<!--HTML-->Information-theoretic quantities such as Renyi entropies show a remarkable universality in their late-time behaviour across a variety of chaotic systems. Understanding how such common features emerge from very different microscopic dynamics remains an important challenge. In this talk, I will address this question in a general class of Brownian models with a variety of different microscopic couplings. In any such model, the Lorentzian time-evolution of the n-th Renyi entropy can be mapped to a transition amplitude with a Euclidean Hamiltonian on 2n copies of the system. Using the structure of the ground states of the Euclidean Hamiltonian, we derive an expression for the saturation value of the n-th Renyi entropy in terms of an equilibrium density matrix, consistent with a general proposal called the equilibrium approximation. In non-integrable models without conserved quantities in (1+1) dimensions, we find that the spectrum of the Euclidean Hamiltonian relevant for the second Renyi entropy and OTOCs has two degenerate ground states, and a well-defined one-particle band consisting of plane waves of domain walls between them. The one-particle band gives rise to the membrane formula for entanglement dynamics, with the velocity-dependent membrane tension determined by its dispersion relation. This structure provides an understanding of entanglement dynamics in terms of a universal set of gapped modes, analogous to the gapless modes which govern hydrodynamic behaviour of conserved charge densities.
<!--HTML-->Information-theoretic quantities such as Renyi entropies show a remarkable universality in their late-time behaviour across a variety of chaotic systems. Understanding how such common features emerge from very different microscopic dynamics remains an important challenge. In this talk, I will address this question in a general class of Brownian models with a variety of different microscopic couplings. In any such model, the Lorentzian time-evolution of the n-th Renyi entropy can be mapped to a transition amplitude with a Euclidean Hamiltonian on 2n copies of the system. Using the structure of the ground states of the Euclidean Hamiltonian, we derive an expression for the saturation value of the n-th Renyi entropy in terms of an equilibrium density matrix, consistent with a general proposal called the equilibrium approximation. In non-integrable models without conserved quantities in (1+1) dimensions, we find that the spectrum of the Euclidean Hamiltonian relevant for the second Renyi entropy and OTOCs has two degenerate ground states, and a well-defined one-particle band consisting of plane waves of domain walls between them. The one-particle band gives rise to the membrane formula for entanglement dynamics, with the velocity-dependent membrane tension determined by its dispersion relation. This structure provides an understanding of entanglement dynamics in terms of a universal set of gapped modes, analogous to the gapless modes which govern hydrodynamic behaviour of conserved charge densities.
Information-theoretic quantities such as Renyi entropies show a remarkable universality in their late-time behavior across a variety of chaotic many-body systems. Understanding how such common features emerge from very different microscopic dynamics remains an important challenge. In this work, we address this question in a class of Brownian models with random time-dependent Hamiltonians and a variety of different microscopic couplings. In any such model, the Lorentzian time evolution of the n th Renyi entropy can be mapped to evolution by a Euclidean Hamiltonian on 2 n copies of the system. We provide evidence that in systems with no symmetries, the low-energy excitations of the Euclidean Hamiltonian are universally given by a gapped quasiparticlelike band. The eigenstates in this band are plane waves of locally dressed domain walls between ferromagnetic ground states associated with two permutations in the symmetric group S n . These excitations give rise to the membrane picture of entanglement growth, with the membrane tension determined by their dispersion relation. We establish this structure in a variety of cases using analytical perturbative methods and numerical variational techniques and extract the associated dispersion relations and membrane tensions for the second and third Renyi entropies. For the third Renyi entropy, we argue that phase transitions in the membrane tension as a function of velocity are needed to ensure that physical constraints on the membrane tension are satisfied. Overall, this structure provides an understanding of entanglement dynamics in terms of a universal set of gapped low-lying modes, which may also apply to systems with time-independent Hamiltonians.
There is mounting evidence that entanglement dynamics in chaotic many-body quantum systems in the limit of large subsystems and long times is described by an entanglement membrane effective theory. In this paper, we derive the membrane description in a solvable chaotic large-N model, the Brownian SYK chain. This model has a collective field description in terms of fermion bilinears connecting different folds of the multifold Schwinger-Keldysh path integral used to compute Rényi entropies. The entanglement membrane is a traveling wave solution of the saddle point equations governing these collective fields. The entanglement membrane is characterised by a velocity v and a membrane tension E(v) that we calculate. We find that the membrane has finite width for v<vB (the butterfly velocity), however for v>vB, the membrane splits into two wave fronts, each moving with the butterfly velocity. Our results provide a new viewpoint on the entanglement membrane and uncover new connections between quantum information dynamics and scrambling.
We give a detailed theory for the leading coarse-grained dynamics of entanglement entropy of states and of operators in generic short-range interacting quantum many-body systems. This includes operators spreading under Heisenberg time evolution, which we find are much less entangled than "typical" operators of the same spatial support. Extending previous conjectures based on random circuit dynamics, we provide evidence that the leading-order entanglement dynamics of a given chaotic system are determined by a function E(v), which is model-dependent, but which we argue satisfies certain general constraints. In a minimal membrane picture, E(v) is the "surface tension" of the membrane and is a function of the membrane's orientation v in spacetime. For one-dimensional (1D) systems this surface tension is related by a Legendre transformation to an entanglement entropy growth rate Γ(∂S/∂x) which depends on the spatial "gradient" of the entanglement entropy S(x,t) across the cut at position x. We show how to extract the entanglement growth functions numerically in 1D at infinite temperature using the concept of the operator entanglement of the time evolution operator, and we discuss possible universality of E at low temperatures. Our theoretical ideas are tested against and informed by numerical results for a quantum-chaotic 1D spin Hamiltonian. These results are relevant to the broad class of chaotic many-particle systems or field theories with spatially local interactions, both in 1D and above.
The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs. However, the full theoretical origin of this picture is not fully established yet. In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations. We apply the universal framework of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach. Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dynamics of Rényi entanglement entropies can be fully derived from the BFT. In particular, we emphasise that long-range correlations develop after quantum quenches, and accounting for these explain the structure of the entanglement growth. We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier. The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.
Entanglement membrane theory is an effective coarse-grained description of entanglement dynamics and operator growth in chaotic quantum many-body systems. The fundamental quantity characterizing the membrane is the entanglement line tension. However, determining the entanglement line tension for microscopic models is in general exponentially difficult. We compute the entanglement line tension in a recently introduced class of exactly solvable yet chaotic unitary circuits, so-called generalized dual-unitary circuits, obtaining a nontrivial form that gives rise to a hierarchy of velocity scales with v E < v B . For the lowest level of the hierarchy, L ¯ 2 circuits, the entanglement line tension can be computed entirely, while for the higher levels the solvability is reduced to certain regions in spacetime. This partial solvability enables us to place bounds on the entanglement velocity. We find that L ¯ 2 circuits saturate certain bounds on entanglement growth that are also saturated in holographic models. Furthermore, we relate the entanglement line tension to temporal entanglement and correlation functions. We also develop methods of constructing generalized dual-unitary gates, including constructions based on complex Hadamard matrices that exhibit additional solvability properties and constructions that display behavior unique to local dimension greater than or equal to three. Our results shed light on entanglement membrane theory in microscopic Floquet lattice models and enable us to perform nontrivial checks on the validity of its predictions by comparison to exact and numerical calculations. Moreover, they demonstrate that generalized dual-unitary circuits display a more generic form of information dynamics than dual-unitary circuits. Published by the American Physical Society 2024
In certain analytically tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables reduces to an effective theory of an "entanglement membrane" in spacetime.These tractable systems involve an average over random local unitaries defining the dynamical evolution.We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries.Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits.This viewpoint provides a framework for applying ideas of coarse graining to dynamical quantities in chaotic systems.We apply the approach to some translationally invariant Floquet spin chains studied in the literature.We show that a consistent line tension may be defined for the entanglement membrane and that there are qualitative differences in this tension between generic models and "dualunitary" circuits.These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to nonrandom Floquet models.We also provide an efficient numerical algorithm for determining the entanglement line tension in 1 þ 1D.