Consider the Sachdev-Ye-Kitaev model with $N$ Majorana fermions and interaction order $q$. For even $4\leq q\ll \sqrt{N}$, we present a microscopic framework that determines the free energy limit at every fixed inverse temperature $β>0$. In particular, this gives the first rigorous derivation of the Schwinger-Dyson thermodynamics of the model. For all even $q\in \{4,6,...,N-4\}$, we further determine the asymptotic location of the largest eigenvalue, which answers a question posed by Feng-Tian-Wei \cite{FengTianWei1}. Consequently, we obtained the free energy limit along every sequence $β\equiv β_N\to\infty$.
Consider the Sachdev-Ye-Kitaev model with N Majorana fermions and interaction order q. For even 4≤q≪N, we present a microscopic framework that determines the free energy limit at every fixed inverse temperature β>0. In particular, this gives the first rigorous derivation of the Schwinger-Dyson thermodynamics of the model. For all even q∈{4,6,...,N−4}, we further determine the asymptotic location of the largest eigenvalue, which answers a question posed by Feng-Tian-Wei \cite{FengTianWei1}. Consequently, we obtained the free energy limit along every sequence β≡βN→∞.
We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian HSYK on n Majorana modes with k-body interactions, and prove that E∥HSYK∥op=(1−o(1))⋅2n/k for super-constant k≤o(n), where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an O(1)-multiplicative factor for all k<n/4.
Our key technical idea is identifying an explicit, deterministic linear operator x such that a fixed quadratic form of x2ℓ exactly equals the expected trace moments of the SYK Hamiltonian for every n and k. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of x, which we show is dominated by the spectrum of a natural (kn)-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on x and transform it into a certificate of a lower bound on the largest quadratic form on HSYK.
Beyond the SYK Hamiltonian, our techniques also apply to the k−local quantum spin glass (QSG), yielding E∥Hk−QSG∥op≤O(n/k) for all 1≤k≤n/2, thus improving the standard O(n) bound by a factor of k.
We present an exact solution in the large-N limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent μ∈[0,2]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-N Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-q and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-q Green's function and extracting the Lyapunov exponent from the 4-point function. The parameter μ continuously interpolates between a free theory at μ=0 and the conventional, maximally chaotic Gaussian SYK model at μ=2, with non-maximal chaos persisting throughout the intermediate regime 0<μ<2. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.
We solve numerically the large-N Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching ${10}^{\ensuremath{-}36}$ accuracy. Using this we compute the energy of the SYK model at low temperatures $T\ensuremath{\ll}J$ and obtain its series expansion up to T7.54. While it was suggested that the expansion contains terms T3.77 and T5.68, we find that the first noninteger power of temperature is T6.54, which comes from the two-point function of the fermion bilinear operator ${O}_{{h}_{1}}=\ensuremath{\chi}{\ensuremath{\partial}}_{\ensuremath{\tau}}^{3}\ensuremath{\chi}$ with scaling dimension ${h}_{1}\ensuremath{\approx}3.77$. The coefficient in front of T6.54 term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.
We derive an approximate analytical formula for the spectral density of the q-body Sachdev-Ye-Kitaev (SYK) model obtained by summing a class of diagrams representing leading intersecting contractions. This expression agrees with that of Q-Hermite polynomials, with Q a nontrivial function of $q\ensuremath{\ge}2$ and the number of Majorana fermions N. Numerical results, obtained by exact diagonalization, are in excellent agreement with this approximate analytical spectral density even for relatively small $N\ensuremath{\sim}8$. For $N\ensuremath{\gg}1$ and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to ${\ensuremath{\rho}}_{\mathrm{asym}}(E)=\mathrm{exp}[2{\mathrm{arcsin}}^{2}(E/{E}_{0})/\mathrm{log}\ensuremath{\eta}]$, where $\ensuremath{\eta}(N,q)$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions and E0 is the ground-state energy per particle. This spectral density reproduces the known result for the free energy in the large-q and large-N limit at arbitrary values of the temperature. In the infrared region, where the SYK model is believed to have a gravity dual, the analytical spectral density is given by $\ensuremath{\rho}(E)\ensuremath{\sim}\mathrm{sinh}[2\ensuremath{\pi}\sqrt{2}\sqrt{(1\ensuremath{-}E/{E}_{0})/(\ensuremath{-}\mathrm{log}\ensuremath{\eta})}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low-energy nuclear excitations. Results for level statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to modeling certain aspects of the nuclear dynamics.
Abstract In this paper we explain the relation between the free energy of the SYK model for N Majorana fermions with a random q-body interaction and the moments of its spectral density. The high temperature expansion of the free energy gives the cumulants of the spectral density. Using that the cumulants are extensive we find the p dependence of the 1/N 2 correction of the 2p-th moments obtained in [1]. Conversely, the 1/N 2 corrections to the moments give the correction (even q) to the β 6 coefficient of the high temperature expansion of the free energy for arbitrary q. Our result agrees with the 1/q 3 correction obtained by Tarnopolsky using a mean field expansion. These considerations also lead to a more powerful method for solving the moment problem and intersection-graph enumeration problems. We take advantage of this and push the moment calculation to 1/N 3 order and find surprisingly simple enumeration identities for intersection graphs. The 1/N 3 corrections to the moments, give corrections to the β 8 coefficient (for even q) of the high temperature expansion of the free energy which have not been calculated before. Results for odd q, where the SYK “Hamiltonian” is the supercharge of a supersymmetric theory are discussed as well.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.