A simple model for Hawking radiation

We consider $n$ free Majorana fermions probing a SYK system comprising of $N$ Majorana fermions. We solve the full system in deep infrared and in large $N$ (as well as large $n$) limit. The essential physics of the SYK system is not affected by the probe Majoranas, except addition of another tower of primaries. The SYK system is seen to induce maximal chaos as well as the whole spectrum of primaries, on to the probe system. The renormalization of soft mode action is computed. We comment on features in common with Hawking radiation.


Introduction
Holography in two dimensions has proven to be rather subtle. This is because AdS 2 space can not support any finite energy excitations, leaving only the option for trivial AdS 2 /CF T 1 correspondence. Two dimensional holography is also intimately related to the physics of black holes [1][2][3] since the near horizon geometry of extremal black holes develop an AdS 2 factor. This entails symmetries of an extremal black hole as a quantum system are those of AdS 2 space. Triviality of AdS 2 /CF T 1 correspondence is manifested in this context in the fact that the microscopic description of extremal black holes is a gapped system [4,5] and therefore flows to a trivial CF T 1 .
More interesting possibilities emerge as one moves away from extremality. As has been suggested by Kitaev [6], in this case one can instead look for near-AdS 2 /near-CF T 1 duality.
Given these developments, it is only natural to wonder if one can build models describing interaction of such a black hole with a probe by similar models. Such a model might be obtained by adding "probe fields" to a model describing a black hole and must qualify the following conditions. Firstly, in order to be qualified as "probe" fields, addition of such fields must not affect the dynamics of original fields, to leading order. Secondly, one does not expect a probe to alter near horizon geometry significantly. This implies addition of probe fields should not change the emergence and breaking pattern of conformal symmetry. Lastly, since the black hole saturates the chaos bound [35], it might be expected to inflict high degree of chaos in the probe system. This should manifest itself through near-maximal Lyapunov exponents of various out of time order (OTOC) correlation functions.
A class of tensor models for black hole probe, satisfying the above conditions, was recently proposed in [36] (also see [37][38][39][40]). However it is desirable to develop disordered models for black hole probe as well. Although not fully quantum mechanical, the original SYK model [6] has the key advantage over Gurau-Witten model [25] that it is much easier to rewrite the theory in terms of bilocal variables for SYK model 1 . One can make educated guess about disordered model for black hole probe by making the following observation about D brane realisation of black holes.
Black hole can be realised as a bound state of large number of D0 branes, and a probe can be thought of as a smaller set of D0 branes, probing this condensate. The important point is that the black hole and the probe both are made of same constituents. This suggests that adding some more fermions to SYK model, interacting with original SYK system in a SYK-ish manner might lead to a model for black hole probe. In this paper, we present a such a model. The paper is organised as follows. In section 2, we set out by briefly reviewing the SYK model. In section 3 we present a disordered model for black hole probe and solve it in large N limit and deep infrared region, compute two point function, four point functions and analyse chaos. In section 4 we perform the disorder average to get a quantum mechanical system of bilocal fields. We derive previously derived results from this formalism and further compute subleading corrections to certain correlators. In section 5 we discuss simplifications that occur for large q. Finally in section 6 we discuss future directions.

A lightening review of the SYK Model
The SYK model [6] contains N Majorana fermions ψ i , i = 1, . . . , N , where N is a large number.
The Hamiltonian is taken to be (1) This model turns out to be solvable at large N limit, in deep infrared region. In this limit, leading diagrams are "melonic" and can be summed over without much difficulty.
One starts by noting that the contribution, shown in fig 1, to the propagator G(t) := This equation is invariant under the conformal 2 transformations However the solution to this equation spontaneously breaks this symmetry down to SL(2, R). This pattern of symmetry breaking is analogous to that of AdS 2 space, whose asymptotic symmetry group contains all reparameterizations of the boundary circle whereas the AdS 2 metric preserves only a SL(2, R) subgroup.
2 In 1 dimension any reparameterization is a conformal transformation.
Given that AdS 2 emerges as near horizon geometry of extremal black holes, one might wonder if SYK model can be thought of as a model for extremal black holes. However analysis of four point functions shows that the conformal symmetry is further broken explicitly. This is the pattern of symmetry breaking of nearly AdS 2 spaces, which in turn arises as near horizon geometry of near extremal black holes. Apart from resemblance in pattern of symmetry breaking, SYK model shares another non-trivial property of black holes, namely maximal chaos. Altogether SYK model stands some chance of being a good model for near-extremal black holes (or more optimistically an example of so called near-AdS 2 /near-CF T 1 duality).
Coming to four point functions, "gauge invariant" four point function has the following where  The blue lines represent a ψ field propagating. In the remaining part of the paper, blue lines will continue to represent ψ field. Any new field would be represented by a different color.
A ladder with n rungs is denoted as F ψ n and can be obtained from F ψ n−1 by acting with the kernel K ψ : The kernel K commutes with SL(2, R) generators. Given any generator J of SL(2, R), one has Here J i acts at time t i . Using (7) one can sum up the ladder diagrams to obtain SL(2, R) symmetry (8) of the kernel K and the fact that preserves the SL(2, R) symmetry, seems to suggest that F ψ preserves SL(2, R) symmetry as well.
However this is not the case, since K ψ happens to have an eigenvalue 1, implying F ψ (t 1 , t 2 ; t 3 , t 4 ) diverges in the strict conformal limit. To take care of this divergence, it is necessary to move away slightly from the conformal point corresponding to explicit breaking of conformal symmetry.
F ψ (χ) can be evaluated utilising conformal symmetry [7]. Of particular interest, is the χ → 0 limit, in which one has Here h m -s are the roots of the equation α 0 is a numerical constant 1 (q−1)J 2 0 b q and 2 F 1 -s are hypergeometric functions. Now we present a class of models, describing a probe interacting with a black hole. The black hole is described by the Hamiltonian (1) of SYK model and interaction of the probe is represented by a new piece H probe . This piece contains original ψ fields, representing degrees of freedoms of the black hole, as well as new fields κ x , x = 1, . . . , n, representing the probe degrees of freedom.
n is a large number, which however is much smaller than N . We would consider physics of this model up to leading order in 1/N and 1/n.
2 ≤ p < q is an even number. j i 1 ...i q−p ;x 1 ...xp are random couplings and have to be averaged over.
Disorder average is specified by H probe is a special case of the generalised SYK models considered in [18]. However the generalisation of H probe considered in [18] was the full Hamiltonian by itself, whereas in the present case H probe describes a probe to SYK system.
To see that κ x can really be thought of as probes, consider the simplest contributions of  It is easy to check that the left diagram is O(N ) and the right one is O(n). This indicates that introduction of κ fields does not alter thermodynamic properties of the system in n/N << 1, N → ∞ limit. Same holds for correlation functions since many of them are obtained by cutting lines from the vacuum melons. We will see this explicitly in what follows.

Propagators
To leading order the propagator G ψ (t 1 , t 2 ) continues to be given by (5). This can be understood

G κ :
The simplest leading contribution to G κ :  Following arguments similar to those used in deriving G, we have the following Schwinger-Dyson equation which coincides with (3) if we replace J 2 1 (G κ ) p by J 2 G ψ p . This implies the solution to (15) is where G c is given in (5). (16) implies that G κ follows the same pattern of spontaneous breaking of reparameterization symmetry as G ψ .

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. contribution to F ψ and is therefore subleading. It is actually possible to sum the contributions up at this order, which in turn dominates over 1/N corrections to F ψ coming purely from (1).

ψψψψ
We will revisit this issue in section 4.

κκκκ
Gauge invariant four point function of κ fields has the following structure:  Clearly, F κ has same structure as in F ψ and therefore Using, The factor of p−1 q−1 is of much significance. It implies that the conformal symmetry is not explicitly broken by F κ . This is because K ψ does not admit an eigenvalue q−1 p−1 . Thus to leading order, and there are no conformal symmetry breaking piece. Lack of breaking of conformal symmetry also implies lack of maximal chaos.
In χ → 0 limit, one has where h m -s are solutions to k c (h) = q−1 p−1 , with k c (h) being given by (12). One can check h m never takes integer values. In the limit p → q, one has h m → h m and consequently F κ (χ) → F ψ (χ).

3.2.3
ψψκκ Gauge invariant mixed four point function has the following structure Simplest leading contribution to F ψκ comes from the left diagram of fig 6. Afterwards one can go on adding appropriate rungs in both sides of the ladder to obtain other leading diagrams.
Summing these diagrams up, one has F ψκ has the following structure: F ψκ (χ) can be evaluated explicitly. Of particular interest is where c m , h m , h m , c m -s are given previously. This expression simplifies considerably in the limit p → q. Defining p−1 q−1 =: 1 − ∆, one has Using this we have The right hand side is not quite conformal due to the appearance of a ln χ term. This term should be clubbed with the non-conformal part of the correlator. Then the conformal part in this limit reads

Chaos
For early times, quantum chaos is encoded in exponential growth in OTOC-s, whereas for ultra large times, chaos shows up in level statistics. Here, we are concerned only about chaos in early times. To this end we consider the following OTOC-s: where repeated indices are summed over and y = ρ(β) 1/4 , ρ(β) being the thermal density matrix at inverse temperature β.
First we consider F ψ . It is given by an infinite sum of ladder diagrams, with rungs given by where G ψ R is the retarded Green's function and G ψ lr is a Wightman correlator: (25) R on F ψ one gets back F ψ except for the 0 th piece. However, for large t 1 , t 2 , the 0 th piece can be neglected. Thus in this limit F ψ can be approximated by eigenfunction of K ψ R with eigenvalue 1. In the conformal limit, eigenfunctions of K ψ R are given by with eigenvalues and hence λ ψ L = 2π/β. Thus F ψ is maximally chaotic. Now let us consider F κ . This is also given by an infinite sum of ladder diagrams, with rungs given by K κ R = p−1 q−1 K ψ R . Following the same line of logic, for large time F κ can be well approximated by an eigenfunction of K κ R with eigenvalue 1. This entails for large times where h is the solution to the following equation Apart from the case p = 2, where the solution is h = 0, this equation always has a solution with negative h, leading to a Lyapunov exponent, that is a finite fraction of the maximal value.
Lastly coming to F ψκ , double ladder structure of the diagrams imply that in appropriate limits, F ψκ can be expressed as a linear combination of eigenfunctions of K ψ R with eigenvalue 1 and that of K κ R with eigenvalue 1. Thus the leading behaviour of F ψκ is captured by Lyapunov exponent λ ψκ = 2π/β.

Bilocal dynamics
Performing disorder average over the path integral and then integrating original Majoranas out give an effective action in terms of bilocal fields. This is particularly interesting achievement, because bilocal fields, being gauge invariant, are analogs of Wilson lines in 0 + 1 dimension and thus this amounts to rewriting the theory in terms of Wilson lines. Such a rewriting, although physically desirable, has not been achieved in higher dimensions, to the best of our knowledge.
Therefore achievability of the same, although in certain limits, is quite significant.
Upon disorder averaging, one has where represents disorder average. The first factor can be borrowed from literature and reads, to leading order in 1/N where G ψ (t 1 , t 2 ) = 1 N i ψ i (t 1 )ψ(t 2 ) (not to be confused with the two point function G ψ ) is a bilocal field and Σ ψ (t 1 , t 2 ) is a Lagrange multiplier field. The second factor is given by where G κ (t 1 , t 2 ) = 1 n x κ x (t 1 )κ x (t 2 ) (not to be confused with the two point function G κ ) is a bilocal field and Σ κ (t 1 , t 2 ) is a Lagrange multiplier field. There are two large numbers in the game, n and N , with N >> n.
Saddle point equations and their solutions are given by Let us make the ansatz for the saddle point values. Then we have In deep infrared, equations of motions of Σ ψ and Σ κ read respectively Putting (31) and (32) into (33) yields two equations for G ψ , which are the same except different over all factors. For consistency these factors must be same. This gives Putting this in (32), we have Defining we see the solutions to (33) are The above equations can not be entirely trustable. This is because when expanded in powers of n/N , (37) contains corrections to arbitrarily large orders, which at some point would be smaller than "quantum" corrections and therefore can not be the whole story up to that order. In order to decide to what extent we should trust (37), or any other correlator for that matter, we note that a saddle point evaluation corresponds to vacuum graphs of O(N ) and O(n). Any diagram obtainable from these leading graphs, can be trusted. After cutting appropriate lines, O(N ) vacuum graphs gives leading (i.e. O(1)) contributions to G ψ (equivalently G saddle ψ ) and F ψ . Similarly O(n) vacuum graphs give leading (i.e. O(1)) contributions to G κ (equivalently G saddle κ ), F κ , F ψκ and subleading (i.e. O(n/N )) contribution to G ψ , F ψ .
To this end, we expand (37) to O(n/N ), to obtain In appendices A.1, A.2 we derive the same results using diagrammatic techniques. Matching of O(n/N ) terms in G κ does not follow from the above argument though. It is desirable to have a better understanding of this curious feature.
In order to obtain the four point functions, we expand the effective action around the saddle.
To this end let us define One can check dG ψ dΣ ψ = dg ψ dσ ψ and dG κ dΣ κ = dg κ dσ κ . Thus the measure in the path integral over bi-local fields can be replaced by Dσ ψ Dσ κ Dg ψ Dg κ .
(40) also implies that In order to evaluate these, we set out by noting that to quadratic order,
After integrating σ ψ and σ κ out, one has where This can be recovered from the diagrammatic methods as well (A.3). Second kind of contributions read where δG ψ = γG ψ . This can also be recovered from diagrams (A.3). Note that this matching does not depend on the exact value of γ.
It is straightforward to derive the remaining two four-point functions F κ and F ψκ , both of which should be evaluated only up to leading order using (44). Upon doing so, one promptly recovers (19) and (21) respectively.
5 Large q limit q → ∞ limit offers more analytical control over the problem. We will however restrict to the discussion of chaos in this limit. Other details of this limit closely parallel that of SYK model.
We refer the reader to [7] for detail and collect some bare essentials in the following.
Firstly in order to have well defined self energy, one holds the combination J := √ q J 2 (q−1)/2 fixed as q → ∞. Now we directly jump to the retarded kernel (24) relevant for chaos, where v is defined by βv cos πv 2 = βJ .
Following arguments given in section 3.3, for large times, we have In order to solve this, we make an ansatz Putting (47) and (50) in (49), and then acting ∂ t 1 ∂ t 2 on it, we get This is Schrödinger equation for the potential −2/cosh 2 x, which is known to have only one bound state with energy −1. This entails Next let us consider F κ R . We note K κ we go to q → ∞ with (q − p) fixed. Following the same steps one finds that the potential in the Schrödinger problem is changed from −2/cosh 2 x to (−2/cosh 2 x + 2∆/cosh 2 x). The eigenvalue problem can not be solved exactly anymore, so we use first order perturbation theory to find Smallest possible choice for q − p is 2. In that case Lastly, we come to F ψκ R . Arguments used in section 3.3 to determine λ ψκ L , imply that in appropriate regime, F ψκ can be expressed as a linear combination of two pieces, with Lyapunov exponents λ ψ L and λ κ L respectively. Thus the leading Lyapunov exponent for F ψκ R is λ ψκ L = λ ψ L .

Discussion and Future Directions
In this work we have presented a model for probes interacting with a near extremal black hole.
As one would physically expect for such a system, the probe system does not alter either the symmetries or the dynamics of the black hole to leading order in 1/N expansion. Further the probe develops high degree of chaos, quantified by near-maximal Lyapunov exponent, due to its interaction with the black hole. The number of degrees of freedom in the probe system provides a new large number n. The interplay between two large numbers makes the first subleading corrections to occur at order O(n/N ) and quite a few correlators can be computed up to this subleading order.
Explicating aspects of thermalisation in this system is a natural next step [42]. This model (as well as the ones considered in [36]) raises general curiosity about infliction of chaos during thermalisation process. In particular it would be interesting to explore if one can make any universal statement for infliction of chaos.  To first subleading order, Schwinger Dyson equation reads where G ψ = G ψ l + G ψ sl + . . . ; Σ ψ = Σ ψ l + Σ ψ sl + . . . and • denotes convolution product. We already know There are two classes of contributions to Σ ψ sl . First class of contributions can be summed up into fig 1, but one of the internal propagators replaced by G ψ sl . This gives the following contribution to Σ ψ sl (t, t ) J 2 0 (q − 1)G q−2 c (t, t )G ψ sl (t, t ) . J 2 1 q p − 1 (G κ (t, t )) p (G c (t, t )) q−p−1 .
This matches the saddle point answer derived from disorder average (38).
A.2 G κ : Making the ansatz G κ sl = νG c and using the expressions for G ψ sl and G κ l , we have the following Schwinger Dyson equation Solving this equation one gets µ = (q − p) 2 p 2 q (J 0 /J 1 ) 2/p , and therefore to O(n/N ), This matches the saddle point result (39).

A.3 F ψ :
There are two types of subleading corrections to F ψ . First type of subleading corrections come from a class of diagrams, that can be summed over to give (see fig 8) F ψ sl,1 = n N q − p − 1 q − 1 This matches the corresponding correction (45) computed using saddle point methods. Second type of corrections are somewhat less non-trivial and come essentially from the subleading correction to G ψ , which enters F ψ through F ψ 0 and K ψ . Using (58), we have F ψ sl,2 = γ pqN . This matches the corresponding correction (46) computed using saddle point methods.