Tensor Models for Black Hole Probes

The infrared dynamics of the SYK model, as well as its associated tensor models, exhibit some of the non trivial features expected of a holographic dual of near extremal black holes. These include developing certain symmetries of the near horizon geometry and exhibiting maximal chaos. In this paper we present a generalization of these tensor models to include fields with fewer tensor indices and which can be thought of as describing probes in a black hole background. In large $N$ limit, dynamics of the original model remain unaffected by the probe fields and the four point functions of the probe fields exhibit maximal chaos, a non trivial feature expected of a black hole probe. Interestingly probe primaries have the same dimensions as primaries of the original fields.


Introduction
The study of quantum mechanical models dual to gravitational systems in two dimensions remains a fascinating and difficult arena of research. Quite notably, simple and solvable examples of this duality have proved to be difficult to construct. Somewhat recently however, the fermionic quantum mechanics of SYK model [1] has been proposed as a system holographically dual to gravity and this has been studied extensively [2][3][4][5][6][7][8][9][10][11][12][13][14]. A key motivating factor for proposing the SYK model as a holographic dual of a black hole background is the fact that the time out of order four-point correlation functions saturate the so-called maximal chaos bound [15] which has been shown to hold in the bulk.
Another important feature of the SYK model is that the emergent conformal symmetry is both spontaneously and explicitly broken, which suggests that it is dual to a near AdS2 background.
In an interesting development, it has been shown [16] that to leading order in the large N expansion, the SYK model (which is disordered, hence not fully quantum mechanical) is identical to the fermionic tensor model of [17,18]. This has been subsequently generalized in a number of interesting directions [19][20][21][22][23][24][25].
In this paper we couple the Klebanov-Tarnopolsky model [19] and Gurau-Witten model [16] to lower-index fields and interpret the resulting quantum mechanical model as holographically dual to probes in a black hole background. This is in part motivated by examples where matrix models have been coupled to vector matter, where the latter can be considered as probes in a background described by the former 1 . The models we study are obtained by adding interactions between the original tensors of [16,19] and tensors of lower rank 2 . To leading order in the 1/N expansion, the additional tensors do not 1 See [26], [27] for an interesting recent example relevant for black hole physics, although this model is not maximally chaotic [28] 2 A model which couples rank three and rank one tensor fields has been considered in [24] but in our affect the physics of original tensors, thus they can rightfully be thought of as probes.
Furthermore, we find that all the four point functions exhibit maximal chaos and it is this feature which qualifies these models as toy models for probes in a black hole background.
We find that our models have the curious feature that the dimensions of primaries made of out of probe tensors, are identical to those of the original primaries.
The rest of the paper is organized as follows. In 2, we give brief introduction to the Klebanov-Tarnopolsky model [19] and discuss a class of possible modifications, that remain solvable in large N limit and in deep infrared. In 3, we consider simplest such model and discuss the propagators, four point functions, primaries and Lyapunov coefficients.
We also comment on possible further modifications of this model, that retain the necessary physics. In 4, we discuss similar modifications of Gurau-Witten model [16]. Finally in 5 we discuss future directions.
2 Interactions in the D=3 uncolored model

A lightening review of the KT Model
The KT model [19] contains a single real fermionic tensor ψ abc of rank 3. Each index transforms as a vector under SO(N ). To differentiate the three copies ofSO(N ) we write the first as SO(N ) 1 , the second as SO(N ) 2 and the third as SO(N ) 3 . The Hamiltonian is taken to be whose diagrammatic representation is given in fig 1. model we preserve the interactions purely between the three-index fields (D index fields for color D). This model can be obtained by "uncoloring" [29] the D=3 Gurau-Witten model [16].
The large N limit [17], [18] is defined as taking N → ∞ while keeping g 0 fixed. In this limit it is the melonic grpahs which contribute to leading order in 1/N and the simplest correction to the propagator comes from the melonic graph in fig 2. A factor of N 3 , coming from fields propagating in loops, cancels the 1/N 3 coming from the two vertices, giving an overall factor of N 0 . Additional melonic corrections to the propagator of the same order are obtained by replacing any of the internal or external propagators in the diagram by this melonic diagram itself. In the large N limit this class of diagrams 3 constitute the complete leading corrections to the free propagator. They can be summed up to give the exact propagator in deep IR as follows. Denoting G(t 1 , t 2 ) to be the propagator and Σ(t 1 , t 2 ) to be the 1PI two point function to leading order in 1/N , 3 Joining the ends of a propagator gives a vacuum diagram. Thus this class of diagrams also give leading contributions to free energy in large N limit. Joining the ends turns the external lines into internal ones and thus one gets an extra factor of N 3 . Since the propagators were O(N 0 ), this means that free energy scales as N 3 , which is good since the number of fields scales as N 3 . one has By definition and in the deep infrared, iω can be ignored. In position space, one obtains giving the following Schwinger-Dyson equation in deep IR. The result (5) is invariant under the conformal transformations 4 t → f (t) Thus the system develops an emergent conformal symmetry in the deep IR. The solution to (5) is which spontaneously breaks the conformal symmetry to SL(2, R).
Next one considers the "gauge invariant" four point function, which has the following structure 1 N 6   A ladder with n rungs is denoted as F n and can be obtained from F n−1 by acting with the kernel K: The kernel K commutes with SL(2, R) generators. Given any generator J of SL(2, R), one has Here J i acts on time t i . Using (9) one can sum up the ladder diagrams to obtain Combining (10) with the fact that preserves the SL(2, R) symmetry, we see that one can use SL(2, R) symmetry to evaluate (11), although the K = 1 subspace requires special care and ultimately results in the breaking of the conformal symmetry.

A class of solvable models
The KT model can be thought of as a toy model for near extremal black holes. A model which includes probes of this black hole should have additional fields which preserve the property of "maximal chaos" and we will present such a model later in 3. In this section we investigate some modifications of KT model, which remain solvable at large N .  E.g. the field κ (12) will have one red and one green lines, the field η (3) will have one blue line and so on. Permuting colors will give other vertices of same class.

Uncolored probe model
In this section we present a class of models motivated by our discussion in section 2.2 and compute the four point functions and Lyapunov coefficient. The Hamiltonian for the simplest of our models is given by where we have introduced factors of N in the interactions such that both g 0 and g 1 are We refer to the first term as KT term, which can be thought of as describing dynamics of a black hole, whose effective degrees of freedoms are captured in the field ψ. We refer to κ (which is same as κ (12) in last section) as the "probe field" and to the second term of 14 as the "probe term", which is to be thought of as describing the interaction between a black hole and the probe 5 . Interactions involving only probe fields are subleading and not considered in this work.
The simplest vacuum diagrams for (14)   The upper diagram, coming from KT term, scales as N 3 , whereas the lower diagram, coming from the probe term, scales as N 2 and thus gives subleading contributions to free energy. This implies that in the large N limit, thermodynamic properties of the system is entirely determined by the KT term. There are also subleading diagrams arising purely from KT interactions as well. For example the diagram in figure 9 also scales as N 2 . One can continue doing this and following logic similar to ψ propagators, one gets the following Schwinger Dyson equation for G κ whose solution is We emphasize that G κ is of same order as G ψ in the 1/N expansion but not in a g 0 expansion 6 . At the level of propagators there is an emergent conformal symmetry which is spontaneously broken to SL(2, R).
0 . This dependence is unlike G which scales as

Four point functions
We now consider four point functions involving various combinations of ψ and κ fields.
All connected graphs which contribute to these four point functions can be obtained by cutting two lines from appropriate melon diagrams.

ψψψψ
Like the propagator, four point functions of ψ fields are not affected by the presence of new interactions to leading order. Therefore the leading contribution to F, the connected , comes from summing up the ladder diagrams shown in fig 3 and is given by (11), which we repeat for convenience 7 where, Following [2] we define in the conformal limit, the normalized four point function: and note that F(χ) was evaluated in [2].
A central point of [2] is that in the strict conformal limit, the four point function of the SYK-model diverges. In addition to the finite and conformally invariant component of which diverges and breaks conformal symmetry (thus cannot be expressed as a function of χ alone). This non-conformal component of F ψ is denoted F ψ h=2 due to the manner in which the eigenvalues of K are parameterized in [2]. The unit eigen-subspace of K 7 From now on, we will use K for K ψ , F for F ψ and F 0 for F ψ 0 .
corresponds to h = 2 and from (17) we see that it is this subspace which leads to the divergence. One proceeds by regulating the spectrum of K and the first non-trivial correction to the unit eigenvalue of K is of order 1 βg 0 , which in turn gives a contribution of order where the ellipsis represent lower order terms in the expansion in βg 0 . Subject to certain assumptions, the first subleading term in (20) has been computed in [2].
For our later discussion of the spectrum, it will be useful to mention that in the short time limit χ → 0 we have

κκκκ
The gauge invariant four point function of the probe fields has the following structure: and obtained by cutting more complicated melons although it is easier to think of these ladder diagrams as being obtained by successively adding rungs constructed only from ψ-fields. 8 We define h 0 = 2 Figure 10: Diagrams contributing to 4 point function of κ The sum of ladders in fig 10 has the same structure as F ψ and one gets where we have used Similarly to (19) we define in the strict conformal limit and then (24) along with (16) implies that the conformally invariant component of the four point function has the same normalization as F ψ : From (24) we see that F κ has a conformally invariant part F κ h =2 and a component F κ h=2 which spontaneously breaks conformal symmetry. In addition, as for F ψ the Lyapunov coefficient is maximal. Interestingly, (24) implies that the probe fields make a copy of the spectrum of primaries of the theory with just the ψ-fields.

ψψκκ
The mixed four point function is The simplest contribution to F ψκ comes from a diagram obtained by cutting a ψ line and a κ line in the lower melon of fig 8. This is a ladder with a single rung as shown in fig 11a, where the uncontracted lines on the left correspond to ψ fields and thus have three unresolved components while those on the right correspond to κ fields and have two unresolved lines. We can express it as Figure 11: a) Basic vertex contributing to ψψκκ . b) One can continue adding rungs on both side of the first diagram to obtain this structure at all orders Now to generate the set of graphs which contribute to F ψκ at leading order in the 1/N expansion, one should add appropriate rungs on both sides of the given rung as presented in figure 11b. All the rungs on both the left and the right correspond to ψ-fields being exchanged as do all the side-rails except for two internal rails and the two uncontracted lines on the right, which represent κ-fields. The final result follows immediately from figure 11b is Similarly as for F ψ we have a component which depends only on the conformal cross ration χ and an additional component which breaks conformal symmetry: The conformal part F ψκ h =2 (χ) has a very similar structure to the conformal component of F ψ or F κ and we evaluate this in appendix A. Since the kernel K commutes with the generators of the conformal group, this computation utilizes essentially the same techniques as used in the computation of the conformal part of F ψ .
To regulate the divergence in the non-conformal h = 2 subspace one must compute the four point function taking into account broken conformal symmetry and further corrections in 1 βg 0 and 1 βg 1 . It remains a difficult task to precisely evaluate F κ h=2 along the lines of the strategy in [2] for computing F ψ h=2 and in the current work will only note the leading scaling behavior. Following a similar argument to that which we outlined in section 3.2.1 for the leading correction to F ψ h=2 , the double pole in (29) implies that after regulating the eigenspace of K, the leading contribution to F κ h=2 scales as which is a higher scaling in βg 0 that the leading contribution to F ψ h=2 of F κ h=2 .

Chaos
In a chaotic quantum system, out of time order correlation functions grow exponentially 9 .
In the present case, there are three out of time order correlators that one may look at in order to diagnose chaos, namely where, y = ρ(β) 1/4 , ρ(β) being the density matrix at inverse temperature β and repeated indices are summed over.
The out of time ordered correlators F ψ and F κ have the same ladder structure as corresponding correlator in SYK model. In both correlators, rungs are given by where G R is the retarded Green's function and G lr is a Wightman correlator: When F ψ (or F κ ) is acted upon by K R , one gets back F ψ (or F κ ), except the 0 th piece.
But this piece has a G R (t 1 )G R (t 2 ) in it, which is negligible for large t 1 , t 2 . So, in this limit, F ψ (or F κ ) is an eigenfunction of K R with eigenvalue 1. In order to solve for this, one can make an ansatz (we will write for F ψ , F κ has same expression up to an over all λ ψ L being the Lyapunov coefficient for F ψ (which is same as λ κ L , the Lyapunov coefficient for F κ ). Looking at eigenfunctions of K R , one finds they are given by in the conformal limit. It is then easy to see that only solution for k R (h) = 1 is h = −1.
In large t 1 , t 2 , one has F ψ (t 1 , t 2 ) = e π(t 1 +t 2 )/β cosh π β t 12 Comparing with (35), we see which is maximal Lyapunov coefficient [15] and black holes are known to saturate this bound. One also has λ κ L = 2π/β along similar lines, thus mimicking the chaotic dynamics of Hawking radiation.
For the remaining correlator F ψκ , a slightly modified version of the above argument holds. We define a new kernel then observe that in the limit t 1 , t 2 >> 1 Thus in this limit F ψκ (t 1 , t 2 ) is an eigenfunction of K R , with eigenvalue 1. Now since K R commutes with K R , we deduce from (41) that Thus the desired eigenfunction is essentially −1 (t 1 , t 2 ) given in (36) and F ψκ also has maximal Lyapunov coefficient λ ψκ L = 2π/β.

Spectrum
The SYK model (and its tensorial cousins) are widely believed to constitute an example of so called N AdS 2 /N CF T 1 [2]. Conformal primaries of the SYK model correspond to bulk fields whose masses are determined by the scaling dimensions of the primaries. To compute these, one first expresses the gauge invariant fermion bilinear as (we write it for the KT model) where h n is the conformal dimension of n th primary, i.e.
. To compute h n -s and c ψ,n -s, one notes in the limit t 12 → 0, t 34 → 0, Again in the same limit, one has a,b,c; a ,b ,c where c 2 m are given in (21). Comparing (45) and (46), one has h ψ,n = h n , c ψ,n = c n .
In the dual bulk theory, corresponding to a boundary primary of dimension h n one gets a bulk field φ n of mass m 2 n = h n (h n − 1).
In the present case, there are extra primaries due to κ-fields. We restrict to the simplest case of (14). Then similarly one gets These primaries lead to a new set of bulk fields φ n with same masses m 2 N = h n (h n − 1).
To leading order we have O m,ψ O n,κ = 0 ∀m, n. Thus for every mass level n, there is an independent U (1) symmetry that rotates O n,ψ and O n,κ . This symmetry is broken though by the following O(1/ √ N ) correction 10 coming from the ψ 2 κ 2 four point function: Now one can choose the following combination of primaries, that are orthonormal up to this order:

Adding additional fields
In section 2.2, we considered a general class of models which are are obtained by adding additional fields in KT model but remain solvable at large N in the deep infrared. We then considered a particular case in section 2. Here we shed some light on the general case.
First we restore permutation symmetry between the three copies of SO(N ). This amounts to introducing three new fields to the original KT model: κ (12) , κ (13) , κ (23) . This is implemented by replacing the probe term in (14) by the following term It is easy to check that G κ 12 = G κ 13 = G κ 23 = G κ , where G κ is same as (16).
Coming to four point functions, again it is easy to check that ψψκ (ij) κ (ij) is the same for all three (ij)-s and has the same expression as in (27). Similarly the four point is also same for all three (ij)-s and same as (23). Along with these, now there is a new four point function κ (ij) κ (ij) κ (jk) κ (jk) . This is the same for all three choices of (ij), (jk) by permutation symmetry. As an example, we consider the case (ij) = (12), (jk) = (23). It has the following structure and the simplest contribution to F κ is shown in figure 12. Figure 12: Simplest leading contribution to κ (12) κ (12) κ (23) κ (23) One can now continue adding rungs in the left, right and center to generate other contributions to F κ . The final answer is and the conformal part of F κ is computed in A. Conformal symmetry is broken by h = 2 subspace. Following the argument of section 3.2.3, we note that Generalizing the chaos computations of section 3.3, it follows that the corresponding also has maximal Lyapunov coefficient λ L = 2π/β.
Next, one can add fields carrying a single index. This corresponds to introducing the following interaction in the Hamiltonian: Starting with propagators, first we note all three G η i (i = 1, 2, 3) are the same by permutation symmetry and we will call them G η . The Schwinger Dyson equation for G η turns out to be very similar to G ψ and G κ and has the following solution The four point functions involving only ψ and κ are not affected by V 2 to leading order.
There are quite a few new four point functions involving η. It can be checked that they all show the same pattern of conformal symmetry breaking and exhibit maximal chaos.

Discussion and Future Directions
In this paper we have presented various tensor models which couple tensors of rank one and two to the tensor models of [16][17][18]. We have argued that these describe the interaction of probes with a near extremal black hole, in particular these models exhibit maximal chaos.

A Conformal part of various four point functions
All the four point functions has the following form First we note that for m = 1, the conformal piece is sum of residues at various simple poles 11 of 1/(1 − K), where k l = 1. Thus the case for all l are similar to l = 0, which is the case for original SYK. Thus we refer the reader to [2].
For m = 1, the situation is different from that of original SYK. We start with m = 2.
Let P h=2 be the projector onto h = 2 subspace and P h =2 its complement. Then We are interested in the conformal piece. Since K is diagonal in |h basis, the conformal piece reads As function of the SL(2, R) invariant variable χ this reads The reader would note the similarity between the right hand side of the above equation and the corresponding one in [2]. Only difference is that we have h|F h =2 where [2] had 11 There is a double pole at h = 2, but it cancells with the finite pieces coming from regulated F 2 [2]. h|F 0 . Since F h =2 is well behaved, just like F 0 , the computation runs parallel to that of F h =2 in [2] and one gets where k c (h m ) = 1. Using and h|h = π 2 tan πh 2h − 1 δ(h − h ) for the continuum tower h = 1/2 + is , we have tan πh m tan 2 πhm 2 Ψ hm (χ) .
In short time limit, i.e. χ → 0, this boils down to We have not taken 12 the double pole at h = 2. We hope this would cancel with finite contributions coming from regulated F 2 , just as it does for SYK model. But we do not attempt to compute it here.
To move on to m > 2, we note that we did not use the details of F h =2 to get to 64.
Thus if we denote the conformal part of h =2 , then we have the recursion relation If F (m)

B Four point functions with maximal chaos
For any four point function with form F m,n = K m (1 − K) n F 0 , the corresponding OTOC F m,n (t 1 , t 2 ) will have maximal Lyapunov coefficient. We note that if we act F m,n by K R m,n ≡ 1 − (1 − K R ) n /K m R , we get back F m,n except for a piece that can be ignored in large t 1 , t 2 limit. Thus in this limit F m,n (t 1 , t 2 ) is an eigenfunction of the operator K R m,n with eigenvalue 1. Now the operator K R m,n clearly commutes with K R and thus has same eigenfunctions, labelled by h with eigenvalues given by It is clear that k m,n (h) = 1 ⇒ k R (h) = 1. We have previously mentioned (see 3.3) that k R (h) = 1 is satisfied for h = −1 with the eigenfunction e −1 (t 1 , t 2 ) given in 36. This essentially implies that in long time limit F m,n = e −1 and therefore has maximal Lyapunov coefficient λ L = 2π/β.