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Solving Lévy Sachdev-Ye-Kitaev Model

Budhaditya Bhattacharjee, William. E. Salazar, Alexei Andreanov, Dario Rosa

Abstract

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 $μ\in [0, 2]$. 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.

Solving Lévy Sachdev-Ye-Kitaev Model

Abstract

We present an exact solution in the large- 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 . Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large- Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large- and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large- Green's function and extracting the Lyapunov exponent from the -point function. The parameter continuously interpolates between a free theory at and the conventional, maximally chaotic Gaussian SYK model at , with non-maximal chaos persisting throughout the intermediate regime . 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.

Paper Structure

This paper contains 18 sections, 1 theorem, 135 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian and Partition Function
  3. Bosonic Oscillators
  4. Static saddles
  5. Fluctuating saddles
  6. The Schwinger-Dyson Equations
  7. Large$-q$ solution
  8. The Infrared Regime
  9. Chaos exponents at large-$q$
  10. Thermodynamics
  11. Large$-q$
  12. Finite$-q$
  13. Comments on the Bulk Dual
  14. Conclusions
  15. Author contributions
  16. ...and 3 more sections

Key Result

Theorem 1

$\blacktriangleleft$$\blacktriangleleft$

Figures (17)

  • Figure 1: Phase diagram of the Lévy Sachdev-Ye-Kitaev model. Top: for $\mu=0$ the model is free. Chaoticity increases with $\mu$ reaching maximum for $\mu=2$, the standard Sachdev-Ye-Kitaev model. Bottom: sparsity pattern of the model, as explained in the main text.
  • Figure 2: Schematic representation of the Bosonic oscillator approach. Each colored bubble represents a collective mode $V(G_I) = \int \psi_{i_1}\cdots \psi_{i_q} \mathrm{d}\tau$ which are linked to Bosonic creation-annihilation pair $a_I, a^\dagger_I$. The hard spheres represent the Majorana fermions.
  • Figure 3: Numerical solution of the Schwinger-Dyson equations (\ref{['eq:sd-eqn-1']}-\ref{['eq:sd-eqn-2']}): the Green function $G(\theta=\frac{2\pi \tau}{\beta})$ as a function of $\mu$ (a) and inverse temperature $\beta$ (b), $J=1$. Left, (a): $G(\theta)$ tends towards the free theory solution (independent of $\beta$) for any $\beta$ as $\mu$ decreases. The red curve is the Gaussian SYK result. Right, (b): $G(\theta)$ changes strongly with $\beta$ for $\mu$ close to $2$. Away from $\mu=2$ the $\beta$-dependence of $G$ is weaker.
  • Figure 4: Solution $\nu$\ref{['eq:eff-int-mu-s']} as a function of $\mathrm{J}_a$ and $\mathrm{J}_b$. The non-trivial dependence of $\nu$ on $\mu$ present on the plot vs. $\mathrm{J}_a$ weakens, e.g. different curves almost collapse, on the plot vs. $\mathrm{J}_a$.
  • Figure 5: The scaling of $\nu$\ref{['eq:eff-int-mu-s']} with $\beta$ (left) and $\beta^{\mu/2}$ (right) vs $\beta$ and $\mathrm{J}_b\sim \beta^{\mu/2}$. A nearly perfect collapse is observed for the plot of $\nu/\beta^{\nu/2}$ vs $\mathrm{J}_b$. The solid black line $\mathcal{J}$ is the limit $\lim_{\beta\rightarrow0}\nu/\beta$ for the Gaussian SYK.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Theorem : Stochastic Representation of Lévy Distribution