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.
A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions
d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions
d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.
This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.
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.
In a previous paper, we showed that the problem of tunneling in quantum wires was integrable in the isotropic case ${g}_{\ensuremath{\sigma}}=2$. In the present work, we continue the exploration of the general phase diagram by looking for other integrable cases. Specifically, we discuss in detail the manifold ${g}_{\ensuremath{\rho}}{+g}_{\ensuremath{\sigma}}=2$, where the associated ``double sine-Gordon'' model is integrable. Transport properties are exactly computed. Surprisingly, the IR fixed points, while having complete reflection of charge and spin currents, do not correspond to two separate leads. Their main characteristic is that they are approached along irrelevant operators of dimension ${1+(1/g}_{\ensuremath{\rho}})$ and ${1+(1/g}_{\ensuremath{\sigma}})$, corresponding to transfer of one electron charge but no spin, or one spin
1/2 but no charge.
We investigate the problem of Dirac fermions living in a background of nematic fluctuations. In our scheme, the local nematic order is carried by Ising spins, and these gauge-like variables are given their own dynamics in the form of a two-dimensional transverse field Ising model. On the basis of quantum Monte Carlo simulations, we present a complete phase diagram, in which the tuning parameters are the applied transverse field and the coupling between the fermions and Ising spins. A careful analysis reveals that, at sufficiently high coupling, the conventional (
2+1)-dimensional Ising universality class of the
T=0 quantum critical point gives way to a first-order phase transition. There is an additional phase boundary along which the Dirac points smear out to form nodal lines in the Brillouin zone. This boundary extends up from the small-transverse-field region of the phase diagram until it eventually merges into the first-order transition line of the Ising spins at large coupling and large transverse field. Along this boundary, collective bosonic fluctuations with SO(4) symmetry are strongly enhanced; across it, the Dirac points' Berry phase switches.