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arXiv:2303.14132 · Journal of Physics A Mathematical and Theoretical

Shannon entropy in quasiparticle states of quantum chains

Wentao Ye, Jiaju Zhang

Published Aug 27, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract We investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. For single-particle and double-particle states, we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to certain magnon excited states in the XXX chain in the scaling limit. We also calculate numerically the Shannon entropy and mutual information for triple-particle and quadruple-particle states in bosonic, fermionic, and XXX chains. We discover that Shannon entropy, unlike entanglement entropy, typically does not separate for quasiparticles with large momentum differences. Moreover, in the limit of large momentum difference, we obtain universal quantum bosonic and fermionic results that are generally distinct and cannot be explained by a semiclassical picture.

DOI
10.1088/1751-8121/ad7429

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Abstract We investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. For single-particle and double-particle states, we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to certain magnon excited states in the XXX chain in the scaling limit. We also calculate numerically the Shannon entropy and mutual information for triple-particle and quadruple-particle states in bosonic, fermionic, and XXX chains. We discover that Shannon entropy, unlike entanglement entropy, typically does not separate for quasiparticles with large momentum differences. Moreover, in the limit of large momentum difference, we obtain universal quantum bosonic and fermionic results that are generally distinct and cannot be explained by a semiclassical picture.
We investigate the quasiparticle excited state R\'enyi entropy of single, double and more intervals in the fermionic chain, bosonic chain and XY chain. When the gap of the theory is large or all the momenta of the quasiparticles are large, the R\'enyi Entropy takes a super universal form, which is independent of, among others, the model, the quasiparticle momenta, and the subsystem connectedness. We calculate analytically the R\'enyi Entropy in the extremely gapped limit by writing the excited states in terms of the local excitations and find different additional contributions to the super universal R\'enyi entropy in various models. The additional contributions to the super universal R\'enyi entropy cannot be neglected when the momentum differences of the excited quasiparticles are small, however, they are negligible when all the momentum differences are large. The R\'enyi entropy with additional terms derived in the extremely gapped limit is universal in the sense that it is valid in the slightly gapped and critical models as long as all the momenta of the excited quasiparticles are large. In the case of double interval in the XY chain we find new exact super universal as well as the universal results. In the case of the bosonic chain in the extremely massive limit we find analytically a novel formula, written as a permanent of a certain matrix, for the R\'enyi entropy. We support all of our analytical results with numerical calculations.
The Rényi entanglement entropies of quasiparticle excitations in the many-body gapped systems show a remarkable universal picture which is independent of the model, the quasiparticle momenta, and the connectedness of the subsystem. In this letter we calculate exactly the single-interval and double-interval Rényi entanglement entropies of quasiparticle excitations in the many-body gapped fermions, bosons, and XY chains and find additional contributions to the universal Rényi entanglement entropy in the excited states with quasiparticles of different momenta. The additional terms are different in different models, depend on the momentum differences of the quasiparticles, and are different for the single interval and the double interval. We derive the analytical Rényi entanglement entropy in the extremely gapped limit, matching perfectly the numerical results as long as either the intrinsic correlation length of the model or all the de Broglie wavelengths of the quasiparticles are small. When the momentum difference of any pair of distinct quasiparticles is small, the additional terms are non-negligible. We argue that the derived formulas can be applied to a wider range of models than those discussed here.
Abstract We investigate the subsystem Schatten distance, trace distance and fidelity between the quasiparticle excited states of the free and the nearest-neighbor coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain. The results support the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. From the quasiparticle picture, we get the universal subsystem distances that are valid when both the large energy condition and the large momentum difference condition are satisfied, by which we mean each of the excited quasiparticles has a large energy and the momentum difference of each pair of the excited quasiparticles is large. In the free fermionic and bosonic chains, we use the subsystem mode method and get efficiently the subsystem distances, which are also valid in the coupled fermionic and bosonic chains if the large energy condition is satisfied. Moreover, under certain limit the subsystem distances from the subsystem mode method are even valid in the XXX chain. We expect that the results can be also generalized for other integrable models.
Abstract We calculate exactly the entanglement content of magnon excited states in the integrable spin-1/2 XXX and XXZ chains in the scaling limit. In particular, we show that as far as the number of excited magnons with respect to the size of the system is small one can decompose the entanglement content, in the scaling limit, to the sum of the entanglement of particular excited states of free fermionic or bosonic theories. In addition we conjecture that the entanglement content of the generic translational invariant free fermionic and bosonic Hamiltonians can be also classified, in the scaling limit, with respect to the entanglement content of the fermionic and bosonic chains with the number operator as the Hamiltonian in certain circumstances. Our results effectively classify the entanglement content of wide range of integrable spin chains in the scaling limit.
Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
We calculate the typical bipartite entanglement entropy ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{N}$ in systems containing indistinguishable particles of any kind as a function of the total particle number NNN, the volume VVV, and the subsystem fraction f=VA/Vf={V}_{A}/Vf=VA​/V, where VA{V}_{A}VA​ is the volume of the subsystem. We expand our result as a power series ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{N}=afV+b\sqrt{V}+c+o(1)$, and find that ccc is universal (i.e., independent of the system type), while aaa and bbb can be obtained from a generating function characterizing the local Hilbert space dimension. We illustrate the generality of our findings by studying a wide range of different systems, e.g., bosons, fermions, spins, and mixtures thereof. We provide evidence that our analytical results describe the entanglement entropy of highly excited eigenstates of quantum-chaotic spin and boson systems, which is distinct from that of integrable counterparts.