QMUL-PH-22-31 What lies beyond the horizon of a holographic p-wave superconductor Lewis Sword and David Vegh

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QMUL-PH-22-31

What lies beyond the horizon of a holographic p-wave superconductor

Lewis Sword and David Vegh

Centre for Theoretical Physics, Department of Physics and Astronomy

Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK

Email: l.sword@qmul.ac.uk,d.vegh@qmul.ac.uk

October 12, 2022

Abstract

We study the planar anti-de Sitter black hole in the p-wave holographic supercon-

ductor model. We identify a critical coupling value which determines the type of phase

transition. Beyond the horizon, at speciﬁc temperatures ﬂat spacetime emerges. Numer-

ical analysis close to these temperatures demonstrates the appearance of a large number

of alternating Kasner epochs.

Contents

1 Introduction 2

2 Holographic p-wave superconductor 3

2.1 Action and equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Boundaryconditions................................ 4

3 Thermodynamics 6

3.1 Phasetransitions.................................. 6

3.2 Grand potential and entropy analysis . . . . . . . . . . . . . . . . . . . . . . . 8

4 Black hole interior 14

4.1 Josephsonoscillations ............................... 14

4.2 Kasnerregime.................................... 15

4.3 Near-oscillatory Kasner epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Conclusion 22

arXiv:2210.01046v2 [hep-th] 11 Oct 2022

1 Introduction

The AdS/CFT correspondence [1, 2, 3], otherwise known as gauge-gravity duality, intro-

duced a method to inspect strongly coupled theories. From it emerged a dictionary relating

ﬁelds in bulk spacetime to operators on its boundary. Using this correspondence, the grav-

itational dual of a superconductor, known as a holographic superconductor, was discovered

[4, 5, 6]. By spontaneously breaking an abelian gauge symmetry of a charged scalar ﬁeld

in Schwarzschild-AdS spacetime, one produces a non-trivial expectation value for the scalar

ﬁeld, which corresponds to a non-zero condensate developing on the boundary1. These origi-

nal models used a simple U(1) charged scalar boson to introduce a condensate (scalar “hair”),

however other bosonic condensate models are possible such as p-wave superconductors where

a charged vector ﬁeld is employed utilising SU(2) gauge theory2. This takes a U(1) sub-

group as electromagnetism and allows the gauge bosons that are charged under the U(1) to

condense. The original p-wave model was presented in [8] followed by a top-down, string

theory approach in [9, 10, 11] and backreaction on the metric was accounted for in [12, 13].

In both vector and scalar condensate cases, the identiﬁcation of the gravitational counterparts

of superconductors is attributed to a condensing ﬁeld and a black hole spacetime with a given

Hawking temperature in the bulk. This topic has garnered great interest since its inception.

The exploration of the black hole interior began with [14, 15] and was extended to the full

scalar ﬁeld holographic superconductor model in [16]. Numerous interesting phenomena

were observed in the interior including Kasner geometries3, collapse of the Einstein-Rosen

(ER) bridge and Josephson oscillations. The interior has subsequently been subjected to

further study. This includes the introduction of additional ﬁeld content and variation of

coupling parameters [19, 20, 21, 22, 23], use of alternative black hole solutions [24, 25, 26],

analysis of RG ﬂows [27, 28], as well as the construction of “no inner-horizon” theorems [29].

Investigation of the interior solutions for the p-wave model are now also being explored, with

the analogous changes in geometry and matter ﬁelds being observed [30].

This paper aims to show the interesting changes in interior geometry by exploring the pa-

rameter space of the p-wave superconductor. The key result is that for a special selection

of parameters, the numerical solutions appear to imply that the interior geometry becomes

ﬂat. Not only that, but either side of this parameter selection, the geometry becomes almost

oscillatory in Kasner universes.

The outline of the paper is as follows. Section 2 introduces the holographic p-wave super-

conductor model. Here the equations of motion are established, followed by details of the

numerical procedure to solve them. We also state the equations’ scaling symmetries, as well

as the horizon and UV series expansions as governed by the necessary boundary conditions.

Section 3 puts the numerical solutions to use, by exploring the exterior of the black hole. We

ﬁrst analyse the ﬁeld content in the exterior conﬁrming that the solutions satisfy the correct

boundary conditions. Under vanishing condensate, we enter normal phase as veriﬁed by the

metric returning to that of a Reissner-Nordstr¨om spacetime. Following this, upon various

choices of the model parameters, the phase diagrams for the holographic superconductor are

produced. The phase curves imply that a critical gYM coupling exists, which diﬀerentiates

between ﬁrst and second order transitions. This is conﬁrmed by analysis of the grand poten-

tial derived from the Euclidean action as well as the entropy. Section 4 explores the interior

1On the boundary the U(1) symmetry that is spontaneously broken in the superconducting phase is a

global symmetry, hence is more accurately described as a superﬂuid.

2In addition to the scalar and vector condensates, d-wave superconductors based on spin-2 condensates

have also been the subject of similar holographic analysis [7].

3In the context of cosmology, Kasner regimes have also been investigated in [17, 18].

and presents our main ﬁndings. We begin by studying the interior ﬁeld content close to the

horizon, which demonstrates typical behaviour previously seen such as the Josephson oscilla-

tions and ER bridge collapse. The focus then turns to speciﬁc points in the parameter space.

Here the key ﬁnding is that at certain values the interior geometry appears to become ﬂat

while slight deviations away from this point lead to a highly oscillatory geometry comprised

of individual Kasner universes for a given bulk radius. Section 5 presents a summary of our

ﬁndings and discusses possible future endeavours.

At the time of review, it has come to our attention that [31] discusses inﬁnite oscillations in

a scalar ﬁeld model and [32] presents results for transitions between diﬀerent Kasner epochs

using a top-down, scalar holographic superconductor model.

2 Holographic p-wave superconductor

2.1 Action and equations of motion

The model used is a (3 + 1)-dimensional, SU (2) Yang-Mills theory with Einstein-Hilbert

and cosmological constant terms allowing us to obtain asymptotically anti-de Sitter (AdS)

geometry. The action is

I=1

κ2

(4) Zd3+1x√−gR−2Λ −1

4Tr [Fµν Fµν ]=Zd3+1x√−gL,(1)

with Lagrangian Land ﬁeld strength

µν =∇µAa

ν− ∇νAa

µ+gYMabcAb

µAc

ν.(2)

Here, Ris the Ricci scalar, Λ = −3/L2is the cosmological constant with Lthe radius of

curvature of AdS, gis the determinant of the metric, κ(4) is the four-dimensional gravitational

constant, gYM = ˆgYM/κ(4) where ˆgYM is the standard Yang-Mills coupling and abc is the Levi-

Civita symbol. Aa

µrepresent the Lie-algebra valued gauge ﬁelds, deﬁned in form notation as

A=Aa

µτadxµwhere τaare the generators of the SU (2) algebra deﬁned by the Pauli matrices,

σa, as τa=σa/2i. In the above, “Tr” refers to the trace over Lie indices and in the convention

used, Tr[τaτb] = 1

2δab. For example 1

4Tr [Fµν Fµν ] = 1

8PaFa

µν Faµν , where for SU(2), the

three generators are denoted by a= 1,2,3 and satisfy Lie bracket [τa, τb] = abcτc. Under

the identiﬁcation of ˆ

Aµ=Aµ/gYM we may think of gYM as a measure of the backreaction:

for large gYM we enter the probe limit. This limit essentially scales away the eﬀect of the

gauge ﬁeld such that it has negligible contributions to the gravitational equations of motion.

Varying the action of (1) with respect to the metric, gµν , and the Yang-Mills gauge ﬁeld,

Aµ, the resulting equations of motion are

Rµν −1

2gµν R+ Λgµν =Tµν ,(3)

DµFµν = 0 ,(4)

where

Tµν =1

2Tr [Fµγ Fνγ]−1

8gµν Tr [FγρFγρ].(5)

Here we deﬁne the gauge covariant derivative as

Dµ=∇µ+gYM[Aµ,·].(6)

In explicit Lie index form, equation (4) is

DµFaµν =∇µFaµν +gYMabcAb

µFcµν = 0.(7)

In order to solve the equations of motion, we adopt the following radial direction (labelled

coordinate z) ﬁeld ans¨atze for the gauge ﬁeld

A=Aa

µτadxµ=A3

tτ3dt +A1

xτ1dx =φ(z)τ3dt +ω(z)τ1dx , (8)

and also the metric

ds2=1

z2−f(z)e−χ(z)dt2+1

f(z)dz2+h(z)2dx2+1

h(z)2dy2.(9)

Inserting these ans¨atze into equations (3) and (4) produces ﬁve individual equations of motion

for the ﬁelds

φ00 =ω2φg2

fh2−χ0φ0

2(10)

ω00 =−eχωφ2g2

f2−f0ω0

f+2h0ω0

h+χ0ω0

2(11)

h00 =eχω2z2φ2g2

8f2h−f0h0

f+h0χ0

2+2h0

z+(h0)2

h−z2(ω0)2

8h(12)

f0=eχω2z3φ2g2

8fh2+fz3(ω0)2

8h2+fz (h0)2

h2+3f

z−3

L2z+1

8eχz3φ02(13)

χ0=eχω2z3φ2g2

8f2h2+f0

f+3

fL2z−eχz3(φ0)2

8f+z3(ω0)2

8h2+z(h0)2

h2−3

z(14)

A non-trivial ω(z) proﬁle is responsible for introducing the condensate, hJx

1i, to the model

since it breaks the U(1) subgroup symmetry associated to rotations around τ3. In other

words, a non-zero ω(z) picks out the xdirection as special and breaks the rotational symmetry

in the x−yplane. Naturally, the metric function h(z) accounts for this symmetry breaking in

the dual description. The chemical potential is associated with the U(1) symmetry generated

by τ3.φ(z) can be thought as the ﬁeld dual to the chemical potential, appearing as the ﬁeld

charged under this U(1) symmetry [8, 12, 13].

2.2 Boundary conditions

The equations of motion using this ansatz enjoy the following scaling symmetries

L→α1L , f →1

α2

f , gYM →1

α1

gYM , eχ→1

α2

eχ.(15a)

z→α3z , ω →1

α3

ω , φ →1

α3

φ . (15b)

eχ→α2

2eχ, φ →1

α2

φ . (15c)

ω→α4ω, h →α4h . (15d)

The symmetries (15a) and (15b) allow us to take zh=L= 1. The others also allow us to scale

the χand hﬁelds such that they take their necessary boundary values: χ(z= 0) = 0 and

h(z= 0) = 1. At the boundary we return to AdS spacetime (i.e. we have an asymptotically

AdS bulk spacetime) which deﬁnes these conditions.

The procedure of numerically obtaining the ﬁeld solutions from the equations of motion

begins by producing series expansions of the ﬁelds at the horizon, z=zh= 1, and at the

UV boundary, z= 0. To be precise we only integrate up to a small cut-oﬀ value of zfor the

UV solutions, denoted as . Analogously we integrate to z= 1 −δat the horizon, for small

δ. The horizon series takes the following form

f=fh1(z−zh) + fh2(z−zh)2+. . .

χ=χh0+χh1(z−zh) + χh2(z−zh)2+. . .

h=hh0+hh2(z−zh)2+. . . (16)

ω=ωh0+ωh2(z−zh)2+ωh3(z−zh)3+. . .

φ=φh1(z−zh) + φh2(z−zh)2+. . .

Here fvanishes at z=zhby deﬁnition of the black hole event horizon, as does φto ensure

we have a ﬁnite norm of the gauge ﬁeld strength squared. Substituting these series solutions

into the equations of motion and solving order by order, the ﬁeld functions are completely

determined by four parameters at the horizon: φh1,ωh0,hh0,χh0. Using symmetry (15c),

we choose χh0= 1 throughout and rescale the necessary quantities when required, to ensure

χ(z= 0) = 0 i.e. we set α2=e−χb0/2with χb0deﬁned below in (17b). Repeating the same

idea for the UV boundary expansion around z= 0, we ﬁnd

f= 1 + fb3z3+O(z4) (17a)

χ=χb0+O(z4) (17b)

h=hb0+hb3z3+O(z4) (17c)

ω=ωb0+ωb1z+O(z2) (17d)

φ=φb0+φb1z+O(z2) (17e)

All higher order terms in zhave coeﬃcients that are constructed from the eight UV coeﬃcient

parameters listed in equations (17a)-(17e).

With the χscaling symmetry allowing us to set χh0= 1, we now look to set the horizon

parameters φh1and hh0such that we are left with a one-dimensional parameter space of

solutions to explore, those being controlled by ωh0. This requires two additional conditions.

First, we require a vanishing source of the ωﬁeld, and in the chosen quantisation provided

by the AdS/CFT correspondence, this corresponds to ωb0= 0 from the UV expansion. The

correspondence also implies that the expectation value of the dual operator is identiﬁed as

hJx

1i=ωb1. This is our condensate. Secondly, since we return to AdS spacetime at the

boundary cut-oﬀ, we require that the anisotropy function h(z) become unity there4. These

two conditions serve as shooting parameters and root ﬁnding algorithms in Mathematica [33]

for example, readily produce solutions. Additionally, the AdS/CFT dictionary states that

the chemical potential, µ, and charge density, ρ, are identiﬁed with coeﬃcients of the φUV

expansion such that µ=φb0,ρ=φb1.

To establish where the condensate becomes non-trivial, the temperature must be deﬁned.

This is the Hawking temperature of the black hole described by equation (9) and can be

4In our numerical practice, we make use of the χsymmetry equation but not the hsymmetry equation,

instead choosing to “shoot” for the hfunction’s boundary value.

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QMUL-PH-22-31Whatliesbeyondthehorizonofaholographicp-wavesuperconductorLewisSwordandDavidVeghCentreforTheoreticalPhysics,DepartmentofPhysicsandAstronomyQueenMaryUniversityofLondon,327MileEndRoad,LondonE14NS,UKEmail:l.sword@qmul.ac.uk,d.vegh@qmul.ac.ukOctober12,2022AbstractWestudytheplanaranti-deSitt...

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