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Towards a theory of the integer quantum Hall transition: From the nonlinear sigma model to superspin chains
Author(s) -
Zirnbauer Martin R.
Publication year - 1994
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.19945060702
Subject(s) - physics , hamiltonian (control theory) , quantum hall effect , quantum mechanics , degenerate energy levels , quantum , sigma model , condensed matter physics , mathematical physics , nonlinear system , magnetic field , mathematics , mathematical optimization
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 /2 h (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 1 d 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.