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arXiv:cond-mat/9404010 · Physical review. B, Condensed matter

Network models of quantum percolation and their field-theory representations

D. H. Lee

Published Oct 15, 1994• 57 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.50.10788

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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 fundamental property of the quantum Hall effect is the presence of edge states at the boundary of the sample, robust against localization and responsible for the perfect quantization of the Hall conductance. The transition between integer quantum Hall plateaus is a delocalization transition, which can be identified as a strong-coupling fixed point of a 1 + 1-dimensional supersymmetric sigma model with topological theta-term. The conformal field theory describing this transition displays unusual features such as non-unitarity, and resisted any attempt of solution so far. In this thesis we investigate the role of edge states at quantum Hall transitions using lattice discretizations of super sigma models. Edge states correspond to twisted boundary conditions for the fields, which can be discretized as quantum spin chains or geometrical (loop) models. For the spin quantum Hall effect, a counterpart of the integer quantum Hall effect for spin transport (class C), our techniques allow the exact computation of critical exponents of the boundary conformal field theories describing higher plateaus transitions. Our predictions for the mean spin conductance are validated by extensive numerical simulations of the related localization problems. Envisaging applications to transport in network models of 2 + 1-dimensional disordered electrons, and to quenches in one-dimensional gapless quantum systems, in this thesis we have also developed a new formalism for dealing with partition functions of critical systems in rectangular geometries. As an application, we derive formulas for probabilities of self-avoiding walks.
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 solve the problem of the spin quantum Hall transition on random networks using a mapping to classical percolation that focuses on the boundary of percolating clusters. Using tools of two-dimensional quantum gravity, we compute critical exponents that characterize this transition and confirm that these are related to the exponents for the regular (square) network through the KPZ relation. Our results demonstrate the relevance of the geometric randomness of the networks and support conclusions of numerical simulations of random networks for the integer quantum Hall transition.
A careful study of the supersymmetric version of Pruisken’s nonlinear σ model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e 2 /2h (mod e 2 /h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.
We study the spin quantum Hall effect and transitions between Hall plateaus in quasi-two-dimensional network models consisting of several coupled layers. Systems exhibiting the spin quantum Hall effect belong to class C in the symmetry classification for Anderson localization, and for network models in this class there is an established mapping between the quantum problem and a classical one involving random walks. This mapping permits numerical studies of plateau transitions in much larger samples than for other symmetry classes, and we use it to examine localization in systems consisting of nnn weakly coupled layers. Standard scaling ideas lead one to expect nnn distinct plateau transitions, but in the case of the unitary symmetry class this conclusion has been questioned. Focusing on a two-layer model, we demonstrate that there are two separate plateau transitions, with the same critical properties as in a single-layer model, even for very weak interlayer coupling.
A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the \ensuremath{\theta} point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent $\ensuremath{\nu}=\frac{4}{3}.$ We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.