Spatially Modulated Phases in Holography

Jerome GauntlettAristomenis Donos

Christiana Pantelidou

Spatially Modulated Phases

• In condensed matter there is a variety of phases that are spatially modulated, spontaneously breaking translation invariance. For example: charge density waves and spin density waves

• The modulation is fixed by an order parameter associated with non-zero momentum .

• The modulation can be in various configurations such as stripes (as in the cuprates), in checkerboards and hexagonal lattices. Helical structure in chiral nematic liquid crystals and helimagnets e.g.

• There is a growing set of holographic examples described by exotic black holes (probe brane studies initiated by [Domokos,Harvey])

�O(k)� �= 0

MnSi

• D=5 helical current phases [Nakamura,Ooguri,Park][Donos,Gauntlett]

• D=5 helical superconducting phases [Donos,Gauntlett]

• D=4 stripes [Donos,Gauntlett][Donos][Withers][Rozali,Smyth,Sorkin,Stang]

x3

x1

�Jx1� = 0

�Jx2� ∝ cos(kx1)

�Jx3� ∝ sin(kx1)

x2

�Jt� = const

�Jx� = 0�Jt� − Jt ∝ cos(2kx)�O� ∝ cos kx

�Jy� ∝ sin kx

• Instabilities of unbroken phase black holes driven by mass squared becoming sufficiently negative

• For a single mode spatially modulation is suppressed

ω2 = mass2 + k2

k

Tc

• However, instabilities for non-zero k can easily be achieved if there is a mixing of modes since one diagonalises a mass matrix

• The above examples achieve this via Chern Simons couplings and axions and one finds

Tc

kkc

• These examples all spontaneously break P and T

• Not necessary: there are D=4,5 black holes dual CDWs that don’t break P and T [Donos,Gauntlett]

• The above examples are for CFTS at . There are also examples of spatially modulated phases in uniform magnetic fields [Bolognesi,Tong][Donos,Gauntlett,Pantelidou][Jokela,Lifschytz,Lippert][Cremonini,Sinkovics][Almuhairi,Polchinski][Ammon,Erdmenger,Shock,Strydom]....

• Closely related examples - but distinct physics - where translations broken explicitly [.....][Horowitz,Santos,Tong][Donos,Hartnoll][Vegh][Erdmenger,Ge,Pang][...]

• Many more examples of spatially modulated black holes to be found

• Finding them is one way of finding new ground states

T, µ

• General results on the thermodynamics of periodic AdS black branes

• Competing superconducting p-wave and (p+ip)-wave orders

Plan

Thermodynamics of Periodic AdS Black Branes

with Aristomenis Donos

Complementary work by [Domokos,Hoyos,Sonnenschien]

• Consider asymptotic AdS black holes with planar topology.

• We are interested in solutions that are periodic in the spatial directions

Assume the black holes have a Killing horizon, dual to configurations in thermal equilibrium

Some of the configurations might be translationally invariant, independent of some or all of the spatial coordinates

If the translation symmetry is broken it can either be spontaneous orexplicit

• For simplicity consider bulk theory with metric and gauge-field g A

gmn ↔ Tµν Am ↔ Jµ

Behaviour at the AdS boundary:

ds2 → γµνdxµdxν

• Bulk black brane solution is periodic in the globally defined with period . Might be compact or non-compact

xi

Li

• The black hole has a Killing horizon and the Killing vector on the boundary

xi

→ ∂t

The solutions might be independent of some

If solutions depend on the symmetry breaking can either be explicit if the sources or depend periodically on or spontaneous if not

xi

γµν xi

A → aµdxµ

xµ = (t, xi)• Introduce coordinates for d-dimensional boundary

aµ

Euclideanise with and

We are interested in minimising free energy density

t → −iτ τ = τ +∆τ T−1 ≡ ∆τ

w = w (γµν , aµ;∆τ, Li) =1

∆τΠIOS (γµν , aµ;∆τ, Li)

Π ≡ ΠiLi

Variation with respect to and gives, as usual:γµν aµ

= − 1

Π

� {Li}

0dd−1x

√γ�12T

µν δγµν + Jµ δaµ�

Focus on the non-compact case.

δw = − 1

∆τΠ

� ∆τ

0

� {Li}

0dτdd−1x

√γ�12T

µν δγµν + Jµ δaµ�

To get variation with respect to periods, scale coordinates:

τ = ∆τ τ xi = Li xi

with

IOS (γµν , aµ;∆τ, Li) = IOS (γµν , aµ; 1, 1)

and

ds2 → γµν dxµ dxν

≡ γττ (∆τ)2 dτ2 + 2γτxi∆τLidτdxi + γxixjLiLjdx

idxj

≡ aτ ∆τ dτ + axi Li dxi

A → aµdxµ

Variation with respect to periods via chain rule

where

Comments:

1. For black branes invariant under translations in direction we have and this gives Smarr-type relation(s)Xi = 0

xi

c.f [El-Menoufi, Ett, Kastor,Traschen]

Xi = w +Π−1

� {L}

0dd−1x

√−γ (T txi

γtxi +�

j

T xixj

γxixj + Jxi

axi)

δw =−Π−1

� {L}

0dd−1x

√−γ

�12T

µν δγµν + Jµ δaµ�

− sδT −�

i

δLi

LiXi

s ≡ − 1

T

w +Π−1

� {L}

0dd−1x

√−γ(T ttγtt +

�

j

T txj

γtxj + J tat)

2. Black branes that are spatially modulated in the direction, dependon wave-numbers . One has

xi

ki ≡ 2π/Li

kiδw

δki= Xi

The fact that this variation is simply related to and was obscure in a number of examples

Tµν Jµ

The thermodynamically preferred spatially modulated black branes in the non-compact case and where symmetry breaking is spontaneoushave Xi = 0

3. The expression for was a definition. It can be related to the area of the event horizon using bulk equations e.g. for two-derivative gravity it is1/4 of the area of the event horizon e.g [Papadimitriou,Skenderis]

S

= w +Π−1

� {L}

0dd−1x

√−γ(T txi

γtxi +�

j

T xixj

γxixj + Jxi

axi)

Further relations can be obtained!

The euclidean boundary is a dimensional torus. The circle with tangent is singled out by the black hole in the bulk

d

Sl(d,Z)The transformation

τ = τ + αixi, no sum on i

xi = xi

with parametrises the same torus with same periods. αi =∆τ

Li

Also, ∂τ = ∂τ

∂τ

Can repeat the entire analysis in the barred coordinates - one should get the same result

Remarkably, this implies

� {L}

0dd−1x

√−γ

�T xitγtt +

�

k

T xixk

γxkt + Jxi

at

�= 0

By considering other transformations preservingSl(d,Z) ∂τ

i �= j

� {L}

0dd−1x

�− det γ

�T xitγtxj +

�

k

T xixk

γxkxj + Jxi

axj

�= 0

E.g. consider CFTs in flat spacetime, , γµν = ηµν aµ = (µ,0)

Assume spontaneous symmetry breaking in the direction.x1

Conservation of andTµν Jµ

T x1t, T x1x1

, . . . , T x1xd−1

, Jx1are constants

alsoδw = −J tδµ− sδT +

δk

k

�w + T x1x1

�

w = −T x2x2

= · · · = −T xd−1xd−1

w = −Ts− J tµ+ T tt

−T xit + Jxi

µ = 0

T xixj

= 0 i �= j �= 1

and

T x1xi

= 0

Bars refer to quantities averaged over a period

• Results should generalise to other holographic scaling solutions such as Lifshitz

• Results can be generalised to include more matter fields

Some of these were not obvious from numerics!

Eg For thermodynamically preferred spatially modulated black branes

T x1x1

= −T x2x2

= · · · = −T xd−1xd−1

Superconducting p-wave phases

withAristomenis Donos

Christiana Pantelidou

Two approaches to study it holographically:

1. D=4,5 use SU(2) gauge fields.

Take the background to be charged with respect to and then spontaneously break the U(1) using the charged vector bosons [Gubser][Roberts,Hartnoll][Ammon,Erdmenger,Grass,Kerner,O’Bannon]

U(1) ⊂ SU(2)

In the specific setting it was shown that p-wave is preferred over (p+ip)-wave [Gubser, Pufu]

p-wave superconductivity is seen in a number of different systems e.g. , organic superconductors, He3 Sr2RuO4

2. D=5 use charged self-dual two-forms [Aprile,Franco,Rodriguez,Russo]

Extend p-wave studies of [Donos,Gauntlett] and develop (p+ip)-wave

• The Charged two-form model in D=5

• The model admits a unit radius vacuum solution with which is dual to a d=4 CFT.

• The CFT has a global U(1) symmetry and is dual to the conserved current

• The two-form satisfies a self duality equation

and is dual to a d=4 self-dual tensor operator with charge and

L = (R+ 12) ∗ 1− 12 ∗ F ∧ F − 1

2 ∗ C ∧ C − i2mC ∧ H ,

∆(OC) = 2 +m

∗H = imC,

JµA

A = C = 0

F = dA, H = dC + ieA ∧ C .

e

AdS5

• Such models arise in string theory. For example type IIB on can be truncated to this model with ,

• Electrically charged AdS-RN black brane describes the spatially homogeneous and isotropic phase of the CFT at , and high

e = 1/√3 m = 1

S5

UV: r → ∞ AdS4

IR: black hole horizon topology and temp

r → r+

At = µ(1− r+r)

Electric flux

R2

T

µ T

• This model has superconducting instabilities if

• The simplest way to see this is to find modes that violating the BF bound of the T=0 limit of the AdS-RN black brane

• To determine the critical temperature at which the instability sets in, one looks for linearised normalisable zero modes in the full AdS-RN black brane solution

• Both p-wave and (p+ip)-wave instabilities depending on wave-number

• They spontaneously break the U(1) and some of the other translations and rotations....

AdS2

e2 >m2

2

k

c3 = c03 +O (r − r+)

p-wave instabilities:

c3(r) ∼ cc3r−|m| + . . .

δC = · · ·+ c3(r) dx1 ∧ dx3

Order parameter points in the direction

Three translations preserved Rotations in the plane preserved

k = 0

(x1, x3)

x2

Helical p-wave instabilities:

δC = · · ·+ c3(r) [sin(kx1)dx1 ∧ dx2 + cos(kx1)dx1 ∧ dx3]

x3

x1

k �= 0

preserved

This is the homogeneous Bianchi symmetry group

∂x2 ∂x3and

∂x1 + k(x3∂x2 − x2∂x3) preserved

V II0

x2

x2

p+ip-wave instabilities

The phase of the order parameter is now involved

Three translations preserved Rotations in combined with gauge transformation preserved

δC = · · ·+ e−ikx1ic3(r)dx1 ∧ (dx2 − idx3)

δC = · · ·+ ic3(r)dx1 ∧ (dx2 − idx3)

p+ip-wave instabilities

Same symmetry as

(x2, x3)

k �= 0

k = 0

- translations in now combined with gauge transformations x1

k = 0

m = 2P and (p+ip)-wave instabilities of AdS-RN for

�2 �1 0 1 2 3k

0.02

0.04

0.06

0.08

0.10

0.12T

e = 2

e = 2.8

e = 3.5

Highest T instability has k �= 0

Each point under bell curve corresponds to the existence of a back reacted p-wave and (p+ip)-wave black hole

For both cases we should find the preferred black holes which minimises the free energy density with respect to . Then we should compare p and (p+ip)

k

• Ansatz

• Where are functions of and we have used the Bianchi invariant one-forms

• Solve ODEs for functions with suitable boundary conditions at the black hole horizon and at the AdS boundary. Analyse the thermodynamics....

ds2 = −g f2 dt2 + g−1dr2 + h2 ω21 + r2

�e2α ω2

2 + e−2α ω23

�

C = (i c1 dt+ c2dr) ∧ ω2 + c3 ω1 ∧ ω3

A = a dt

ω1 = dx1

ω2 = cos (kx1) dx2 − sin (kx1) dx3,

ω3 = cos (kx1) dx2 + sin (kx1) dx3

V II0

g, f, h,α, ci, a r

Back reacted Helical p-wave black holes

0.0 0.5 1.0 1.5 2.0k

0.002

0.004

0.006

0.008

0.010

0.012

0.014

T

e = 2

0.02 0.04 0.06 0.08 0.10 0.12T0.0

0.2

0.4

0.6

0.8

1.0

cc3

Ttt = 3M + 8ch

Tx1x1 = M + 8ch

Tx2x2 = M + 8cα cos(2kx1)

Tx3x3 = M − 8cα cos(2kx1)

Tx2x3 = −8cα sin(2kx1)

Tx1x1 = Tx2x2 = Tx3x3

⇒ ch = 0

preferred branch:

∂kw = 0

⇒

0.0 0.5 1.0 1.5 2.0k

0.002

0.004

0.006

0.008

0.010

0.012

0.014

T

e = 2

0.02 0.04 0.06 0.08 0.10 0.12T0.0

0.2

0.4

0.6

0.8

1.0

cc3

Ttt = 3M

Tx1x1 = M

Tx2x2 = M + 8cα cos(2kx1)

Tx3x3 = M − 8cα cos(2kx1)

Tx2x3 = −8cα sin(2kx1)

• As we find that , with The solution interpolates between AdS5 in the UV and a helical scaling solution in the far IR:

Emergent helical scaling in the IR at T=0

T → 0

t → λzt, x2,3 → λx2,3, x1 → x1

r → λ−1r

ds2 = −(f20L

−2)r2zdt2 + L2 dr2

r2+ (k2h2

0)dx21 + r2

�e2α0 ω2

2 + e−2α0 ω23

�

C = . . .

A = . . .

s → T 2/z

where are constantsf0, L, h0,α0

Scaling

z ∼ 2

Similar to ground states of [Iuzuka,Kachru et al]

�1 0 1 2 3k

0.01

0.02

0.03

0.04

0.05

0.06

0.07T

e = 2.8

Similar to the previous case, but now as we have

r → λ−1r, t → λzt, x1,3 → λ1−γx1,3, x2 → λ1+γx2

Similar to ground states of [Taylor]

T → 0 k → 0

ds2 = −(f20L

−2)r2zdt2 + L2 dr2

r2+ α−2

0 r2(1−γ)�dx2

1 + dx23

�+ α2

0r2(1+γ)dx2

2

New anisotropic scaling solution:

�2 �1 0 1 2 3k

0.02

0.04

0.06

0.08

0.10

0.12T

e = 3.5 Similar to previous cases, but now we see the phenomenon of pitch inversion at Such pitch inversion is observed in chiral nematics and helimagnets.

Metric approaches AdS5 in the IR!

Similar behaviour was seen in s-wave superconductor [Horowitz, Roberts] and also in p-wave black holes for su(2) model [Basu]

Has been described as emergent conformal symmetry. However, correlators do not approach those of CFT for small frequency.

T �= 0

ds2 = −gf2dt2 + g−1dr2 + h2(dx1 +Qdt)2 + r2(dx22 + dx2

3)

A = adt+ bdx1

C = e−ikx1(ic1dt+ c2dr + ic3dx1) ∧ (dx2 − idx3)

Time translations preserved, but stationary metric

Three space translations preserved (plus gauge)

Rotations in preserved (plus gauge)

Back reacted (p+ip)-wave black holes

(x2, x3)

e = 2

0.02 0.04 0.06 0.08 0.10 0.12T0.0

0.2

0.4

0.6

0.8

1.0

cc3

Tx1x1 = Tx2x2 = Tx3x3

⇒ ch = 0

preferred branch:

∂kw = 0

⇒

Ttt = 3M + 8ch ,

Ttx1 = 4cQ ,

Tx1x1 = M + 8ch ,

Tx2x2 = M ,

Tx3x3 = M , Two extra Smarr relations

⇒ cQ = 0

0.0 0.5 1.0 1.5 2.0k

0.005

0.010

0.015T

e = 2

0.02 0.04 0.06 0.08 0.10 0.12T0.0

0.2

0.4

0.6

0.8

1.0

cc3

Ttt = 3M

Ttx1 = 0

Tx1x1 = M

Tx2x2 = M

Tx3x3 = M

0.0 0.5 1.0 1.5 2.0k

0.005

0.010

0.015T

e = 2.8

No analogue of pitch inversion

�1 0 1 2k

0.01

0.02

0.03

0.04

0.05

0.06

0.07T

�2 �1 0 1 2 3k

0.02

0.04

0.06

0.08

0.10

0.12T

e = 3.5

For all values of the charge we find as (in some cases at very low temperatures)

e s → 0 T → 0

The nature of the ground states is unclear....

Competing orders

0.2 0.4 0.6 0.8 1.0T�Tc

�0.096

�0.094

�0.092

�0.090

�0.088

�0.086

�0.084

w

0.2 0.4 0.6 0.8 1.0T�Tc�0.30

�0.25

�0.20

�0.15

�0.10

we = 3.5

e = 2

p-wave preferred(p+ip)-wave preferred

0.0 0.2 0.4 0.6 0.8 1.0T�Tc�0.18

�0.17

�0.16

�0.15

�0.14

�0.13

�0.12

�0.11

w

e = 2.8(p+ip)-wave then first order transition to p-wave

Final Comments

Rich classes of spatially modulated black holes exist with interesting properties and novel ground states

More to be discovered