#### Fermionic spectral functions in backreacting p-wave superconductors at finite temperature

Received: October
Fermionic spectral functions in backreacting p-wave
G.L. Giordano
N.E. Grandi
A.R. Lugo
Open Access
c The Authors.
We investigate the spectral function of fermions in a p-wave superconducting nite both temperature and gravitational coupling, using the AdS=CF T correspondence and extending previous research. We found that, for any coupling below a critical value, the system behaves as its zero temperature limit. By increasing the coupling, the \peak-dip-hump" structure that characterizes the spectral function at xed momenta disappears. In the region where the normal/superconductor phase transition is rst order, the presence of a non-zero order parameter is re ected in the absence of rotational symmetry in the fermionic spectral function at the critical temperature.
superconductors; at; AdS-CFT Correspondence; Gauge-gravity correspondence; Holography and
Contents
1 Introduction 2 Bosonic sector: the holographic p-wave superconductor 2.1
The bulk gravitational solution
Vanishing Beckenstein-Hawking entropy
Non-vanishing Beckenstein-Hawking entropy
The boundary QFT interpretation
Dirac spinors in the gravitational background
Ingoing boundary conditions for vanishing Beckenstein-Hawking
en
Ingoing boundary conditions for non-vanishing Beckenstein-Hawking
The ultraviolet behavior
Normal modes and the Dirac cones
Fermionic operators in the QFT
Numerical results
A About the positivity of the spectral function
B On the spectral function in conformally
at backgrounds
Introduction
eld theory
the \large N limit".
systems. Roughly, it describes a d-dimensional quantum
eld theory in the strong coupling
In this setup, to model superconductors at
nite temperature and chemical
potential, a dynamical charged
eld describing the condensate needs to be added to the bulk
superconductor models.
symmetry due to the scalar condensate [4, 5, 11{13].
condensate is some component of a vector
eld. In order for it to be charged, a
nonnon-zero gravitational coupling and nite temperature [7].
regarding the spectral function are added at the end.
Bosonic sector: the holographic p-wave superconductor
The bulk gravitational solution
0; 1; 2; 3 and signature (
+ ++), and gauge group SU(2) with Hermitean generators a
preserve the diagonal subgroup of U(1)gauge
SO(2)rotation can be obtained. However they always have free
cosmological constant through
strength is de ned by F MaN
derived from this action read
S(bos) =
@N AaM + abc AbM AcN . The equations of motion
GMN +
GMN =
rSF MaN + abc AbS F McN = 0 ;
FSaM FNaS
where \r" stands for the usual geometric covariant derivative.
We are interested in solutions of the form [6, 14, 17, 34]
G =
A =
f (z) s(z)2 dt2 +
(z) dt 0 + W (z) dx 1 :
To understand this ansatz, we rst observe that a non-vanishing
(z) breaks the SU(2)
the equations (2.2) we get
s f g2 W 0 0 =
s0 =
2 W 2
where we have de ned the dimensionless coupling2
scaling symmetries
2The coupling constant gYM in [15] and ours are related by,
= 1=(p2 gYM).
some of the elds.
(x; y ; g; W )
(x ; z ; ; W )
(z) =
z + : : : ;
W (z) = WUV + W U0V z + : : : ;
f (z) = 1 + f3 z3 + : : : ;
s(z) = 1
g(z) = 1 + g3 z3 : : : ;
2 W 02UV z4 + : : : ;
nite Beckenstein-Hawking entropy cases in the forthcoming sections.
Now, by taking
T =
s(zIR) jf 0(zIR)j
Vanishing Beckenstein-Hawking entropy
(z) =
W (z) = wIR
f (z) = 1
s(z) = sIR +
g(z) = gIR
4 s20 gI2R wIR
2 I2R gIR wIR z3 e 2 gIR wIR z + : : : ;
2 I2R gIR wIR z3 e 2 gIR wIR z + : : : ;
values of the coupling
are displayed. We see that rotational and gauge symmetries are
signals the breaking of the U(1) 0 remaining gauge symmetry.
(corresponding to gYM
0:71) a horizon at zh
the extremal AdS-Reissner-Nordstrom black hole3
2:462 forms and the solution tends to
W (z) = 0 ;
(z) = 1
f (z) =
g(z) = s(z) = 1 ;
For couplings above this quantum critical value
only the AdS-Reissner-Nordstrom
solution is present.
Non-vanishing Beckenstein-Hawking entropy
0. The
(z) =
zh) + : : : ;
W (z) = WIR + WI0R (z
zh) + : : : ;
f (z) = fI0R (z
s(z) = sIR + s0IR (z
g(z) = gIR + gI0R (z
zh) + : : : ;
zh) + : : : ;
zh) + : : : :
W (z) = 0 ;
g(z) = s(z) = 1 ;
(z) =
f (z) = 1
black hole solution
3Strictly speaking, the AdS-Reissner-Nordstrom black hole is de ned for 0
zh, see next subsection.
Reissner-Nordstrom black hole solution.
where Q2
condition implies that the relation
numerically found,
0:995 and zh
6 should hold, consistent with the values
T =
zh =
electrostatic forces.
The boundary QFT interpretation
When 0
critical line).
When c <
interpreted as a nite chemical potential
for the particles charged under such symmetry,
three regions of the gravitational coupling, namely
0:62 the system has a second order phase transitions from the
0:995 the phase transitions become rst order (red critical line).
of the unstable branch (T2) respectively.
the symmetry broken phase ceases to exist.
respectively, for three di erent typical couplings
= 0:62 =
c (where Tc = 0:0252852 ) and
= 0:80 >
c (where Tc = 0:0103462 ).
function changes over the whole phase diagram.
T plane. The black point
indicates the critical coupling
= 0:40 < c
= 0:62 =
c (green),
= 0:80 >
c (blue). (1) (2) (3) The free energy for the values of
results from the QFT point of view.
Dirac spinors in the gravitational background
generators f
g = f
3=2; 1=2; 2=2g where 1; 2
; 3 are the Pauli matrices. To
ABg, and a spin connection f!ABg,
!03 =
!03 = +!30 =
!13 = +!13 =
!31 =
S(fer) =
and to introduce the projectors on eigenstates of 3 given by P
3) with
eigenvalprinciple [37, 38]. If we x the boundary value of the Dirac
elds to be the left
(right)handed part
), the action to be considered is
y i 0 and h = h (x) dx dx
resulting equations of motion read
is the induced metric on the boundary. The
= 0 :
the spinors as
( detG Gzz) 1=4 ei k x
k(z) = z3=2f 4 s 21 ei k x
1
k(z) where
as usual
!23 = +!23 =
Local gamma-matrices obey f
A DA
where the 2+1 dimensional
two-dimensional spinors
The equations of motion (3.4) then reads
4The generators of the Lorentz subgroup in 2 + 1 dimensions are,
k =
k+ =
i 1=2 the boosts generators and
y)-plane. The left/right
colored spinors k (z) are thus Dirac spinors of the boundary theory.
Now we can make explicit the color indices by writing
follows to introduce the four-tuples ~uk(z) and ~vk(z) as
k = ( k
i) where
and the 4
4 real matrix
g W=2
g W=2
g W=2
g W=2
In terms of this notation, equations (3.7) are compactly written as
+~u0k(z) + i U(z) ~vk(z) = 0 ;
~vk0(z) + i U(z) ~uk(z) = 0 ;
+ at the
~uk(z) = C(z; z0) ~u(0)
k
~vk(z) = i S(z; z0) ~u(0) + C(z; z0) ~vk(0) ;
k
C(z; z0) = P cosh
S(z; z0) = P sinh
dz0 U(z0)
dz0 U(z0)
P eRzz0 dz0 U(z0) + P e Rzz0 dz0 U(z0) ;
P e Rzz0 dz0 U(z0) : (3.12)
to analyze the behavior of the solutions in the IR and UV limits.
at zIR ! 1. The matrix U(z) is constant there
U(1) = BB sIR
) satisfying
2 = gI2R kx
Let us take z0 = zir
operators read
1, and let us consider the region z > zir
1. There the evolution
C(z; zir) ' P
S(z; zir) ' P
cosh( +(z
zir)) 12 2
sinh( +(z
zir)) 12 2
cosh( (z
zir)) 12 2
sinh( (z
zir)) 12 2
D P t UIR ; (3.16)
i D~ P t UIR ~vk(ir) +
P t ~u(ir) + i D~ P t UIR ~vk(ir) ;
k
P t ~v(ir) + i D~ P t UIR ~u(kir) +
k
P t ~v(ir) i D~ P t UIR ~u(kir) : (3.17)
k
2. The condition that the solution
z limit we have
Let us rst consider
2 > 0, and then
yielding the solution
implying the behaviors
~u(ir) and ~u
C(z; zir) ' 1 ;
S(z; zir) ' 0 ;
~uk(z) ' ~u(kir) ;
~vk(z) ' ~vk(ir) :
~uk(z) ' P
~vk(z) ' P
where the initial values of the elds satisfy the linear relation
~u(ir) = i P D~ P t UIR ~vk(ir) :
k
the case
Similar result is obtained when
boundary conditions, with the replacement
2 < 0 and/or
2 < 0 under the imposition of ingoing
i sign(!) p
2 in (3.18). Regarding
4 T (1 z) C ; C
horizon region zir < z
1. The evolution operators (3.12) at leading order then read
1, and let us look at the near
C(z; zir) ' cos
4 T ln 1 1
From (3.11) they yield the solution
~uk(z) ' cos
~vk(z) '
4 T ln ! 4 T
S(z; zir) ' i sin
4 T ln 1 1
~u(ir) + sin
C ~u(kir) + cos
4 T 1 1 ln
~v(ir), as
After imposing this condition the near horizon solution reads,
~uk(z) ' e i 4 !T ln 11 zzir ~u
~vk(z) ' e i 4 !T ln 11 zzir ~v
The ultraviolet behavior
boundary z ! 0. There the matrix U(z) goes to a constant
U(0) = BBB
and the evolution operators (3.12) at leading order are
Plugging back into (3.11), the leading behavior results
C(z; 0) ' 1 ;
S(z; 0) ' UUV z :
~vk(z) ' ~vk(UV) + i z UUV ~u(kUV) :
Normal modes and the Dirac cones
to de ne fermionic normal modes as regular solutions with ~u(UV)
~uk(zUV) = ~0. Due to
~u(UV) = C(zUV; zir) ~u(ir)
k k
i S(zUV; zir) ~v
~v(UV) = i S(zUV; zir) ~u(ir) + C(zUV; zir) ~v
k k
i L ~u(kir) holds where,
S(zUV; zir) L) ~u(kir) ;
C(zUV; zir) L) ~u(kir) ;
exist a non trivial solution i
C(!; kx; ky)
det (C(zUV; zir)
S(zUV; zir) L) = 0 :
of this equation:
S(zir; 0) '
UUV zir. In such case, equation (3.32) reads
that the eigenvalues of UUV, given by ( +;
vanish. The equations
They de ne the two UV Dirac cones
=2; 0; 0), and that satisfy
being essentially blind to the UV part of the geometry. Then for
2 = 0 we can
this time becomes
left with the case 2 = 0, where
2 = gI2R kx
2 = 0. This de nes the
that rotational symmetry in the momentum plane is broken in the IR.
representation of SU(2).
The two point correlation functions of such operator can be
the retarded fermionic correlator closely following references [15].
So(fner)shell =
(2 )3 ~u(kUV)y C ~v(UV)
k
of the elds; from (3.31) we get
~v(UV) = Mk ~u(UV)
k k
Then the on-shell action is
i (S(zUV; zir)
C(zUV; zir) L) (C(zUV; zir)
So(fner)shell =
(2 )3 ~u(kUV)y C Mk ~u(kUV) :
S =
i hO~~(k) O~~y(0)icjret
3 3(k) GR(k) = C
From here, using equation (3.39), we can read the correlation function
GR(k) = Mk C :
fermionic dynamics is the spectral function (k), de ned as
Im Tr GR(k) =
Im Tr (Mk C) :
is singular for them.
(b) (kx; ky) = (0:2; 0:1) .
(c) (kx; ky) = (0:2; 0:4) .
0:20 Tc and
UV (in violet) Dirac cones.
Numerical results
contained in the
gures displayed in this section. We plotted the spectral function as a
xed values of (kx; ky), for di erent values of
and T .
(kx; ky) and low
c, and a low temperature T < Tc. It is seen that the intersections
perature. Each
gure corresponds to a xed
and !. In all cases, as the temperature
with gure 5 sub gure (f), or gure 7 sub gure (b) with
gure 6 sub gure (f), we see that
correspond to three
xed values of , smaller, equal and bigger than c
. Each
gure in
The value of ! decreases from
gure to
gure inside each set. It is evident by comparing
the spectral function is milder for
> c than in the cases
c. Notice also that even
(kxF ; kyF ) =
lifetime. In
it matches with the value of the normal phase.
Discussion
extending previous work in the literature.
(d) (kx; ky) = (k ; 0).
(e) (kx; ky) = (0:2; 0:1) .
(f ) (kx; ky) = (0:2; 0:4) .
(a) (kx; ky) = (k ; 0).
(b) (kx; ky) = (0:2; 0:1) .
(c) (kx; ky) = (0:2; 0:4) .
(g) (kx; ky) = (k ; 0).
(h) (kx; ky) = (0:2; 0:1) .
(i) (kx; ky) = (0:2; 0:4) .
ing (purple) phases, at
xed (kx; ky), for three di erent temperatures and couplings:
(b) (c) T = 0:20 Tc ;
= 0:40 <
c ; (d) (e) (f) T = 0:28 Tc ;
= 0:62 =
c ; (g) (h) (i)
T = 0:40 Tc ;
= 0:80 > c .
the normal modes.
gravitational coupling is increased.
the critical temperature of the superconducting transition.
(a) T = Tc.
(b) T = 0:85 Tc.
(c) T = 0:60 Tc.
(d) T = 0:43 Tc.
(e) T = 0:20 Tc.
(f ) T = 0.
= 0:40 <
c, ! = 0:25 . It
(a) T = Tc.
(b) T = 0:85 Tc.
(c) T = 0:60 Tc.
(d) T = 0:43 Tc.
(e) T = 0:20 Tc.
(f ) T = 0.
= 0:40 <
c, ! = 0:18 . It
(a) T = Tc.
(b) T = 0:85 Tc.
(c) T = 0:60 Tc.
(d) T = 0:43 Tc.
(e) T = 0:20 Tc.
(f ) T = 0.
= 0:40 <
c, ! = 0:12 . It
(a) T = Tc.
(b) T = 0:79 Tc.
(c) T = 0:60 Tc.
(d) T = 0:38 Tc.
(e) T = 0:28 Tc.
(f ) T = 0.
= 0:62 = c, ! = 0:25 . It
(a) T = Tc.
(b) T = 0:79 Tc.
(c) T = 0:60 Tc.
(d) T = 0:38 Tc.
(e) T = 0:28 Tc.
(f ) T = 0.
= 0:62 =
c, ! = 0:18 . It
(a) T = Tc.
(b) T = 0:79 Tc.
(c) T = 0:60 Tc.
(d) T = 0:38 Tc.
(e) T = 0:28 Tc.
(f ) T = 0.
= 0:62 =
c, ! = 0:12 .
(a) T = 0:80 Tc.
(b) T = 0:40 Tc.
(c) T = 0.
= 0:80 >
c, ! = 0:25 .
represent the UV and IR Dirac cones respectively.
(a) T = 0:80 Tc.
(b) T = 0:40 Tc.
(c) T = 0.
= 0:80 >
c, ! = 0:18 .
represent the UV and IR Dirac cones respectively.
Acknowledgments
Salam ICTP for kind hospitality and
nancial support during some stages of the work.
CONICET, Argentina.
About the positivity of the spectral function
(a) T = 0:80 Tc.
(b) T = 0:40 Tc.
(c) T = 0.
= 0:80 >
c, ! = 0:12 .
represent the UV and IR Dirac cones respectively.
The starting point is the (on-shell) conserved current
J A(x; z)
J A(z; k) = i s pf
DAJ A(z; k) =
Q0(z; k) = 0
2 Im
k+(z) k (z) =
2 Im ~uk(z)y C ~vk(z)
2 Im ~uk(z)y S(z) 1 ~u0k(z)
Here the matrix S(z)
last equality we used the equations of motion (3.10).
can evaluate it in known limits.
the relation (3.38) and the de nition (3.41) we obtain
QUV(k)
Q(0; k) = i C ~u(kUV) y
GR(k)
GR(k)y
If we assume that QUV(k)
we conclude that,
that the hermitian matrix i GR(k)
GR(k)y is non-negative. From the de nition (3.42)
0, and being the vector ~u(UV) arbitrary, we must conclude
(!; kx; ky)
GR(k)
GR(k)y
(a) ! = 0, T = Tc.
(b) ! = 0, T = 0:95 Tc.
(c) ! = 0, T = 0:85 Tc.
(d) ! = 0, T = 0:20 Tc.
Figure 14. The spectral function at xed = 0:40 <
di erent temperatures. Right: (1)
gure; (2)
(a) ! = 0, T = Tc.
(b) ! = 0, T = 0:97 Tc.
(c) ! = 0, T = 0:79 Tc.
(d) ! = 0, T = 0:41 Tc.
di erent temperatures. Right: (1)
gure; (2)
(a) ! = 0, T = Tc.
(b) ! = 0, T = 0:80 Tc.
(c) ! = 0, T = 0:40 Tc.
= 0:80 >
di erent temperatures. Right: (1)
a function of ! at (kx; ky)=
= (0:20; 0:10); (3)
as a function of ! at (kx; ky)=
= (0:20; 0:40).
= 0:4.
~uk(z) ' e
z BB b4 CC
B@ b3 AC
not relevant here. From (A.2) we get
QIR(k)
Q(z; k)jz!1
= lim BB2 Im( +) e
+ z 2 BB b2 CC SIR1BBB b2 CC
B
z 2 BB b4 CC SIR1BBB bb43 CCAC
B@ b3 AC @
) z BB b4 CC
B@ b3 AC
SIR1 BBB@ bb43 CCACCCCCACCCCA
SIR
Sjz!1 =
3 (A.7)
S2 are real and given respectively by,
except when we are outside both IR Dirac cones, i.e.
2 > 0, in which case QIR(k) = 0.
e ectively (k)
0, being zero outside both IR Dirac cones or in other words,
is strictly
positive inside some IR Dirac cone and null outside both of them.
where ~b is read from (3.26) in terms of ~u(kir). From (A.2) we get,
~uk(z) ' exp i 4 !T ln(1 z) ~b
QIR(k) = 2~by ~b > 0
where in space of momenta.
On the spectral function in conformally
at backgrounds
G =
A = dt
dt2 + d~x2 + dz2
+ dx1 W1 + dx2 W2
and after introducing
Green function
~vk(z) = i
2 =
and take as b.c. a
when z ! 1. With this b.c. the solution can be written
2 = ky
de ned by
U =
0 =
1 = kx
The general solution of (3.10) is
~uk(z) =
Let us de ne,
GR(k) = @
The spectral function is then obtained from (3.42),
(!; kx; ky) = 2
of the right signs in the de nition of the roots of
2 in the region where they are negative,
positivity of the spectral function. And when
2 < 0 considered in subsection 3.1.1.
of smoothness after shifting ! ! ! + i with
! 0+; in fact if we assume that
we have,
=)
j!!!+i
and then the exponential will be well-behaved at z ! 1 if (B.5) holds [15].
Open Access.
This article is distributed under the terms of the Creative Commons
any medium, provided the original author(s) and source are credited.
(2008) 015 [arXiv:0810.1563] [INSPIRE].
033 [arXiv:0805.2960] [INSPIRE].
Lect. Notes Phys. 843 (2012) 1 [arXiv:1002.3823] [INSPIRE].
superconductors, Rev. Mod. Phys. 75 (2003) 473 [INSPIRE].
Mod. Phys. 63 (1991) 239 [INSPIRE].
Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
holographic superconductors, arXiv:1012.5312 [INSPIRE].
Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
[hep-th/9802150] [INSPIRE].
Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
[hep-th/0505189] [INSPIRE].
78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
JHEP 11 (2014) 066 [arXiv:1405.3714] [INSPIRE].
superconductors, JHEP 11 (2010) 085 [arXiv:1002.4416] [INSPIRE].
phase transitions, arXiv:1110.3814 [INSPIRE].
[INSPIRE].
charged black holes, JHEP 07 (2015) 172 [arXiv:1501.04033] [INSPIRE].
Oxford U.K., (1981).
[2] A. Damascelli , Z. Hussain and Z.X. Shen , Angle-resolved photoemission studies of the cuprate [3] M. Sigrist and K. Ueda , Phenomenological theory of unconventional superconductivity , Rev.
[4] S.A. Hartnoll , C.P. Herzog and G.T. Horowitz , Building a holographic superconductor , Phys.
[5] S.A. Hartnoll , C.P. Herzog and G.T. Horowitz , Holographic superconductors, JHEP 12 [6] S.S. Gubser and S.S. Pufu , The gravity dual of a p-wave superconductor , JHEP 11 ( 2008 ) [7] S.S. Gubser , TASI lectures: collisions in anti-de Sitter space , conformal symmetry and [8] J.M. Maldacena , The large-N limit of superconformal eld theories and supergravity, Int . J.
[9] S.S. Gubser , I.R. Klebanov and A.M. Polyakov , Gauge theory correlators from noncritical [10] E. Witten , Anti-de Sitter space and holography, Adv. Theor. Math. Phys . 2 ( 1998 ) 253 [11] S.A. Hartnoll , Lectures on holographic methods for condensed matter physics, Class. Quant.
[12] S.S. Gubser , Phase transitions near black hole horizons , Class. Quant. Grav . 22 ( 2005 ) 5121 [13] S.S. Gubser , Breaking an Abelian gauge symmetry near a black hole horizon , Phys. Rev . D [14] M. Ammon , J. Erdmenger , V. Grass , P. Kerner and A. O'Bannon , On holographic p-wave super uids with back-reaction , Phys. Lett . B 686 ( 2010 ) 192 [arXiv:0912.3515] [INSPIRE].
[15] S.S. Gubser , F.D. Rocha and A. Yarom , Fermion correlators in non-Abelian holographic [16] R.E. Arias and I.S. Landea , Backreacting p-wave superconductors , JHEP 01 ( 2013 ) 157 [17] C.P. Herzog , K.-W. Huang and R. Vaz , Linear resistivity from non-Abelian black holes , [18] S.-S. Lee , A non-Fermi liquid from a charged black hole: a critical Fermi ball , Phys. Rev . D [19] H. Liu , J. McGreevy and D. Vegh , Non-Fermi liquids from holography , Phys. Rev. D 83 [20] T. Faulkner , H. Liu , J. McGreevy and D. Vegh , Emergent quantum criticality, Fermi surfaces and AdS2 , Phys . Rev . D 83 ( 2011 ) 125002 [arXiv:0907.2694] [INSPIRE].
[21] M. Cubrovic , J. Zaanen and K. Schalm , String theory, quantum phase transitions and the emergent Fermi -liquid, Science 325 ( 2009 ) 439 [arXiv:0904. 1993 ] [INSPIRE].
[22] T. Faulkner , N. Iqbal , H. Liu , J. McGreevy and D. Vegh , Holographic non-Fermi liquid xed points , Phil. Trans. Roy. Soc. A 369 (2011) 1640 [arXiv:1101.0597] [INSPIRE].
[23] T. Faulkner , N. Iqbal , H. Liu , J. McGreevy and D. Vegh , From black holes to strange metals , [24] T. Faulkner , N. Iqbal , H. Liu , J. McGreevy and D. Vegh , Charge transport by holographic [25] N. Iqbal , H. Liu and M. Mezei , Lectures on holographic non-Fermi liquids and quantum [26] J. Zaanen , Y.W. Sun , Y. Liu and K. Schalm , The AdS/CMT manual for plumbers and electricians, Universiteit Leiden , Leiden The Netherlands , 15 October 2012 .
[27] J.-W. Chen , Y.-J. Kao and W.-Y. Wen , Peak-dip-hump from holographic superconductivity , [28] P. Basu , J. He , A. Mukherjee , M. Rozali and H.-H. Shieh , Comments on non-Fermi liquids [29] T. Faulkner , G.T. Horowitz , J. McGreevy , M.M. Roberts and D. Vegh , Photoemission [30] S.S. Gubser , F.D. Rocha and P. Talavera , Normalizable fermion modes in a holographic [31] M. Ammon , J. Erdmenger , M. Kaminski and A. O'Bannon , Fermionic operator mixing in [32] N. Iqbal and H. Liu , Real-time response in AdS/CFT with application to spinors , Fortsch.
[35] A.R. Lugo , E.F. Moreno and F.A. Schaposnik , Holography and AdS4 self-gravitating dyons , JHEP 11 ( 2010 ) 081 [arXiv:1007.1482] [INSPIRE].
[36] A.R. Lugo , E.F. Moreno and F.A. Schaposnik , Holographic phase transition from dyons in an AdS black hole background , JHEP 03 ( 2010 ) 013 [arXiv:1001.3378] [INSPIRE].
[37] M. Henningson and K. Sfetsos , Spinors and the AdS/CFT correspondence, Phys . Lett . B 431 [38] M. Henneaux , Boundary terms in the AdS/CFT correspondence for spinor elds , in Mathematical methods in modern theoretical physics, Tbilisi Georgia , ( 1998 ), pg. 161 [39] L.D. Landau , E.M. Lifshitz and L.P. Pitaevskii , Statistical physics, part 2, Pergamon Press, [40] X. Arsiwalla , J. de Boer , K. Papadodimas and E. Verlinde , Degenerate stars and