We study the entanglement spectrum (ES) of a finite XXZ spin- chain in the limit Δ → −1 + for both open and periodic boundary conditions. At Δ =− 1 (ferromagnetic point) the model is equivalent to the Heisenberg ferromagnet and its degenerate ground state manifold is the S U (2) multiplet with maximal total spin. Any state in this so-called 'symmetric sector' is an equal weight superposition of all possible spin configurations. In the gapless phase at Δ >− 1 this property is progressively lost as one moves away from the Δ =− 1 point. We investigate how the ES obtained from the states in this manifold reflects this change, using exact diagonalization and Bethe ansatz calculations. We find that in the limit Δ → −1 + most of the ES levels show divergent behavior. Moreover, while at Δ =− 1 the ES contains no information about the boundaries, for Δ >− 1 it depends dramatically on the choice of boundary conditions. For both open and periodic boundary conditions the ES exhibits an elegant multiplicity structure for which we conjecture a combinatorial formula. We also study the entanglement eigenfunctions, i.e. the eigenfunctions of the reduced density matrix. We find that the eigenfunctions corresponding to the non-diverging levels mimic the behavior of the state wavefunction, whereas the others show intriguing polynomial structures. Finally we analyze the distribution of the ES levels as the system is detuned away from Δ =− 1.
We study the entanglement spectrum (ES) of a finite XXZ spin- chain in the limit Δ → −1 + for both open and periodic boundary conditions. At Δ =− 1 (ferromagnetic point) the model is equivalent to the Heisenberg ferromagnet and its degenerate ground state manifold is the S U (2) multiplet with maximal total spin. Any state in this so-called 'symmetric sector' is an equal weight superposition of all possible spin configurations. In the gapless phase at Δ >− 1 this property is progressively lost as one moves away from the Δ =− 1 point. We investigate how the ES obtained from the states in this manifold reflects this change, using exact diagonalization and Bethe ansatz calculations. We find that in the limit Δ → −1 + most of the ES levels show divergent behavior. Moreover, while at Δ =− 1 the ES contains no information about the boundaries, for Δ >− 1 it depends dramatically on the choice of boundary conditions. For both open and periodic boundary conditions the ES exhibits an elegant multiplicity structure for which we conjecture a combinatorial formula. We also study the entanglement eigenfunctions, i.e. the eigenfunctions of the reduced density matrix. We find that the eigenfunctions corresponding to the non-diverging levels mimic the behavior of the state wavefunction, whereas the others show intriguing polynomial structures. Finally we analyze the distribution of the ES levels as the system is detuned away from Δ =− 1.
We study the entanglement spectrum (ES) of a finite XXZ spin 1/2 chain in the limit \Delta -> -1^+ for both open and periodic boundary conditions. At \Delta=-1 (ferromagnetic point) the model is equivalent to the Heisenberg ferromagnet and its degenerate ground state manifold is the SU(2) multiplet with maximal total spin. Any state in this so-called "symmetric sector" is an equal weight superposition of all possible spin configurations. In the gapless phase at \Delta>-1 this property is progressively lost as one moves away from the \Delta=-1 point. We investigate how the ES obtained from the states in this manifold reflects this change, using exact diagonalization and Bethe ansatz calculations. We find that in the limit \Delta ->-1^+ most of the ES levels show divergent behavior. Moreover, while at \Delta=-1 the ES contains no information about the boundaries, for \Delta>-1 it depends dramatically on the choice of boundary conditions. For both open and periodic boundary conditions the ES exhibits an elegant multiplicity structure for which we conjecture a combinatorial formula. We also study the entanglement eigenfunctions, i.e. the eigenfunctions of the reduced density matrix. We find that the eigenfunctions corresponding to the non diverging levels mimic the behavior of the state wavefunction, whereas the others show intriguing polynomial structures. Finally we analyze the distribution of the ES levels as the system is detuned away from \Delta=-1.
We investigate the entanglement entropy (EE) and entanglement spectra (ES) of critical $\mathrm{SU}(N) (2\ensuremath{\le}N\ensuremath{\le}4)$ spin chains and other integrable models of finite length with the density matrix renormalization group method. For all models under investigation, we find a remarkable agreement of the level spacings and the degeneracy structure of the ES with the spectrum of the corner Hamiltonian (CS), defined as the generator of the associated corner transfer matrix. The correspondence holds between ES(n) at the nth cut position from the edge of the spin model, and the spectrum CS(n) of the corner Hamiltonian of length n, for all values of n that we have checked. The cut position dependence of the ES shows a period-N oscillatory behavior for a given SU(N) chain, reminiscent of the oscillatory part of the entanglement entropy observed in the past for the same models. However, the oscillations of the ES do not die out in the bulk of the chain, in contrast to the asymptotically vanishing oscillation of the entanglement entropy. We present a heuristic argument based on Young tableaux construction that can explain the period-N structure of the ES qualitatively.
At large values of the anisotropy Δ, the open-boundary Heisenberg spin- chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this ‘edge-locking’ phenomenon in the entire Δ > 1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large Δ there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as Δ is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, or both free either as single magnons or as bound composites corresponding to ‘string’ solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of Δ (‘exceptional points’), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy Δ, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ = 1) and edge-locking becomes stable in the whole Δ > 1 region.
Abstract: We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions in sectors with a fixed number of down spins. This forces the existence of a droplet of down spins in the system. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We also prove that the droplet states, which were known to be separated by an energy gap from the ground state, are also separated by an energy gap from the remainder of the spectrum, i.e., they form an isolated band. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
We study the phase diagram of the spin-1/2 XXZ Heisenberg chain in a staggered magnetic field. An exact solution is obtained along the phase boundary, where the exchange anisotropy and the staggered-field strength satisfy an exact analytical relation, yielding a highly degenerate manifold of ground states with off-diagonal long-range order. We demonstrate that these exact ground states lie on the boundary between the ferromagnetic phase and a gapless Luttinger liquid phase. Upon further decreasing the anisotropy, the gapless phase eventually gives way to a gapped quantum-disordered phase. Large-scale density-matrix renormalization group calculations confirm this phase structure and indicate that the transition across the exact boundary is of first order. Our results provide an exact benchmark for understanding quantum phase transitions driven by staggered magnetic fields in one-dimensional quantum spin systems.