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arXiv:cond-mat/9409090 · Physical review. B, Condensed matter

Semiclassical description of the frustrated antiferromagnetic chain

D. Allen, David Sénéchal

Published Mar 1, 1995• 27 citations
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Abstract

The antiferromagnetic Heisenberg model on a chain with nearest- and next-nearest-neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N. We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.

DOI
10.1103/physrevb.51.6394

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The antiferromagnetic Heisenberg model on a chain with nearest- and next-nearest-neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N. We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.
The Heisenberg antiferromagnetic spin chain with both dimerization and frustration is studied. The classical ground state has three phases (a Neel phase, a spiral phase and a colinear phase), around which a planar spin-wave analysis is performed. In each phase, we discuss a non-linear sigma model field theory describing the low energy excitations. A renormalization group analysis of the SO(3) matrix-valued field theory of the spiral phase leads to the conclusion that the theory becomes SO(3)×SO(3)SO(3) \times SO(3)SO(3)×SO(3) and Lorentz invariant at long distances. This theory is analytically known to have a massive spin-1/2 excitation. We also show that Z2 Z_2 ~Z2​  solitons in the field theory lead to a double degeneracy in the spectrum for half-integer spins.
We study the antiferromagnetic spin chain with both dimerization and frustration. The classical ground state has three phases, a Néel phase, a spiral phase and a collinear phase, which we study through a nonlinear sigma-model approach. In the spiral phase, the field theory becomes and Lorentz invariant at long distances, a model which is exactly solvable. The low-energy spectrum is doubly degenerate with massive `elementary' spin-1/2 particles and `two-particle' triplet and singlet physical excitations. The field theory also supports solitons which lead to a double degeneracy of the ground state for half-integer spins (when there is no dimerization).
Effective nonlinear field theory for a semiclassical long-wavelength description of one-dimensional S=1/2 Heisenberg antiferromagnets with dimerized ground states is derived within an approach using two-site coherent states. The method is applied to bond-alternating anisotropic spin chains in presence of external magnetic field and next-nearest-neighbor interactions. Linear and nonlinear excitations in singlet and magnetic phases are studied. Soliton solutions are found and their properties under semiclassical quantization are discussed. \textcopyright{} 1996 The American Physical Society.
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+2{\mathbf{S}}_{\mathrm{i}+2}Si+2​, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.
A large-S approach to the description of chiral phases in frustrated antiferromagnetic spin-S chains with easy-plane anisotropy is proposed. The approach is valid in the vicinity of the classical Lifshitz point j=1/4, where j is the relative strength of the frustrating next-nearest-neighbor coupling. The effective theory turns out to be the nonlinear σ-model with the additional fourth-order term in spatial derivatives. The topological term persists for j>1/4, which leads to disappearance of the chiral gapped phase for half-integer S, in accordance with the recent numerical results.