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arXiv:1911.10171 · Journal of High Energy Physics

SYK correlators for all energies

Alexandre Streicher

Published Feb 1, 2020• 32 citations
arXivHTMLPDF

Abstract

Abstract The Sachdev-Ye-Kitaev (SYK) model, a theory ofNMajorana fermions withq-body interactions (a.k.a.q-local), becomes in the largeqlimit a conformally-broken Liouville field theory. Taking this limit preserves many interesting properties of the model, yet makes the theory as a whole much more tractable. Accordingly, we produce novel expressions for the two and four-point correlators at arbitrary temperature and find the surprising result they take a universal closed form. We note that these expressions correctly match onto and interpolate between previously-obtained low-energy results and simple high-energy perturbative checks. We find that the time-ordered four-point correlators are always determined by finite temperature OPEs into the identity and Hamiltonian, while the out-of-time-order four-point correlators remain nontrivial and always scramble. This has only been established in the conformal limit, so to find that it holds for largeqat all temperatures/couplings is a nontrivial result. Finally, we determine the system’s thermalization and scrambling rates and find that they always agree, regardless of temperature. This adds to the increasing body of evidence that there exists simple underlying structures in the dynamics ofq-local systems, such as those formed by SYK’s epidemic operator growth.

DOI
10.1007/jhep02(2020)048

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Abstract The Sachdev-Ye-Kitaev (SYK) model, a theory ofNMajorana fermions withq-body interactions (a.k.a.q-local), becomes in the largeqlimit a conformally-broken Liouville field theory. Taking this limit preserves many interesting properties of the model, yet makes the theory as a whole much more tractable. Accordingly, we produce novel expressions for the two and four-point correlators at arbitrary temperature and find the surprising result they take a universal closed form. We note that these expressions correctly match onto and interpolate between previously-obtained low-energy results and simple high-energy perturbative checks. We find that the time-ordered four-point correlators are always determined by finite temperature OPEs into the identity and Hamiltonian, while the out-of-time-order four-point correlators remain nontrivial and always scramble. This has only been established in the conformal limit, so to find that it holds for largeqat all temperatures/couplings is a nontrivial result. Finally, we determine the system’s thermalization and scrambling rates and find that they always agree, regardless of temperature. This adds to the increasing body of evidence that there exists simple underlying structures in the dynamics ofq-local systems, such as those formed by SYK’s epidemic operator growth.
Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large N theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to N=60 Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, λ, across a wide range in temperatures, including in the regime where λ approaches the universal bound, λ=2π/β.
A bstract In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, which can be measured by certain out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. In this article we develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all q -body interactions. We derive the full operator growth structure in the large q limit at all temperatures. We see that its temperature dependence has a remarkably simple form consistent with the slowing down of scrambling as temperature is decreased. Furthermore, our finite-temperature scrambling results can be modeled by a modified epidemic model, where the thermal state serves as a vaccinated population, thereby slowing the overall rate of infection.
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.
We present a diagrammatic expansion for thermal expectation values of low-weight Majorana strings in a fixed realization of the NNN-fermion Sachdev--Ye--Kitaev (SYK) model. At each order in the expansion, the observable is expressed as a finite sum of polynomials in the couplings multiplied by temperature-dependent kernels built from the large-NNN melonic propagator. At fixed βJβJβJ, the expansion is organized in powers of N−1/2N^{-1/2}N−1/2, and each order can be efficiently evaluated with classical resources polynomial in NNN. The expansion also extends to correlations at arbitrary imaginary times and sufficiently short real times. For the four-body SYK model, we show that the diagrammatic predictions give an excellent account of exact diagonalization results at sizes up to N=24N=24N=24. We also apply the method to the Maldacena--Qi (MQ) model formed from two identical copies of a fixed SYK realization. This enables the identification of a nearby SYK-like Hamiltonian whose thermofield double state has higher fidelity with the Maldacena--Qi ground state than the thermofield double of the original SYK Hamiltonian.
We solve numerically the large-NNN 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{T}^{7.54}T7.54. While it was suggested that the expansion contains terms T3.77{T}^{3.77}T3.77 and T5.68{T}^{5.68}T5.68, we find that the first noninteger power of temperature is T6.54{T}^{6.54}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{T}^{6.54}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 investigate the nonequilibrium dynamics of complex Sachdev-Ye-Kitaev (SYK) models in the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where q/2q/2q/2 denotes the order of the random Dirac fermion interaction. We extend previous results by Eberlein et al. [Phys. Rev. B 96, 205123 (2017)] to show that a single SYK $q\ensuremath{\rightarrow}\ensuremath{\infty}$ Hamiltonian for $t\ensuremath{\ge}0$ is a perfect thermalizer in the sense that the local Green's function is instantaneously thermal. The only memories of the quantum state for t<0t<0t<0 are its charge density and its energy density at t=0t=0t=0. Our result is valid for all quantum states amenable to a 1/q1/q1/q expansion, which are generated from an equilibrium SYK state in the asymptotic past and acted upon by an arbitrary combination of time-dependent SYK Hamiltonians for t<0t<0t<0. Importantly, this implies that a single SYK $q\ensuremath{\rightarrow}\ensuremath{\infty}$ Hamiltonian is a perfect thermalizer even for nonequilibrium states generated in this manner.