A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
Recently an elegant and quite powerful finite-system approach to determine the exponents ηx and ηz from simple spectral properties has been proposed. For critical systems, the two exponents can be expressed in terms of finite-size spectral gaps as follows: η(N)x=2ΔE01(N)/ΔE(N), η(N)z=2ΔE00(N)/ΔE(N). Here ΔE(N) is the finite-size gap between the ground state (SzT=0,k=0) and the lowest excitation at k=2π/N; ΔE01(N) is the gap to the lowest ‖SzT‖=1 excitations (at k=π), and ΔE00(N) is the gap to the next lowest SzT=0 excited state. The η(N) sequence is then extrapolated to N→∞. For XY models, differences between s=1/2 and s≥1 appear. For s=1/2, the excitations which determine ΔE00(N) and ΔE(N) are degenerate, which implies that ηz=1/2, in agreement with the exact analytic result. For spin-1, however, the next lowest SzT=0 state is located at k=2π/N instead of k=π, and is therefore identical to the state which determines the gap ΔE. The resulting equality ΔE=ΔE00 implies ηz=2, as in the spin-1/2 case. In fact, our result corresponds to power-law decay for all s, and hence we differ from Schulz and Ziman, who claim the out-of-plane correlation function decays exponentially for s>1/2. For the in-plane correlation function, the spectral gap method again agrees with the exact result ηx=0.5 for s=1/2. The consensus of this and other numerical methods for s=1 gives a value ηx≂0.20, considerably different from the case of s=1/2. Hence it is tempting to conjecture that ηx is s dependent, implying that XY models belong to different universality classes for different s. However, a finite-size study of the conformal anomaly produces the result that c=1, independent of s. This situation is further discussed.
Abstract Applying the (infinite) density-matrix renormalisation group technique, we explore the effect of an explicit dimerisation on the ground-state phase diagram of the spin-1 XXZ chain with single-ion anisotropy D. We demonstrate that the Haldane phase between large-D and antiferromagnetic phases survives up to a critical dimerisation only. As a further new characteristic the dimerisation induces a direct continuous Ising quantum phase transition between the large-D and antiferromagnetic phases with central charge c=1/2 c = 1 / 2 , which terminates at a critical end-point where c=7/10 c = 7 / 10 . Calculating the critical exponents of the order parameter, neutral gap and spin–spin-correlation function, we find β=1/8 β = 1 / 8 (1/24), ν=1 ν = 1 (5/9), and η=1/4 η = 1 / 4 (3/20), respectively, which proves the Ising (tricritical Ising) universality class in accordance with field-theoretical predictions.
A physically motivated approach is used to study the linear chain XXZ spin-1 Hamiltonian in the antiferromagnetic regime. A variational Hamiltonian is obtained restricting the basis set to states with local antiferromagnetic environments separated by sites with Sz=0 acting as domain walls. Ground-state properties of this variational Hamiltonian are obtained and, in particular, an order–disorder transition is found with critical behavior corresponding to the two-dimensional Ising model universality class, in agreement with Haldane conjecture. Excellent agreement with finite chain data concerning the position of Haldane’s point, ground-state energy, and elementary excitations is found.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results: Δc2=0 for the XY-Haldane phase transition and Δc3=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be c≃1.0 at Δc2=0 and c≃0.5 at Δc3=1.185, respectively, which shows that the XY-Haldane phase transition at Δc2=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at Δc2=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.