum relaxation
Received: May
Ward identity and Homes' law in a holographic superconductor with momentum relaxation
Kyung Kiu Kim 0 1 2 3 5 6
Miok Park 0 1 2 4 6
KeunYoung Kim 0 1 2 5 6
0 85 Hoegiro , Seoul 130722 , Korea
1 50 Yonseiro , Seoul 120749 , Korea
2 123 Cheomdangwagiro , Gwangju 61005 , Korea
3 Department of Physics, College of Science, Yonsei University
4 School of Physics, Korea Institute for Advanced Study
5 School of Physics and Chemistry, Gwangju Institute of Science and Technology
6 Open Access , c The Authors
We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we nd that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding di eomorphism from ory perspective. We con rm them by numerically computing all twopoint functions from holographic perspective. Second, we investigate Homes' law and Uemura's law for various hightemperature and conventional superconductors. They are empirical and (material independent) universal relations between the super uid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes' law does not hold but the Uemura's law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is nite for a neutral scalar instability while it is in nite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected. ArXiv ePrint: 1604.06205
superconductor; with; momentum; relaxation

Holography and condensed matter physics (AdS/CMT), Gaugegravity
correspondence
Ward identities: constraints between conductivities
Homes' law and Uemura's law
Conclusion and discussions
A Twopoint functions related to the real scalar operator
Contents
1 Introduction 2 3 5
Equilibrium state
AC conductivities
Charged scalar case (q 6= 0)
Neutral scalar case (q = 0)
General expression for super uid density
Analytic derivation: eld theory
Numerical con rmation: holography
Conductivities at small !
AC conductivities: holographic model and method
Conductivities with a neutral scalar hair instabitliy
Introduction
reviews and references, we refer to [2, 3, 9, 10].
in section 2.
developed [42{51].
We explain each issue in the following.
formation of the scalar hair
may be understood as the coupling of the charged scalar
to the charge of the black hole through the covariant derivative DM
iqAM , which gives an e ective mass of
su ciently negative near the horizon to destabilize the scalar
eld. Based on this
it does not break a U(1) symmetry, but at most breaks a Z2 symmetry
electric conductivity will be
nite contrary to the case with a complex scalar hair
conductivity that derived from the neutral hairy black hole is indeed nite.
may be
two constraints relating three transport coe cients, ; , and
[3, 55]. The
contwopoint functions: ,
and three twopoint functions related to the operator
can be obtained algebraically
is computed numerically. This is why only
is presented in the literature [2].
alone cannot determine
and , which must
the Ward identities for twopoint functions analytically from
eld theory perspective.
of our numerical method.
right above the transition temperature T~c.
~s(T~ = 0) = C DC(T~c) T~c ;
limit C
where ~s, T~c and
universal constant: C
DC are scaled to be dimensionless, and C is a dimensionless
4:4 or 8:1. They are computed in [47] from the experimental
8:1. Notice that momentum relaxation is essential here because without
momentum relaxation
DC is in nite. There is another similar universal relation,
Uemura's law, which holds only for underdoped cuprates [56, 57]:
~s(T~ = 0) = B T~c ;
relaxation region, related to coherent metal regime.
identities giving constraints between conductivities analytically from
eld theory
perspecin our model. In section 6 we conclude.
AC conductivities: holographic model and method
Equilibrium state
S =
1 F 2
1 X2 (@ I )
eld A, the complex scalar
mass m, two massless scalar elds,
The action (2.1) yields equations of motion
1 F 2
1 X2 (@ I )
1 X2 @M I @N I ;
rM F MN =
= 0 ;
2 I = 0 ;
for which we make the following ansatz:
A = At(r)dt +
I = ( x; y) ;
ds2 =
+ r2(dx2 + dy2) :
renormalization, which are explained in [28, 31, 53, 54, 61] in more detail.
order parameter, condensate. Near boundary (r !
two undetermined coe cients J
, which are identi ed with the source and
condensate respectively. The dimension
of the condensate is related to the bulk mass
of the complex scalar by m2 =
2 to perform numerical analysis.
3). In this paper, we take m2 =
I is introduced to give momentum relaxation e ect
is the parameter for the strength of momentum relaxation. For
= 0, the
and Horowitz (HHH) [7, 8].
related to the boundary stress tensor of the model (2.1) when
= 0 [28]:
i =
where ;
are indices in the eld theory x
U(r) = r2
At = n
(r) = 0;
2rh + n24+rhB2 , and n is interpreted as charge density. It is the dyonic black brane [62]
2
modi ed by
due to
I [54]. The thermodynamics and transport coe cients(electric,
the case without magnetic eld, see [31]. Next, if
a superconducting state with
nite condensate and its analytic formula is not available.3
= 0 and in [46] for
consider this case and refer to [2, 63, 64].
3A nonzero
(r) induces a nonzero
(r), which changes the de nition of `time' at the boundary so eld
theory quantities should be de ned accordingly.
momentum (say
= x),
P_ xi = ExhJ ti
F tx and assume a nite density system without current, i.e.
First, if
analytic formula is given by
6= 0.
(a) (r)= 2
(c) At(r)=
(b) (r)
(d) U(r)= 2
= 0:5). The black curves are
for normal phase ( O
= 0). In
black curves agree to the analytic formula in (2.9), where
enters only into U(r).
AC conductivities
B 6= 0 see [31] for normal phase.
Ai(t; r) =
gti(t; r) =
i(t; r) =
e i!tr2hti(!; r) ;
the background are4
Near boundary (r ! 1) the asymptotic solutions are
t h0tx = 0 ;
0 = 0 ;
htx = ht(x0) +
ax = a(x0) +
x = x(0) +
The onshell quadratic action in momentum space reads
Sr(e2n) =
1 Z d! hJ a !Aab(!)J!b + J a !Bab(!)R!bi ;
0a(x0)1
J a = B@h(t(0x0))CA ;
0a(x1)1
Ra = B@h(t(3x3))CA ;
A = B@0 2U (1) 0C ;
A
B = B@0
density. The index ! in J a and Ra are suppressed.
uctuations in momentum space by
a collectively. i.e.
a = ( ai ; hti ; i) :
a(r) = (r
1) 4i!T +na ('a + '~a(r
motion are doubled too.
Every 'ia yields a solution
ia(r), which is expanded near boundary as
where a
1 and the leading terms Sia are the sources of ith solutions and Oia are the
constructed from a basis solution set f iag:
with arbitrary constants ci's. For a given J a, we always can nd ci,5
so the corresponding response Ra may be expressed in terms of the sources J b,
a(r) =
Ra = Oiaci = Oia(S 1)i J b :
b
basis 'ia, (i = 1; 2;
= BBB ... ... .
With (2.22), the action (2.13) becomes
1 Z
1 Z
Sr(e2n) =
J ! Aab(!) + BacOic(S 1)ib(!) J b i
!
retarded Green's functions are explicitly denoted as
B GT J GT T GT S AC :
GSJ GST GSS
grx = 0 [53].
(a) Electric conductivity
(b) Thermoelectric conductivity
(c) Thermal conductivity
AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal
conductivity( (!)) for
GJ J + GJ T
GJ J + GT J
GT T (! = 0)
GJ T + GT J
Conductivities with a neutral scalar hair instabitliy
derivative DM
Charged scalar case (q 6= 0)
thermoelectric conductivity ( (!)), and thermal conductivity ( (!)) for
= 1 and
part of conductivities.
One feature we want to focus on in
phase the DC conductivity is nite due to momentum relaxation.
for the complex scalar hair
may be understood as the coupling of the charged scalar to
the charge of the black hole through the covariant derivative DM
other words, the e ective mass of
de ned by m2
q2jgttjAt2 can be compared
with the BreitenlohnerFreedman (BF) bound. The BF bound for AdSd+1 is
iqAM . In
may be su ciently negative near the horizon to destabilize the
The e ective mass m2
instability conditions can be summarized by one inequality [46]
me2 = 4m2
= m2BF ;
which reproduces the result for
= 0 in [2]:
me2 =
= m2BF :
but at most breaks a Z2 symmetry
momentum relaxation (
. Therefore, it would be interesting to see if the
this issue properly.
7We thank SangJin Sin for suggesting this.
(a) Electric conductivity
(b) Thermoelectric conductivity
(c) Thermal conductivity
AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal
conductivity( (!)) for
= 1. For q = 0, m2
of gure 3 from
= 1 +
4 Q ;
= 4
= 4
(a) Electric conductivity
(b) Thermoelectric conductivity
(c) Thermal conductivity
AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal
conductivity( (!)) for
(rh))=2.
merical values very well. For a special case with
= 0, in
gure 5, we see that (!) = 1,
conducting case.
General expression for super uid density
rM F MN =
the xdirection, the xcomponent of the Maxwell equation reads
gF xr = @tp
= lim p
gF tx + iq
Ax =
gtx = r2
x =
a^x(!; r)e i!t ;
i 4!T h^x(!; r)e i!t + i!r2 e i!t ;
h^x(!; r)
A0 + A1(r
be expanded near horizon as
term goes to zero as8
With the following sourcevanishingboundary conditions9
x = lim
e i!t = 0 ; (3.11)
except Ax, the current (3.6) can be interpreted as
J x =
i! xx(!) Axjr=1 :
8More explicitly, by using (3.9) and (2.6), the rst term of (3.8) is expressed as
p gF xr = p g Fxrgrrgxx
+ i! A0te
(r rh) (3.10). Thus,
O(!)e i!t. Similarly, by using (3.9) and (2.6), the second term of (3.8) is expressed as
dr@tp gF tx =
drp ggttgxx@t2 Ax = !2
which is of order !2.
in counting order ! near horizon as shown in footnote 8.
lim !Im[ ] = lim
!!0 Ax(r = 1)
= lim
= lim
!!0 Ax(r = 1) rh
!!0 Ax(r = 1) rh
This shows how the hairy con guration
con rms our numerical analysis.
Ward identities: constraints between conductivities
eld
thedensity and normal uid density and the relation between them.
regarding di eomorphism from
eld theory perspective. In our eld theoretic derivation
the stressenergy tensor T
, respectively.
scalar operator.
Our main results are (4.44){(4.45) for
nite magnetic
eld (B 6= 0)
and (4.56){(4.58) for zero magnetic
perspective, by solving bulk equations numerically.
where h , A , I , , and
are the nondynamical external sources of the stressenergy
tensor T
, U(1) current J , real scalar operators OJ , and complex operators O y; O
J (x)i =
(x)i = 2
OI (x) =
; hO (x)i =
(Pt) twopoint functions:
Analytic derivation:
eld theory
therein to the case with real and complex scalar elds, which are I and
in (4.1). Our
nal results are (4.44){(4.45) and (4.56){(4.58).
eW [h ;A ; I ; ; ] =
DXe S[X;h ;A ; I ; ; ]
(y)) = 4
(x)J (y))i = 2
(x)OI (y)) = 2
(x)O (y)) = 2
hPt(J (x)J (y))i =
Pt(J (x)OI (y)) =
Pt(J (x)O (y)) =
Pt(OJ (x)OI (y)) =
Pt(OJ (x)O (y)) =
Pt(O (x)O
y(y)) =
h (x) h (y)
h (x) A (y)
A (x) A (y)
We consider the generating functional W [h ; A ; I ; ;
] invariant under di
eomorphism, x
, and the variation of the elds can be expressed in terms of a Lie
10See [66] for a holographic derivation.
derivative with respect to the vector eld
= (L h)
= r
For di eomorphism invariance, the variation of W should vanish:
W =
A = (L A) =
I = (L I ) =
= (L
) =
) = 0 ;
di eomorphism.
= 0 ;
U(1) gauge symmetry, which is summarized in footnote 12.
(x) . Here we also used the Ward identity for
By taking a derivative of (4.18) with respect to either h (y), A (y), J (y) or
we obtain the Ward identities for the twopoint functions:
D hPt(J (y)T (x))i + F
hPt(J (y)J (x))i
y) + h @
Pt(T (y)T (x)) + (x
T (x) + h
Pt(T (y)J (x))
@ I Pt(T (y)OI (x))
+ 2Re h @
Pt(J (y)O (x))
= 0 ;
Pt(T (y)O (x))Eo = 0 ;
Pt(OJ (y)J (x)) + h
Pt(OJ (y)OI (x)) @ I
o = 0 ;
Pt(O (y)T (x))
Pt(O (y)J (x)) + h
Pt(O (y)OI (x)) @ I
Pt(O (y)O (x)) @
= 0 ;
where the covariant derivatives act only on the operators of x.
GE
0 =
0 =
I GE
iG~JE;I (k)
iG~E;I (k)
0 =
0 = k
identities (4.19){(4.22) read
From here we consider a at space, h
, and assume external elds such as
, are constant in spacetime. We further assume translation invariance is
, should be constant in spacetime. In momentum space, the Ward
0 =
0 = k
functions. Thus, the Ward identities (4.23){(4.26) become
0 = k G~JR; (k) + iF
0 =
G~R; + iG~R;I (k)
constant expectation values for the energymomentum and current
J i = (n; 0; 0) ;
nite or zero condensate
external magnetic eld with a background scalar I :
F = Bdx ^ dy ;
I = ( x; y) :
Under these conditions the Ward identities (4.27){(4.29) becomes
I GR
iG~JR;I (k)
!G~JR;0j + iB ij G~JR;i
iB ikG~jR;i + ! jkn + i IkG~jR;I = 0 ;
= 0 ;
Ij = 0 ;
= 0 ;
= 0 ;
= 0 ;
B) hJ Qi + !(
B) hSJ i
= 0 ;
= 0 ;
= 0 :
Finally, using the Kubo formulas for conductivities11
we obtain the relations between the conductivities:
de ned as
iG~xR;0y ;
iG~0Rx;0y ;
G~IR=1;0x
iG~IR=1;0y :
With this notation, (4.33){(4.35) can be rewritten as
Ward 1 :
Ward 2 :
Ward 3 :
where we rede ned
to subtract a counter term and 0
hT T i ;!=0 =
=2 [3].
hT T i ;!=0. In normal phase, if
= 0 and B 6= 0,
11The complexi ed conductivities are denoted by X
Xxy iXxx, where X = ; ; ; .
= 0 ;
= 0 ;
= 0 ;
B) hSJ i
hT T i ;!=0 ;
! hJ T i + ! n + i hJ Si = 0 ;
! hT T i + !
+ i hT Si = 0 ;
! hST i + i hSSi = 0 ;
Using the Kubo formulas
we obtain the relations between the conductivities:
hQQi +
hJ Si) = 0 ;
hJ J i + n + i ! hJ Si = 0 ;
hSQi +
hSJ i + n + i ! hSSi = 0 :
Ward 4 :
Ward 6 :
Ward 5 :
hJ Si = 0 ;
hJ Si = 0 ;
hST i + i hSSi = 0 ;
= 0, hT T i!=0 =
Numerical con rmation: holography
phase, hT T i!=0 =
and =2 for
= 0 and
+ hT T i!=0. In normal
and (4.56){(4.58) to check if they add up to zero or not.
were reported in [54] and reproduced in gure 2.
Here in
scalar operator, hJ Si, hQSi, and hSSi. Contrary to
gure 2 there is no divergence at
other cases: 1) B = 0 and =
= 0:1, 2) B = 0 and
= 0, 3) B 6= 0. It turned out that all
cases in the appendix A.
!G1;t + kG1;x = 0.
12If W is invariant under U(1) gauge transformations, A
, the Ward identity for
onegreen, blue).
(a) Ward 4: (4.56)
(b) Ward 5: (4.57)
(c) Ward 6: (4.58)
temperatures shown in gure 6 all together. They are almost zero, less than 10 15.
Conductivities at small !
model of superconductor we check Homes' law and Uemura's law.
Re[ ] =
Re[ ] =
Im[ ] +
Im[ ] =
Im[ ] + Im[ ] =
for superconducting phase. By these relations, once
is obtained,
are completely
phase there is another contribution due to condensate.
Re[ ] +
Re[ ] =
Im[ ] +
Im[ ] =
Re[ ] +
Re[ ] =
Im[ ] + Im[ ] =
hQSi +
hSSi = 0 ;
values linear to !. Ks is introduced as a strength of the pole of Im[ ],
numerical data.
Ks = lim !Im[ ] ;
J . Contrary to the case of
= 0,
are not determined by
only, because there
once we know
, , and , we can read o
hJ Si, hQSi, and hSSi by the Ward identities.
part of , , and
= 0). At
small !, it is inferred that Re[hJ Si]
Also Re[hQSi]
!2 from (4.62) and Im[hQSi]
!2 from (4.60) and Im[hJ Si]
! from (4.61).
! from (4.63).13 Finally, the small !
(see for example the solid curves in
gure 2), unlike normal phase, Im[ ] and Im[ ] have
parts. In summary, the small ! behaviours can be written as
0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/Tc
0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/Tc
= 3; 5; 7; 10 (green, blue, purple,
hQSi, hSSi) diverge when
agree to the numerical results in
gure 6, 12 and 13.
If we de ne a normal uid density (Kn) as
goes to zero at small !. We have con rmed that (4.65){(4.70)
temperature in our numerics.
superconductor phase.
Ks + Kn. In gure 8, we plot
between the dotted and solid curve at a given
zero,14 Kn vanishes for =
. 2 ( gure 8(a)).
. 2 and incoherent state
without a Drude peak for =
. 2.15 In coherent state, the normal uid density Kn can be
Homes' law and Uemura's law
DC is in nite and Homes' law cannot be satisi ed.
56{58]. Uemura's law appearing in underdoped cuprates is
~s(T~ = 0) = B T~c ;
and Homes' law satis ed in a broader class of materials is
~s(T~ = 0) = C DC(T~c) T~c ;
and q. To check this it is
law respectively.
To compute B and C,
B =
s =
C =
the super uid density ~s(=
= 2 in our model [28].
results of ~s; T~c and
DC for q = 3 are shown in gure 9.
gure 9 we may expect that there is a linear relation between ~s and T~c at
least for large
small =
nd that Uemura's law
holds only for =
& 2, of which data are red dots. Interestingly, the parameter regime
behaviour. They correspond to
gure 8(a) and there is a gap between charge density
nd that
Uemura's law is satis ed for large
but with a di erent constant B. For example,
for q = 2, B
6:87 and for q = 6, B
4:64 in the regime of =
& 2 ( gure 10(c)).
Since Uemura's law is observed in underdoped regimes, if
can be interpreted as a doping
parameter our result will be consistent with phenomena.
DC for q = 3.
q=2
q=3
q=6
(a) B(= ~s=T~c) , q = 3
(b) B(= ~s=T~c) for q = 2; 3; 6
= 0:3; 0:4; 0:5; 0:7; 1).
In (a) the black line is drawn for B
5:47, and in (b) the black lines are drawn for B
(a) C(= ~s=( DCT~c)), q = 3
(b) ~s vs DCT~c, q = 3
regime ( =
= 2; 3; 5; 7; 10). In
Based on our results on Uemura's law ( gure 10(a)) and
DC ( gure 9(c)), we may
anticipate if Homes' law is satis ed. If
DC is quickly decreasing function approaching to
constant for
& 2 we may have a chance to obtain Homes' law. However, our
does not show that behaviour. Therefore, as shown in
gure 11, Home's law does not hold
in both coherent regime (red dots) and incoherent regime (blue dots). In
gure 11(a), for
representation, a plot of ~s versus
relation between between ~s and
DCT~c, where it is also clear that there is no linear
satis ed for di erent values of q either.
temperature (Planckian dissipation):
nN (Tc) ;
( =
2), where momentum relaxation is weak. In
gure 8(b), all curves coincide and it
Homes' law. The relaxation time
for our model can be written as
f (T = ; = ; q)
f is not universal near Tc. Furthermore, we may induce that f
= 2 because Tc
gure 9(b) and
model is isotropic four dimensional.
Uemura's law.
as expected.
rM F tM = iq
we may de ne the charge density of hair outside the horizon, nhair, as
gF trjr=1
gF trjr=rh = iq
Acknowledgments
of POSCO TJ Park Foundation.
Twopoint functions related to the real scalar operator
for other cases too: 1) B = 0; =
= 0:1, 2) B = 0;
= 0, 3) B 6= 0. For completeness,
we show here the numerical data of hJSi, hQSi, hSSi for (1) and (2) in
gure 12 and 13
gure 14 we show the numerical results of Ward identites for (3).
green, blue).
red, orange, green, blue, purple).
(a) Ward 1: (4.44)
(b) Ward 2: (4.45)
(c) Ward 3: (4.46)
led case: we plotted all components of the Ward identities (4.44){(4.46)
= 0; 0:5; 1; 1:5.
Open Access.
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