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arXiv:0801.2780 · Symmetry Integrability and Geometry Methods and Applications

Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

Teunis C. Dorlas

Published Jan 17, 2008• 0 citations
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Abstract

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.

DOI
10.3842/sigma.2008.007

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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.
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).
Second-rank tensor order parameters appear for liquid crystals and crystallographic transitions. The simplest model with a disordered high-temperature state has interactions invariant under cubic spin transformations. It is shown that for spin 1 this leads to the 3-state Potts model. It is suggested that this model is therefore relevant to the understanding of phase transitions with such an order parameter. A formalism for series expansions and a new proof of the Kramers-Wannier symmetry on the square lattice are presented. Pade approximants for the series expansions including a procedure explicitly utilizing the symmetry are discussed. The negative interaction 'orthogonal' model analogous to the Ising antiferromagnet is also discussed and shown to have a highly degenerate ground state. On the square lattice this is the degeneracy of Lieb's ice. Relationships to ferroelectric models and the three-colour problem are also discussed.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.
Exact ground states of a spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with Heisenberg intradimer and Ising interdimer couplings are found by two independent rigorous procedures. The first method uses a unitary transformation to establish a mapping correspondence with an effective classical spin model, while the second method relies on the derivation of an effective hard-core boson model by continuous unitary transformations. Both methods lead to equivalent effective Hamiltonians providing a convincing proof that the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice exhibits a zero-temperature magnetization curve with just two intermediate plateaus at one-third and one-half of the saturation magnetization, which correspond to stripe and checkerboard orderings of singlets and polarized triplets, respectively. The nature of the remarkable stripe order relevant to the one-third plateau is thoroughly investigated with the help of the corresponding exact eigenvector. The rigorous results for the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice are compared with the analogous results for the purely classical Ising and fully quantum Heisenberg models. Finally, we discuss to what extent the critical fields of SrCu2(BO3)2{\mathrm{SrCu}}_{2}{(\mathrm{BO}}_{3}){}_{2}SrCu2​(BO3​)2​ and (CuCl)Ca2Nb3O10{(\mathrm{CuCl})\mathrm{Ca}}_{2}{\mathrm{Nb}}_{3}{\mathrm{O}}_{10}(CuCl)Ca2​Nb3​O10​ can be described within the suggested Ising-Heisenberg model.
Similar quantum phase diagrams and transitions are found for three classes of one-dimensional models with equally spaced sites, singlet ground states (GS), inversion symmetry at sites and a bond order wave (BOW) phase in some sectors.The models are frustrated spin-1/2 chains with variable range exchange, half-filled Hubbard models with spin-independent interactions and modified Hubbard models with site energies for describing organic charge transfer salts.In some range of parameters, the models have a first order quantum transition at which the GS expectation value of the sublattice spin S A 2  of odd or even-numbered sites is discontinuous.There is an intermediate BOW phase for other model parameters that lead to two continuous quantum transitions with continuous S A 2 .Exact diagonalization of finite systems and symmetry arguments provide a unified picture of familiar 1D models that have appeared separately in widely different contexts.(