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arXiv:1612.04109 · Nuclear Physics B

Gaussian free fields at the integer quantum Hall plateau transition

Roberto Bondesan, Dagmar Wieczorek, Martin R. Zirnbauer

Published Feb 28, 2017• 29 citations
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Abstract

In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker–Coddington network model and a supersymmetric vertex model.

DOI
10.1016/j.nuclphysb.2017.02.011

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•Jan 1, 1994•31 citations
In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker–Coddington network model and a supersymmetric vertex model.
Stationary wave functions at the transition between plateaus of the integer quantum Hall effect are known to exhibit multifractal statistics. Here we explore this critical behavior for the case of scattering states of the Chalker-Coddington network model with point contacts. We argue that moments formed from the wave amplitudes of critical scattering states decay as pure powers of the distance between the points of contact and observation. These moments in the continuum limit are proposed to be correlation functions of primary fields of an underlying conformal field theory. We check this proposal numerically by finite-size scaling. We also verify the conformal field theory prediction for a three-point function involving two primary fields.
The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model (NL σ M). Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4 . Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be Δ q = q ( 1 − q ) / 4 . The Lagrangian of the RG-fixed point theory for r retarded and r advanced replicas is proposed to be the GL ( r | r ) n = 4 Wess-Zumino-Witten model deformed by a truly marginal perturbation. The latter emerges from the NL σ M by a natural scenario of spontaneous symmetry breaking.
We show that the localization transition in the integer quantum Hall effect as described by the Chalker-Coddington network model is quantum critical. We first map the anisotropic network model to the problem of diagonalizing a one-dimensional non-Hermitian noncompact supersymmetric lattice Hamiltonian of interacting bosons and fermions. Its behavior is investigated numerically using the density matrix renormalization group method, and critical behavior is found at the plateau transition. This result is confirmed by a generalization of the Lieb-Schultz-Mattis theorem.
Multifractals arise in various systems across nature whose scaling behavior is characterized by a continuous spectrum of multifractal exponents Δ_{q}. In the context of Anderson transitions, the multifractality of critical wave functions is described by operators O_{q} with scaling dimensions Δ_{q} in a field-theory description of the transitions. The operators O_{q} satisfy the so-called Abelian fusion expressed as a simple operator product expansion. Assuming conformal invariance and Abelian fusion, we use the conformal bootstrap framework to derive a constraint that implies that the multifractal spectrum Δ_{q} (and its generalized form) must be quadratic in its arguments in any dimension d≥2.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
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.