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arXiv:2112.12109 · Physical Review Research

Lindbladian dissipation of strongly-correlated quantum matter

Lucas Sá, Pedro Ribeiro, Tomaž Prosen

Published Jun 30, 2022• 60 citations
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Abstract

We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos.It describes N strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by M quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators.Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour.By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy).In the large-N, large-M limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state.We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation.In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.

DOI
10.1103/physrevresearch.4.l022068

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We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos.It describes N strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by M quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators.Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour.By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy).In the large-N, large-M limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state.We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation.In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.
In interacting quantum many-body systems, relaxation toward equilibrium reflects a competition between internal chaotic dynamics and environmental dissipation. While conventional Markovian baths typically produce exponential decay, non-Markovian dissipation can give rise to more intricate behavior, including algebraic relaxation. We study an open Sachdev-Ye-Kitaev (SYK) model coupled to a pseudogapped fermionic bath, using the Keldysh formalism to compute steady-state correlations in the large-NNN limit. Our results uncover a rich dynamical phase diagram, with regimes of bath-driven power-law relaxation, chaos-driven exponential decay, and an intermediate pre-relaxation phase where exponential decay crosses over to algebraic decay. These findings demonstrate that non-Markovian environments can qualitatively reshape relaxation mechanisms in strongly correlated quantum many-body systems.
We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Of particular interest for us is the time evolution of the dissipative form factor, which quantifies the average overlap between the initial and time-evolved density matrices as an open quantum generalization of the Loschmidt echo. We find that the dissipative form factor exhibits dynamical quantum phase transitions. We analytically demonstrate a discontinuous dynamical phase transition in the limit of a large number of fermion flavors, which is formally akin to the thermal phase transition in the two-coupled SYK model between the black-hole and wormhole phases. We also find continuous dynamical phase transitions that do not have counterparts in the two-coupled SYK model. While the phase transitions are sharp in the limit of a large number of fermion flavors, their qualitative signatures are present even for a finite number of fermion flavors, as we show numerically.
We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are nonrandom, while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large NNN, where NNN is the number of Majorana fermion operators, and also in the limit of large NNN and MMM, where MMM is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite NNN, we also study the distribution of the eigenvalues of the SYK Lindbladians.
A bstract We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q -body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q . With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q , leading to classical dynamics of the operator size growth with dissipation.
Understanding the far-from-equilibrium dynamics of dissipative quantum systems, where dissipation and decoherence coexist with unitary dynamics, is an enormous challenge with immense rewards. Often, the only realistic approach is to forgo a detailed microscopic description and search for signatures of universal behavior shared by collections of many distinct, yet sufficiently similar, complex systems. Quantum chaos provides a powerful statistical framework for addressing this question, relying on symmetries to obtain information not accessible otherwise. This thesis examines how to reconcile chaos with dissipation, proceeding along two complementary lines. In Part I, we apply non-Hermitian random matrix theory to open quantum systems with Markovian dissipation and discuss the relaxation timescales and steady states of three representative examples of increasing physical relevance: single-particle Lindbladians and Kraus maps, open free fermions, and dissipative Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries, correlations, and universality of many-body open quantum systems, classifying several models of dissipative quantum matter. From a theoretical viewpoint, this thesis lays out a generic framework for the study of the universal properties of realistic, chaotic, and dissipative quantum systems. From a practical viewpoint, it provides the concrete building blocks of dynamical dissipative evolution constrained by symmetry, with potential technological impact on the fabrication of complex quantum structures. (Full abstract in the thesis.)
Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large N theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to N=60 Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, λ, across a wide range in temperatures, including in the regime where λ approaches the universal bound, λ=2π/β.