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arXiv:cond-mat/9811376 · Journal of Physics A Mathematical and General

Coexistence of long-range order for two observables at finite temperatures

N Macris, C-A Piguet

Published Jan 1, 1999• 1 citation
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Abstract

We give a criterion for the simultaneous existence or non-existence of two long-range orders for two observables, at finite temperatures, for quantum lattice many body systems. Our analysis extends previous results of Tian (1997 30 841) limited to the ground state of similar models. The proof involves an inequality of Dyson-Lieb-Simon (1978 J. Stat. Phys. 18 335) which connects the Duhamel two-point function to the usual correlation function. An application to the special case of the Holstein model is discussed.

DOI
10.1088/0305-4470/32/5/007

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We give a criterion for the simultaneous existence or non-existence of two long-range orders for two observables, at finite temperatures, for quantum lattice many body systems. Our analysis extends previous results of Tian (1997 30 841) limited to the ground state of similar models. The proof involves an inequality of Dyson-Lieb-Simon (1978 J. Stat. Phys. 18 335) which connects the Duhamel two-point function to the usual correlation function. An application to the special case of the Holstein model is discussed.
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.
The existence of long-range order (LRO) at low temperature in the isotropic XY\mathrm{XY}XY model with spin $S\ensuremath{\geqq}1$ on simple cubic lattice in three or more dimensions is shown. This completes the proof of the existence of LRO at low temperature in the isotropic XY\mathrm{XY}XY model with arbitrary spin size in three or more dimensions, as the proof for S=12S=\frac{1}{2}S=21​ was given earlier by Dyson, Lieb, and Simon. The existence of LRO in the ground state is shown for $S\ensuremath{\geqq}\frac{3}{2}$ in two dimensions.
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 consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range orders.