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

Abstract

The nonlinear supermatrix σ-model is widely used to understand the physics of Anderson localization and the level statistics in noninteracting disordered electron systems. Here we develop a supersymmetry 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 superbosonized description of the model paves the way for the precise calculation of the window of universality in which random matrix theory is applicable to the chaotic SYK system, and for the derivation of the corresponding Wigner-Dyson eigenvalue statistics.

From operator statistics to wormholes

“…It provides little insight into how to obtain the spectral properties of a physical unitary system. An integral over the elements of the Hamiltonian in (2.6) provides an analytical handle to compute an average over such physical observables described above, [48][49][50][51]. The RMT was originally developed to model the strongly interacting Hamiltonians that arise in nuclear and atomic systems.…”

“…In relation to the SYK model itself, a relevant question that is studied in [51] concerns the relation between the more widely familiar collective field treatment of the SYK model (originally developed by [34,70,71]) to the σ-model approach discussed in the present work. A unified approach will be quite useful to develop a coherent picture.…”

Exact generating function of a zero-dimensional supersymmetric non-linear sigma model

“…Supersymmetry is a powerful tool for the study of random systems, ranging from random matrices theory (RMT), quantum chaotic and disordered systems [1][2][3], with connection to string theory and the SYK model [4][5][6]. In particular, it has proven to be most useful in the study of Anderson localization in the weak disorder limit.…”

Optical lattice platform for the SYK model

“…2 and 3, has attracted much interest as a strongly interacting system, which exhibits many prominent properties of modern theoretical physics including non-Fermi liquid behavior, AdS/CFT duality, fully chaotic behavior, and aspects of integrability. The model is an excellent building block for systems where all these properties reveal themselves and lead to fascinating physics that can be studied effectively [4][5][6][7][8][9][10][11][12][13][14][15] . The model consists of N fermions with all to all random interactions.…”