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arXiv:cond-mat/0404251

Effects of anisotropy on a quantum Heisenberg spin glass in three dimensions: a renormalization-group study

J. Ricardo de Sousa, B. Boechat, Claudette E. Cordeiro, N. S. Branco

Published Apr 10, 2004• 0 citations
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Abstract

We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction ($J_{ij}$), Gaussian and uniform, are used, with zero mean and width $\bar{J}$. The $(kT/\bar{J}) \times \Delta_0$ phase diagram is obtained, where $T$ is the temperature, $\Delta_0$ is the first moment of the probability distribution for the uniaxial anisotropy, and $k$ is the Boltzmann constant. For the Ising model ($\Delta_0=1$), there is a spin-glass phase at low temperatures (high $\bar{J}$) and a paramagnetic phase at high temperatures (low $\bar{J}$). For the isotropic Heisenberg model ($\Delta_0=0$), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with $\Delta_0$, as expected, but goes to zero at a finite value of the anisotropy parameter, namely $\Delta_0 = \Delta_c \sim 0.59$. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for $\Delta_c \leq \Delta_0 < 1$, belongs to the same universality class as the transition for the Ising spin glass.

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We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (JijJ_{ij}Jij​), Gaussian and uniform, are used, with zero mean and width Jˉ\bar{J}Jˉ. The (kT/Jˉ)×Δ0(kT/\bar{J}) \times \Delta_0(kT/Jˉ)×Δ0​ phase diagram is obtained, where TTT is the temperature, Δ0\Delta_0Δ0​ is the first moment of the probability distribution for the uniaxial anisotropy, and kkk is the Boltzmann constant. For the Ising model (Δ0=1\Delta_0=1Δ0​=1), there is a spin-glass phase at low temperatures (high Jˉ\bar{J}Jˉ) and a paramagnetic phase at high temperatures (low Jˉ\bar{J}Jˉ). For the isotropic Heisenberg model (Δ0=0\Delta_0=0Δ0​=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with Δ0\Delta_0Δ0​, as expected, but goes to zero at a finite value of the anisotropy parameter, namely Δ0=Δc∼0.59\Delta_0 = \Delta_c \sim 0.59Δ0​=Δc​∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for Δc≤Δ0<1\Delta_c \leq \Delta_0 < 1Δc​≤Δ0​<1, belongs to the same universality class as the transition for the Ising spin glass.
We study the anisotropic Heisenberg spin-glass model on a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (Jij)({J}_{ij})(Jij​), Gaussian and uniform, are used, with zero mean and width J‾\overline{J}J. The $(kT∕\overline{J})\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Delta}}_{0}$ phase diagram is obtained, where TTT is the temperature, ${\ensuremath{\Delta}}_{0}$ is the first moment of the probability distribution for the uniaxial anisotropy, and kkk is the Boltzmann constant. For the Ising model $({\ensuremath{\Delta}}_{0}=1)$, there is a spin-glass phase at low temperatures (high J‾\overline{J}J) and a paramagnetic phase at high temperatures (low J‾\overline{J}J). For the isotropic Heisenberg model $({\ensuremath{\Delta}}_{0}=0)$, our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with ${\ensuremath{\Delta}}_{0}$, as expected, but goes to zero at a finite value of the anisotropy parameter, namely ${\ensuremath{\Delta}}_{0}={\ensuremath{\Delta}}_{c}\ensuremath{\sim}0.59$. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for ${\ensuremath{\Delta}}_{c}<{\ensuremath{\Delta}}_{0}<1$, belongs to the same universality class as the transition for the Ising spin glass.
We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.
We perform a finite-size scaling study of the three-dimensional Heisenberg spin glass in the presence of weak random anisotropic interactions, up to lattice sizes L=32L=32L=32. Anisotropies have a major impact on the phase transition. The chiral-glass susceptibility does not diverge due to a large anomalous dimension. It follows that the anisotropic spin glass belongs to a Universality Class different from the isotropic model, which questions the applicability of the chirality scenario.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
We characterize the phase diagram of anisotropic Heisenberg spin glasses, finding both the spin and the chiral glass transition. We remark on the presence of strong finite-size effects in the chiral sector. On the spin glass sector, we find that the universality class is that of Ising spin glasses. Our data are compatible with a unique phase transition for the chiral and spin glass sector. We focus on keeping finite-size effects under control, and we stress that they are important to understand experiments. Thanks to large GPU clusters we have been able to thermalize cubic lattices with up to 643{64}^{3}643 spins, over a vast range of temperatures (hence, of relaxation times).