Supersymmetry method for interacting chaotic and disordered systems: the SYK model

Abstract

The nonlinear supermatrix $\sigma$-model is widely used to understand the physics of Anderson localization and the level statistics in noninteracting disordered electron systems. In contrast to the general belief that the supersymmetry method applies only to systems of noninteracting particles, we adopt this approach to the disorder averaging in the interacting models. In particular, we apply supersymmetry to study the Sachdev-Ye-Kitaev (SYK) model, where the disorder averaging has so far been performed only within the replica approach. We use a slightly modified, time-reversal invariant version of the SYK model and perform calculations in real-time. As a demonstration of how the supersymmetry method works, we derive saddle point equations. In the semiclassical limit, we show that the results are in agreement with those found using the replica technique. We also develop the formally exact superbosonized representation of the SYK model. In the latter, the supersymmetric theory of original fermions and their superpartner bosons is reformulated as a model of unconstrained collective excitations. We argue that the supersymmetry description of the model paves the way for precise calculations in SYK-like models used in condensed matter, gravity, and high energy physics.

17 pages, RevTex

References in corpus (16)

• An Investigation of AdS${}_2$ Backreaction and Holography
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Chaos, Complexity, and Random Matrices
• Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
• More on Supersymmetric and 2d Analogs of the SYK Model
• Higher Dimensional Generalizations of the SYK Model
• Competition between Chaotic and Non-Chaotic Phases in a Quadratically Coupled Sachdev-Ye-Kitaev Model
• Dispersive SYK model: band structure and quantum chaos
• Charge frustration and quantum criticality for strongly correlated fermions
• A note on the SYK model with complex fermions
• Diffusion in higher dimensional SYK model with complex fermions
• Bosonization for disordered and chaotic systems
• Differentiable potentials and metallic states in disordered one-dimensional systems
• Localization-delocalization transition in the quasi-one-dimensional ladder chain with correlated disorder
• Toda lattice representation for random matrix model with logarithmic confinement
• Superbosonization in disorder and chaos: The role of anomalies

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• Statistic physics in bound state in 3D fermi gas system
• Statistics of local level spacings in single- and many-body quantum chaos
• Exact generating function of a zero-dimensional supersymmetric non-linear sigma model
• Replica Symmetry Breaking in Random Non-Hermitian Systems