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 a Dirac fermion model with three kinds of disorder as well as a marginal interaction which forms the critical line of
c=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.
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.
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.
We study by the strong disorder renormalization group (RG) method the low-energy properties of the one-dimensional Hubbard model with random-hopping matrix-elements
tmin<t<tmax, and with random onsite Coulomb repulsion terms $0\ensuremath{\leqslant}{U}_{min}<U<{U}_{max}$. There are two critical phases, corresponding to an infinite randomness spin random singlet for strong interactions
(Umin>tmax) and to an orbital infinite randomness fixed point for vanishing interactions $({U}_{max}∕{t}_{max}\ensuremath{\rightarrow}0)$. To each critical infinite randomness fixed point is connected a Griffiths phase, the correlation length, and dynamical exponent of which have well defined asymptotic dependences on the corresponding quantum control parameter. The theoretical predictions for the scaling in the vicinity of the critical points compare well to numerical RG simulations.
It has long been known that particles with short-range repulsive interactions in spatial dimension
d=1 form universal quantum liquids in the low density limit: all properties can be related to those of the spinless free Fermi gas. Previous renormalization-group (RG) analyses demonstrated that this universality is described by a RG fixed point, infrared stable for
d<2, of the zero density gas. We show that for
d>2 the same fixed point describes the universal properties of particles with short-range attractive interactions near a Feshbach resonance; the fixed point is now infrared unstable, and the relevant perturbation is the detuning of the resonance. Some exponents are determined exactly, and the same expansion in powers of $(d\ensuremath{-}2)$ applies for scaling functions for
d<2 and
d>2. A separate exact RG analysis of a field theory of the particles coupled to ``molecules'' finds an alternative description of the same fixed point, with identical exponents; this approach yields a $(4\ensuremath{-}d)$ expansion which agrees with the recent results of Nishida and Son [Phys. Rev. Lett. 97, 050403 (2006)]. The existence of the RG fixed point implies a universal phase diagram as a function of density, temperature, population imbalance, and detuning; in particular, this applies to the crossover between the Bose-Einstein condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) superfluid of
s-wave paired fermions. Our results open the way towards computation of these universal properties using the standard field-theoretic techniques of critical phenomena, along with a systematic analysis of corrections to universality. We also propose a
1∕N expansion [based upon models with
Sp(2N) symmetry] of the fixed point and its vicinity, and use it to obtain results for the phase diagram.