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arXiv:cond-mat/9409049 · Physical review. B, Condensed matter

Chiral and spin order in the two-dimensional ±JXYspin glass: Domain-wall scaling analysis

Muriel Ney‐Nifle, H. J. Hilhorst

Published Apr 1, 1995• 14 citations
arXivHTMLPDF

Abstract

This is an analytic study of the two-dimensional XY spin glass with \ifmmode\pm\else\textpm\fi{}J disorder. The Hamiltonian has a continuous spin symmetry and a discrete chiral symmetry, and therefore possesses, potentially, two different order parameters and correlation lengths. The cost of breaking the symmetries is probed by comparing the ground-state energy under periodic (P) boundary conditions with the one under antiperiodic (AP) and under reflecting (R) boundary conditions. Two energy differences (domain-wall energies) appear, \ensuremath{\Delta}${\mathit{E}}^{\mathrm{AP}}$ and \ensuremath{\Delta}${\mathit{E}}^{\mathit{R}}$, whose scaling behavior with system size is nontrivially related to the correlation length exponents. For a specific distribution of the \ifmmode\pm\else\textpm\fi{}J disorder we show that the chiral, and spin correlation lengths diverge with the same exponent as T goes down to 0. The common exponent has a common cause, viz. the low reversal energy of domains of chiral variables. For general disorder we give a heuristic argumentation in terms of droplet excitations that leads to spin ordering on a longer, or equal, scale than chiral ordering. These results are in contrast with intepretations of Monte Carlo simulations.

DOI
10.1103/physrevb.51.8357

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This is an analytic study of the two-dimensional XY spin glass with \ifmmode\pm\else\textpm\fi{}J disorder. The Hamiltonian has a continuous spin symmetry and a discrete chiral symmetry, and therefore possesses, potentially, two different order parameters and correlation lengths. The cost of breaking the symmetries is probed by comparing the ground-state energy under periodic (P) boundary conditions with the one under antiperiodic (AP) and under reflecting (R) boundary conditions. Two energy differences (domain-wall energies) appear, \ensuremath{\Delta}EAP{\mathit{E}}^{\mathrm{AP}}EAP and \ensuremath{\Delta}ER{\mathit{E}}^{\mathit{R}}ER, whose scaling behavior with system size is nontrivially related to the correlation length exponents. For a specific distribution of the \ifmmode\pm\else\textpm\fi{}J disorder we show that the chiral, and spin correlation lengths diverge with the same exponent as T goes down to 0. The common exponent has a common cause, viz. the low reversal energy of domains of chiral variables. For general disorder we give a heuristic argumentation in terms of droplet excitations that leads to spin ordering on a longer, or equal, scale than chiral ordering. These results are in contrast with intepretations of Monte Carlo simulations.
The ordering of the three-dimensional isotropic XY\mathit{XY}XY spin glass with nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. To investigate the ordering of the spin and the chirality, we compute several independent physical quantities including the glass order parameter, the Binder parameter, the correlation-length ratio, the overlap distribution, and the non-self-averageness parameter, etc., for both the spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of spin-chirality decoupling, i.e., that the CG and SG order occur at two separate temperatures, 0<TSG<TCG0<{T}_{\mathrm{SG}}<{T}_{\mathrm{CG}}0<TSG​<TCG​, is obtained from the glass order parameter, and is fully corroborated by the Binder parameter. By contrast, the CG correlation-length ratio yields a rather pathological and inconsistent result in the range of sizes we studied, which may originate from the finite-size effect associated with a significant short-length dropoff of the spatial CG correlations. Finite-size-scaling analysis yields the CG exponents ${\ensuremath{\nu}}_{\mathrm{CG}}=1.{36}_{\ensuremath{-}0.37}^{+0.15}$ and ${\ensuremath{\eta}}_{\mathrm{CG}}=0.{26}_{\ensuremath{-}0.26}^{+0.29}$, and the SG exponents ${\ensuremath{\nu}}_{\mathrm{SG}}=1.{22}_{\ensuremath{-}0.06}^{+0.26}$ and ${\ensuremath{\eta}}_{\mathrm{SG}}=\ensuremath{-}0.{54}_{\ensuremath{-}0.52}^{+0.24}$. The exponents obtained are close to those of the Heisenberg SG, but are very different from those of the Ising SG. The chiral overlap distribution and the chiral Binder parameter exhibit the feature of a continuous one-step replica-symmetry breaking (1RSB), consistently with previous reports. Such a 1RSB feature is again like that of the Heisenberg SG, but is different from the Ising SG, which may be the cause of the difference in the CG critical properties from those of the Ising SG despite the common Z2{Z}_{2}Z2​ symmetry.
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We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.
The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 ( L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T × , the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T × to be ν SG = 0.9 ± 0.2 and ν CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T × .
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for L×LL \times LL×L square lattices with L≤20L \le 20L≤20, and xxx = 0.25 and 0.5, where xxx is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy EdwE_{dw}Edw​ equal to 0 or 4. The probability distribution of the entropy, SdwS_{dw}Sdw​, is found to depend strongly on EdwE_{dw}Edw​. The results for SdwS_{dw}Sdw​ when Edw=4E_{dw} = 4Edw​=4 agree with the prediction of the droplet model. Our results for SdwS_{dw}Sdw​ when Edw=0E_{dw} = 0Edw​=0 agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of Edw=0E_{dw} = 0Edw​=0 domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.