Chaos, entanglement, and non-equilibrium dynamics in Hermitian and non-Hermitian quantum many-body systems

Abstract

Quantum chaos and entanglement have appeared as two key intertwined concepts to characterize fundamental quantum correlations of quantum many-body states and to understand the emergence of thermal and non-thermal steady states in unitarily evolving isolated or non-unitary open quantum systems, e.g., subjected to repeated measurements. These topics have been the subject of enormous interest in quantum condensed matter and high-energy physics, as well as in quantum information sciences, with potential experimental verification and promise in emerging technologies.

First, I will discuss the impact of quantum fluctuations, symmetry breaking, and associated non-trivial dynamics on quantum chaotic growth, as captured by the out-of-time-ordered correlator (OTOC), in the quantum p-spin glass model, a paradigmatic model of quantum glass. We make a non-trivial extension of the formalism for calculating the OTOC to the replica-symmetry-breaking spin glass phase. Our findings show that quantum fluctuations affect quantum chaos differently in the paramagnetic and spin glass phases. We demonstrate a curious non-monotonic behavior of quantum chaos as we approach the classical limit near the glass transition temperature.

Calculating entanglement entropy is inherently challenging due to the exponential size of the Hilbert space in many-body systems. In this talk, I will introduce a new path integral formulation for Renyi entanglement entropy in generic Fermionic systems, enabling us to address entanglement entropy using standard many-body techniques, such as the dynamical mean field theory (DMFT) for the strongly correlated Hubbard model. Our findings show that Renyi entropy in the metallic state within DMFT in the two-dimensional Hubbard Model aligns well with the conformal field theory (CFT) predicted entanglement-to-thermal crossover formula. Additionally, I will discuss mutual information across the Mott metal-insulator transition.

Calculating entanglement entropy in generic non-interacting non-Hermitian systems presents a significant challenge, unlike in Hermitian systems. I will show that a new path integral formulation, similar to that for entanglement, can help overcome these challenges in non-Hermitian systems. Our approach allows us to directly access long-time non-equilibrium steady states, offering advantages over the existing numerical methods. Using this approach, I will demonstrate the effects of measurements on entanglement dynamics and steady-states, as captured by the non-Hermitian Hamiltonian evolution that originates in the "no-click limit" of the quantum measurement trajectory.

The Sachdev-Ye-Kitaev (SYK) model and its variants, which have holographic connections to black holes in quantum gravity, have also emerged as a new paradigm for understanding the non-Fermi liquid and other strongly correlated metallic phases in condensed matter systems. I will discuss the exact numerical computation of the entanglement entropy in this model for finite systems and compare it with exact saddle-point calculations using the path integral formalism in the large-N limit. Our study reveals interesting physics captured by Renyi entropy in these models.

Finally, I will discuss the fate of a non-Fermi liquid to Fermi liquid quantum phase transitions (QPT) at finite N in a variant of the SYK model, which is solvable in the large-N limit, where N represents the number of fermion sites. We will probe this QPT using different measures and also analyze the approach to thermalization in the two phases following a sudden quench. Finally, I will conclude with a future outlook.