The excitation spectra of ‘bound’ states with a pair of complex wave numbers in the antiferromagnetic chain are derived for an arbitrary spin state S from the Bethe equations. The solutions in general contain those of both Categories II and III. The spectra are found to have E = ∑Li=1(π/2)|sin Ki|, where L = 2S + 2, 0 < Ki < π, and the total wave number K is given by K = ∑Li=1Ki. The spectra are 2S + 1-foldly degenerate.
The excitation spectra of ‘bound’ states with a pair of complex wave numbers in the antiferromagnetic chain are derived for an arbitrary spin state S from the Bethe equations. The solutions in general contain those of both Categories II and III. The spectra are found to have E = ∑Li=1(π/2)|sin Ki|, where L = 2S + 2, 0 < Ki < π, and the total wave number K is given by K = ∑Li=1Ki. The spectra are 2S + 1-foldly degenerate.
We compare the excitation spectra in the presence of a magnetic field of a number of integrable (exactly solvable) and nonintegrable quantum spin chains of various spin value s. The archetypal Bethe-ansatz integrable model is the s= 1/2 Heisenberg antiferromagnet (HB AFM). The excitation spectra are characterized by a soft mode which tracks across the Brillouin zone as the field increases to its saturation value. A class of Bethe-ansatz integrable models with SU(2) symmetry and the general spin s display excitation spectra qualitatively similar to the spin- 1/2 model above, for all s. A second class of Bethe-ansatz integrable models has SU(n) symmetry, where n=2s+1. Like the SU(2) integrable chains, these models have gapless excitation spectra, but the basic Brillouin zone changes from k=±2π/(2s+1)a. Studies show that periodicity of the SU(3) member of the class changes (increases) as the field increases to saturation. For both classes of integrable models, there is a single type of excitation pattern which is generically similar for all s. In the case of the other models, on the other hand, numerical studies show that the excitations divide into at least two distinct classes. In the case of the s=1 HB AFM, at high fields (corresponding to SzT=N,N−1, . . .,N/2) the excitations map approximately onto the complete set of excitations for s= 1/2 , whereas at low fields (SzT=N/2,N/2−1,. . .,0) the excitations have notable classical character. In the case of the s=1 model with pure biquadratic exchange, one set of excitations, corresponding to SzT even (SzT=N,N−2,. . .,2,0), again shows an approximate mapping to the complete excitation set for s= 1/2 . The second class of excitations, corresponding to SzT odd, are very different. They are symmetric about k=±π/2a for all SzT, i.e., correspond to a basic Brillouin zone of ±π/2a.
Short chains of up to 12 spins are studied exactly to obtain the excitation spectrum of an antiferromagnetic hamiltonian H=1/2J Sigma i=1N sigma i, sigma i+1+1/4J Sigma i=1N sigma i, sigma i+2 (J>0, even, N+1 identical to 1, N+2 identical to 2). As well as antiferromagnetic spin waves of total spin S=1, there are S=0 bound magnon states lying partly below the spin wave branch. The susceptibility shows the usual peak at low temperatures, and the zeros of the partition function lie on the negative real axis in the complex mu =exp(-mHe/kBT) plane.
We investigate the entanglement properties of the excited states of the spin- Heisenberg ( XXX ) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. We consider eigenstates obtained from both real and complex solutions ('strings') of the Bethe equations. Physically, the former are states of interacting magnons, whereas the latter contain bound states of groups of particles. We first focus on the situation with few particles in the chain. Using exact results and semiclassical arguments, we derive an upper bound S MAX for the entanglement entropy. This exhibits an intermediate behaviour between logarithmic and extensive, and it is saturated for highly-entangled states. As a function of the eigenstate energy, the entanglement entropy is organized in bands. Their number depends on the number of blocks of contiguous Bethe–Takahashi quantum numbers. In the presence of bound states a significant reduction in the entanglement entropy occurs, reflecting that a group of bound particles behaves effectively as a single particle. Interestingly, the associated entanglement spectrum shows edge-related levels. At a finite particle density, the semiclassical bound S MAX becomes inaccurate. For highly-entangled states S A ∝ L c , with L c the chord length, signalling the crossover to extensive entanglement. Finally, we consider eigenstates containing a single pair of bound particles. No significant entanglement reduction occurs, in contrast with the few-particle case.
Recently there has been a renewed interest in the spectra and role in dynamical properties of excited states of the spin-1/2 Heisenberg antiferromagnetic chain in longitudinal magnetic fields associated with Bethe strings. The latter are bound states of elementary magnetic excitations described by Bethe-ansatz complex non-real rapidities. Previous studies on this problem referred to finite-size systems. Here we consider the thermodynamic limit and study it for the isotropic spin-1/2 Heisenberg XXX chain in a longitudinal magnetic field. We confirm that also in that limit the most significant spectral weight contribution from Bethe strings leads to (k,ω)-plane gapped continua in the spectra of the spin dynamical structure factors S+−(k,ω) and Sxx(k,ω)=Syy(k,ω). The contribution of Bethe strings to Szz(k,ω) is found to be small at low spin densities m and to become negligible upon increasing that density above m≈0.317. For S−+(k,ω), that contribution is found to be negligible at finite magnetic field. We derive analytical expressions for the line shapes of S+−(k,ω), Sxx(k,ω)=Syy(k,ω), and Szz(k,ω) valid in the (k,ω)-plane vicinity of singularities located at and just above the gapped lower thresholds of the Bethe-string states's spectra. As a side result and in order to provide an overall physical picture that includes the relative (k,ω)-plane location of all spectra with a significant amount of spectral weight, we revisit the general problem of the line-shape of the transverse and longitudinal spin dynamical structure factors at finite magnetic field and excitation energies in the (k,ω)-plane vicinity of other singularities. This includes those located at and just above the lower thresholds of the spectra that stem from excited states described by only real Bethe-ansatz rapidities.
We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.