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Physical review. B, Condensed matter

Critical exponent of the random flux model on an infinite two-dimensional square lattice and anomalous critical states

Viktor Z. Cerovski

Published Sep 28, 2001• 14 citations
Publisher

Abstract

The random flux model (RFM) on a square lattice with random flux $\ensuremath{\Phi}\ensuremath{\in}(\ensuremath{-}p\ensuremath{\pi},p\ensuremath{\pi}),0<p<~1$ per plaquette has been studied by analyisis of high-precision transfer-matrix calculations, as well as the exact numerical diagonalization in the vicinity and at the band center. Interpreting thus obtained results using the recent findings for the Anderson bond-disordered (ABD) model from Phys. Rev. B 62, 12 775 (2000), the importance of chiral symmetry is further emphasized and localization properties explained in terms of the scaling associated with the central two states of the spectrum. The results support the existence of two critical states at the band center of the infinite system. These states exhibit a different kind of critical behavior, named anomalous, allowing one to calculate the critical exponent which has been done for $p=0.5$ and $1.$ Some differences between the ABD and RFM models are also discussed, the most important one being a broken symmetry at the critical point of the former.

DOI
10.1103/physrevb.64.161101

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The random flux model (RFM) on a square lattice with random flux \ensuremathΦ\ensuremath∈(\ensuremath−p\ensuremathπ,p\ensuremathπ),0<p< 1\ensuremath{\Phi}\ensuremath{\in}(\ensuremath{-}p\ensuremath{\pi},p\ensuremath{\pi}),0<p<~1\ensuremathΦ\ensuremath∈(\ensuremath−p\ensuremathπ,p\ensuremathπ),0<p< 1 per plaquette has been studied by analyisis of high-precision transfer-matrix calculations, as well as the exact numerical diagonalization in the vicinity and at the band center. Interpreting thus obtained results using the recent findings for the Anderson bond-disordered (ABD) model from Phys. Rev. B 62, 12 775 (2000), the importance of chiral symmetry is further emphasized and localization properties explained in terms of the scaling associated with the central two states of the spectrum. The results support the existence of two critical states at the band center of the infinite system. These states exhibit a different kind of critical behavior, named anomalous, allowing one to calculate the critical exponent which has been done for p=0.5p=0.5p=0.5 and 1.1.1. Some differences between the ABD and RFM models are also discussed, the most important one being a broken symmetry at the critical point of the former.
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Gade [Nucl. Phys. B 398,398,398, 499 (1993)] has shown that the density of states for a particle hopping on a two-dimensional bipartite lattice in the presence of weak disorder and in the absence of time-reversal symmetry (chiral unitary universality class) is anomalous in the vicinity of the band center $\ensuremath{\varepsilon}=0$ whenever the disorder preserves the sublattice symmetry. More precisely, using a nonlinear $\ensuremath{\sigma}$ model that encodes the sublattice (chiral) symmetry and the absence of time-reversal symmetry she argues that the disorder average density of states diverges as $|\ensuremath{\varepsilon}{|}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}c|\mathrm{ln}\ensuremath{\varepsilon}{|}^{\ensuremath{\kappa}})$ with c some nonuniversal positive constant and $\ensuremath{\kappa}=1/2$ a universal exponent. Her analysis has been extended to the case when time-reversal symmetry is present (chiral orthogonal universality class) for which the same exponent $\ensuremath{\kappa}=1/2$ was predicted. Motrunich et al. [Phys. Rev. B 65,65,65, 064206 (2002)] have argued that the exponent $\ensuremath{\kappa}=1/2$ does not apply to the density of states in the chiral orthogonal universality class. They predict that $\ensuremath{\kappa}=2/3$ instead. We confirm the analysis of Motrunich et al. within a field theory for two flavors of Dirac fermions subjected to two types of weak uncorrelated random potentials: a purely imaginary vector potential and a complex valued mass potential. This model is the naive continuum limit of a model describing a particle hopping on a square lattice in the background of a $\ensuremath{\pi}$-flux phase and subjected to weak disorder that preserves the sublattice symmetry and time-reversal invariance. By commonly held universality arguments, this model is believed to belong to the chiral orthogonal universality class. Our calculation relies in an essential way on the existence of infinitely many local composite operators with negative anomalous scaling dimensions.
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.
We present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.
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