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arXiv:cond-mat/9512057 · Physical review. B, Condensed matter

Variational study of the random-field XY model

Thomas Garel, Giulia Iori, Henri Orland

Published Feb 1, 1996• 21 citations
arXivHTMLPDF

Abstract

A disorder-dependent Gaussian variational approach is applied to the $d$-dimensional ferromagnetic $\mathrm{XY}$ model in a random field. The randomness yields a nonextensive contribution to the variational free energy, implying a random mass term in correlation functions. The Imry-Ma low-temperature result, concerning the existence ($d>4$) or absence ($d<4$) of long-range order is obtained in a transparent way. The physical picture which emerges below $d=4$ is that of a marginally stable mixture of domains. We also calculate within this variational scheme, disorder-dependent correlation functions, as well as the probability distribution of the Imry-Ma domain size.

DOI
10.1103/physrevb.53.r2941

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A disorder-dependent Gaussian variational approach is applied to the ddd-dimensional ferromagnetic XY\mathrm{XY}XY model in a random field. The randomness yields a nonextensive contribution to the variational free energy, implying a random mass term in correlation functions. The Imry-Ma low-temperature result, concerning the existence (d>4d>4d>4) or absence (d<4d<4d<4) of long-range order is obtained in a transparent way. The physical picture which emerges below d=4d=4d=4 is that of a marginally stable mixture of domains. We also calculate within this variational scheme, disorder-dependent correlation functions, as well as the probability distribution of the Imry-Ma domain size.
Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures (TTT) greater than zero; the Lorentzian-squared terms vanish at T=0T=0T=0. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.
We propose a soluble quantum spherical XY ferromagnet with a random field in the boson space. We obtain a general expression of the critical temperature below which the ordered ferromagnet phase appears. The Imry–Ma result concerning the lower critical dimension is recovered, and the critical exponents near the critical temperature are calculated. We show that the random-field fluctuations rather than the quantum fluctuations dominate the phase transition and critical behavior of the system. The entropy vanishes as at low temperatures, contrary to the classical spherical model.
The quantum version of the random-field spherical XY\mathrm{XY}XY model on a ddd-dimensional hypercubic lattice is presented in the boson space, and solved exactly. We recover the Imry-Ma result concerning the lower critical dimension dc=4,{d}_{c}=4,dc​=4, and calculate the critical exponents near the critical temperature. In particular, we obtain a general phase diagram of the model with arbitrary dimension ddd when d>dc.d>{d}_{c}.d>dc​. The ferromagnetic ordering is reduced by quantum fluctuations, and destroyed completely by random-field fluctuations with sufficiently large values of the random-field variance \ensuremath{\sigma}. The entropy and the specific heat vanish as Td/2{T}^{d/2}Td/2 at low temperatures. Since the model is equivalent to a Bose system, we show a superfluid-Mott insulator transition at a critical chemical potential ${\ensuremath{\mu}}_{c}.$
Beyond the Imry-Ma Length: Scaling Behavior in the 3D Random Field XYXYXY Model
We have performed studies of the 3D random field XYXYXY model on L×L×LL \times L \times LL×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of hrh_rhr​ = 1.875, for L=L =L= 64, 96 and 128, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, S(k)S ( k )S(k) at TTT = 1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The magnetization jump per spin scales with size roughly as L−3/4L^{- 3/4}L−3/4, while the energy jump per spin scales like L−3/2L^{- 3/2}L−3/2. The results also indicate the existence of an approximately logarithmic divergence of S(k)S ( k )S(k) as k→0k \to 0k→0. The magnetic susceptibility, χ(k⃗=0)χ(\vec{\bf k} = 0 )χ(k=0), on the other hand, seems to have a value of about 14.2 under these conditions. This suggests the absence of a ferromagnetic phase, and that the lower critical dimension for long-range order in this model is three. Similar results are found for LLL = 64 samples at hrh_rhr​ = 2.0 and TTT = 0.875. We expect that the behavior is qualitatively similar along the entire phase boundary, but the scaling exponents may not be universal. These results appear to be related to recent work on quantum disorder.
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