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arXiv:2203.09158 · Journal of Statistical Mechanics Theory and Experiment

Thermodynamic symmetry resolved entanglement entropies in integrable systems

Lorenzo Piroli, Eric Vernier, Mario Collura, Pasquale Calabrese

Published Jul 1, 2022• 28 citations
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Abstract

Abstract We develop a general approach to compute the symmetry-resolved Rényi and von Neumann entanglement entropies (SREE) of thermodynamic macrostates in interacting integrable systems. Our method is based on a combination of the thermodynamic Bethe ansatz and the Gärtner–Ellis theorem from large deviation theory. We derive an explicit simple formula for the von Neumann SREE, which we show to coincide with the thermodynamic Yang–Yang entropy of an effective macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin chain, we test our result against iTEBD calculations for thermal states, finding good agreement. As an application, we provide analytic predictions for the asymptotic value of the SREE following a quantum quench.

DOI
10.1088/1742-5468/ac7a2d

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Abstract We develop a general approach to compute the symmetry-resolved Rényi and von Neumann entanglement entropies (SREE) of thermodynamic macrostates in interacting integrable systems. Our method is based on a combination of the thermodynamic Bethe ansatz and the Gärtner–Ellis theorem from large deviation theory. We derive an explicit simple formula for the von Neumann SREE, which we show to coincide with the thermodynamic Yang–Yang entropy of an effective macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin chain, we test our result against iTEBD calculations for thermal states, finding good agreement. As an application, we provide analytic predictions for the asymptotic value of the SREE following a quantum quench.
Abstract We study the behaviour of Rényi entropies in a generic thermodynamic macrostate of an integrable model. In the standard quench action approach to quench dynamics, the Rényi entropies may be derived from the overlaps of the initial state with Bethe eigenstates. These overlaps fix the driving term in the thermodynamic Bethe ansatz (TBA) formalism. We show that this driving term can be also reconstructed starting from the macrostate’s particle densities. We then compute explicitly the stationary Rényi entropies after the quench from the dimer and the tilted Néel state in XXZ spin chains. For the former state we employ the overlap TBA approach, while for the latter we reconstruct the driving terms from the macrostate. We discuss in full detail the limits that can be analytically handled and we use numerical simulations to check our results against the large time limit of the entanglement entropies.
We study the behaviour of Renyi entropies in a generic thermodynamic macrostate of an integrable model. In the standard quench action approach to quench dynamics, the Renyi entropies may be derived from the overlaps of the initial state with Bethe eigenstates. These overlaps fix the driving term in the thermodynamic Bethe ansatz (TBA) formalism. We show that this driving term can be also reconstructed starting from the macrostate's particle densities. We then compute explicitly the stationary Renyi entropies after the quench from the dimer and the tilted Neel state in XXZ spin chains. For the former state we employ the overlap TBA approach, while for the latter we reconstruct the driving terms from the macrostate. We discuss in full detail the limits that can be analytically handled and we use numerical simulations to check our results against the large time limit of the entanglement entropies.
Abstract In this paper, we investigate symmetry-resolved entanglement entropy (SREE) in free bosonic quantum many-body systems. Utilizing a lattice regularization scheme, we compute symmetry-resolved Rényi entropies for free complex scalar fields and a specific class of non-local field theories, where entanglement entropy (EE) exhibits volume-law scaling. We present effective and approximate eigenvalues for the correlation matrix used in computing SREE and demonstrate their consistency with numerical results. Furthermore, we explore the equipartition of EE, verifying its effective behavior in the massless limit. Finally, we comment on EE in non-local quantum field theories and provide an explicit expression for the symmetry-resolved Rényi entropies.
Entropy is a fundamental concept in equilibrium statistical mechanics, yet its origin in the nonequilibrium dynamics of isolated quantum systems is not fully understood. A strong consensus is emerging around the idea that the stationary thermodynamic entropy is the von Neumann entanglement entropy of a large subsystem embedded in an infinite system. Also motivated by cold-atom experiments, here we consider the generalization to R\'enyi entropies. We develop a new technique to calculate the diagonal R\'enyi entropy in the quench action formalism. In the spirit of the replica treatment for the entanglement entropy, the diagonal R\'enyi entropies are generalized free energies evaluated over a thermodynamic macrostate which depends on the R\'enyi index and, in particular, is not the same state describing von Neumann entropy. The technical reason for this perhaps surprising result is that the evaluation of the moments of the diagonal density matrix shifts the saddle point of the quench action. An interesting consequence is that different R\'enyi entropies encode information about different regions of the spectrum of the postquench Hamiltonian. Our approach provides a very simple proof of the long-standing issue that, for integrable systems, the diagonal entropy is half of the thermodynamic one and it allows us to generalize this result to the case of arbitrary R\'enyi entropy.
Abstract We study the Rényi entropies in the spin- 1 / 2 anisotropic Heisenberg chain after a quantum quench starting from the Néel state. The quench action method allows us to obtain the stationary Rényi entropies for arbitrary values of the index α as generalised free energies evaluated over a calculable thermodynamic macrostate depending on α . We work out this macrostate for several values of α and of the anisotropy Δ by solving the thermodynamic Bethe ansatz equations. By varying α different regions of the Hamiltonian spectrum are accessed. The two extremes are α → ∞ for which the thermodynamic macrostate is either the ground state or a low-lying excited state (depending on Δ) and α = 0 when the macrostate is the infinite temperature state. The Rényi entropies are easily obtained from the macrostate as function of α and a few interesting limits are analytically characterised. We provide robust numerical evidence to confirm our results using exact diagonalisation and a stochastic numerical implementation of Bethe ansatz. Finally, using tDMRG we calculate the time evolution of the Rényi entanglement entropies. For large subsystems and for any α , their density turns out to be compatible with that of the thermodynamic Rényi entropies.
We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Renyi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z. Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large-z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.