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arXiv:cond-mat/9811401 · Europhysics Letters (EPL)

Localization of fermions in an anisotropic random magnetic field

Yong Baek Kim

Published Jul 15, 1999• 0 citations
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Abstract

We study the localization of fermions in an anisotropic random magnetic field in two dimensions. It is assumed that the randomness in a particular direction is stronger than those in the other directions. We consider a network model of zero-field contours, where there are two types of randomness—the random tunneling matrix element at the saddle points and the unidirectional random variation of the number of fermionic states following zero-field contours. After averaging over the random complex tunneling amplitude, the problem is mapped to an SU (2 N ) random exchange quantum spin chain in the N → 0 limit. We suggest that the fermionic state becomes critical in an anisotropic fashion.

DOI
10.1209/epl/i1999-00378-5

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We study the localization of fermions in an anisotropic random magnetic field in two dimensions. It is assumed that the randomness in a particular direction is stronger than those in the other directions. We consider a network model of zero-field contours, where there are two types of randomness—the random tunneling matrix element at the saddle points and the unidirectional random variation of the number of fermionic states following zero-field contours. After averaging over the random complex tunneling amplitude, the problem is mapped to an SU (2 N ) random exchange quantum spin chain in the N → 0 limit. We suggest that the fermionic state becomes critical in an anisotropic fashion.
We consider a multichannel network model to study the localization problem of noninteracting fermions in a random magnetic field with zero average. We argue that the number of channels M is even. After averaging over the randomness, the network is mapped onto M coupled SU(2N) spin chains in the N\ensuremath{\rightarrow}0 limit. In the large conductance limit g=M(e2{\mathit{e}}^{2}e2/2\ensuremath{\pi}\ensuremath{\Elzxh}) (M\ensuremath{\gg}2), it turns out that this system is equivalent to a particular representation of the U(2N)/U(N)\ifmmode\times\else\texttimes\fi{}U(N) sigma model (N\ensuremath{\rightarrow}0) without a topological term. The beta function \ensuremath{\beta}(1/M) of this sigma model in the 1/M expansion is consistent with the previously known \ensuremath{\beta}(g) of the unitary ensemble. These results and further arguments support the conclusion that all the states are localized.
One of the points of interest in the study of random systems is the nature of the eigenstates, i.e., exponentially localized or extended, and the density of states. To identify the disorder-induced effects, we consider the one-dimensional s=1/2 xy models with random exchange, or random field. Using the transformation to spinless fermions, the problem is reduced to a nonlinear map, providing accurate numerical estimates for the integrated density of states and the exponential localization length. The results clearly reveal: (i) appearance of disorder-induced tails in the integrated density of states at the bottom and top of the spectrum, yielding to a characteristic field dependence of magnetization and susceptibility at zero temperature; and (ii) important differences between random-exchange and random-field models. In the random-exchange case, the middle of the band is found to be extended, while in the random-field case, corresponding to the standard Anderson model, all states are exponentially localized. This difference also affects the temperature dependence of zero-field specific heat and susceptibility as T→0.
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We present some exact results for the effect of disorder on the critical properties of an anisotropic XY\mathrm{XY}XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real-space renormalization-group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the x direction to a ferromagnet in the y direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a nonzero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground-state energy has an essential singularity. The results obtained for the dynamical critical exponent, typical correlation length, and for the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.
We develop a recently proposed mapping of the two-dimensional Ising model with random exchange (RBIM) via the transfer matrix, to a network model for a disordered system of noninteracting fermions. The RBIM transforms in this way to a localization problem belonging to one of a set of nonstandard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalization transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to M3,{M}^{3},M3, and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM the free energy, typical correlation lengths in quasi-one dimension for both the spin and the disorder operators, and the even powers of spin-spin correlation functions and their disorder averages. We examine in detail the square-lattice, nearest-neighbor $\ifmmode\pm\else\textpm\fi{}J$ RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability $1\ensuremath{-}p.$ Studying temperatures T> 0.4J,T>~0.4J,T> 0.4J, we obtain precise coordinates in the $p\ensuremath{-}T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point.