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Journal of Physics A Mathematical and General

Two order parameters in quantum XZ spin models with Gibbsian ground states

T. C. Dorlas, W. I. Skrypnik

Published Jun 17, 2004• 3 citations
Publisher

Abstract

We describe a family of quantum spin models which are generators of a discrete Markovian process. We show that there exists an explicit expression for the ground state of such models and give a simple argument for the existence of two types of long-range order in such systems. Two special examples of these systems are analysed in detail.
DOI
10.1088/0305-4470/37/26/002

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A class of general spin 1 2 lattice models on hyper-cubic lattice Z d , whose Hamiltonians are sums of two functions depending on the Pauli matrices S 1 , S 2 and S 3 , respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously for large values of the parameter α playing the role of the inverse temperature.It is shown that the ferromagnetic order in x direction exists for all dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple).
Quantum models on the hyper-cubic d-dimensional lattice of spin-1 2 particles interacting with linear oscillators are shown to have three ferromagnetic ground state order parameters.Two order parameters coincide with the magnetization in the first and third directions and the third one is a magnetization in a continuous oscillator variable.The proofs use a generalized Peierls argument and two Griffiths inequalities.The class of spinoscillator Hamiltonians considered manifest maximal ordering in their ground states.The models have relevance for hydrogen-bond ferroelectrics.The simplest of these is proven to have a unique Gibbsian ground state.
The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.
We prove the existence of long-range order (LRO) in the ground state of the 2D quantum spin-1/2 XXZ-model for all anisotropies 0≤Δ leq0.220 \leq \Delta\ leq 0.220≤Δ leq0.22 and Δ≥1.47. We achieve this extension of previous proofs by developing an improved variant of the Dyson-Lieb-Simon proof of LRO in quantum spin systems and by deriving improved upper and lower bounds for the ground state energy of the XXZ-model.