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arXiv:2108.09866 · Quantum

Eigenstate entanglement in integrable collective spin models

Meenu Kumari, Alvaro Martin Alhambra

Published Apr 27, 2022• 12 citations
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Abstract

The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis {|j,m⟩}m=−jj in any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to 1/2 of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.

DOI
10.22331/q-2022-04-27-701

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The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis {|j,m⟩}m=−jj in any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to 1/2 of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.
We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.
Abstract We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of type. We evaluate in closed form the reduced density matrix of a block of L spins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as when L tends to infinity, where the coefficient a is equal to ( m − k )/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when their Rényi entropy R q becomes independent of the parameter q . We have also computed the Tsallis entanglement entropy of the ground state of these generalized Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when . Finally, in the case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of . This is also true in the case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an ( m + 1)-simplex in whose vertices are the weights of the fundamental representation of .
Bipartite and global entanglement are analyzed for the ground state of a system of NNN spin-1∕21∕21∕2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick Hamiltonian. Under certain conditions, which include the special case of supersymmetry, the ground state can be constructed analytically. In the case of antiferromagnetic coupling and for an even number of particles, the system has a finite energy gap and the ground state undergoes a smooth transition, as a function of the continuous anisotropy parameter $\ensuremath{\gamma}$, from a separable $(\ensuremath{\gamma}=\ensuremath{\infty})$ to a maximally entangled state $(\ensuremath{\gamma}=0)$. From the analytic expression for the ground state, the bipartite entanglement between two subsets of spins as well as the global entanglement are calculated. Despite the absence of a quantum phase transition a discontinuous change of the scaling of the bipartite entanglement with the number of spins in the subsystem is found at the isotropy point $\ensuremath{\gamma}=0$: While at $\ensuremath{\gamma}=0$ the entanglement grows logarithmically with the system size with no upper bound, it saturates for $\ensuremath{\gamma}\ensuremath{\ne}0$ at a level only depending on $\ensuremath{\gamma}$.
We discuss the behaviour of the entanglement entropy of the ground state in various collective systems. Results for general quadratic two-mode boson models are given, yielding the relation between quantum phase transitions of the system (signalled by a divergence of the entanglement entropy) and the excitation energies. Such systems naturally arise when expanding collective spin Hamiltonians at leading order via the Holstein–Primakoff mapping. In a second step, we analyse several such models (the Dicke model, the two-level Bardeen–Cooper–Schrieffer model, the Lieb–Mattis model and the Lipkin–Meshkov–Glick model) and investigate the properties of the entanglement entropy over the whole parameter range. We show that when the system contains gapless excitations the entanglement entropy of the ground state diverges with increasing system size. We derive and classify the scaling behaviours that can be met.
We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m − k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).
Understanding the emergence of chaos in many-body quantum systems away from semi-classical limits, particularly in spatially local interacting spin Hamiltonians, has been a long-standing problem. In these intrinsically quantum regimes, quantum chaos has been primarily understood through the correspondence between the eigensystem statistics of midspectrum eigenstates and the universal statistics described by random matrix theory (RMT). However, this correspondence no longer holds for finite-temperature eigenstates. Here we show that the statistical properties of finite-temperature eigenstates of quantum chaotic Hamiltonians can be accurately described by pure random states constrained by a local charge, with the average charge density of the constrained random state ensemble playing the same role as the average energy density of the eigenstates. By properly normalizing the energy density using a single Hamiltonian-dependent parameter that quantifies the typical energy per degree of freedom, we find excellent agreement between the entanglement entropy statistics of eigenstates and that of constrained random states. Interestingly, in small pockets of Hamiltonian parameter phase space which we previously identified as `maximally chaotic' [PRX 14, 031014 (2024)], we find excellent agreement not only at the level of the first moment, including O(1) corrections, but also at the level of statistical fluctuations. These results show that notions of maximal chaos -- in terms of how much randomness eigenstates contain -- can still be defined at finite temperature in physical Hamiltonian models away from semi-classical and large-NNN limits.