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Physical Review B

Ground state and low-lying excitations in a Heisenberg spin-glass model in three dimensions

Fumitaka Matsubara, Takayuki Shirakura, S. Takahashi, Y. Baba

Published Nov 11, 2004• 7 citations
Publisher

Abstract

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 $V$ 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.

DOI
10.1103/physrevb.70.174414

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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 genetic algorithm (GA) in the Heisenberg model that combines a triadic crossover and a parameter-free genetic algorithm. Using the algorithm, we examined the ground-state stiffness of the ±J\pm J±J Heisenberg model in three dimensions up to a moderate size range. Results showed the stiffness constant of θ=0θ= 0θ=0 in the periodic-antiperiodic boundary condition method and that of θ∼0.62θ\sim 0.62θ∼0.62 in the open-boundary-twist method. We considered the origin of the difference in θθθ between the two methods and suggested that both results show the same thing: the ground state of the open system is stable against a weak perturbation.
We investigate the nature of the low-energy, large-scale excitations in the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings and free boundary conditions, by studying the response of the ground state to a coupling-dependent perturbation introduced previously. The ground states are determined exactly for system sizes up to 123{12}^{3}123 spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.
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We performed Monte Carlo simulation of the three dimensional ±J Heisenberg model in a Parisi state. That is, having started with one of Parisi states A, we calculated the magnetization MA(t) projected on A. We found that at low temperatures (T ≪ TSG) MA(t) for small t exhibits a relaxation similar to that of the ferromagnetic phase, where TSG is the spin glass transition temperature estimated before. From this behavior, we estimated temperature dependence of the order parameter of that Parisi state.
We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a two-leg ladder geometry using the density-matrix renormalization-group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=sin\ensuremath{\phi}$ and $J=cos\ensuremath{\phi}$ with an angle parameter $\ensuremath{\phi}$. Based on the results for several types of order parameters, excitation gaps, string order parameter, and entanglement spectra, the $\ensuremath{\phi}$-dependent ground state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range-ordered states, namely rung singlet (analog to N\'eel in three dimensions), zigzag, ferromagnetic, and stripy, and the presence of gapped spin liquids around the exactly solvable Kitaev points $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\ensuremath{\pi}$-flux state in order to establish how the gapped Kitaev spin liquid extends away from the $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.