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arXiv:cond-mat/0508218 · Indian Journal of Physics

A solvable quantum antiferromagnet model

Bikas K. Chakrabarti, Jun‐ichi Inoue

Published Aug 9, 2005• 2 citations
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Abstract

We introduce a quantum antiferromagnet model, having exactly soluble thermodynamic properties. It is an infinite range antiferromagnetic Ising model put in a transverse field. The free energy gives the ground state energy in the zero temperature limit and it also gives the low temperature behaviour of the specific heat, the exponential variation of which gives the precise gap magnitude in the excitation spectrum of the system. The detailed behaviour of the (random sublattice) staggard magnetisation and susceptibilities are obtained and studied near the Néel temperature and the zero temperature quantum critical point.

DOI
10.48550/arxiv.cond-mat/0508218

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We introduce a quantum antiferromagnet model, having exactly soluble thermodynamic properties. It is an infinite range antiferromagnetic Ising model put in a transverse field. The free energy gives the ground state energy in the zero temperature limit and it also gives the low temperature behaviour of the specific heat, the exponential variation of which gives the precise gap magnitude in the excitation spectrum of the system. The detailed behaviour of the (random sublattice) staggard magnetisation and susceptibilities are obtained and studied near the Néel temperature and the zero temperature quantum critical point.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
We consider the quantum XY spin chain with quasi-periodic two-valued exchange couplings and a uniform transverse field. The equivalence of the system with a free fermionic model permits a detailed analysis of its thermodynamics. The zero-temperature magnetization is a Cantor function of the applied field. The zero-field specific heat and susceptibility have a power-law behaviour at low temperature, modulated by periodic amplitudes. The critical exponents interpolate continuously between those of the periodic and of the random model.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature Tpeak{\mathit{T}}_{\mathrm{peak}}Tpeak​\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
The ground-state properties of the triangular antiferromagnetic Ising model with Ising constant J in a transverse field h are investigated by a real-space renormalisation group method. For h=0, the known properties (energy=-J, finite entropy, no order) are recovered. For h not=0, the ground state remains highly degenerate, no order exists and the correlation function exhibits power-law behaviour up to (h/J)c approximately=1.41, where a phase transition takes place to a non-degenerate ground state. At h/J=(h/J)c, power-law behaviour is still observed for the correlation function but with a different exponent. Just above this region a gap Delta opens ( Delta approximately ((h/J)-(h/J)c)s with s=0.8) and the coherence length of the correlation function xi diverges as xi varies as ((h/J)-(h/J)c)-v with v=0.8. This corresponds to a three-dimensional classical model where h is replaced by the temperature. In this type of system, where the ground state has a degeneracy proportional to N2/3 it is predicted that a transition will occur at a finite T c to a low-temperature phase without order but with power-law behaviour for the correlation function, as for the classical XY model in two dimensions.
Using the integral equation method, we have got the analytic expressions of the internal energy and specific heat for the one-dimensional Ising model with nearest-neighbour antiferromagnetic repulsion Js and infinitely long-range exponential (Kae-Helfand-type) ferromagnetic attraction J1 acting only on even-numbered neighbours. Under the conditions of J1=1 and Js being 0.05, 0.1 and 0.3 respectively, we have drawn the curves of the internal energy and specific heat varying with temperature in zero external magnetic field, and pointed out its critical phase transition point. Under the conditions of J1=1,Js=0.1, we have drawn the curves of influence of the external magnetic field upon the internal energy and specific heat. Because there exist double solutions for the system under low temperatures, zero or weak external magnetic field and weaker antiferromagnetic interactions, it is inferred that there should exist another quasi-stable state of the model under the given conditions.
We study quantum criticality in the infinite range transverse-field Ising model. We find subtle differences with respect to the well-known single-site mean-field theory, especially in terms of gap, entanglement and quantum criticality. The calculations are based on numerical diagonalization of Hamiltonians with up to a few thousand spins. This is made possible by the enhanced symmetries of the model, which divide the Hamiltonian into many block-diagonal sectors. The finite temperature phase diagram and the characteristic jump in heat capacity closely resemble the behavior in mean-field theory. However, unlike mean-field theory where excitations are always gapped, the excitation gap in the infinite range model goes to zero from both the paramagnetic side and from the ferromagnetic side on approach to the quantum critical point. Also, contrary to mean-field theory, at the quantum critical point the quantum Fisher information becomes large, implying long-range multipartite entanglement. We find that the main role of temperature is to shift statistical weights from one conserved sector to another. However, low energy excitations in each sector arise only near the quantum critical point implying that low energy quantum fluctuations can arise only in the vicinity of the quantum critical field where they can persist up to temperatures of order the exchange constant.