We study the consequences of the random mass, random scalar potential, and random vector potential on the line of fixed points between integer and/or fractional quantum Hall states and an insulator. This line of fixed points was first identified in a clean Dirac fermion system with both Chern-Simon coupling and Coulomb interaction [Phys. Rev. Lett. 80, 5409 (1998)]. By performing a renormalization-group analysis in
1/N (N is the number of species of Dirac fermions) and the variances of three disorders ${\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V},{\ensuremath{\Delta}}_{A},$ we find that ${\ensuremath{\Delta}}_{M}$ is irrelevant along this line, and both ${\ensuremath{\Delta}}_{A}$ and ${\ensuremath{\Delta}}_{V}$ are marginal. With the presence of all three disorders, the pure fixed line is unstable. Setting Chern-Simon interaction to zero, we find one nontrivial line of fixed points in the $({\ensuremath{\Delta}}_{A},w)$ plane with dynamic exponent
z=1 and continuously changing \ensuremath{\nu}; it is stable against small $({\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V})$ in a small range of the line
1<w<1.31, therefore it may be relevant to integer quantum Hall transition. Setting ${\ensuremath{\Delta}}_{M}=0,$ we find a fixed plane with
z=1, the part of this plane with $\ensuremath{\nu}>1$ is stable against small ${\ensuremath{\Delta}}_{M},$ therefore it may be relevant to fractional quantum Hall transition. Although we do not find a generic fixed point with all the couplings nonvanishing, we prove that the theory is renormalizable to the order ${(1/N)}^{2},(1/N)\ensuremath{\Delta},{\ensuremath{\Delta}}^{2},$ and we explore the interesting processes which describe the interferences between the Chern-Simon interaction, the Coulomb interaction, and the three kinds of disorders.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
We argue that massless Dirac particles in two spatial dimensions with
1∕r Coulomb repulsion and quenched random gauge field are described by a manifold of fixed points which can be accessed perturbatively in disorder and interaction strength, thereby confirming and extending the results of Herbut, Juricic, and Vafek [arXiv:0707.4171 (unpublished)]. At small interaction and small randomness, there is an infrared stable fixed curve which merges with the strongly interacting infrared unstable line at a critical endpoint, along which the dynamical critical exponent
z=1.
Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\ensuremath{\epsilon}$-expansion in the spatial dimension $d=2\ensuremath{-}\ensuremath{\epsilon}$. In two dimensions $(\ensuremath{\epsilon}=0)$, the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the
(1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.
We study the effects of quenched disorder in a class of quantum chains with (
p+1)-multispin interactions exhibiting a free fermionic spectrum, paying special attention to the case
p=2. Depending if disorder couples to (i) all the couplings or just to (ii) some of them, we have two distinct physical scenarios. In case (i), we find that the transitions of the model are governed by a universal infinite-randomness critical point surrounded by quantum Griffiths phases similarly as happens to the random transverse-field Ising chain. In case (ii), we find that quenched disorder becomes an irrelevant perturbation: the clean critical behavior is stable and Griffiths phases are absent. Beyond the perturbative regime, disorder stabilizes a line of finite-randomness critical points (with nonuniversal critical exponents), that ends in a multicritical point of infinite-randomness type. In that case, quantum Griffiths phases also appear surrounding the finite-disorder transition point. We have characterized the correlation functions and the low-temperature thermodynamics of these chains. Our results are derived from a strong-disorder renormalization-group technique and from finite-size scaling analysis of the spectral gap computed exactly (up to sizes $\ensuremath{\sim}{10}^{7}$) via an efficient new numerical method recently introduced in the literature [Phys. Rev. B 104, 174206 (2021)].
We use the recently proposed all orders beta functions to study localization problems in two dimensions.A classification of random Dirac hamiltonians is presented.Beta functions for the Chalker-Coddington network model are presented and it is described how the renormalization group flows are toward a true singularity.The main subject of this talk is localization-delocalization transitions in two dimensions.This is a subject with many challenging and unsolved problems, the most important being finding a theoretical description of the critical properties of the quantum Hall transition.The Chalker-Coddington network model is perhaps the simplest model believed to be in the universality class of the quantum Hall transition [1].It can be formulated as a theory of random Dirac fermions, and it is believed that the disorder averaged effective theory should possess an infra-red fixed point conformal field theory capable of predicting the critical exponents governing the transition, e.g. the delocalization length exponent[2, 3].I will describe an attempt at finding the expected fixed point based on a proposed all orders beta function [5,6].The all-orders beta functions for the network models are based on general formulas found in [4].These beta-functions generally describe current-current interactions in two dimensions.Though they remain conjectural, several strong checks of the formula were performed for the simplest su(2) case in [7].As I will describe, for reasons that are not understood at the time, but appear to be specific to the application of the general theory to disordered systems, the renormalizaton group flows toward a true singularity, and thus we did not find the expected fixed point.The plan of the talk is as follows.I will first describe a classification of Random Dirac Fermions in 2d.In the next section I explain how supersymmetric Effective Actions are obtained upon disorder averaging.These generally involve current-current interactions and in the next section I describe the renormalization group (RG) for general interactions of this kind.Using arguments independent of the detailed beta functions, I then propose a connection between the possible fixed points and the Virasoro Master equation.In the last sections these methods are applied to the network models for the quantum Hall transtions.Unfortuanately, as I will describe, the all orders beta functions that appear to be correct for su(2) predict flows toward true singularities in the disordered problems.