Aspects of Lifshitz Holography - ap.ioap.io/archives/talks/bh8/bh8.pdf · t ! zt xi! xi Real-world experiments are done at very small temperatures. Theory must explain physical properties

Context: AdS/Condensed MatterProblem: Finding a Candidate Gravity DualInterpretation: Thermodynamic Properties

Summary

Aspects of Lifshitz Holography

Amanda Peet, Ben Burrington, Ida G. Zadeh,and Gaetano Bertoldi1

1Department of PhysicsUniversity of Toronto

BH VIII, Niagara Falls, Fri 13 May 2011

Slides: http://amandapeet.ca/talks/bh8/

Based on: 0905.3183, 0907.4755, 1007.1464, 1101.1980

Amanda Peet, Ben Burrington, Ida G. Zadeh, Gaetano Bertoldi Aspects of Lifshitz Holography – BH VIII – 13 May 2011 1 / 20

http://amandapeet.ca/talks/bh8/

http://arxiv.org/abs/0905.3183





Summary

Lest We Forget

RememberAndrewChamblin fromAmarillo,Texas?It would havebeen his 42ndbirthday today,May 13th.Rest in peace,mate ... we(I) miss you.


http://andrewchamblin.org/

http://andrewchamblin.org/


Summary

Outline

1 Context: AdS/Condensed MatterQuantum Phase Transitions in Non-Relativistic Field TheoriesAdS/CFTAdS/Condensed Matter

2 Problem: Finding a Candidate Gravity DualSetup: Equations of Motion and SymmetriesLimits: Connecting to Known SolutionsAnalytics: Perturbation TheoryNumerics: How Solutions Map Out Parameter Space

3 Interpretation: Thermodynamic PropertiesThermodynamics from Horizon PhysicsBlack Li/AdS Brane NumericsLifshitzy Degrees of Freedom



Summary

Quantum Phase Transitions in Non-Relativistic Field TheoriesAdS/CFTAdS/Condensed Matter

References

K. Copsey, R. Mann, 1011.3502.J. Blaback, U. H. Danielsson, T. Van Riet, 1001.4945.K. Balasubramanian, K. Narayan, 1005.3291.H. Singh, 1009.0651.H. Singh, 1011.6221.A. Donos, J. P. Gauntlett, 1008.2062.R. Gregory, S. L. Parameswaran, G. Tasinato,I. Zavala, 1009.3445.A. Donos, J. P. Gauntlett, N. Kim, O. Varela,1009.3805.S. Kachru, X. Liu, M. Mulligan, 0808.1725.M. Taylor, 0812.0530.A. Adams, A. Maloney, A. Sinha, S. E. Vazquez,0812.0166. R. G. Cai, Y. Liu, Y. W. Sun, 0909.2807.Y. S. Myung, Y. W. Kim, Y. J. Park, 0910.4428.M. C. N. Cheng, S. A. Hartnoll, C. A. Keeler,0912.2784. E. Ayon-Beato, A. Garbarz, G. Giribet,M. Hassaine, 1001.2361.C. M. Chen, D. W. Pang, 1003.5064.U. H. Danielsson, L. Thorlacius, 0812.5088.R. B. Mann, 0905.1136.D. W. Pang, 0905.2678.M. H. Dehghani, R. B. Mann, 1006.3510.

K. Balasubramanian, J. McGreevy, 0909.0263.E. J. Brynjolfsson, U. H. Danielsson, L. Thorlacius,T. Zingg, 0908.2611. M. H. Dehghani, R. B. Mann,1004.4397.S. F. Ross, O. Saremi, 0907.1846.Y. Liu, Y. W. Sun, 1006.2726.K. B. Fadafan, 0912.4873.S. J. Sin, S. S. Xu, Y. Zhou, 0909.4857.M. Cadoni, G. D’Appollonio, P. Pani, 0912.3520.F. Aprile, S. Franco, D. Rodriguez-Gomez, J. G. Russo,1003.4487.C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis,R. Meyer, 1005.4690.B. H. Lee, S. Nam, D. W. Pang, C. Park, 1006.0779.E. Perlmutter, 1006.2124.S. A. Hartnoll, P. K. Kovtun, M. Muller, S. Sachdev,0706.3215.W. Y. Wen, 1009.3952.V. G. M. Puletti, S. Nowling, L. Thorlacius, T. Zingg,1011.6261.S. A. Hartnoll, P. Petrov, 1011.6469.K. Goldstein, S. Kachru, S. Prakash, S. P. Trivedi,0911.3586.



Summary


Quantum Phase Transitions

A quantum phase transition is a transition between differentphases of matter at zero temperature. It results from quantumfluctuations.The ground state energy has nonanalytic behaviour.Characteristic energy [gap] scale ∆ and (coherence) lengthscale ξ behave with particular scaling near quantum critical point:

ξ ∼ (g − gc)−ν ∆ ∼ (g − gc)zν

At quantum critical point, system invariant under “anisotropic”scaling of space and time. Boost invariance is broken.

t → λz t x i → λx i

Real-world experiments are done at very small temperatures.Theory must explain physical properties of finite-T system insidethe quantum critical region.



Summary


Lifshitz Field Theory

Toy model describing quantum phase transitions in somecondensed matter systems (e.g. high-Tc superconductors) isLifshitz field theory:

S =

∫dtd2x

((∂tφ)2 − κ(∇2φ)2

)

This has “anisotropic” scale invariance with z = 2.High-Tc superconductors are believed to be strongly coupled inthe vicinity of quantum phase transitions. Traditionalsuperconductivity theories are not well-equipped to describe thephysics there.AdS/CFT Correspondence gave us an entirely new method ofanalyzing the strong-coupling behaviour of large-N gauge fieldtheories. Can we find a dual?



Summary


The AdS/CFT Correspondence

AdS/CFT was discovered via careful analysis of near-horizonbehaviour of spacetimes exerted by collections of N D-branes.Spacetime is closed string physics. Open strings? Gauge fieldtheory lives on D-branes. Valid approximation at low energy.Most famous example: D3-branes. Duality relates string theoryon AdS5×S5 to N = 4 supersymmetric SU(N) gauge fieldtheory. Very generally, AdS/CFT is a string/gauge duality.String and gauge parameters are related by

gs ∼ g2YM

√α′/RAdS ∼ (gsN)−1/4

Field theory expansion parameters (’t Hooft):

1/N 1/λ ≡ (g2YMN)−1

Expansion in string coupling gs and in stringiness α′ is a doubleexpansion for quantum gravity.



Summary


RG Flow

AdS/CFT: symmetries, states, operators, correlation functions ofthe two theories are physically equivalent. Work out dictionary.Radial direction in AdS interpreted as energy scale in gauge fieldtheory. Seeing this is simple: consider an open string probe.Energy cost of straight string stretched distance r is

E = T · r =1

2πα′r

Other probes such as gravitons also yield E ∝ r in string units.Suppose we perturb CFT to RG flow from CFT in UV to lesssymmetric theory in IR. How is AdS geometry changed?Finite temperature in field theory↔ energy above extremality.Chemical potential in field theory↔ charged [D = 5] black branegeometry.



Summary


Scaling Symmetry

Lifshitz field theory and others possess scaling symmetrypreserving SO(d) but affecting time and space in (d + 1)dimensions differently: t → λz t , xi → λxi .Any candidate gravity dual should possess this “anisotropic”scaling symmetry. By analogy with AdS (which has z = 1), thissuggests a metric of the form proposed by KLM (z = 2):

ds2 = L2(−r2zdt2 + r2dx idx jδij +

dr2

r2

), i ∈ 1,2, . . . ,d

where Λ = −d(d + 1)/(2L2).Gravity dual’s radial coordinate must scale too: r → λ−1r .Candidate gravity duals to Lifshitz-type field theories at finite Tand µ will be “Li/AdS” black branes carrying charge.



Summary


Seeking a Lifshitz-like Gravity Dual

How to embed Lifshitz gravityduals in string/M theory? Thisis not an easy problem. (e.g.some candidate geometrieshave a naked singularity.)We insist on AdS asymptoticsto provide the UV completion,while hunting for Lifshitz-likebehaviour in the interior.

Seek solutions with horizons,for finite-T physics.Use a phenomenologicalU(1) dilaton gravity modelwith arbitrary z and d .Mateos & Trancanelli recentlyreported [GLS] studying avery similar model with NfD7-branes smeared in one offield theory directions tocreate a spatial anisotropy.“Relevant to RHIC”. (But is itSUSYic? Stable?)

Amanda Peet, Ben Burrington, Ida G. Zadeh, Gaetano Bertoldi Aspects of Lifshitz Holography – BH VIII – 13 May 201110 / 20


Summary

Setup: Equations of Motion and SymmetriesLimits: Connecting to Known SolutionsAnalytics: Perturbation TheoryNumerics: How Solutions Map Out Parameter Space

The Model and the Gravity Dual Ansatz

Bulk action in (d + 2) dimensions:

S =1

16πGd+2

∫dd+2x

√−g(

R − 2Λ− 2(∇φ)2 − e2αφG2)

Ansatz for candidate gravity dual:

ds2 = e2A(r)dt2 + e2B(dx idx jδij) + e2C(r)dr2

φ = φ(r) A = eG(r)dt

i.e., only radial coordinate dependence is permitted. G = dA.Reduced 1-dimensional action:

L1D = d(d−1)2 eA+dB−C(∂B)2 + e−A+dB−C+2G+2αφ(∂G)2

+deA+dB−C∂A∂B − eA+dB−C(∂φ)2 − eA+dB+CΛ

Consistent trunction provided C EOM obeyed. It is a Lagrangemultiplier imposing the Hamiltonian constraint.



Summary


Equations of Motion and 3 Symmetries

Eliminating C’s derivatives gives the 1D equations of motion:12d(d − 1)eA+dB−C(∂B)2 + e−A+dB−C+2G+2αφ(∂G)2

+deA+dB−C∂A∂B − eA+dB−C(∂φ)2 + eA+dB+CΛ = 0,d [2∂φ+ α (∂A− ∂B)] eA+dB−C = D0,

eA+dB−C∂φ+ αe−A+dB−C+2G+2αφ∂G = P0,

e−A+dB−C+G+2αφ∂G = Q.

3 conserved quantities: D0,P0,Q. 3 symmetries?rescaling of t and x i leaving field theory (d + 1)-volume invariant:

(A,B,C, φ,G)→ (A + dδ1,B − δ1,C, φ,G + dδ1)

global part of U(1) gauge symmetry: eG → eG + δ3redefining the gauge coupling by shifting the dilaton:

(A,B,C, φ,G)→ (A,B,C, φ+ δ2,G − αδ2)



Summary


Limits: Asymptotically AdS and Li Black Branes

aAdS black branes:

A(r) = ln(

Lr√

1−( rh

r

)d+1)

B(r) = ln(Lr)

C(r) = ln

(L

r√

1−( rhr )

d+1

)

φ(r) = φbA = gbdt

Scaling: s ∝ T d .

aLi black branes:

A(r) = ln(

LLaLr z√

1−( rh

r

)z+d)

B(r) = ln(Lr),

C(r) = ln

(LL

r√

1−( rhr )

z+d

)

2αφ(r) = ln(r−2d Φ

),

G(r) = ln

((z−1)Ld aLr z+d

(1−( rh

r )d+z)

2Q

)

where α =√

2d/(z − 1).Scaling: s ∝ T d/z .



Summary


Perturbing About Horizon and About AdS

Perturb analytically out from the horizon, and in from infinity.Then use numerics to connect them.Horizon:

eG1 = a0[g0(r − rh) + g1(r − rh)2 + · · ·

]

eA1 = a0

[(r − rh)

12 + a1(r − rh)

32 + · · ·

]

eC1 = c0

(r−rh)12

+ c1(r − rh)12 + · · ·

(in P0 = 0 gauge, B does not get perturbed.)AdS: metric and dilaton perturbations contain r−(d+1), r−2d

terms. Gauge perturbation contains r−(d−1).Constants in each perturbation depend on: α,d , rh,D0,Q, c0.First two of these are theory parameters. Other four?Fix rh. Can use 2 remaining symmetries to set D0,Q at will.So then c0 parametrizes a family of solutions.



Summary


Parametrizing Li/AdS Black Branes

We can parametrize all our Li/AdS black branes by this onequantity c0 that remained unfixed by symmetry.How does this work? Symmetries are key. It turns out that

temperature T is determined by D0;chemical potential µ is controlled by gb.

Work in P0 = 0 gauge. Fix rh = 1. Use 2 remaining symmetriesto ensure that, upon numerical integration out to AdS infinity, getcanonically normalized metric functions for AdS.For any given horizon size, say rh = 1, we can therefore map outsolutions with various (T , µ).T/µ should occur only once for a given horizon radius ifsolutions with different parameters are in fact unique.Limit T µ is the close-to-AdS regime.Limit T µ is the close-to-Lifshitz regime.



Summary

Thermodynamics from Horizon PhysicsBlack Li/AdS Brane NumericsLifshitzy Degrees of Freedom

Thermodynamic Equation of State

From perturbation analysis about the [flat] horizon, read off

T =r2h a0

4πLc0=

D0

αdrdh 2πLd+1

s =rdh

4Gd+2

n =Q

4πGd+2Ld−1 µ =gb

L2

Energy density E? Use background subtraction with AdS (inPoincaré patch) as reference spacetime.

E =1

16πGd+2

2(d + 1)

(D0 + 2dαQgb)

αLd+1

The above matches with scaling arguments (!), giving

E = [d/(d + 1)] (Ts + µn)

For Lifshitz-like regime, coefficient is instead [d/(d + z)].



Summary


Black Li/AdS Numerics: No Phase Transitions

Lifshitz black branes havec0 =

√α2 + 2

√rh/(d + 1).

AdS black branes havec0 =

√rh/(d + 1). Our Li/AdS

guys have c0 in between.So... is the graph of T/µ vs c0monotonic?Yes. Smooth interpolationbetween Li-like scaling andAdS scaling.No discontinuous phasetransition occur here for anyd , z we studied (up to d = 9).

(a) d = 3. (b) d = 5.

(c) d = 7. (d) d = 9.

Figure 1: The plots of ln (4Gd+2s) versus ln(LT ) for fixed µ = 1. Figures (1a), (1b),(1c) and (1d) correspond to d = 3, 5, 7, and 9, respectively. The different curvesin each plot correspond to different values of α, with α = 4 green (solid), α = 2magenta (long-dashed), α = 1 cyan (dot-dashed), and α = 0.75 blue (dashed). α isrelated to z via α =

2d/(z − 1).

13



Summary


Entropy of Lifshitzy Degrees of Freedom

In Li-like regime where T µ

s = c(z,d)(Lµ)d

4Gd+2

(Tµ

)d/z

Normalize using L, then plot svs T on log-log plot. Givesln(c(d , z)) as intercept.c parametrizes # d.o.f, inanalogy to N2 for N = 4 SYM.Numerics: c(d , z) goes tod-dep. constant at large z.Small-z non-monotonicity isunsurprising (AdS is z = 1).Figure 2: The plot of ln (c(z, d)) as a function of ln(z) for fixed value of µ = 1.

The different curves correspond to different dimension with d = 3 black (solid),d = 5 brown (long-dash), d = 7 red (dot-dash), and d = 9 coral (dash). Curvesfor d = 2, 4, 6, 8 behave similarly. First of all, notice that the curves’ intercept atln(z) = 0 is given by the value ln

(4π)d/(d + 1)d

which is non-monotonic in d.

Secondly, we find it interesting that the tails become flat out at z → ∞ and that thelarge z behaviour for various d is monotonic in d. Given these two facts, it may beno surprise that the curves ln(c(z, d)) for fixed d are generically non-monotonic forsmall values of z, and that in fact they cross each other.

above the ground state. For such a case, the “legal” wave numbers may be written

k =

n1/n2/

...

. (47)

Further we assume Lifshitz scaling symmetry and rotational symmetry in the spatialdimensions (broken only by the presence of the “box”). The only consistent relation

15



Summary


Tracking Lifshitzy Degrees of Freedom

Tracking how # d.o.f. depend on d and z?Dilute-gas calculation gives rough estimate:

s = δ(z,d)−d/zκ(z,d)Ωd−1

(2π)d4Gd+2

(d + z)

z2 (Lµ)d Γ

(dz

)(Tµ

)d/z

where δ, κ are quantities of O(1).In the large-z limit,

limz→∞

Γ(d/z)(d + z)/z2 = 1/d

which is independent of z, agreeing with our numerics. So forT µ at large z we get the same scaling as gravity:

s = c(d)T d/zµ(z−1)d/z



Summary

Conclusions

Context: AdS/CMT duality.Scope: limited. (We don’t claim to embed Li in ST.)Seek charged dilatonic black brane geometries with horizon,Lifshitz-like region, and AdS asymptotics.Li/AdS black brane numerics show no discontinuous phasetransitions occur in going from T µ from T µ.Analytics yield thermodynamic equation of state:

E =d

d + 1(Ts + µn)

as we should expect for an AdS-embedded Li-like gravity dual.

Correlation functions and transport properties?Explicit string embeddings?Modelling experimentally desirable features - cond-mat/hep-th?


Documents

Aspects of Lifshitz Holography - ap.ioap.io/archives/talks/bh8/bh8.pdf · t ! zt xi! xi Real-world experiments are done at very small temperatures. Theory must explain physical properties