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Journal of Statistical Mechanics Theory and Experiment

Decay of correlations in N-vector ferromagnetic quantum models with long-range interactions

Alina Kargol

Published Oct 1, 2014• 3 citations
Publisher

Abstract

A lattice model of N -vector quantum anharmonic oscillators with long-range attraction is studied. This model is characterized by the critical temperature T C ( N ) ⩾ 0. By means of the Hammersley–Lieb–Simon inequality and a path integral representation of the two-point correlation function g N , we prove that, for T > T C (1) ⩾ T C ( N ), up to a constant this function is bounded from above by the interaction intensity. The decay of the two-point correlation function of the corresponding classical model is also discussed.

DOI
10.1088/1742-5468/2014/10/p10006

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We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.
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A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.