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Nuclear Physics B

Exact critical exponents for quantum spin chains, non-linear σ-models at θ=π and the quantum hall effect

Ian K. Affleck

Published Mar 1, 1986• 519 citations
Publisher
DOI
10.1016/0550-3213(86)90167-7

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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.

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On the infrared limit of the O(3) nonlinear $σ$-model at $θ= π$

Martin R. Zirnbauer

2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.

arXiv:2408.12215•Aug 22, 2024•1 citation
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.
On the infrared limit of the O(3) nonlinear σσσ-model at θ=πθ= πθ=π
2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.