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arXiv:1801.06871 · Physical review. D/Physical review. D.

Large q expansion in the Sachdev-Ye-Kitaev model

Grigory Tarnopolsky

Published Jan 15, 2019• 33 citations
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Abstract

We consider the Sachdev-Ye-Kitaev (SYK) model, where interaction involves $q$ fermions at a time. We find the $1/{q}^{2}$ correction to the thermal two-point function in the large $q$ expansion. Using this result, we find the $1/{q}^{3}$ correction to the SYK free energy.

DOI
10.1103/physrevd.99.026010

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We consider the Sachdev-Ye-Kitaev (SYK) model, where interaction involves qqq fermions at a time. We find the 1/q21/{q}^{2}1/q2 correction to the thermal two-point function in the large qqq expansion. Using this result, we find the 1/q31/{q}^{3}1/q3 correction to the SYK free energy.
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 describe the nonequilibrium quench dynamics of the Sachdev-Ye-Kitaev models of fermions with random all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that, at long times, the fermion two-point function has a thermal form at a final temperature determined by energy conservation, and the numerical analysis is consistent with a thermalization rate proportional to this temperature. We also obtain an exact analytic solution of the quench dynamics in the large qqq limit of a model with qqq fermion interactions: in this limit, the thermalization of the two-point function is instantaneous.
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.
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 investigate the non-equilibrium dynamics of complex Sachdev-Ye-Kitaev (SYK) models in the q→∞q\rightarrow\inftyq→∞ 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→∞q\rightarrow\inftyq→∞ Hamiltonian for t≥0t\geq 0t≥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→∞q\rightarrow\inftyq→∞ Hamiltonian is a perfect thermalizer even for non-equilibrium states generated in this manner.
The Sachdev-Ye-Kitaev (SYK) model is a model of qqq interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large NNN expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in NNN. We then derive the form of the effective action to all orders.