Backreacting p-wave Superconductors

0.28 MB·

by  Raúl E. Arias

Backreacting p-wave Superconductors
Raúl E. Arias †1 and Ignacio Salazar Landea ‡†2
†IFLP-CONICET and Departamento de Física
Facultad de Ciencias Exactas, Universidad Nacional de La Plata
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1 CC 67, 1900, La Plata, Argentina
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t ‡Abdus Salam International Centre for Theoretical Physics,
c
O ICTP-IAEA Sandwich Training Educational Programme
Strada Costiera 11, 34151, Trieste, Italy
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3

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Abstract
t
-
p Westudythegravitationalbackreactionofthenon-abeliangaugefieldonthegravitydualtoa2+1p-wave
e superconductor. We observe that as in the p+ip system a second order phase transition exists between a
h
[ superconducting and a normal state. Moreover, we conclude that, below the phase transition temperature

2 Tc the lowestfree energyis achievedby the p-wavesolution. In orderto probe the solution,we compute the
v
holographicentanglemententropy. For bothp andp+ipsystems the entanglemententropysatisfiesanarea
3
2 law. For any given entangling surface, the p-wave superconductor has lower entanglement entropy.
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2
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i
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1rarias@fisica.unlp.edu.ar
[email protected]
1 Introduction
The AdS/CFT correspondence [1, 2, 3] in its original form relates a conformal field theory in d dimensions
with type II string theory on AdS . The power of the correspondence arises from the fact that it is a
d+1
weak/strong coupling duality i.e. it relates the strong coupling regime of the field theory with the weak
coupling regime of the string theory and viceversa. After the works [4, 5], the gauge/gravity conjecture
begun to be an useful tool to study condensed matter physics. In particular, it has been applied to study
stronglycorrelatedcondensedmattersystemsthroughtheanalysisofasemiclassicaldualgravitytheory(see
[7, 8] for a review).
In the present work we analyze the backreaction of the gravity dual to a p-wave superconductor3 in
3+1 dimensions [6] (see [9, 10] for a similar treatment in 4+1 dimensions). Along the way we rederive the
backreactionof the colorful p+ip superconductors previouslystudied in [11]. We use the prescriptiongiven
in [12, 13, 14] to compute the entanglement entropy from the holographic point of view for both gravity
duals. Similar computations of entanglement entropy in backgrounds duals to condensed matter systems
can be found in [15, 16, 17, 18].
p-wavesuperconductivity is a phase of matter produced when electrons with relative angular momentum
j = 1 form Copper pairs and condense. In other words, the operator that condense is a vector, charged
under a U(1) symmetry. This kind of superconductivity is supposed to originate from "strongly correlated"
electrons and therefore the BCS theory is not the correctapproachto study its microscopic dynamics. This
phenomena is a challenge for theoretical physics, and due to the fundamental property of the gauge/gravity
duality mentioned above one could envisage the study of such systems through their weak gravitydual. We
are going to introduce the minimal ingredients that one needs on the gravity side in order to reproduce
the dynamics of the superconductor, this kind of approach aims to reproduce the properties of a condensed
matter system without trying to explain their microscopicalorigin.
Stringtheoryembeddingsofp-wavesuperconductorswerestudiedin[19,20,21,22,23,24]. Thenecessary
minimal ingredients on the bulk to have finite temperature, chemical potential and spontaneous symmetry
breaking (SSB) are: a black hole geometry and a non-Abelian gauge field [25, 26]. The solutions we will
consider are asymptotically AdS backgrounds with a SU(2) non-Abelian gauge field. The SSB is realized
on the bulk side as a non-trivialasymptotics (hair) for the gauge field. The chemicalpotential and the SSB
arisebyturningontwoindependentdirectionsinsidethenon-Abeliangaugegroup. Thesymmetrybreaking
occurs on the gravity side through the formation of a condensate outsides the horizon.
The entanglement entropy (EE) between a subsystem and it’s complement is the von Neumann
A B
entropy
= Tr (ρ lnρ ). (1)
A A A A
S −
Here ρ =Tr (ρ) is the density matrix obtained by tracing the density matrix of the whole system ρ over
A B
the subsystemdegreesoffreedom. Roughly speakingS measureshow muchinformationis hidden inside
A
B
3The tiny difference between a superconductor and a superfluid arises in the fact that although both effects are produced
byaspontaneously symmetrybreaking, inthe firstcasethere isalocalsymmetry whichisspontaneously broken whileinthe
second case is a global symmetry. We are goingto use the terms superfluid and superconductor interchangably here. For the
consideredphenomena thisdistinctiondoesnotmakeanydifference.
1
whenwe subdivide the system. From the pointof view of the dualgravitytheorythe EE wasconjectured
B
[12] to be proportionalto the bulk minimal area surface, γ , whose boundary at infinity coincides with the
A
boundary of (see [14] for a review)
A
2πArea(γ )
A
= . (2)
SA κ2
Hereκisthebulkgravitationalconstant. Notethatthestandardthermalentropyisobtainedasaparticular
case of the EE, when the region is the whole system. In [27] the authors provide a demonstration of
A
thisholographictechniquetocomputethe entanglemententropyforsphericalsurfacesandzerotemperature
CFTs. Onthis workwe computethis quantityfora stripgeometryinthe backgroundsdualto a p-waveand
to a colorful p+ip superconductor.
This paper is organized as follows: in section 2 we compute the backreaction of a 3+1 gravity dual to a
p-wave superconductor in 2+1 dimensions and analyze its thermodynamic properties. In order to compare
withthe colorfulsuperconductor,wereviewinsubsection2.2the backreactionofthe gravitydualofap+ip
superconductor. On section 3 we compute the holographic entanglement entropy for a strip geometry in
both systems,as a function of the temperature andthe length of the strip. The conclusionsare summarized
in section 4.
2 p and p + ip holographic superconductors
As mentioned in the introduction, the gravity dual to a p-wave superconductor is modeled by an Einstein-
Yang-Mills (EYM) theory. In [6, 28], the 3+1 dimensional gravity theory dual to a p-wave superconductor
hasbeencomputedintheprobelimit. Moreovertheauthorsshowedthatthep+ipsuperconductorgeometry
studiedin[11]wasunstableundersmallfluctuations,andthatthestableconfigurationwasthatofthep-wave
solution. In this section we compute the backreaction of the non-Abelian gauge field on the geometry dual
to a p-wave superconductor in 3+1 dimensions and compare the results with those for the p+ip case.
WewillworkinthesimplestsetupandconsiderSU(2)asthegaugegroup. Inthep+ipcase,theansatzfor
thegaugefieldissuchthatitbreakstheU(1)subgroupoftheinternalgaugeSU(2)andthespatialrotational
SO(3) group symmetries into a diagonal subgroup of them. Instead, the the p-wave superconductor, breaks
both U(1)symmetries completely. The gravitysolutionthat describesthe strongcoupling dynamicsof both
kinds of superconductors is as follows: a charged superconducting layer develops outside the horizon due to
the interplay between the electric repulsion (with the charged black hole) and the gravitationalpotential of
the asymptotically AdS geometry. At high enough temperatures there is no hair outside the black hole and
the solutionisjustanAdS-Reissner-Nordström(AdSRN)blackhole. BelowacriticaltemperatureT anon-
c
trivial gauge field with non-vanishing chemical potential on the boundary of the geometry and a sourceless
non-vanishing condensate in the bulk appears, originating a breaking of the SU(2) gauge symmetry.
2
2.1 p-wave superconductor in 2+1 dimensions
2.1.1 Solution
We start from 3+1 SU(2) Yang Mills Theory in AdS gravity (see [29] for a review about solutions for this
theory), the Lagrangiandensity is
1
κ2 =R 2Λ Tr(F Fµν) (3)
(4)L − − 4 µν
where Λ = 3 , κ is the gravitational constant in four dimensions and the field strength of the SU(2)
−Rˆ2 (4)
gauge field is written as
Fa =∂ Aa ∂ Aa +g ǫabcAbAc (4)
µν µ ν − ν µ YM µ ν
withg = gˆYM the parameterthatmeasuresthe backreactionandgˆ theusualYang-Millscoupling. We
YM κ(4) YM
use latin letters for SU(2) indexes and greek letters for the space-time coordinates. By scaling the gauge
field as A˜ = A we see that the large g limit corresponds to the probe (non-backreacting) limit of the
g YM
YM
gauge field. Roughly one can think that 1 counts the degrees of freedom of the dual field theory that
gˆ2
YM
are charged under the SU(2) gauge group. Moreover, 1 counts the total number of degrees of freedom.
κ2
(4)
Consideringbackreactionof the gaugefiled amountsto saythat the number ofchargedstates is of the same
order as the number of degrees of freedom of the system.
The equations of motion following from the action are
1 3 1 g
G =R g R = g + Tr[F Fγ] µνTr[F Fγρ] (5)
µν µν − 2 µν R2 µν 2 µγ ν − 8 γρ
D Fµν = 0 (6)
µ
we propose the following ansatz [9, 30]
1
ds2 = M(r)σ(r)2dt2+ dr2+r2h(r)2dx2+r2h(r)−2dy2, (7)
− M(r)
for the background geometry, and
A=φ(r)τ3dt+ω(r)τ1dx. (8)
for the gauge field. Here we use the matrix-valued notation A = Aaτadxµ with τa = σa and σa the usual
µ 2i
Pauli matrices, the SU(2) generators satisfy [τa,τb] = ǫabcτc. A solution developing ω = 0 in the gauge
6
field ansatz (8) breaks the U(1) gauge symmetry associatedwith rotations aroundτ3 (usually called U(1) )
3
and a h=0 in the metric breaks U(1) symmetry associatedto rotations on the xy plane. At high enough
xy
6
temperatures we expect no hair outside the black hole and the solution with no condensate is AdSRN with
ω(r) = 0,
h(r) = 1
σ(r) = 1,
r
h
φ(r) = µ 1 ,
− r
(cid:16) µ2r2(cid:17) µ2 r
M(r) = r2+ h +r2 h. (9)
r2 − 8 h r
(cid:18) (cid:19)
3
Replacing the ansatz inyo the EYM equations of motion results into five equations, three of them are
second order differential equations, and the remaining two are first order constraints
3r 1 g2 φ2ω2 1 rh′2 ω′2
M′ = YM +rφ′2 M + +
Rˆ2 − 8σ2 rh2M − r h2 8rh2
(cid:18) (cid:19) (cid:18) (cid:19)
σ ω′2 g2 φ2ω2
σ′ = rh′2+ + YM ;
h2 8r 8rM2h2σ
(cid:18) (cid:19)
1 g2 φ2ω2 2 h′ M′ σ′
h′′ = ω′2+ YM h′ + + ;
8r2h − M2σ2 − r − h M σ
(cid:18) (cid:19) (cid:18) (cid:19)
g2 φ2ω 2h′ M′ σ′
ω′′ = YM +ω′ ;
− M2σ2 h − M − σ
(cid:18) (cid:19)
g2 φω2 2 σ′
φ′′ = YM φ′ . (10)
r2h2M − r − σ
(cid:18) (cid:19)
This system of equations enjoys four scaling symmetries that become useful when numerically solving it,
they are
1. σ λσ, φ λφ
→ →
2. ω λω, h λh
→ →
3. M λ−2M, σ λσ, g λ−1g , Rˆ λRˆ
→ → YM → YM →
4. M λ2M, r λr, φ λφ, ω λω
→ → → →
Using these scaling symmetries we can set R = r = 1 and fix the boundary value of the metric functions
h
σ( ) = h( ) = 1. The geometry and the gauge field must be regular at the horizon which implies the
∞ ∞
following expansion in the IR (small r)
M = M (r r )+M (r r )2+...
1 h 2 h
− −
h = h +h (r r )2+...
0 2 h
−
σ = σ +σ (r r )+σ (r r )2+...
0 1 h 2 h
− −
ω = ω +ω (r r )2+ω (r r )3+...
0 2 h 3 h
− −
φ = φ (r r )+φ (r r )2+... (11)
1 h 2 h
− −
On other hand in the UV (large r) the desired behavior is:
Mb (ωb)2+ρ2
M = r2+ 1 + 1 +...
r 8r2
hb (ωb)2
h = 1+ 3 1 +...
r3 − 32r4
(ωb)2
σ = 1 1 +...
− 32r4
ωb g2 µ2ωb
ω = ωb+ 1 YM 1 +...
0 r − 6r3
ρ g2 µ2ωb
φ = µ+ + YM 1 +... (12)
r 12r4
4
To achieve SSB, we look for solutions where the non-normalizable component vanishes ωb = 0. Standard
0
AdS/CFT dictionary instruct us to interpret the boundary and sub-leading values of φ as the chemical
potentialµandthechargedensityρofthedualfieldtheory[31]. Moreover,thesub-leadingcoefficientMb in
1
the boundaryexpansionofg coincideswiththeregularizedEuclideanon-shellaction[4]. The normalizable
tt
coefficient in ω is dual to the vacuum expectation value of the current J1 ωb and serves as an order
h xi ∝ 1
parameter for the system.
Solutions of the system (10) depend on the four IR coefficients φ ,ω ,h ,σ and the backreaction pa-
1 0 0 0
rameter g . All other coefficients in (11) can be written in terms of those. We proceed to integrate the
YM
equations of motion numerically out from the horizon using a shooting method in order to get the desired
asymptotic behavior. We explore the range g [0.85,24] and observe that the behavior of the functions
YM ∈
doesnotchangequalitativelyasg isvaried. Infigure1and2wegivetheplotofthesolutionsof (10)with
YM
boundary conditions (12). We use µ to adimensionalize whenever needed. This means that we are working
in the grand canonical ensemble.
1.0000 1.4 M HrL
20
1.2
ΩHrL
0.9995
1.0
ΦHrL
ΣHrL 0.8
0.9990
hHrL 0.6
0.4
0.9985
0.2
0.0 r
r 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1.5 2.0 2.5 3.0 3.5 4.0
Figure 2: The dimensionless metric function M(r)
Figure 1: The dimensionless metric functions σ(r)
and the gauge field functions ω(r) and φ(r) for
and h(r) for g =2 and T =0.2312µ.
YM g =2,T =0.2312µ.
YM
2.1.2 Thermodynamics
Inthissectionwecomputethethermodynamicquantitiesassociatedwiththesolutions. Asweshallseefrom
thestudyofthepotentialfunctioninthegrandcanonicalensemble4 wehaveasecondorderphasetransition
between a superconducting and normal symmetric phases.
The temperature of the dual theory is given by the Hawking temperature of the black hole
M 1
T = 1 = 24σ2 φ2 r (13)
2π 16π 0− 1 h
(cid:0) (cid:1)
wherethesecondequalitycomesfromtheconsistencyoftheseriesexpansion(11)thatrelatesthecoefficient
4Togofromthegrandcanonicalensemble(fixedµ)withfreeenergyΩ,tothecanonicalensemble(fixedρ)withfreeenergy
F,weshouldaddaboundarytermtotheEuclideanaction. ThischangesthevariationalproblemandimpliestheknownGibbs
relationF =Ω+µρ.
5
M with σ and φ . The area of the horizon, A , gives the entropy
1 0 1 h
2π 2π2VT2 122
S = A = (14)
κ2 h κ2 (24σ2 φ2)2
(4) (4) 0− 1
where V = dxdy. In figure 3 we plot the order parameter ωb(i.e. the VEV of the current J1 ) as a
1 h xi
function of the temperature. Note that at T = T the condensate vanishes showing the disappearance of
R c
the superconducting state for T > T . From our numerical results we find Jx (1 T )1/2 near T and
c h 1i ∝ − Tc c
therefore the critical exponent takes the value 1/2. Figure 4 shows the behavior of the Bekenstein-Hawking
entropy (14) as function of the temperature for our solution and the AdSRN black hole.
Ω1b
Μ2
0.6
0.5
0.4
0.3
0.2
0.1
T
0.10 0.15 0.20 Μ
Figure3: Theplotshowsthenormalizablecoefficientoftheωfunctionwhichisproportionaltothecondensate
J1 . The black, greenand blue lines refers to solutions with g =1,1.5,2and T =0.0749,0.1565,0.2312
h xi YM c
respectively. Note that the condensate vanishes for T >T .
c
Κ2S
Μ2 Κ2W
2.5 VΜ3
-0.05
2.0
-0.10
1.5
-0.15
1.0 -0.20
0.5 -0.25
T T
0.10 0.15 0.20 0.25 Μ
0.10 0.15 0.20 0.25 Μ
Figure 4: The entropy as a function of the tem- Figure5: ThepotentialfunctionΩcomputedfrom
perature. The blue line is for the superconducting (22) as a function of T for g = 2. The red line
YM
phasewithg =2andtheredlineforthenormal is the potential for the RN solution and the blue
YM
phase(AdSRNgeometry). Thereisasecondorder line is for the superconductor case.
phase transition at T =T =0.2312.
c
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The gauge/gravitycorrespondenceidentifies the Euclidean on-shellgravity actionS times the tempera-
E
ture T as the grand canonical potential function Ω of the system. To compute it we continue to Euclidean
signature, time being compactified with period 1 to avoid singularities. The on-shell action has a factor 1
T T
due to time integration, writing S = S˜bulk one has
on−shell T
S˜ = dxdydr√ g (15)
bulk
− − L
Z
where the lagrangian density is given by (3). The yy component of the stress tensor is proportional to the
metric and then the Einstein equations (5) implie that
r2
G = κ2 R (16)
yy 2h2 (4)L−
(cid:16) (cid:17)
Then we have
1
Gµ = R=Gr+Gt+Gx+ κ2 R (17)
µ − r t x 2 (4)L−
and from this we obtain (cid:16) (cid:17)
2 r3Mσ h ′ ′
= (18)
L r2σκ2(4) " h (cid:18)r(cid:19)#
where ′ denotes derivative with respect to the holographic coordinate r. Then, the bulk contribution to the
on shell action (15) can be written as
2V r3Mσ h ′
S˜ = dxdydr√ g = (19)
bulk −Z − L −κ2(4) " h (cid:18)r(cid:19)#r=r∞
where r is the boundary of the space. As usual, in order to have a well defined variational problem when
∞
imposingDirichletboundaryconditionsonthemetricweneedtoaddtotheactionaGibbons-Hawkingterm
1 V M′ σ′ 2
S˜ = dxdy√ g nµ = r2σ +M + , (20)
GH −κ2 − ∞∇µ −κ2 2 σ r
(4) Z (4) (cid:20) (cid:18) (cid:19)(cid:21)r=r∞
wherenµdx =√Mdristheoutwardpointingunitnormalvectortotheboundaryandg isthedeterminant
µ ∞
of the induced metric on the boundary. Precisely at r =r (20) divergesand therefore must be regularized
∞
adding the intrinsic boundary counter-term
1 V
S˜ = dxdy√ g = r2√Mσ (21)
ct κ2(4) Z − ∞ κ2(4) h ir=r∞
Finally the dual thermodynamic potential Ω results
Ω = lim S˜
on−shell
r∞→∞
= lim (S˜ +S˜ +S˜ ) (22)
bulk GH ct
r∞→∞
Upon regularizing the action the potential Ω results to coincide with the sub-leading value of the g
tt
component of the background metric i.e. Ω = Mb [4]. We have verified our numerical solution computing
1
Ω in both ways finding an excellent agreement. In figure 5 we plot the potential (22) as function of the
temperature. AswementionedaboveasecondorderphasetransitiondevelopsatT =T : thegrandpotential
c
and the entropy are continuous but S is not differentiable. Below T the system is in the superconducting
c
phase,asweincreasethetemperatureaboveT the AdSRNgeometrydominatesthe freeenergy,thismodels
c
a transition from a superconducting to a normal phase.
7
2.2 p+ip wave superconductors
Here, we review the results of [11] and compare them with the results of the previous section. We will find
that at T =T the system has a second order phase transition and for all ranges of temperatures the grand
c
potential of the p-wave solutionfound in previous section is lower than that of the p+ip, implying that the
stable phase of the system is the p-wave phase in accordance with the stability analysis [6].
2.2.1 Solution
The backgroundand gauge field ansatz for model a p+ip-wave solution are
dr2
ds2 = M(r)dt2+r2h(r)2(dx2+dy2)+ (23)
− M(r)
A = φ(r)τ3dt+ω(r)(τ1dx+τ2dy). (24)
One important difference with the p-wave superconductor of the previous section arises in the choice of the
gauge field ansatz that now breaks the U(1) U(1) into a diagonal combination. The p-wave case fully
3 xy
×
breaks the U(1) U(1) . This allows us to use a metric ansatz that is totationally symmetric in the
3 xy
×
xy-plane.
The equations of motion obtained for this ansatz are four second order differential equation plus a first
order constraint arising from the rr component of the Einstein equations
h 1 3 M′ φ′2 ω′2 h′ 6 h′ M′ g2 ω2 φ2 ω2
h′′ = + + + + + YM +
−2 r2 − Rˆ2M rM 8M 4r2h2 − 2 r h M − 8r2hM M 2r2h2
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
3 M M′ 1 ω′2 h′ h′ 2 3 g2 ω2 φ2 3ω2
M′′ = + + + M′ + φ′2+ YM +
Rˆ2 r −M r 4rh2 − h − h − r 8 4r2h2 M 2r2h2
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
g2 ω ω2 φ2 M′ω′
ω′′ = YM
M r2h2 − M − M
(cid:20) (cid:21)
2g2 φω2 1 h′
φ′′ = YM 2φ′ +
r2h2M − r h
(cid:20) (cid:21)
3 M ω′2 M′ h′ 2 h′ 1
0 = + 1 + r + M + +M′ + φ′2.
−Rˆ2 r2 − 4h2 M h r h 8
(cid:20) (cid:21) (cid:20) (cid:18) (cid:19) (cid:21)
The equations have three scaling symmetries that will help us to numerically solve the system. They are
1. ω λω, h λh
→ →
2. M λ−2M, φ φ, Rˆ λRˆ, g gYM
→ → λ → YM → λ
3. M λ2M, h h, φ λφ, r λr
→ → λ → →
and allows us to set R = r = 1 and the value of h(r) at the boundary to h( ) = 1. The IR behavior of
h
∞
8
these equations are those of a charged black hole
M = M (r r )+M (r r )2+...
1 h 2 h
− −
h = h +h (r r )+h (r r )2+...
0 1 h 2 h
− −
ω = ω +ω (r r )+ω (r r )2+...
0 1 h 2 h
− −
φ = φ (r r )+φ (r r )2+... (25)
1 h 2 h
− −
where as before we impose the Maxwell potential φ to vanish at the horizon in order to have a well defined
gauge field in the Euclidean continuation. On the UV we demand
Mb 8hbMb+ρ2+2(ωb)/3
M = r2+2hbr+(hb)2+ 1 + − 1 1 1 +...
1 1 r 8r2
hb (ωb)2
h = 1+ 1 1 +...
r − 48r4
ωb hbωb
ω = 1 1 1 +...
r − r2
ρ ρhb
φ = µ+ 1 +... (26)
r − r2
Note that for SSB we do not allow for a non-normalizable piece in ω. As before the scaling symmetries (3)
allowsto fix hb =1. Infigure6weplotthe behaviorofthe solutionsandfigure7 showsthe orderparameter
0
J1 > ωb as function of the temperature. For T =T both condensates vanish and a second order phase
h xi ∝ 1 c
transition onsets. Note that the values of the condensate for the p+ip case are lower than those in the
p-wave case.
Ω1b
Μ2
0.6
M HrL 0.5
0.10 25
104hHrL
0.4
0.08 107ΩHrL
ΦHrL 0.3
0.06
0.2
0.04
0.1
0.02
T
0.0
r 0.10 0.15 0.20 Μ
1.5 2.0 2.5 3.0 3.5 4.0
Figure7: ThedualtheoryVEV J1 ωb asafunc-
Figure 6: Behavior of the dimensionless functions h xi∝ 1
tion of temperature for the case of the p-wave (blue
M(r),h(r),ω(r)andφ(r),plottedforg =2,T =
YM line) and colorful (orange line) superconductors for
0.2312µ.
g = 2. Its vanishing for T > T = 0.2312, sug-
YM c
gestingaphasetransitionbetweenasuperconducting
and a normal state.
9