Various results concerning the spectra of the Ising model in a strong transverse field and the anisotropic Heisenberg antiferromagnet are proved. It is proved that the ground state energy of the Ising model in a strong transverse field in all dimensions converges to its infinite volume limit exponentially with a specific power law correction. It is also proved that this model in all dimensions has continuous spectrum in the infinite volume limit. For the anisotropic Heisenberg antiferromagnet it is proved that in dimensions of at least two, the energy spectrum contains a continuous part in the infinite volume limit. All results are obtained by perturbation theory using polymer expansions.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.
Botet and Jullien [Phys. Rev. B 27, 613 (1983)] have conducted a finite-chain scaling analysis on the spin-1 antiferromagnetic Heisenberg-Ising chain. They claim that the ground-state properties are completely different from the case of spin
21. Using finite-size scaling techniques, they conclude that between the
XY gapless phase and the doublet-ground-state N\'eel phase, a new phase appears in an extended anisotropy range. This phase, which encompasses the isotropic Heisenberg model, is characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap. We perform a similar finite-chain scaling analysis on the spin-
21 antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-
41 system is known rigorously not to exhibit the type of behavior proposed by Botet and Jullien, their analysis is open to doubt. The possibility arises that a strong singularity at the Heisenberg point for spin 1 (similar to the essential singularity for spin
21) may be giving rise to misleading results. The
T=0 phase behavior of the spin-1 Heisenberg-Ising antiferromagnetic chain, therefore, remains in question and specific additional numerical studies are proposed to clarify the matter.
Abstract Exact calculations are presented for the ground state of linear antiferromagnetic Heisenberg chains with open ends. The wave function and the energy are given for chains with N = 4, 6, and 8, and quantum number S = 1/2. It is shown that in the limit for N → ∞ the ground state is probably nondegenerate and the long‐range order tends to zero. The short‐range order shows a strongly oscillating character due to end effects persisting over relatively long distances. A comparison with closed rings is made and the connection with the observation of a reduction of the zero‐point deviation in the neighbourhood of non‐magnetic impurities in more‐dimensional systems is pointed out.
Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with a boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.