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arXiv:cond-mat/9711109 · Physical Review Letters

Termination of Multifractal behavior for Critical Disordered Dirac Fermions

Jean-Sébastien Caux, Nobuhiko Taniguchi, Alexei M. Tsvelik

Published Feb 9, 1998• 43 citations
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Abstract

We consider Dirac fermions interacting with a disordered non-Abelian vector potential. The exact solution is obtained through a special type of conformal field theory including logarithmic correlators, without resorting to the replica or supersymmetry approaches. It is shown that the proper treatment of the conformal theory leads to a different multifractal scaling behavior than initially expected. Moreover, the previous replica solution is found to be incorrect at the level of higher correlation functions.

DOI
10.1103/physrevlett.80.1276

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We consider Dirac fermions interacting with a disordered non-Abelian vector potential. The exact solution is obtained through a special type of conformal field theory including logarithmic correlators, without resorting to the replica or supersymmetry approaches. It is shown that the proper treatment of the conformal theory leads to a different multifractal scaling behavior than initially expected. Moreover, the previous replica solution is found to be incorrect at the level of higher correlation functions.
We reexamine in detail the problem of fermions interacting with a non-Abelian random vector potential. Without resorting to the replica or supersymmetry approaches, we show that in the limit of infinite disorder strength the theory possesses an exact solution which takes the form of a logarithmic conformal field theory. We show that the proper treatment of the locality conditions in the SU(2) theory leads to the termination of the multifractal spectrum, or in other words to the termination of the infinite hierarchies of negative-dimensional operators that were thought to occur. Based on arguments of logarithmic degeneracies, we conjecture that such a termination mechanism should be present for general SU(N). Moreover, our results lead to the conclusion that the previous replica solution of this problem yields incorrect results.
We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wave functions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavors, are obtained by deriving from conformal field theory an effective Gaussian model for the wave function amplitudes and mapping it to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wave functions remain delocalized in the presence of strong disorder.
We obtain the disorder averaged (critical) four-point correlation functions for N species of (two-dimensional Euclidean) Dirac fermions subject to a (Gaussian) random su(N) gauge field. The replica approach and the strong disorder approach yield identical results, as one might expect in the light of recent developments in the N=2 case [1]. We establish a connection with `dressed' Z_N twist fields in the c=-2 logarithmic conformal field theory (LCFT), thereby extending the recent connection to `dressed' Z_2 twists [1,2]. We draw attention to the fact that `dressed' Z_2 twist operators have previously appeared in the description of half flux-quantum quasiholes in the bulk excitations of the Haldane--Rezayi Quantum Hall state [3].
The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang et al., arXiv:2209.07103]. Here we study, by means of large-scale quantum Monte Carlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.
Recently, a concept of generalized multifractality, which characterizes fluctuations and correlations of critical eigenstates, was introduced and explored for all 10 symmetry classes of disordered systems. Here, by using the nonlinear sigma-model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) field theory, we extend the theory of generalized multifractality to boundaries of systems at criticality. Our numerical simulations on two-dimensional systems of symmetry classes A, C, and AII fully confirm the analytical predictions of pure-scaling observables and Weyl symmetry relations between critical exponents of surface generalized multifractality. This demonstrates the validity of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ for the description of Anderson-localization critical phenomena, not only in the bulk but also on the boundary. The critical exponents strongly violate generalized parabolicity, in analogy with earlier results for the bulk, corroborating the conclusion that the considered Anderson-localization critical points are not described by conformal field theories. We further derive relations between generalized surface multifractal spectra and linear combinations of Lyapunov exponents of a strip in quasi-one-dimensional geometry, which hold under the assumption of invariance with respect to a logarithmic conformal map. Our numerics demonstrate that these relations hold with an excellent accuracy. Taken together, our results indicate an intriguing situation: the conformal invariance is broken but holds partially at critical points of Anderson localization.
We study a Dirac fermion model with three kinds of disorder as well as a marginal interaction which forms the critical line of c=1c=1c=1 conformal field theory. Computing scaling equations by the use of a perturbative renormalization group method, we investigate how such an interaction affects the universality classes of disordered systems with non-interacting fermions. We show that some specific fixed points are stable against an interaction, whereas others are unstable and flow to new random critical points with a finite interaction.