holography and non-relativistic systems - school of...

Holography and non-relativistic systems

Marika Taylor

Institute for Theoretical PhysicsUniversity of Amsterdam

IPM, April 2010

Marika Taylor Non-relativistic systems

Holography

Gauge/gravity duality increasingly dominates the high energyliterature:

It is used as a tool to explore strong coupling dynamics of QFTs.On the other hand it provides a qualitatively new paradigm forgravitational physics: spacetime is emergent, reconstructed fromgauge theory data.

In this talk we will focus on non-relativistic realizations of holographywhich may have applications to condensed matter physics.


Outline

1 Why use holography for condensed matter systems?2 Non-relativistic systems: introduction3 Massive vector models with z < 14 Schrödinger backgrounds


Holography: a tool for quantum critical points

Given asymptotically AdS spacetimes we can compute:N -point functions of chiral gauge invariant operators;Thermodynamic quantities eg free energy→ phase structure;Non-local operators eg Wilson loops using probe strings.

These tools allow us to explore strongly interacting quantum fieldtheories. Moreover:

The natural computables in holography are the desirablecomputables for condensed matter systems, whilst scaleinvariance is realistic (cf QCD).


Applications to condensed matter

Holography computes phase structure, correlation functions andhence transport properties in neighborhoods of strongly coupledquantum critical points.

Explosion of interest in modeling:

Quantum Hall systems

(Non-)Fermi liquids

Superconductivity (of globalcurrents)

but very early days...


Applications to condensed matter

We have tools and landscapes...

But what key questions can weaddress?


Outline



Non-relativistic scale invariant systems

Inspired by cold atom physics, Son initiated holographicmodeling of non-relativistic systems, with scale invariance:

x→ λx; t→ λzt.

with z the dynamical scaling exponent.The case z = 1 has full relativistic symmetry, but z > 1 is relevantfor cold atom systems, perhaps even for high Tcsuperconductivity? (Sachdev)


Schrödinger symmetry

Son discussed the case of z = 2, in which one can add specialconformal symmetry C to the algebra to form the Schrödingergroup, along with a central term related to particle number.He noticed that one could write down a metric in (D+3)dimensions to geometrically realize Schrödinger:

ds2 = −b2(dx+)2

r4+

2dx+dx− + dxidxi + dr2

r2.

with xi being D-dimensional spatial coordinates.The (D + 1)-dimensional field theory in a background withcoordinates (x+, xi) was proposed to be dual to this background,with the charge associated with compact x− being particlenumber.


Schrödinger solution of massive vector model

Schrödinger backgrounds arise as solutions to variousgravitational theories.Son used gravity coupled to a massive vector:

S =1

16πGN

∫dD+3x

√−g(R− 2Λ− 1

2FµνF

µν − 12m2AµA

µ)

with m2 = 2(D + 2).The previous metric, together with a vector field A+ = b/r2,solves the equations of motion.


Schrödinger solution of TMG

3d Schrödinger backgrounds also arise as solutions totopologically massive gravity (TMG).Recall that the TMG action is

S =∫d3x

(√−g(R− 2Λ) +

12µ

(ΓdΓ + Γ3))

where Γ is the 1-form Christoffel symbol.The Schrödinger metric solves the equations of motion whenµ = 3 (= null warped AdS).


Schrödinger holographic duality

What strongly correlated systems are we actually describing?

Y. Shin et al , Phase diagram of a two-component Fermi

gas with resonant interactions, Nature 451, 689 (2008)

Son was motivated by cold fermionsat unitarity in 3d.Note however that the bulkspacetime has two extra dimensions,one radial (r) and one compactlightcone coordinate (x−).


Schrödinger holographic duality

Maldacena et al (2008) observed that these spacetimes areobtained via TsT transformations of AdS solutions.This suggested that the holographic dual is a 4d null dipoletheory, living on (x+, x−, xi) with x− compactified.Lightcone compactification introduces a spacetime singularity,and corresponding quantum problems in the dual field theory.

In this talk we will use precision holography techniques to explore theduality for Schrödinger spacetimes.


References

1 Monica Guica, Kostas Skenderis, M.T. and Balt van ReesWhat is the dual to Schrödinger backgrounds?arXiv:1004.xxxx

2 Ricardo Caldeira-Costa, M.T.Holographic duality for chiral modelsarXiv:1004.xxxx

as well as related work in progress.


Lifshitz symmetry

Holographic duals have also been constructed for systems withLifshitz symmetry (Kachru et al, ’08)

ds2 = dr2 − e2zrdt2 + e2rdxidxi,

with z = 1 AdS.The prototype free Lifshitz scalar system in D + 1 dimensions(with z = 2)

S =12

∫dtdDx

[(∂tφ)2 − µ(∇2φ)2

]where ∇ = ∂i∂i is the spatial Laplacian. No special conformalsymmetry!


Lifshitz symmetry

These backgrounds are supported by an anisotropic stressenergy tensor, which violates the strong energy condition.For this reason it is hard to find realizations in terms of gravitycoupled to matter, which can be embedded in string theory cf deSitter. (Hartnoll et al 2009)In what follows we will focus on the Schrödinger case: Lifshitzcan be analyzed analogously, but is qualitatively different...


Outline



The massive vector model

Recall the massive vector model:

S =1

2κ2d+1

∫dd+1x

√−g[R+ Λ− 1

4FmnF

mn − 12m2BmB

m

],

with Λ = d(d− 1) and m2 = z(z + d− 2).Generalizing our z = 2 solution to arbitrary z, we find:

ds2 =dr2

r2+

1r2

(σ2r2−2z(dx+)2 + 2dx+dx− + dxidxi

);

B+ = br−z,

with b2 = 2σ2(1−z)z .


Features of the massive vector model

Note that:1 Anti-de Sitter space is the limit σ2 = 0, z = 1.2 The background admits a scaling symmetry for all values of the

dynamical exponent z, both critical slowing down z > 1 andcritical speeding up z < 1.

3 Only at z = 1, 2 does the background admit in addition specialconformal symmetries.

4 For 0 < z < 1, the background is asymptotically anti-de Sitter,whilst for z > 1 the asymptotic behavior of the metric as r → 0violates this property.


The z < 1 case

Consider the background at z < 1:

ds2 =dr2

r2+

1r2


);

B+ = br−z.

Since the spacetime is asymptotically AdS, the dual theory is arelativistic CFT in the UV.Expand around b = σ = 0 AdS solution: the vector field B is dualto a vector operator of scaling dimension (z + d− 1) in the CFT,which is relevant for z < 1!


Dual holographic interpretation

From the explicit expression for the vector field, we can show thatb acts as a null source for the dual vector operator Vµ.Thus the dual theory is a CFT deformed by a relevant vectoroperator with ∆ = (z + d− 1):

SCFT → SCFT +∫ddxbV− + · · ·

The resulting theory no longer has Lorentz invariance, since thesource explicitly distinguishes a null direction.Remarkably, however, the resulting theory still admits ananisotropic scale invariance in which:

x+ → λzx+; x− → λ2−zx−; xi → λxi.

as well as residual translational and rotational symmetries.


Holographic duality for chiral models

The massive vector model for z < 1 provides dualdescriptions of strongly interacting anisotropic scaleinvariant systems with z < 1.The duality relates (as usual) a (d+ 1)-dimensionalbackground to a field theory in d dimensions, (x+, x−, xi).Compactifying x− and letting x+ = t, the resulting"non-relativistic" (d− 1)-dimensional theory will exhibitcritical speeding up in which t→ λzt whilst xi → λxi.


Some highlights of these models

A precise holographic dictionary be set up, and correlationfunctions computed. For example, for scalar operators of scalingdimension ∆ in 2d one finds:

〈O∆(x+, x−)O∆(0, 0)〉 =1|x+|∆

F

(|x+|2−z

|x−|z

).

which is of the required form. However, F is a priori an arbitraryfunction, whilst for operators coupling to minimal scalars onlyspecific functions F appear.These models involve vectors of mass m2 = z(z + d− 2) with0 < z < 1 which can occur in Sasaki-Einstein compactifications.It would be interesting to find a consistent truncation of aSasaki-Einstein compactification which includes this sector...


Relation to condensed matter

Dynamical scaling with z = 2 and z = 3 is arguably of the mostinterest for condensed matter (cold atoms, superconductivity), butscale invariance with z < 1 has also appeared in condensed matterliterature.

Theoretical: Cardy, 1992 considered anisotropic scale invariancewith z = 4/5 as a continuum limit of the chiral Potts model, usedto describe melting transitions.Experimental: phase transitions in xenon.


Outline



Massive vector models with z > 1

The background solution is:

ds2 =dr2

r2+

1r2

(−σ2r2−2z(dx+)2 + 2dx+dx− + dxidxi

);

B+ = br−z.

For z > 1, the g++ term in the metric blows up fastest at theboundary r → 0 as r−2z > r−2.This implies that the metric is not asymptotically AdS, andactually has a degenerate conformal boundary, which isone-dimensional.


Backgrounds with z > 1

Since these backgrounds are not asymptotically AdS, this setsthis case apart from other cases of precision holography, basedupon the notion of (weakly) AlAdS spacetimes.Computation in the Schrödinger case has mostly proceeded byanalogy with AdS, but one cannot assume standard resultsconcerning the holographic dual for such spacetimes.For example, there are no guarantees that the dual descriptionhas the form of an ordinary local, renormalizable quantum fieldtheory.


Holographic duality for z > 1

We will argue that the correct dual description is a d-dimensional non-local field theory with anisotropic scale in-variance, a null dipole theory, and set up precision holographyfor this duality.


Irrelevant deformation of a CFT

Consider background solution when b (and hence σ) areinfinitesimal, so g++ term negligible:

ds2 =dr2

r2+

1r2


);

B+ = br−z,

From AdS/CFT dictionary, we see that B is dual to an irrelevantvector operator of dimension (z + d− 1).When b is infinitesimal, the AdS/CFT dictionary implies that bacts as a source for this irrelevant operator.



Thus, just as for z < 1 the dual theory can be thought of as adeformation of a CFT:

SCFT → SCFT +∫ddxbV− + · · ·

The dimension of the operator guarantees that the new theoryactually has anisotropic scale invariance.However, deforming a theory by an irrelevant operator introducesnew UV divergences and changes the UV behavior; the ellipsesdenote additional counterterms generated in removing thedivergences.Working perturbatively in the source for the irrelevant operatorsuffices for correlation functions in the CFT but not for the caseat hand here (finite b).



Let us consider a scalar operator O∆ of dimension ∆ in the CFT,which is dual to a bulk scalar field φ of mass M .The two point function of this operator is then

〈O(x)O(0)〉 =c∆|x|2∆

where x2 = 2x+x− + xixi.What is the form of the two point function at finite b?



It is interesting to first compute this perturbatively in b as follows.First let us Fourier transform with respect to the lightconecoordinate x−:

〈O(x+, k+, xi)O(0,−k+, 0)〉 = c∆(k+)∆−1(x+)−∆eik+xixi

(x+)2 ,

and then compute the corrections perturbatively in terms ofhigher-point functions at the conformal point

δb〈O(x)O(0)〉 = b

∫ddy〈O(x)O(0)V−(y)〉+ · · ·

Computation of the integrals can be carried out using differentialregularization, which derives additional UV countertermsperturbatively in b.


Result for d = 2, z = 2

The two point function expanded in powers of b is:

〈O∆(x+, k+)O∆(0,−k+)〉 =A

|x+|∆(k+)∆−1

×(1 + c1b(k+)2 ln(k+) + c2b

2(k+)4 ln(k+) + · · ·)

where ci are calculable ∆ dependent constants.This expression can actually be resummed to give:

〈O(x+, k+)O(0,−k+)〉 =A(k+)|x+|∆′(k+)

,

where ∆′(k+) = ∆ + b(k+)2 + · · · and A(k+) are knownfunctions of the lightcone momentum k+.This in turn is the general form of a two point function of anoperator of non-relativistic scaling dimension ∆′(k+) and particlenumber k+ in a Schrödinger theory in one dimension x+.


Interpretation

Viewing the theory as an irrelevant anisotropic deformation of atwo dimensional CFT, we recovered the expected form of theSchrödinger two point function.The field theory realization is however two dimensional, with thetheory being non-local in the x− direction.It is non-local because we found an infinite series ofcounterterms involving higher derivatives (equivalently momenta)along this direction.


Null dipole theory

Instead of viewing the theory as irrelevant deformations of a CFT, itcan be viewed as a null dipole theory:

Products of dipole fields Φ are given by

Φ1 ∗ Φ2 ≡ Φ1(x− 12L2)Φ2(x+

12L1).

where the vector Lµ is the dipole length of Φ.Ordinary theories become dipole theories upon this substitution.Spacelike L was developed by Ganor et al, 2000.Here L ∝ b should be null for all fields, with the dipole lengthsdetermined by the global charges of fields in the undeformedtheory.


Null dipole theory

There exists a Seiberg-Witten type map:

Ldipole(Φ) = LCFT (Φ) + Lirrelevant(Φ)

with the dipole fields Φ non-trivially related to the ordinary fieldsΦ via:

Φ(x) = e−ib∫A−(x+L)Φ(x).

One can easily verify perturbatively in b that:

Ldipole(Φ) = LCFT (Φ) + bV−(Φ) +O(b2)

indicating that the dipole theory indeed resums all the irrelevantdeformations.


Null dipole theory

Few results on renormalizability and unitarity of dipole theories;none on null dipole theories.Simple arguments indicate that they are at least as well-definedas lightlike non-commutative theories: in lightcone quantizationthe Hamiltonian associated with x+ translations is Hermitian.Non-locality of the theory would be confined to the x− direction.


Holographic calculation

Now let us try to understand two point functions in the bulk, using thefollowing algorithm:

1 Write down the bulk field equation for the field φ of mass M inthe Schrödinger background:

�φ = M2φ.

2 Find a solution to this equation which is regular throughout thespacetime.

3 Derive the renormalized holographic one point function:

〈O〉 =δIrenδφ(0)

Here Iren is the renormalized onshell scalar action, withboundary counterterms added to remove r → 0 divergences, andφ(0) the source for the dual operator.


Holographic calculation

4 The one point function can be expressed in terms of theasymptotic expansion of the field φ and the two point function isobtained by further differentiation:

〈OO〉 = − δ

δφ(0)〈O〉.

The steps can be carried out exactly as in the asymptotically AdScase....


Non-locality of counterterms

The momentum space result for the two point function ford = z = 2 is

〈O(k−, k+)O(−k−,−k+)〉 = (k−k+)2∆′−2 Γ(1−∆′)Γ(∆′ − 1)

,

with ∆′(k+) = 1 +√

1 +M2 + b2(k+)2. The dependence on(k− ↔ x+) is as in the field theory.However, the boundary counterterms depend explicitly on ∆′ :

I =12

∫d3x√g((∂φ)2 +M2φ2)− 1

2

∫d2x√γφ2(1−∆′) + · · ·

and are thus non-local in the x− direction, as expected.


Holographic dictionary for probe operators

Renormalized correlation functions can be computed from per-turbing around Schrödinger and using holographic "renormal-ization", provided that we allow for non-localities in the x− di-rection.


Asymptotically Schrödinger spacetimes

One also wants to understand the metric sector, namelyholography for asymptotically Schrödinger spacetimes.For example, there are known solutions for Schrödinger blackholes:

ds2 =dr2

r2f(r)+

1r2

(σ2g(r)r2

(dx−)2 +h(r)β2

(dx+)2 + 2dx+dx−)

where as r → 0 the functions (g(r), f(r))→ 1 and h(r)→ 0.These should correspond to thermal states in the dual theory, buthow do we show this?


Energy momentum tensor

So far, we have worked with holographic setups in which theasymptotic structure is:

ds2 =dr2

r2+

1r2gij(x, r)dxidxj

with the dual stress energy tensor being related to terms in theexpansion of gij(x, r).One might have thought that here we could define anasymptotically Schrödinger structure in terms of asymptotics ofgij(x, 0), and hence calculate a dual 〈Tij〉.

Leaving aside the fact that gij(x, 0) is divergent and degenerate, themetric is actually not the correct asymptotic data...


Non-relativistic stress energy tensor

The conserved stress energy tensor of a non-relativistic theory isnecessarily non-symmetric: tij 6= tji.

Any theory in Minkowski spacetime that possesses a con-served, symmetric stress energy tensor Tij is Lorentz invariant.

In a non-Lorentz invariant theory one should couple the model tobackground gravity using the vielbein eia.Then by varying the action as:

tij = e−1ebjδS

δeib,

with e the vielbein determinant, one obtains a non-symmetricenergy momentum tensor tij .


Holographic renormalization

The boundary value of the vielbein e(0) acts as a source for thenon-symmetric stress energy tensor.Therefore, the defining relation for holography will now be writtenin terms of the action S[e(0), B(0)], where B(0) indicates theboundary value for the vector field.Note that the bulk terms in the action can be rewritten in terms ofa vielbein as:

S =∫d3xtr

(e ∧R+ 2

Λ3e ∧ e ∧ e+ · · ·

),

where R is the Riemann tensor and e = − ∗3 e.Need to specify appropriate e(0) and derive counterterms whichshould only be non-local in x−.


Linearized level

As a first step, let us consider linearized fluctuations around theSchrödinger background:

ds2 =dr2

r2+

1r2

(−b2r−2(dx+)2 + 2dx+dx− + habdx

adxb)

;

Bm = bδm−r−2 + bm.

We have found the most general solutions for the fluctuations haband bm for both the massive vector model and for TMG; thesehave the maximal number of independent solutions.It is convenient to work the metric equations of motion, but habcan then be converted into a vielbein fluctuation.


Linearized level

Expanding the linearized solutions around r = 0 we can find themost general allowed asymptotic behavior of the vielbein andvector field fluctuations.For example, switching off the fluctuation for the vector field forbrevity:

e+− = r−1(1 + e(0)+−) + re(2)+−

e+− = r−1e(0)+− + re(2)+−

e++ = −b2r−3 + r−1(e(0)++) + re(2)++

e−+ = −b2r−3e(0)+− + r−1e(0)−+ + · · ·

where the fluctuations e(0)ia are independent, and e(2)ia areconstrained.Regularity conditions relate e(0) and e(2) as usual.


Linearized level

Note the leading term in the perturbed vielbein are actually moredivergent than those in the background. Despite this:

Treating e(0) as the sources, one can renormalize the bulk actionusing counterterms with only allowed non-locality in the x−

direction.Varying the renormalized action and using the regular solution ofthe linearized equations gives us the two point functions for tij

〈t++t++〉 =c+

3|x+|4; 〈t−−t−−〉 =

c−3|x−|4

,

which (for TMG) is in agreement with the suggestions ofStrominger et al.The two point function for the operator V dual to the vector fieldcan also be computed, with the scaling dimension of the operatordepending on the lightcone momentum as bk+.


Summary

We can obtain renormalized correlation functions in Schrödingerbackgrounds, provided we allow non-locality of countertermsalong the x− direction.Correlation functions of all operators support the identificationwith a CFT deformed by irrelevant operators.When we work to higher order in fluctuations, the degree ofdivergence of the vielbein as r → 0 and hence of the regulatedonshell action increases.Defining a general notion of asymptotically Schrödingerspacetimes via boundary conditions on eia would be the nextstep.In the field theory the irrelevant deformations could be resummedinto a dipole theory, but the holographic interpretation of thisresummation remains unclear.


Conclusions

Holography provides a very precise framework that can be usedto compute field theory properties from geometries and viceversa.In these talks, we have introduced the notion of precisionholography for asymptotically AdS ×X geometries with orwithout linear dilaton.We then explored how these techniques can be extended toweakly asymptotically AdS spacetimes, and to non-relativisticbackgrounds.


Outlook

Non-relativistic backgrounds

Develop further the notion of holography for cases such as thesewhich have degenerate boundaries.

Applications

Exploit precision holography tools in applications to condensedmatter.


holography and non-relativistic systems - school of...

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