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arXiv:2412.14799 · Journal of High Energy Physics

Nonlinear soft mode action for the large-p SYK model

Marta Bucca, Márk Mezei

Published Mar 12, 2025• 2 citations

Abstract

A bstract The physics of the SYK model at low temperatures is dominated by a soft mode governed by the Schwarzian action. In [1] the linearised action was derived from the soft mode contribution to the four-point function, and physical arguments were presented for its nonlinear completion to the Schwarzian. In this paper, we give two derivations of the full nonlinear effective action in the large p limit, where p is the number of fermions in the interaction terms of the Hamiltonian. The first derivation uses that the collective field action of the large- p SYK model is Liouville theory with a non-conformal boundary condition that we study in conformal perturbation theory. This derivation can be viewed as an explicit version of the renormalisation group argument for the nonlinear soft mode action in [2]. The second derivation uses an Ansatz for how the soft mode embeds into the microscopic configuration space of the collective fields. We generalise our results for the large- p SYK chain and obtain a “Schwarzian chain” effective action for it. These derivations showcase that the large- p SYK model is a rare system, in which there is sufficient control over the microscopic dynamics, so that an effective description can be derived for it without the need for extra assumptions or matching (in the effective field theory sense).

DOI: 10.1007/jhep03(2025)089

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The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.

A bstract The IR dynamics of SYK is that of the Schwarzian theory, the effective theory of broken reparametrization invariance. In the double scaling limit, SYK is completely solvable by chord diagrams, whose generating functional is a bilocal Liouville theory. At low temperatures a set of modes in this description becomes soft. We interpret them as reparametrization of some twisted time coordinates, and show explicitly that they lead to the nonlinear Schwarzian theory. We further consider deformations of the theory in the double scaling limit, giving rise to diagrams with multiple species of chords, and show that the generating functional is now a Liouville theory with multiple fields. These deformations can be tracked to the IR and we discuss how they affect the Schwarzian.

The SYK model has played an important role in recent developments in many-body quantum chaos. We study a spatially local generalisation of it: the SYK lattice. Starting from the nonlinear action of pseudo-Goldstone bosons that dominate its dynamics at low temperatures, in the long wavelength limit we reorganise this action as the effective field theory for fluctuating hydrodynamics, thereby showing how the hydrodynamic degrees of freedom embed into the microscopic description of the model. We compute the hydrodynamic effective action to high orders in the derivative expansion and determine all the corresponding transport coefficients. Hence this work derives hydrodynamics from the microscopic description of a strongly coupled quantum many-body system.