An SYK-Like ModelWithout Disorder

Edward WittenMiami, Dec. 16, 2016

The original Sachdev-Ye model (1993) was a model of spins withrandom couplings, meant to give a soluble model of a spin liquidstate. Kittaev (2015) suggested to use a similar model for aholographic model of a black hole. There have been many recentcontributions, among others by Polchinski and Rosenhaus and byMaldacena and Stanford who filled in many details in Kitaev’sarguments. The model is now usually called the SYK model.

I will begin with an introduction to the SYK model and thenexplain a little about my own paper. Holographic gauge/gravityduality says that an ordinary quantum field theory in D spacetimedimensions can be equivalent to a quantum gravity model in D + 1dimensions. The first nontrivial case of this is D = 1, i.e. ordinaryquantum mechanics, which might be dual to quantum gravity in2 = 1 + 1 dimensions.

One thing that is special to 2 dimensions that in AdS2, spatialinfinity is disconnected

just like the region at infinity of a black hole. This means thatAdS2 is almost a black hole and its behavior under smallperturbations can give a model of a black hole. In higherdimensions, it takes a non-small perturbation of AdS to make ablack hole. But in 1 + 1 dimensions, we should be able to make ablack hole model just by exploring the behavior at very lowenergies.

What sort of quantum mechanics model will be dual to quantumgravity in AdS2? We don’t really know, but it may be that a verylarge class of models will do, maybe in some sense any model thatis sufficiently complicated. Anyway the SYK model in its mostoften studied form (but see for example Gross and Rosenhaus forsome of the possible variations) is a model of N Majorana fermionsin 0 + 1 dimensions with the action

I =

∫dt

(i

N∑k=1

ψkd

dtψk + ji1i2i3i4ψi1ψi2ψi3ψi4

),

where the ji1i2i3i4 are Gaussian random variables with variance〈jI jI ′〉 = 6J2δII ′/N

3. (We will come back later to the question ofwhether a model of this kind really requires random couplings.)

The model is solvable in the large N limit because the dominantFeynman diagrams are generated by a simple procedure:

As a result, one can write an integral equation that determines thepropagator:

As usual, Σ is the self-energy and G = (id/dt − Σ)−1 is thepropagator.

In other words, we get a simple equation

Σ(t, t ′) = J2

((1

i ddt − Σ

)(t, t ′)

)3

.

I’ve written it in a way that may look a little clumsy. On the righthand side, G = 1/(id/dt − Σ) is the propagator, and(

1

id/dt − Σ

)(t, t ′)

is its matrix element from time t ′ to time t.

As I explained before, if there is really an AdS2 dual, we will beable to get a black hole model just from the behavior at very lowenergies, so let us look at that. Something remarkable happens. Inthe integral equation, at low values of the energy ω, we can dropthe term id/dt = ω, because Σ vanishes more slowly than ω atsmall ω. Thus the integral equation can be approximated at low ωas

Σ(t, t ′) = J2((

1

−Σ

)(t, t ′)

)3

,

where I just dropped the id/dt term from the version that I wrotebefore.

When we drop the d/dt term, a simple scaling argument showsthat we must have Σ ∼ ω1/2 and we eventually get an exactsolution for the propagator with G (ω) = kω−1/2, with a certainconstant k. This is a conformally invariant solution with ψ being aprimarily field of dimension 1/4. That value should not come as acomplete surprise, since naively it is the value needed for conformalinvariance of the original interaction

∫dt ψ4.

So far the solution with G (ω) ∼ 1/ω1/2 is a conformally-invariantsolution, which might be what one would expect in the context ofAdS/CFT. Let me stress, since we will see in a moment that thephrase “conformally-invariant” could mean different things in thiscontext, that when I say that the solution for G (ω) isconformally-invariant, what I mean is that it is invariant underSL(2,R) (or more precisely SO(1, 2) = SL(2,R)/Z2, since thecenter Z2 of SL(2,R) acts trivially). This is the analog in D = 1dimension of the special conformal symmetry group SO(D, 2) in Dspacetime dimensions.

But the equation

Σ(t, t ′) = J2((

1

−Σ

)(t, t ′)

)3

,

with the d/dt term dropped, has much more symmetry: it isinvariant under arbitrary reparametrizations of the time. Basicallythe reason that this happened is that the interaction term∫

dt ψ(t)4 is invariant under arbitrary reparametrizations of thetime, if one considers ψ to transform like a field of dimension 1/4.Thus the low energy theory has a full diff S1 symmetry, assumingthat we consider t to be a periodic variable (or diff R if t isreal-valued).

The diff S1 symmetry of the low energy theory may come as asurprise, but from the AdS2 point of view one might have guessedthis. Remember that in general the conformal boundary ofAdSD+1 is a D-manifold M with a conformal metric only. In otherwords, the metric tensor gIJ of M is a D ×D matrix that is definedup to gIJ → eφgIJ for arbitrary φ. In D = 1, a conformal structureisn’t anything at all, since g is a 1× 1 matrix that can becompletely gauge-fixed by multiplying by eφ. So the boundary ofAdS2 should have complete diffeomorphism symmetry. This is theanalog of special conformal invariance SO(D, 2) for D > 2, and ofthe Brown-Henneaux Virasoro symmetry in D = 2.

Although the integral equation with the d/dt term dropped hasthe full diffeomorphism symmetry, the particular solutionG = kω−1/2 does not. This solution only has SO(1, 2) ∼ SL(2,R)symmetry. By acting with diff S1, we can make aninfinite-dimensional family of solutions which is just a copy of thehomogeneous space diff S1/SO(1, 2).

The fact that AdS2 can be understood as a system with anasymptotic diff S1 symmetry that is spontaneously broken down toSO(1, 2) was not appreciated prior to the study of the SYK model.But possibly it should have been. I’ve already told you why thelogic of AdS/CFT leads one to think of AdS2 as a system with anasymptotic diff S1 symmetry. On the other hand, AdS2 itself hasonly SO(1, 2) as a group of symmetries. So we can think of diff S1

as being spontaneously broken to SO(1, 2). See recent papers byAlmheiri and Polchinski; Maldacena, Stanford, and Z. Yang; andEngelsoy, Mertens, and H. Verlinde.

However, in the context of the SYK model, the diff S1 symmetry isnot exact. We had to drop the d/dt term from the equation inorder to get this symmetry. When we incorporate the d/dt term,the diff S1/SO(1, 2) degeneracy is lifted and the associated“Goldstone boson” is turned into a soft mode, a “pseudoGoldstone boson.” This particular pseudo Goldstone boson isdescribed by a very peculiar action that was found by Kitaev.

What is the AdS2 interpretation of the fact that the diff S1

symmetry is explicitly broken by the d/dt term? Suppose we aredescribing a black hole in four dimensions. If it is electricallyneutral, we use the Schwarzschild or Kerr solution. For anelectrically charged black hole, we use Reissner-Nordstrom orKerr-Newman. In general black hole physics is hard. Trying to geta simpler case to study, we can look at an extremal or almostextremal black hole. In four dimensions, the extremal black holehas a near horizon region that is AdS2 × S2. The S2 isn’t veryimportant here and we can just say that AdS2 describes the nearhorizon geometry of an extremal black hole. The embedding ofAdS2 in some bigger spacetime (that has AdS2 for its near horizonregion) explicitly breaks the diff S1 symmetry of AdS2. In short,the explicit breaking of diff S1 means that the SYK modeldescribes a little more than the near horizon AdS2 region.

To summarize, the SYK model appears to be a novel andunexpected example of holographic duality, and maybe it will beuseful in giving an unexpected way to study black holes. Muchmore has been done than I’ve explained and in particular Kitaevand later authors have compared more than I have explained toexpectations from AdS2.

But I will end the general introduction here and spend theremaining time to explain my recent paper (“An SYK-Like ModelWithout Disorder”; see also a more recent paper by Klebanov andTarnopolsky).

A possible drawback of the SYK model is that the elementaryfermions ψk are gauge-invariant operators (trivially – there isn’tany gauge symmetry!) so in a gravitational dual description, theyshould correspond to some bulk fields. In better understandexamples of holographic duality, like N = 4 super Yang-Mills infour dimensions, the analogs of the ψk are not gauge-invariant anddon’t have bulk duals. One can therefore ask if there is some wayto modify the SYK model so that, instead of invoking randomcouplings, one has some specific couplings that have a symmetrythat could be gauged, so that the elementary fields ψk would notbe gauge-invariant and would not have bulk duals.

It turns out that there is a way to do this that is quitestraightforward. The basic idea is due to Gurau and Rivasseau (apaper I found helpful is V. Bonzom, R. Gurau, A. Riello, and V.Rivasseau, Critical Behavior Of Colored Tensor Models In TheLarge N Limit, Nucl. Phys. B853 (2011) 174-195; the idea ofrelating this work to the SYK model was also considered by J.Suh). Their original motivation was to generalize matrix models oftwo-dimensional gravity. In a matrix model, one considers a fieldφij that is a two-index tensor, where i , j = 1, . . . , n and a symmetrygroup SO(N) or U(N) or Sp(N) acts on the i and j indices.(There are somewhat different models with the same group or twodifferent groups acting on the i and j index; if the groups are thesame, one can constrain φij to be symmetric, antisymmetric,hermitian, etc. A hermitian matrix is the most-studied case.)

One then finds that the dominant Feynman diagrams in the largeN limit are planar diagrams.

One can think of aplanar diagram as representing a discrete 2d geometry (forexample, take all the polygons to be regular polygons with sides ofunit length). This proved to be a surprisingly useful way togenerate random 2d geometries and thereby to study models of 2dquantum gravity.

Searching for an analogous approach to quantum gravity aboved = 2, Gurau and Rivasseau (and earlier authors ca. 1991,including Ambjorn, Durhus, and Jonson; M. Gross; and Sasakura)replaced the matrix or 2-index tensor by a tensor with 3 or moreindices. So for example, it could be a field φijk with a separateSO(N) symmetry acting on each index i , j , or k.

It is possible to make a φ4 coupling in which each of the four φfields shares one of its three indices with each of the other fields.Instead of describing this by a formula, it is perhaps clearer todraw a picture:

In this picture, one can perhaps visualize a tetrahedron. This is thesupposed to be the germ of a relation to random 3-geometries.

However, Gurau, Rivasseau, et. al. showed that if one scales theφ4 coupling with N so that the model has a large N limit, thenonly a very restricted class of Feynman diagrams contribute in thelarge N limit. They are precisely the diagrams of the SYK model,the ones that are generated by the recursion procedure that Imentioned at the beginning:

(This is not hard to prove but I will omit the details. My papercontains a summary.)

This class of Feynman diagrams corresponds to a very special setof 3-geometries, which probably causes a problem for using arandom 3-index tensor field to describe quantum gravity in 3dimensions. But it is just what we want for getting an alternativeto the usual SYK model.

It means that instead of, as in the SYK model, considering Nfermions with random couplings, one can do the following. SetN = n3 for some large n and consider a fermion field ψijk ,i , j , k = 1, . . . , n, with an SO(n) symmetry on each index. For sucha field, one can write a quartic coupling ψ4 – in the configurationof the tetrahedron that we discussed before – and a model withthis coupling will have the same large N limit as the SYK model.

Now we can gauge the SO(n)3 symmetry. This does not changethe thermodynamics significantly for large n because ψ has n3

components and the dimension of the gauge group is of order n2.But after gauging, we expect only gauge-invariant fields made fromψ to have bulk duals, and not ψ itself. Thus the model will now bemore like better-established examples of holographic duality.

I’ve wondered if this variant of the SYK model has the followingadditional advantage, but as you will see I am not sure it does.Since the ensemble average of a quantum system with randomcouplings is not really a quantum system, an ensemble-averagedmodel may not suffice for addressing some subtle questions aboutquantum mechanics of black holes. So I thought removing therandomness from the SYK model might be important for thatreason. But I am not sure of this for the following reason.Although an ensemble average of quantum systems is not aquantum system, any typical example of the ensemble is. Morever,if N is large any typical example of the ensemble is very similar tothe ensemble average, for all quantities that can be computed bytaking a large N limit. It may be that any important properties ofa typical example of the ensemble that are lost when one takes theensemble average are also lost, in the other approach, when onetake the large n limit of the system with SO(n)3 symmetry.