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Journal of Physics Condensed Matter

The Haldane phase at finite temperatures: a defect approach

Holger Köhler, R. Schilling

Published Sep 28, 1992• 12 citations
Publisher

Abstract

The authors investigate a one-dimensional, spin s=1 Heisenberg antiferromagnet with exchange and single-ion anisotropy. A self-consistent approach using defects, which was recently proposed by Gomez-Santos (1989), is applied to determine the phase diagram at T=0 K and is generalised to finite temperatures. The phase diagram for exchange anisotropy constant epsilon >or=1 is in reasonable agreement with earlier numerical results; the three different phases (Neel, Haldane and large-D phases) can be characterised by the defect concentration. It is shown that the ground state and the excitations are BCS-like. The Haldane phase can be interpreted as a BCS phase in particular, and the lowest excitations, as a break-up of 'Cooper pairs'. The specific heat C exhibits a maximum. Its position T max increases with increasing gap energy E g ; below T max , there exists a regime with linear temperature dependence.

DOI
10.1088/0953-8984/4/39/005

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The authors investigate a one-dimensional, spin s=1 Heisenberg antiferromagnet with exchange and single-ion anisotropy. A self-consistent approach using defects, which was recently proposed by Gomez-Santos (1989), is applied to determine the phase diagram at T=0 K and is generalised to finite temperatures. The phase diagram for exchange anisotropy constant epsilon >or=1 is in reasonable agreement with earlier numerical results; the three different phases (Neel, Haldane and large-D phases) can be characterised by the defect concentration. It is shown that the ground state and the excitations are BCS-like. The Haldane phase can be interpreted as a BCS phase in particular, and the lowest excitations, as a break-up of 'Cooper pairs'. The specific heat C exhibits a maximum. Its position T max increases with increasing gap energy E g ; below T max , there exists a regime with linear temperature dependence.
We study the ground-state phase diagram and the excitations of a one-dimensional s=1 Heisenberg antiferromagnet including exchange and single ion anisotropy with anisotropy constants \ensuremath{\epsilon} and D, respectively. Starting from the ground state for D=\ensuremath{\infty}, a new class of spin-one defects is introduced. Treating these defects as fermions and performing a mean-field approximation, we find three phases: the N\'eel, the Haldane, and the large-D phase. The corresponding phase diagram is partly in good agreement with existing data. The N\'eel and the string order parameters have the correct behavior in the large-D and the Haldane phase, but not in the N\'eel phase. For a certain variational state, we find a ground-state energy which perfectly agrees with the best quantum Monte Carlo data. Finally we argue that the low-lying one-particle excitations are of fermionic character.
The one-dimensional S =1 anisotropic antiferromagnetic Heisenberg model with next-nearest as well as nearest-neighbor exchange intearction is studied with particular emphasis on the effect of the frustration. The ground-state phase diagram and the low-lying excitations are examined by use of the bosonization method and the self-consistent harmonic approximation. It turns out that the frustration works in stabilizing the Haldane phase against the Neel phase, since its effect on the latter phase is larger. The large- D phase is almost unaffected by the frustration, resulting in its relative stability against other phases. The excitation gap in the Haldane phase (Haldane gap) is found to be reduced by the frustration near the boundary to the large- D phase, while it increases in the vicinity of the Neel phase.
We consider a quasi-one-dimensional system of spin-1 Heisenberg antiferromagnetic chains in two-dimensional and three-dimensional hypercubic lattices with interchain coupling J and uniaxial single-ion anisotropy D. Using large-scale numerical simulations, we map out the J-D phase diagram and investigate the low-lying excitations of the Haldane phase in the J≪1 limit. We also provide direct evidence that the Haldane phase remains a nontrivial symmetry-protected topological state for small but finite J.
We investigate ground-state and low-lying excited states of the S = 1 antiferromagnetic Heisenberg chain with exchange anisotropy, single-ion anisotropy and biquadratic exchange and with free boundary conditions. We present an approximate mapping of the S = 1 model to an effective S = 1/2 chain in an external magnetic field, which can be solved exactly in several limiting cases. We obtain results for the phase diagram, correlation functions and the excitation spectrum. The transition to the Haldane phase is related to the condensation of the lowest soliton modes and the hidden order parameter of the Haldane phase is identified as transverse long-range order in the effective S = 1/2 chain.
In the present work low energy excitations in various antiferromagnetic systems are investigated.The first part is devoted to the antiferromagnetic Heisenberg chain with two different alternating magnetic ions having spins S A and S B .The system can be characterised as a ferrimagnet with a long range ordered ground state.Linear spin wave theory (LSWT) leads to a qualitatively correct description of two excitation branches.The low energy spectrum contains ferromagnetic magnons which are discussed in the introductory chapter.The LSWT results are compared with the numerical data from exact diagonalisation for the case S A =1 and S B =1/2.In the second part of the thesis, the quantum phase transitions from the Néel phase to the Heisenberg phase and from the Néel phase to the Haldane phase for half-odd-integer spins S =1 /2, 3/2 and integer spins S =1 , 2, respectively, are studied in context of the Haldane conjecture.The models are analysed by means of high-order series expansions for the ground state properties and excitation energies.The expansion parameter is the anisotropy λ where the unperturbed case λ =0corresponds to the Ising model.For the integer-S systems the following scenario is obtained: The phase transitions from the Néel phase to the Haldane phase are determined by the condensation of the so-called spin zero defects (SZD) at finite anisotropy.These domain wall type excitations can be characterised as S z =0states embedded in the antiferromagnetic order.The critical exponents β of the order parameter and η of the correlation function are those of the two-dimensional Ising model.In contrast to the above scenario, the half-odd-integer S =1/2 chain undergoes the Kosterlitz-Thouless phase transition at the isotropic point.In the case of the S =3/2 chain the critical behaviour of the static structure factor S (zz) (π) is consistent with the exponentially divergent correlation length at the isotropic point.Finally, kink excitations of the S =1 /2 Heisenberg antiferromagnet chain with nearest and next nearest neighbour interactions (the zigzag chain) are investigated.Here a kink is defined as a free spin S =1 /2 mediating two dimerised ground states.The phase diagram of the zigzag chain contains the Majumdar-Ghosh model as well as the S =1/2 spin ladder and the S =1Heisenberg antiferromagnet.Starting from the Majumdar-Ghosh model, a two-kink variational wave function with S tot =1is constructed.The variational parameter is the characteristic distance between the two kinks.The variational wave function describes the low energy triplet excitation of the gapped zigzag chain in a qualitatively correct way.Within this approach, the elementary excitation of the spin ladder as well as the Haldane triplet of the S =1chain appears as a bound state of two kinks.The phase diagram of the zigzag chain contains the incommensurate phase where the wave vector of the lowest mode k 0 runs from k 0 =0to k 0 = π.At the qualitative level, this behaviour is correctly described by the variational approach.
We study the structure and properties of the ground state of an antiferromagnetic S=1 spin chain with single-ion anisotropy energy d and exchange anisotropy energy \ensuremath{\epsilon}. We apply the nonlocal unitary transformation of Kennedy and Tasaki to take into account the essential correlations and restrict our study to the subspace of spin-zero defects (solitons). We investigate the quantum phase transitions in the (\ensuremath{\epsilon},d) plane in a cluster approximation which takes into account correlations between spins at 3 neighboring sites in the unitarily transformed representation. We obtain quantitative results for the phase diagram, for the values of the hidden order parameter, and for amplitudes that describe the transition from the N\'eel to the Haldane phase in terms of the dissociation of soliton pairs.