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arXiv:2305.11211 · Physical review. B./Physical review. B

Average pure-state entanglement entropy in spin systems with SU(2) symmetry

Rohit Patil, Lucas Hackl, George R. Fagan, Marcos Rigol

Published Dec 1, 2023• 13 citations
arXivHTMLPDF

Abstract

We study the effect that the SU(2) symmetry, and the rich Hilbert space structure that it generates in lattice spin systems, has on the average entanglement entropy of highly excited eigenstates of local Hamiltonians and of random pure states. Focusing on the zero total magnetization sector $({J}_{z}=0)$ for different fixed total spin $J$, we argue that the average entanglement entropy of highly excited eigenstates of quantum-chaotic Hamiltonians and of random pure states has a leading volume-law term whose coefficient ${s}_{A}$ depends on the spin density $j=J/(\mathfrak{j}L)$, with ${s}_{A}(j\ensuremath{\rightarrow}0)=ln(2\phantom{\rule{0.16em}{0ex}}\mathfrak{j}+1)$ and ${s}_{A}(j\ensuremath{\rightarrow}1)=0$, where $\mathfrak{j}$ is the microscopic spin. We provide numerical evidence that ${s}_{A}$ is smaller in highly excited eigenstates of integrable interacting Hamiltonians, which lends support to the expectation that the average eigenstate entanglement entropy can be used as a diagnostic of quantum chaos and integrability for Hamiltonians with non-Abelian symmetries. In the context of Hamiltonian eigenstates we consider spins $\mathfrak{j}=\frac{1}{2}$ and 1, while for our calculations based on random pure states we focus on the spin $\mathfrak{j}=\frac{1}{2}$ case.

DOI
10.1103/physrevb.108.245101

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We study the effect that the SU(2) symmetry, and the rich Hilbert space structure that it generates in lattice spin systems, has on the average entanglement entropy of highly excited eigenstates of local Hamiltonians and of random pure states. Focusing on the zero total magnetization sector (Jz=0)({J}_{z}=0)(Jz​=0) for different fixed total spin JJJ, we argue that the average entanglement entropy of highly excited eigenstates of quantum-chaotic Hamiltonians and of random pure states has a leading volume-law term whose coefficient sA{s}_{A}sA​ depends on the spin density j=J/(jL)j=J/(\mathfrak{j}L)j=J/(jL), with sA(j\ensuremath→0)=ln(2j+1){s}_{A}(j\ensuremath{\rightarrow}0)=ln(2\phantom{\rule{0.16em}{0ex}}\mathfrak{j}+1)sA​(j\ensuremath→0)=ln(2j+1) and sA(j\ensuremath→1)=0{s}_{A}(j\ensuremath{\rightarrow}1)=0sA​(j\ensuremath→1)=0, where j\mathfrak{j}j is the microscopic spin. We provide numerical evidence that sA{s}_{A}sA​ is smaller in highly excited eigenstates of integrable interacting Hamiltonians, which lends support to the expectation that the average eigenstate entanglement entropy can be used as a diagnostic of quantum chaos and integrability for Hamiltonians with non-Abelian symmetries. In the context of Hamiltonian eigenstates we consider spins j=12\mathfrak{j}=\frac{1}{2}j=21​ and 1, while for our calculations based on random pure states we focus on the spin j=12\mathfrak{j}=\frac{1}{2}j=21​ case.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2)\mathrm{SU}(2)SU(2) symmetry, constrained to fixed total spin JJJ and magnetization Jz=0J_z = 0Jz​=0. Focusing on spin-12\tfrac1221​ lattices and subsystem fractions f<12f < \frac{1}{2}f<21​, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume VVV. In addition to the expected leading volume law term, we prove the existence of a 12log⁡V\frac{1}{2}\log V21​logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1)O(1)O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2)\mathrm{SU}(2)SU(2) symmetry shapes average entanglement.
We study the average and the standard deviation of the entanglement entropy of highly excited eigenstates of the integrable interacting spin-1/2 XYZ chain away from and at special lines with U(1) symmetry and supersymmetry. We universally find that the average eigenstate entanglement entropy exhibits a volume-law coefficient that is smaller than that of quantum-chaotic interacting models. At the supersymmetric point, we resolve the effect that degeneracies have on the computed averages. We further find that the normalized standard deviation of the eigenstate entanglement entropy decays polynomially with increasing system size, which we contrast with the exponential decay in quantum-chaotic interacting models. Our results provide state-of-the art numerical evidence that integrability in spin-1/2 chains reduces the average and increases the standard deviation of the entanglement entropy of highly excited energy eigenstates when compared with those in quantum-chaotic interacting models.
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.
To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of U(1) symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions ν of the spectrum are included in the average. We find indications that, for ν→0, the magnitude of the negative O(1) correction is only slightly greater than the one predicted for random pure states. For finite ν, following a phenomenological approach, we derive a simple expression that describes the numerically observed ν dependence of the O(1) deviation from the prediction for random pure states.
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that, while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.
In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.