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Physical review. B, Condensed matter

Formation of the Haldane phase by soliton-pair dissociation: Results from a cluster approximation

Hans-Jürgen Mikeska, Paola Verrucchi

Published Aug 1, 1995• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.52.3571

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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.
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.
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.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and 0.0≲α≤1.00.0\lesssim\alpha \leq 1.00.0≲α≤1.0.
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.
A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy Dtsumj{\mathit{Dtsum}}_{\mathit{j}}Dtsumj​(Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​)2{)}^{2})2 is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value Dc{\mathit{D}}_{\mathit{c}}Dc​=1 is shown in the lower Haldane gap and the correlation functions.
We propose nonlocal unitary transformations on S =1 antiferromagnetic chains in a simple operator form. This is a generalization of the transformation recently introduced by Kennedy and Tasaki to clarify the hidden symmetry in the Haldane phase. The three operators ( T i α ≡ i S i α e i πS i β ) a p p e a r i n g i n t h e s e t r a n s f o r m a t i o n s a r e e x a m i n e d a n d s o m e s i m p l e a l g e b r a i c r e l a t i o n s a r e s h o w n t o h o l d a m o n g T_i^α a n d S_i^α. I t i s a l s o s h o w n t h a t t h r e e d i f f e r e n t f o r m s o f t r a n s f o r m e d H a m i l t o n i a n s a r e t r a n s f o r m e d i n t o e a c h o t h e r b y local u n i t a r y t r a n s f o r m a t i o n s .