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arXiv:cond-mat/9907122 · Physical review. B, Condensed matter

Quasi-long-range order in the random anisotropy Heisenberg model: Functional renormalization group in 4−ε dimensions

Dima E. Feldman

Published Jan 1, 2000• 56 citations
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Abstract

The large-distance behaviors of the random field and random anisotropy $O(N)$ models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg $(N=3)$ model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field $O(N)$ model the correlation length is found to be finite at the arbitrarily weak disorder for any $N>3.$ The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.

DOI
10.1103/physrevb.61.382

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The random field and random anisotropy N -vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg ( N =3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈 m ( r 1 ) m ( r 2 )〉∼¦ r 1 − r 2 ¦ − 0.62ε . The magnetic susceptibility diverges at low fields as χ∼ H −1+0.15ε . In the random field N -vector model the correlation length is finite at arbitrarily weak disorder for any N >3.
We study the long-distance behavior of the O(N)O(N)O(N) model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation xxx using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.
The low-temperature thermodynamic properties of a spin-\textonehalf{} one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H=\ensuremath{\Sigma}{i}^{}{J}_{i}({\ensuremath{\sigma}}_{x}^{i}{\ensuremath{\sigma}}_{x}^{i+1}+{\ensuremath{\sigma}}_{y}^{i}{\ensuremath{\sigma}}_{y}^{i+1}+\ensuremath{\gamma}{\ensuremath{\sigma}}_{z}^{i}{\ensuremath{\sigma}}_{z}^{i+1})$ are studied as a function of disorder and anisotropy. The Ji> 0{J}_{i}>~0Ji​> 0 are independent random variables obeying a probability distribution P(J)P(J)P(J), and $0<~\ensuremath{\gamma}<~\ensuremath{\infty}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\ensuremath{\gamma}=1$), the XY\mathrm{XY}XY case ($\ensuremath{\gamma}=0$). and the Ising case ($\ensuremath{\gamma}=\ensuremath{\infty}$) are fixed points of the RG transformation. It is found that in the XY\mathrm{XY}XY region $\ensuremath{\gamma}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the XY\mathrm{XY}XY limit ($\ensuremath{\gamma}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\ensuremath{\chi}\ensuremath{\sim}\frac{1}{(T{\mathrm{ln}}^{m}(\frac{T}{{T}_{0}}))}$ in the entire region $0<~\ensuremath{\gamma}<~1$ for arbitrary probability distributions. In the Ising region ($\ensuremath{\gamma}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\ensuremath{\rightarrow}0$. Possible relevance of these results to recent experiments on Qn(TCNQ)2{(\mathrm{TCNQ})}_{2}(TCNQ)2​ is discussed.
We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.
We study the ferromagnetic phase transition in a randomly layered Heisenberg magnet using large-scale Monte Carlo simulations. Our results provide numerical evidence for the infinite-randomness scenario recently predicted within a strong-disorder renormalization-group approach. Specifically, we investigate the finite-size scaling behavior of the magnetic susceptibility, which is characterized by a nonuniversal power-law divergence in the Griffiths phase. We also study the perpendicular and parallel spin-wave stiffnesses in the Griffiths phase. In agreement with the theoretical predictions, the parallel stiffness is nonzero for all temperatures T<TcT<{T}_{c}T<Tc​. In contrast, the perpendicular stiffness remains zero in part of the ordered phase, giving rise to anomalous elasticity. In addition, we calculate the in-plane correlation length, which diverges already inside the disordered phase at a temperature significantly higher than Tc{T}_{c}Tc​. The time autocorrelation function within model AAA dynamics displays an ultraslow logarithmic decay at criticality and a nonuniversal power law in the Griffiths phase.
Magnetic phase transition (MPT) to magnetic quasi-long-range order (QLRO) phase in a three-dimensional Heisenberg weak (D/J=4) random anisotropy (RA) model is investigated by Monte Carlo simulation. The isotropic and cubic distributions of RA axes are considered for simple-cubic-lattice systems. Finite-size scaling analysis shows that the critical couplings for the former and latter are Kc=0.704 35(2) and Kc=0.709 98(4), respectively. While the critical exponent 1/ν=1.408 24(0) is the same for both cases. A second-order MPT to the QLRO phase is therefore evidenced to be possible in favor with the existence of the QLRO predicted by recent functional renormalization group theories.