Spatially modulated phases in AdS/CFT Aristomenis Donos Imperial - - PowerPoint PPT Presentation

Spatially modulated phases in AdS/CFT

Aristomenis Donos

Imperial College London Talk at U. of Crete March, 2012

Based on work with J. P. Gauntlett

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

Motivation

The AdS/CFT correspondence is a powerful tool to study strongly coupled (conformal) quantum field theories Interest in application to strongly coupled Condensed Matter Theory systems

One focus: systems with strongly coupled “quantum critical points” - phase transition at zero temperature Another focus: thermally driven symmetry breaking phase transitions e.g. superconductivity / superfluidity

[Gubser; Hartnoll, Herzog, Horowitz]

⇒ What about spatially modulated order? e.g. spin density waves, charge density waves, stripe phase of underdoped cup rate superconductors, FFLO [Nakamura, Ooguri, Park] [AD,

Gauntlett] [Bergman, Jokela, Lifschytz, Lippert]

Normal/broken phase transitions

Normal phase Critical points exhibiting full relativistic conformal invariance could be described by AdS geometries in string or M-theory The boundary field theory at finite temperate is described by black hole (black brane) solutions asymptoting to AdS Finite chemical potential would correspond to a charged black hole (black brane) with the charge carried by a bulk gauge field Phase transition Certain fields can condense due to an instability, at a critical temperature Tc, spontaneously breaking a space-time (density waves, nematic phases) and/or internal symmetry (superfluidity) Emergence of new black hole branch

Bottom-up approach

Study phase transitions at a minimal setting The AdS Reissner-Nordström black hole is the canonical example of charged black hole in Einstein-Maxwell theory to play the role of a normal phase Couple additional fields which become unstable below a critical black hole temperature Tc Advantages Discover new mechanisms for instabilities/condensation Uncovers universal behaviour close to Tc Disadvantages Dual field theory existence not guaranteed Low temperature behavior model dependent

Top-down approach

Consider AdSd × M solution of string/M-theory Compactify on M to generate an infinite tower of KK modes in d dimensions For SUSY compactifications in d = 4, 5 there is a consistent truncation with at least a SUGRA multiplet (gµν, Aµ) → electric RN black hole [Gauntlett, Varela] Also possible to retain additional fields (multiplets) in the consistent truncation e.g. AdS4 × SE 7, AdS5 × SE 5, AdS5 × H2 × S4, AdS4 × H3 × S4 Advantages Guaranteed to have a field theory dual Couplings, scalar potentials are not arbitrary Disadvantages Hard Branches/instabilities outside consistent truncation

[AD, Gauntlett]

Plan

Electric AdS RN black hole (normal phase) solution in string/M-theory Consider perturbative coupling of minimal N = 2 SUGRA in D = 4, 5 to additional fields (multiplets) Study stability of normal phase against perturbations Embed the mechanism in known string/M-theory reductions ❀ SUGRA couplings can in general break translational invariance! ❀ Rich structure of competing orders in string/M-theory

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

Electric AdS4 RN black hole

The Einstein-Maxwell in d = 4 is LEM = 1 2R ∗ 1 + 6 ∗ 1 − 1 2 F ∧ ∗F The electrically charged AdS RN black hole is ds2

4 = −f dt2 + dr2

f + r2 dx2

1 + dx2 2

• A = µ
• 1 − r+

r

• dt

f = 2r2 −

• 2r2

+ + µ2

2 r+ r + µ2 r2

+

2r2

• There is an outer horizon located at r = r+
• Temperature is T = (12 r2

+ − µ2)/(8πr+), entropy is

s = 2πr2

+

⇒ Finite entropy at T = 0

The extremal limit

At T = 0 the near horizon (IR) limit is AdS2 × R2 → 1 dim.

• r the chiral sector of 1 + 1 dim. CFT

The R2 Fourier modes of bulk fields yield a continuum of dual

• perators O

k in the IR CFT

The modes k = 0 break translations in the UV CFT Check unitarity (BF) bound of the IR CFT for all k If for high T RN bh is stable and the IR CFT is unstable, there must be a Tc for the onset of the instability

s-wave superfuids

A prototype instability The AdS2 × R2 limit is (after rescaling) ds2 = −12r2 dt2 + dr2 12r2 + dx2

1 + dx2 2,

A = 2 √ 3 rdt Add minimally coupled complex scalar L = LEM − 1 2

• Dµψ
• 2 − 1

2m2 |ψ|2 Dµψ = (∂µ − ıq Aµ)ψ For modes ψ = φ eı

k x the equation of motion gives

DµD∗

µφ −

k2φ − m2φ = 0 ⇒ m2

eff = −q2 + m2 +

k2 Violates AdS2 BF bound if m2

eff < −3 but lightest mode always at

• k = 0

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

The model

Consider the d = 4 theory of gravity gµν coupled to a gauge field Aµ and a pseudo scalar ϕ

L = 1 2R ∗ 1 − 1 2 ∗ d ϕ ∧ d ϕ − V (ϕ) ∗ 1 − 1 2τ(ϕ)F ∧ ∗F − 1 2ϑ(ϕ)F ∧ F

For perturbative considerations, we are interested in the first few terms

V = −6 + 1 2m2

s ϕ2 + · · · ,

τ = 1 − n 12 ϕ2 + · · · , ϑ = c1 2 √ 3 ϕ + · · ·

Purely electric/magnetic RN black hole still solution d ∗ d ϕ + V ′ ∗ 1 + 1 2τ′F ∧ ∗F + 1 2ϑ′F ∧ F =0 d(τ ∗ F + ϑF) =0

The AdS2 perturbation

Einstein’s equations read Rµν = ∂µϕ∂νϕ − τ 1 4gµνFλρF λρ − FµρFνρ

• + gµνV

In general, perturbations of the gauge field will mix with metric perturbations δAx2 =a(t, r) sin(kx1) δϕ =w(t, r) cos(kx1) δgtx2 =2 √ 3 rhtx2(t, r) sin(kx1) δgx1x2 =hx1x2(t, r) cos(kx1) The function hx1x2 can be eliminated from the equations of motion to yield a second order system

The instability

The second order system in matrix form is ✷AdS2v − M2v = 0, v = (φx1x2, a, w) M2 =    k2

1 √ 3k

24 √ 3k 24 + k2 −c1k −c1k k2 + ˜ m2

s

   , ˜ m2

s = m2 s + n

The mass spectrum is found after diagonalizing to give three AdS2 masses as functions of k The lowest mass matrix eigenvalue m2

min occurs at non-zero k

for sufficiently large c1 It can violate the AdS2 BF bound even with a stable k = 0 sector M-theory embedding Consider skew-whiffed AdS4 × SE 7 and dimensionally reduce to

• btain N = 2 minimal gauged SUGRA coupled to 1 vector mult.

Black hole static modes

To determine Tc we need to construct a static, normalizable mode The AdS2 analysis suggests the perturbation δgtx2 = λ [r (r − r+) h(r) sin(kx1)] δAx2 = λ [a(r) sin(kx1)] δϕ = λ [w(r) sin(kx1)] , λ << 1 The equations of motion lead to a system of three second

• rder ODEs for h, a, and w

Demanding regular perturbation near the black hole horizon leads to the expansion

h = h+ + O(r − r+), a = a+ + O(r − r+), w = w+ + O(r − r+)

With the system being linear we can always choose one of the constants of integration to be equal to one

Asymptotic boundary conditions

In general two constants of integration at infinity h = h0 + · · · + h3 1 r3 + · · · a = a0 + · · · + a1 1 r + · · · w = w1 1 r + · · · + w2 1 r2 + · · · The constant h0 would correspond to a boost “chemical potential” and a0 to a current chemical potential. For spontaneous symmetry breaking we set a0 = h0 = 0 We assumed that m2

s = −4. We choose ∆

Oφ = 2 and set w1 = 0 For a fixed wavenumber k we have a total of 6 free variables 2 (horizon)+ 3 (infinity)+ T ⇒ One solution (Discreet more precisely!)

Fix AdS4 mass m2

s = −4

• 10

20 30 m

• s

2

2 4 6 8 10 c1

0.8 1.0 1.2 1.4 1.6 1.8 2.0 k 0.01 0.02 0.03 0.04 T 0.8 1.0 1.2 1.4 1.6 k 0.002 0.004 0.006 0.008 0.010 0.012 0.014 T

Questions for perturbation theory Higher order perturbative analysis shows that modulated black hole branches exist Can be used to study thermodynamics close to Tc ❀ In general they are continuous transitions (second order) Would still like to ask Transport properties of modulated phases? What is the low temperature behaviour? Modulation persists at low temps? If yes, new emergent IR with modulation? ❀ Easier to answer in some 5D models

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

Helical superconductors

Spatially modulated phases exist in holography Superconducting phases exist in holography Combination of the two? FFLO phase (1964) - a variation of BCS: Construct a situation in which the Fermi momentum of spin up quasi-particles is not equal to that of spin down. If they form Cooper pairs they will have net momentum leading to a spatially modulated superconducting order parameter Perhaps seen in some heavy fermions (eg CeCOIn5) and some

• rganic superconductors

Holographic superfluids

Ingredients: theory of gravity with gauge field which is dual to a conformal field theory with global U(1) symmetry. In addition we need charged fields in the bulk which provide the

• rder parameter for the superconductivity in the dual CFT

s-wave superconductors have l = 0 order parameter → use charged bulk scalar fields. [Gubser][Hartnoll, Herzog, Horowitz] p-wave superconductors have l = 1 order parameter Seen in eg He3, heavy fermions, organics, Sr2RuO4 In D = 4, 5 use SU(2) gauge fields.Take the background to be charged with respect to U(1) ⊂ SU(2) [Gubser] and then spontaneously break the U(1) In D = 5 use a charged first-order two-form

[Aprile,Franco,Rodriguez,Russo]

The charged 2-form model

Consider the D = 5 model L = (R + 12) ∗ 1 − 1 2 ∗ F ∧ F − 1 2 ∗ C ∧ ¯ C − ı 2mC ∧ ¯ H F = dA, H = dC + ı qe √ 3 A ∧ C It is a consistent truncation of D = 5, N = 8 gauged SUGRA for qe = 1, m = 1 The electric RN black brane is a solution with C = 0 To analyze stability of C, simply examine its equation of motion on AdS2 × R3/RN bh background ∗H = ımC and is dual to a charge qe, d = 4 self-dual tensor operator with ∆ (OC) = 4 + m

2-form perturbations

Consistency of the 2-form equation of motion requires to consider δC = dr ∧ (u1 dx1 − v1 dx2) + ı dt ∧ (u2 dx1 − v2 dx2) +dx3 ∧ (u3 dx2 + v3 dx1) Two independent cases

• ui = di cos(kx3),

vi = di sin(kx3), p-wave

• ui = di eıkx3,

vi = ıdi eıkx3, p + ıp-wave Solve for d1 and d2 in terms of d3 The component d3 satisfies a second order equation For the AdS2 × R3 background

• D2 − L2

m2 + k2 + kq 3 √ 2m

• d3 = 0,

D = ∇ + ıq √ 3 A ❀ m2

eff = L2

k2 + m2 − kq 3 √ 2m − q2 18

The order parameter is δC = · · · − d3 [cos(kx3) dx2 + sin(kx3) dx1] ∧ dx3 A helical px-wave structure. Also possible to describe a helical px + ıpy-wave

Static normalizable modes

For fixed chemical potential and m = 1 construct static spontaneous symmetry breaking modes for q = 2, 1.8, 1.7, 1.5

0.5 1.0 1.5 2.0 2.5 3.0 3.5k 0.01 0.02 0.03 0.04 T

The k = 0 mode becomes quantum critical for charge q = 1.8

[Aprile,Franco,Rodriguez,Russo]

The string/M-theory consistent truncation has q = 1 → low temps, numerical error

Helical superconducting black holes

The linearised static mode takes the form δC = (ıc1(r) dt + c2(r) dr) ∧ ω2 + c3 ω1 ∧ ω3 ω1 = dx1 ω2 = cos(kx1) dx2 − sin(kx1) dx3 ω3 = sin(kx1) dx2 + cos(kx1) dx3 Key observation in constructing the back reacted geometries is that the unstable mode preserves the Bianchi VII0 group L1 = ∂x1 + k (x3∂x2 − x2∂x3) L2 = ∂x2 L3 = ∂x3 ❀ LLi ωj = 0 ⇒ LLi δC = 0 Back-reaction should respect this symmetry

Helical superconducting black holes

Non-linear ansatz ds2 = −gf 2 dt2 + g −1 dr2 + h2 ω2

1 + r2

e2α ω2

2 + e−2α ω2 3

• C = (ı c1 dt + c2 dr) ∧ ω2 + c3 ω1 ∧ ω3

A = a dt Where g, f , h, α, ci and a are only functions of r and we also used the Bianchi VII0 invariant one-forms ω1 = dx1 ω2 = cos(kx1) dx2 − sin(kx1) dx3 ω3 = sin(kx1) dx2 + cos(kx1) dx3 Plugging in the 5D equations of motion yields a consistent system of non-linear ODEs with k a parameter

Helical superconducting black holes

The RN black hole At high temperatures we only find the RN black brane solution (normal phase) f = 1, h = r, α = 0, ci = 0 a = µ

• 1 − r2

+

r2

• , g = r2 − r4

+

r2 + µ2 3 r4

+

r4 − r2

+

r2

• with the 5D fields reading

ds2 = −g dt2 + g −1 dr2 + r2 dx2

1 + dx2 2 + dx2 3

• A = a dt,

C = 0

Helical superconducting black holes

Modulated black hole phase Starting at Tc and finite k we find new black hole branches Use shooting method to solve ODEs Demand regularity on the horizon r = r+ g(r+) = a(r+) = 0 with all functions having an analytic expansion at r+ Demand expansion at infinity appropriate for spontaneous symmetry breaking g = r2 1 − M r −4 + · · ·

• ,

f = 1 − ch r −4 + · · · h = r

• 1 + ch r −4

, α = cα r −4 + · · · a = µ + q r −2 + · · · , c3 = cv r −|m| + · · ·

Helical superconducting black holes

Integration constants at infinity associated with physical quantities of the dual field theory The vev of the superconducting order parameter is determined by both cv and k O (k)C ∝ cv In addition we have charge density J0 ∝ q

Helical superconducting black holes

Holographic renormalisation reveals the stress tensor Ttt = 3M + 8ch Tx1x1 = M + 8ch Tx2x2 = M + 8cα cos (2kx1) Tx3x3 = M − 8cα cos (2kx1) Tx2x3 = 8cα sin (2kx1) ❀ Spatially modulated pressure and shear in the x2 − x3 plane Free energy density w vol3 ≡ T [ITot]OS with w = wk (T, µ) = −M

Helical superconducting black holes

A parameter count reveals that, for fixed µ, there exists a two parameter family of black holes labeled by T and k. For fixed T and µ minimise the free energy with respect to k Free energy minimised on the red curve All black holes have cv = 0 and hence are helical superconductors All black holes have whelical < wRN

Helical superconducting black holes

For the thermodynamically preferred k(T)

• 0.0

0.4 0.8 1.2 1.6 2.0 2.4 2.8 102T 0.110 0.105 0.100 0.095 0.090 0.085 w

• 0.4

0.8 1.2 1.6 2.0 2.4 102 T 3 4 5 10 k

• 0.0

0.4 0.8 1.2 1.6 2.0 2.4 102 T 4 8 12 16 20 102 cv

Helical superconducting black holes

Zero temperature limit As T → 0 our black hole solutions approach smooth domain walls interpolating between AdS5 in the UV and a new scaling solution in the IR ds2 = −r2z dt2 + r −2 dr2 + h2

0 ω2 1 + r2

e2α0 ω2

2 + e−2α0 ω2 3

• C =
• ı c1(0) rz+1 dt + c2(0) dr
• ∧ ω2 + c3(0)r ω ∧ ω3

A = a0rz dt z, h0, α0, ci(0), a0 are constants Scaling t → λzt, x2,3 → λx2,3, x1 → x1, r → λ−1r

Outline

1 Motivation / Introduction 2 A diagnostic for BH perturbative instabilities 3 Holographic stripes 4 Helical superconductors

Helical superconducting black holes

5 Final comments

Final comments

Studied instabilities of electrically charged black holes leading to a wide range of spatially modulated black holes Constructed the first black holes dual to spatially modulated phases → helical p-wave superconducting order, novel scaling symmetry in the IR For the values of m and qe we looked at the IR solutions has real scaling dimensions. For other values they can be complex. Stable? Interesting to calculate transport coefficients using linear response they We looked at p-wave order. There is also p + ıp wave order. Which one wins?

Final comments

Other examples:

D = 5 Einstein-Maxwell Chern-Simons : similar story (to appear) D = 5 with SU(2) × U(1): expect a similar story (in progress) Example in D = 4 with axion and gauge-field would involve solving PDEs Modulated instabilities of magnetic branes

Rich set of examples with couplings natural in string/M-theory. Are they generic ground states of deformed CFTs?