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arXiv:cond-mat/0309576 · Journal of Physics A Mathematical and General

Parisi states in a Heisenberg spin-glass model in three dimensions

Fumitaka Matsubara, Takayuki Shirakura, Y Baba

Published Feb 18, 2004• 1 citation
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Abstract

Having developed a hybrid genetic algorithm, we have studied low-lying excited states of the ± J Heisenberg model in two ( d = 2) and three ( d = 3) dimensions. We have found evidence of the occurrence of the Parisi states in d = 3 but not in d = 2. That is, in L d lattices, there exist metastable states with a finite excitation energy of Δ E ∼ O ( J ) for L → ∞, and energy barriers Δ W between the ground state and those metastable states are Δ W ∼ O ( JL θ ) with θ > 0 in d = 3 but with θ < 0 in d = 2. This finding favours the replica-symmetry-breaking or the trivial–nontrivial scenario of the SG phase over the droplet scenario.

DOI
10.1088/0305-4470/37/9/l01

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Having developed a hybrid genetic algorithm, we have studied low-lying excited states of the ± J Heisenberg model in two ( d = 2) and three ( d = 3) dimensions. We have found evidence of the occurrence of the Parisi states in d = 3 but not in d = 2. That is, in L d lattices, there exist metastable states with a finite excitation energy of Δ E ∼ O ( J ) for L → ∞, and energy barriers Δ W between the ground state and those metastable states are Δ W ∼ O ( JL θ ) with θ > 0 in d = 3 but with θ < 0 in d = 2. This finding favours the replica-symmetry-breaking or the trivial–nontrivial scenario of the SG phase over the droplet scenario.
We developed a hybrid genetic algorithm and studied the ground state and low-lying excitations of the $\ifmmode\pm\else\textpm\fi{}J$ Heisenberg model on a three-dimensional $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L(L\ensuremath{\leqslant}11)$ lattice. Our results suggest occurrence of a Parisi state. That is, there exists a metastable state with a finite energy of $\ensuremath{\Delta}E\ensuremath{\sim}O(J)$ in the thermodynamic limit and the energy barrier $\ensuremath{\Delta}W$ between the metastable state and the ground state is $\ensuremath{\Delta}W\ensuremath{\propto}J{L}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\sim}0.53$. Our results also suggest that the energy $\ensuremath{\Delta}E$ of small scale excitations are related to a scale of the excitation VVV in a similar manner to those of the ferromagnetic model, i.e., $\ensuremath{\Delta}E\ensuremath{\propto}J{L}^{{\ensuremath{\theta}}_{L}}{V}^{{\ensuremath{\theta}}_{v}}$ with ${\ensuremath{\theta}}_{L}\ensuremath{\sim}\ensuremath{-}4.7$ and ${\ensuremath{\theta}}_{v}\ensuremath{\sim}2.0$. This fact suggests that similarity exists in the ground state structure between the $\ifmmode\pm\else\textpm\fi{}J$ and the ferromagnetic Heisenberg models, though the spin directions of those two models differ greatly from one another.
We have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures $T\ensuremath{\simeq}{0.7T}_{c}$ and system sizes up to L=32L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}32L=32, we found a P(q)P(q)P(q) as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.
We study the low-energy excitations of the three dimensional spin glass through a large-scale Monte Carlo simulation on lattices up to L=18L=18L=18. We find smooth extrapolations down to zero temperature, which, in the case of the energy and of the link overlap, can be directly -- and favourably -- compared with previous investigations featuring ground states (i.e., at zero temperature). The best fit for the fractal dimension of the excitations is provided by Replica-Symmetry Breaking theory, but we also consider the alternative TNT description. The P(q)P(q)P(q) is found to verify the Parisi-Toulouse temperature scaling. Our data provides a spectacular confirmation of the overlap-equivalence hypothesis.
One of the key predictions of Parisi's broken replica symmetry theory of spin glasses is the existence of a phase transition in an applied field to a state with broken replica symmetry. This transition takes place at the de Almeida-Thouless (AT) line in the h-T plane. We have studied this line in the power-law diluted Heisenberg spin glass in which the probability that two spins separated by a distance r interact with each other falls as 1/r^{2σ}. In the presence of a random vector field of variance h_{r}^{2} the phase transition is in the universality class of the Ising spin glass in a field. Tuning σ is equivalent to changing the dimension d of the short-range system, with the relation being d=2/(2σ-1) for σ<2/3. We have found by numerical simulations that h_{AT}^{2}∼(2/3-σ) implying that the AT line does not exist below six dimensions and that the Parisi scheme is not appropriate for spin glasses in three dimensions.
We re-examine the spin-glass (SG) phase transition of the ± J Heisenberg models with and without random anisotropy D in three dimensions ( d = 3) using two complementary methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T SG ∼ 0.19 J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ SG for D = 0 is obtained using almost the same transition temperature of T SG ∼ 0.18 J . Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.
We have studied the diluted Heisenberg spin glass model in a three-component random field for the commonly used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r^{2σ}, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L^{*}, where L^{*} is large (≈4000-8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L^{*}. We sketch a renormalization group scenario to explain how these features could arise. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.