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arXiv:math-ph/0601060 · Symmetry Integrability and Geometry Methods and Applications

Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States

W. I. Skrypnik

Published Jan 24, 2006• 2 citations
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Abstract

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).

DOI
10.3842/sigma.2006.011

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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 construct exact nontrivial ground states of spin-2 quantum antiferromagnets on the hexagonal lattice. Using the optimum ground state approach we determine the ground state in different subspaces of a general spin-2 Hamiltonian consistent with some realistic symmetries. These states, which are not of simple product form, depend on two free parameters and can be shown to be only weakly degenerate. We find ground states with different types of magnetic order, i.e., a weak antiferromagnet with finite sublattice magnetization and a weak ferromagnet with ferrimagnetic order. For the latter it is argued that a quantum phase transition occurs within the solvable subspace.
We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, in general anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a ≥ 0, b ≥ 0, and the symbols uj, α denote cartesian components of the spins. For some specific values of the two parameters a and b, and on the basis of available theoretical results, one can prove the existence of an ordering transition taking place at finite temperature, and obtain rigorous upper and lower bounds on the transition temperatures. This holds, for example, when n = 2, 3, a > b = 0 (the models studied in our previous paper), as well as for n = 2, a = b > 0 and n = 3, b > a = 0, where a continuous O(2) symmetry of the interaction is involved. We have studied these two latter cases by computer simulation, and made comparison with mean-field treatment; simulation results show a broad qualitative similarity between the four models, and a closer, quantitative one, between pairs of models with the same number of spin components, especially for n = 2.
This thesis investigates properties of classical and quantum spin systems on lattices. These models have been widely studied due to their relevance to condensed matter physics. We identify the ground states of an antiferromagnetic RP2 model, these ground states are very di�erent from the ferromagnetic model and there was some disagreement over their structure, we settle this disagreement. Correlation inequalities are proved for the spin- 1/2 XY model and the ground state of the spin-1 XY model. This provides fresh results in a topic that had been stagnant and allows the proof of some new results, for example existence of some correlation functions in the thermodynamic limit. The occurrence of nematic order at low temperature in a quantum nematic model is proved using the method of reflection positivity and infrared bounds. Previous results on this nematic order were achieved indirectly via a probabilistic representation. This result is maintained in the presence of a small antiferromagnetic interaction, this case was not previously covered. Probabilistic representations for quantum spin systems are introduced and some consequences are presented. In particular, N´eel order is proved in a bilinear-biquadratic spin-1 system at low temperature. This result extends the famous result of Dyson, Lieb and Simon [35]. Dilute spin systems are introduced and the occurrence of a phase transition at low temperature characterised by preferential occupation of the even or odd sublattice of a cubic box is proved. This result is the first of its type for such a mixed classical and quantum system. A probabilistic representation of the spin-1 Bose-Hubbard model is also presented and some consequences are proved.