Disorder in the Sachdev-Yee-Kitaev Model

We give qualitative arguments for the mesoscopic nature of the Sachdev-Yee-Kitaev (SYK) model in the holographic regime with $q^2/N\ll 1$ with $N$ Majorana particles coupled by antisymmetric and random interactions of range $q$. Using a stochastic deformation of the SYK model, we show that its characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity in the opposite regime with $q/N=1/2$, in leading order. The stochastic evolution of the SYK model can be mapped onto that of random matrix theory, with universal Airy oscillations at the edges. A spectral hydrodynamical estimate for the relaxation of the collective modes is made.


I. INTRODUCTION
The understanding of how entropy is produced and developed in heavy-ion collisions at ultra-relativistic energies is the subject of intensive interest at collider energies [1]. Key to this is the concept of thermalization and the time it takes to reach it [2][3][4][5][6]. More generally, there is a wide theoretical interest in the understanding of non-perturbative entropy formation in quantum processes ranging from atomic systems at the unitarity limit [7,8] to string theory using black-holes [9].
The SYK model consists of N quantum mechanical fermions with Gaussian distributed random couplings of rank-q and strength J. The model is solvable at large N and fixed J by a saddle point approximation where a special class of Feynman graphs is selected. Originally, this model was proposed by Sachdev and Yee [10] to describe quantum spin fluids. More recently, Kitaev [11] has suggested the model to shed light on holography, by arguing that the large N limit of the model is dual to a black hole in an emergent AdS 2 space-time. A number of investigations have since followed [12][13][14][15].
The SYK model offers a simple framework for discussing the formation of black holes in quantum mechanics [11]. In the regime q 2 /N 1, numerical analyses [14,15] support the existence of a chaotic regime described by random matrix theory at late times. The chaotic regime signals the onset of a black hole. The purpose of this paper is to stress the mesoscopic nature of the SYK model throughout its ballistic, diffusive and er-godic regimes. In the opposite regime with q 2 /N 1, we make use of spectral determinants, to show how a viscid fluid description emerges with a small spectral viscosity that maps exactly on random matrix theory. This regime is dominated by planar diagrams in leading order.
The organization of the paper is as follows: in section 2 we briefly outline the SYK model and discuss its bulk spectral distribution in the holographic limit with q 2 /N 1. In section 3, we provide a qualitative description of the mesoscopic nature of the model and give a simple estimate for the ergodic time. In section 4, we discuss the opposite limit with q 2 /N 1 using the SYK characteristic determinant. We show that it obeys a viscid Burgers equation which is analogous to the one derived using random matrix theory for the GUE ensemble. The inviscid equation gives rise to a semi-circular distribution, while the viscid equation gives rise to Airy universality at the edge of the spectrum. We use it to estimate the contribution of the edge states to the partition function at low temperature. We also suggest a spectral hydrodynamical estimate for the stochastic relaxation of the collective modes. Our conclusions are in section 5.

II. SYK MODEL
The SYK model consists of N -Majorana fermions with q-interactions in 0 + 1-dimensions, with random and antisymmetric couplings. The corresponding quantummechanical Hamiltonian say for q = 4 is with N -Majorana fermions ψ a of flavor a = 1, ..., N . The couplings J are antisymmetric in all entries and randomly sampled from a Gaussian distribution with zero mean but fixed variance N q = (q −1)!J 2 /N q−1 . All units are set by J which will be set to 1. As operators in a Hilbert space, the flavored Majorana ψ a map onto the representations of the Clifford algebra Cl[ N 2 ] as realized by γ a matrices (2) refers to a sum of sparse matrices with N 4 random weights. (1-2) exhibit particle-hole symmetry which is enforced by an anti-unitary operation. As a result the spectra exhibit some degeneracy for some values of N modulo 8 (Bott periodicity) [14,15].
The Hamiltonian in (2) is bounded and symmetric. The edge states map onto the low-lying N -body excitations close to the ground state, and the central states map onto the high lying N -body excitations. The latters follow a Gaussian distribution [14,15]. Specifically, consider the average partition function where uniform convergence is assumed (this is likely upset at the edges [16]). Formally, the moments in (3) are with typically (j 1 j 2 ...j 2k−1 j 2k ) = (i 1 i 1 ...i k i k ). Since the Γ s need at least one common factor in order to anticommute, and these pairs form only a small fraction of all pairs, we can throw away the anti-commutators in (4) at large N [15,16]. For a fixed sequence (i 1 i 2 ...i k ), the trace in each term of the j contribution is L and there are (2k − 1)!! such contributions. The final sum over the The partition function at high temperature (small β) is with the corresponding bulk entropy The inverse Laplace transform of (6) gives a Gaussian distribution of the central eigenvalues Overall, (8) is in agreement with the arguments and numerics presented in [13][14][15]. Eq. (8) fails at the edges of the spectrum [13][14][15]. For q 2 /N 1 the symmetric edges expand as ±N λ 0 with an exponential growth of states away from the edges given by sinh (2cN |E − N λ 0 |) 1 2 in the triple scaling limit [14].

III. MESOSCOPY
In [14,17] the deformed spectral form factor was defined as and analyzed both analytically and numerically in [14] with a sample of the results shown in Fig. 1. The features shown are generic of mesoscopic systems with multi-fermion induced interactions such as those developed in the disordered QCD vacuum [18] (and references therein).
FIG. 1: Spectral form factor for the SYK model from [14]. We super-imposed on the figure four mesoscopic regimes. An arrow points at the time corresponding to the ergodic (Thouless) time.
A useful formula for discussing mesoscopic systems is the semi-classical form of the spectral form factor at large β or small temperatures [19] for times much smaller than the quantum (Heisenberg) time, t < t H = 1/δ with δ = 2π/L the quantum energy spacing. Here p(t) is the classical return probability, typically of the form The microscopic and quantitative many-body analysis of (11) will be presented elsewhere [20]. Qualitatively, (11) can be thought as the probability of return for a given flavor undergoing anomalous Brownian motion in a linear volume V 1 = L. Each random walk spreads in a time t an effective squared distance X 2 ≈ t ∆ with ∆ the anomalous diffusion exponent (∆ = 1 for the canonical Brownian random walk). A simple estimate of the return probability (11) in this random walk approximation is p(t) ≈ V 1 /X. The ergodic (Thouless) time is reached when the random walks fill out the effective vol- For all super-diffusive random walks with ∆ > 2, (12) is in qualitative agreement with the slope in Fig. 1. In particular, for ∆ = 8, the result (12) is in agreement with the numerical results and estimates in [14,15]. The ballistic time is identified as t B ≈ L 0 below which the left plateau is seen in Fig. 1. In the ergodic regime with t E < t < t H we have p(t) ≈ 1, and grows linearly with time in overall agreement with the rise in Fig. 1. In the Heisenberg regime with t > t H , the spectral form factor is dominated by the self-correlation for a single energy level which is normalized to a deltafunction in energy space g(E → 0) = δ(E), and translates to a constant in time (14) which is the right plateau in Fig. 1. As in mesoscopic systems, we note the hierarchy of times The ergodic regime is universal and follows from random matrix theory and symmetries as observed in [14,15]. In the presence of time-reversal symmetry the counting of paths in (10) is increased by a factor of 2. The particularly short time t S ≈ lnL reported in [21] is of the order of the Ehrenfest time and may be a signal for the loss of quantum coherence at the edge of the ballistic regime.

IV. RANDOM MATRIX LIMIT
In the q 2 /N 1 limit, the SYK model provides a quantum mechanical realization of the holographic principle as discussed by many [12][13][14][15]. By increasing the q-range of the random interaction the model undergoes a transition to a random matrix regime a situation similar to the one encountered in the context of quantum spin glasses [16]. In this section we specialize to the case with maximum randomness with q/N = 1/2 which is opposite to the holographic regime. This regime is dual to random matrix theory and chaotic for all time scales in leading order, as we now show.

A. Ergodic evolution
To streamline the counting for the case with q/N = 1/2, it is more convenient to re-define (9) using the new normalization for the q-range couplings with a typical basis element of rank-p n in the minimal representation of the Clifford algebra Cl([ N 2 ]). There are C pn N such basis elements, and they all satisfy Γ 2 J = 1. The characteristic determinant for the SYK model is defined as with the measure P(τ, α J ) ≈ e − 1 2τ α J i α J i , and P(0, α J ) ≈ δ(α J ). We note that the measure reduces to a deltafunction as τ → 0, and asymptotes a Gaussian as τ > 1 which is the SYK model. Eq. (18) provides a stochastic deformation of the SYK model with vanishing couplings as τ → 0, much like in the random matrix deformation in [24].
To analyze (18) we set N = 2n and specialize to the case p n = n. This is the case with maximum range for the random couplings. With this in mind, we unwind the determinant using Grassmannians, and carry the Gaussian integration over the random couplings α J to obtain We now use a Fierz re-arrangement of the 4-Grassmannian induced interaction with For any large n, we have As a result the third line contributions to the Fierz rearrangement in (20) are all of order 1/n in comparison to the first two lines, and subleading. Therefore (20) simplifies to Since (−1) n/2 Γ 2n squares to one and we can write (23) as The labels ± refer to the positive-negative eigenvalues of the chirality matrix (−1) n/2 Γ 2n . This result is physically expected as the Fierzing in (20) rescinds the 4-Fermi induced interaction into all spin channels in the large Hilbert space. In leading 1/n, all the spin bearing channels wash out, except for the scalar and pseudoscalar channels with each carrying an effective coupling of 1/2L. The chiral copies in (25) reflect on the particlehole symmetry noted in [14,15,22,23]. Therefore at large n, the characteristic determinant (19) splits into two chiral copies with (26) with ν L = 1/L. It follows that each of the chiral copies in (26) close under ergodic evolution (reverse diffusion) (27) Using the complex Cole-Hopf transformation for the characteristic determinant f L = ∂ z lnΨ ± /L withL = L/2, (27) for the SYK model maps onto the viscid Burgers equation with ν L playing the role of a (negative) spectral viscosity [25]. In terms of the (cold) entropy S/N ≈ ln2/2 [13], the spectral viscosity is ν L = 1/e S . We note that (27)(28) map onto the ergodic equation for the characteristic determinant of the unitarity ensemble of random matrix theory of finite size L/2 and β D = 2 [25]. This mapping together with the semi-circular distribution (see below) guarentee that the spectral form factor in (9) is also of the general form shown in Fig. 1 in the random matrix regime with q/N = 1/2 and in leading order.

B. Airy universality
We now focus on one of the two chiral copies and study its spectrum. The formal solution to (27) is (29) which is the convolution of the diffusion kernel with the initial condition in L-space. The L-saddle point approximation to (29) yields the Cole-Hopf transform with 2z + = z+ √ z 2 − 4τ . Eq. (30) acts as a Coulomb-like potential for the macroscopic spectral density of eigenvalues with (L = L/2) which is semi-circular.
Eq. (31) can also be shown to follow from the moment analysis of H, if we were to note that all crossing diagrams are suppressed by powers of the ratio in (22), in comparison to the non-crossing diagrams. As a result only the planar contributions are retained for N = 2n and p n = n at large n, leading to the standard Pastur equation for the resolvent and a semi-circle.
The key feature of the semi-circle are its edges at ± √ 4τ with an accumulation of states of order Lλ 3/2 which suggests the microscopic re-scaling (unfolding) at the origin of the Airy universality (soft-edge universality). This follows from either the rescaled expansion around the saddle point in (29) [24], or the shock analysis of the viscid Burgers equation [25], with the result The characteristic determinant (32) and the inverse characteristic determinant capture the overall depletion of the eigenvalues at the edges [26][27][28]. This depletion is universal and for β D = 2 it also follows from the method of orthogonal polynomials. Either way, the result is [28,29] 1 The universal contribution of (33) to the partition function at low temperature is For large L, the integration is dominated by the large s-asymptotic of theAiry functions The first term yields the leading contribution to the partition function which results in a leading contribution to the entropy at low temperature This is twice the entropy noted in the holographic regime in leading order. In the random matrix regime (q 2 /N 1) the number of random degrees of freedom grows as L 2 and not as L. Finally, we observe that in the holographic regime (q 2 /N 1), the analogue of (37) in the large N limit is with E N /N = e 0 ≈ 0.0406 [13,14]. We note that (38) is of the form suggested in the context of the correspondence between a black hole and a highly excited string, with E N identified as the Rindler energy [30]. In contrast, (37) is dual to a black hole only if the (negative) ground state energy or spectrum edge λ 0 → N ln2/(2π) for q 2 /N 1. This can be checked numerically.

C. Spectral and thermal relaxations
In the random matrix regime, all states are chaotic in leading order, and we may ask about their stochastic relaxation (analogue of quasi-normal modes). For that we note that the correspondence with random matrix theory allows us to map the SYK evolution of eigenvalues to that of a fluid of eigenvalues [36]. The fluid deformation and relaxation are controlled by the local conservation of the density of eigenvalues and Eulerian dynamics in 1+1 dimension. In particular, the local spectral speed of sound can be read from [36] as c s ≈ 4β D / √ 4τ with β D = 1, 2, 4 (Dyson index). The characteristic relaxation time is the time it takes the sound density wave to cross the semi-circle, Finally, in the holographic regime, the observation that (38) is the entropy of a highly excited string on the Rindler horizon, suggests that the approach to thermal equilibrium in the SYK model is captured in the dual picture by the analogue of an in-falling string on a thermal black-hole [31,32]. For the latter, the entropy grows with the longitudinal momentum of the infalling matter at the Rindler horizon S(t) ≈ P (t) ≈ e λ L t with λ L = 2π/β [31,33]. In general, we conclude that the increase in the rate of the logarithm of the entropy is bounded by the Kolmogorov-Sinai (rate) entropy, i.e. dlnS/dt ≤ λ L . This is the chaos bound reported in [33] which is reminiscent of the Bekenstein bound [34]. Both saturates near a black hole. In contrast, a much smaller entropy rate was noted in classical chaotic systems far from equilibrium [35]. For the SYK model in the holographic regime, the time for the loss of quantum coherence (scrambling time) t L ≈ 1/λ L ≈ L 0 is at the edge of the ballistic regime. It is comparable to the thermalization times reported in the (higher dimensional) holographic models [3][4][5].

V. CONCLUSIONS
In the holographic regime with q 2 /N 1, we have presented qualitative arguments in support of the mesoscopic nature of the SYK model. In particular, the prechaotic phase appears super-diffusive. In the opposite regime with q 2 /N 1 the SYK model maps on random matrix theory in leading order. We have shown that in the ergodic phase, the characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity ν L = 1/L. In this regime, all the SYK spectrum is chaotic with universal Airy oscillations at the edges, in leading order. The characteristic spectral relaxation of the low lying modes is controlled by the spectral speed of sound.
While all our analysis was carried out for the complex representations with β D = 2 universality, we expect that the results carry for β D = 1, 4 including the soft edge uni-versality, after careful analysis of the real and quaternion representations of Cl ([ N 2 ]). An open problem is how the planar approximation established in the random matrix regime, can be used to organize the quantum mechanical. Finally, in the holographic regime, the analogy between the emergent black hole and a mesoscopic "quantum dot" offers the intriguing possibility for its realization in other mesocopic systems. The model may provide interesting insights for the estimate of thermalization times in current collider experiments.