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arXiv:cond-mat/9506095 · Physical review. B, Condensed matter

Network model of localization in a random magnetic field

Yong Baek Kim, Akira Furusaki, Derek K. K. Lee

Published Dec 15, 1995• 15 citations
arXivHTMLPDF

Abstract

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(${\mathit{e}}^{2}$/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.

DOI
10.1103/physrevb.52.16646

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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.
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 obtain the field-theory representations of several network models that are relevant to two-dimensional transport in high magnetic fields. Among them, the simplest one, which is relevant to the plateau transition in the quantum Hall effect, is equivalent to a particular representation of an antiferromagnetic SU(2N) (N\ensuremath{\rightarrow}0) spin chain. Since the latter can be mapped onto a \ensuremath{\theta}\ensuremath{\ne}0, U(2N)/U(N)\ifmmode\times\else\texttimes\fi{}U(N) \ensuremath{\sigma} model, and since recent numerical analyses of the corresponding network give a delocalization transition with \ensuremath{\nu}\ensuremath{\approxeq}2.3, we conclude that the same exponent is applicable to the \ensuremath{\sigma} model.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
We use the recently proposed all orders beta functions to study localization problems in two dimensions.A classification of random Dirac hamiltonians is presented.Beta functions for the Chalker-Coddington network model are presented and it is described how the renormalization group flows are toward a true singularity.The main subject of this talk is localization-delocalization transitions in two dimensions.This is a subject with many challenging and unsolved problems, the most important being finding a theoretical description of the critical properties of the quantum Hall transition.The Chalker-Coddington network model is perhaps the simplest model believed to be in the universality class of the quantum Hall transition [1].It can be formulated as a theory of random Dirac fermions, and it is believed that the disorder averaged effective theory should possess an infra-red fixed point conformal field theory capable of predicting the critical exponents governing the transition, e.g. the delocalization length exponent[2, 3].I will describe an attempt at finding the expected fixed point based on a proposed all orders beta function [5,6].The all-orders beta functions for the network models are based on general formulas found in [4].These beta-functions generally describe current-current interactions in two dimensions.Though they remain conjectural, several strong checks of the formula were performed for the simplest su(2) case in [7].As I will describe, for reasons that are not understood at the time, but appear to be specific to the application of the general theory to disordered systems, the renormalizaton group flows toward a true singularity, and thus we did not find the expected fixed point.The plan of the talk is as follows.I will first describe a classification of Random Dirac Fermions in 2d.In the next section I explain how supersymmetric Effective Actions are obtained upon disorder averaging.These generally involve current-current interactions and in the next section I describe the renormalization group (RG) for general interactions of this kind.Using arguments independent of the detailed beta functions, I then propose a connection between the possible fixed points and the Virasoro Master equation.In the last sections these methods are applied to the network models for the quantum Hall transtions.Unfortuanately, as I will describe, the all orders beta functions that appear to be correct for su(2) predict flows toward true singularities in the disordered problems.
Network models of dirty electronic systems are mapped onto an interacting field theory of lower dimensionality by interpreting one space dimension as time. This is accomplished via Feynman's interpretation of antiparticles as particles moving backwards in time. The method developed maps calculation of the moments of the Landauer conductance onto calculation of correlation functions of an interacting field theory of bosons and fermions. The resulting field theories are supersymmetric and closely related to the supersymmetric spin-chain representations of network models recently discussed by various authors. As an application of the method, the two-edge Chalker-Coddington model is shown to be Anderson localized, and a delocalization transition in a related two-edge network model (recently discussed by Balents and Fisher) is studied by calculation of the average Landauer conductance.
Despite a very good understanding of single-particle Anderson localization in one-dimensional (1D) disordered systems, many-body effects are still full of surprises; a famous example is the interaction-driven many-body localization (MBL) problem, about which much has been written, and perhaps the best is yet to come. Interestingly enough the noninteracting limit provides a natural playground to study nontrivial multiparticle physics, offering the possibility to test some general mechanisms with very large-scale exact diagonalization simulations. In this paper, we first revisit the 1D many-body Anderson insulator through the lens of extreme value theory, focusing on the extreme polarizations of the equivalent spin chain model in a random magnetic field. A many-body-induced chain breaking mechanism is explored numerically, and compared to an analytically solvable toy model. A unified description, from weak to large disorder strengths WWW emerges, where the disorder-dependent average localization length $\ensuremath{\xi}(W)$ governs the extreme events leading to chain breaks. In particular, tails of the local magnetization distributions are controlled by $\ensuremath{\xi}(W)$. Remarkably, we also obtain a quantitative understanding of the full distribution of the extreme polarizations, which is given by a Fr\'echet-type law. In a second part, we explore finite interaction physics and the MBL question. For the available system sizes, we numerically quantify the difference in the extreme value distributions between the interacting problem and the noninteracting Anderson case. Strikingly, we observe a sharp ``extreme-statistics transition'' as WWW changes, which may coincide with the MBL transition.