We provide and study complete sets of one‐loop renormalization group equations of several Finkel'stein non‐linear σ models, the effective field theories describing the diffusive quantum fluctuations in correlated disordered systems. We consider different cases according to the presence of certain symmetries induced by the original random Hamiltonians, and we show that, for interacting systems, the Cartan's classification of symmetry classes is not enough to uniquely determine their scaling behaviors. image
We provide and study complete sets of one‐loop renormalization group equations of several Finkel'stein non‐linear σ models, the effective field theories describing the diffusive quantum fluctuations in correlated disordered systems. We consider different cases according to the presence of certain symmetries induced by the original random Hamiltonians, and we show that, for interacting systems, the Cartan's classification of symmetry classes is not enough to uniquely determine their scaling behaviors. image
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.
The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
Generalized multifractality characterizes system size dependence of pure scaling local observables at Anderson transitions in all 10 symmetry classes of disordered systems. Recently, the concept of generalized multifractality has been extended to boundaries of critical disordered noninteracting systems. Here we study the generalized boundary multifractality in the presence of electron-electron interaction, focusing on the spin quantum Hall symmetry class (class C). Employing the two-loop renormalization group analysis within the Finkel'stein nonlinear sigma model, we compute the anomalous dimensions of the pure scaling operators located at the boundary of the system. We find that generalized boundary multifractal exponents are twice larger than their bulk counterparts. Exact symmetry relations between generalized boundary multifractal exponents in the case of noninteracting systems are explicitly broken by the interaction.
Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems; (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals; and (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and nonequilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. A self-contained derivation of the exact flow equations is presented and frequently used truncation schemes are described. Reviewing selected applications it is shown how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.
The interacting disordered electron problem is reviewed with emphasis on the quantum phase transitions that occur in a model system and on the field-theoretic methods used to describe them. An elementary discussion of conservation laws and diffusive dynamics is followed by a detailed derivation of the extended nonlinear sigma model, which serves as an effective field theory for the problem. A general scaling theory of metal-insulator and related transitions is developed, and explicit renormalization-group calculations for the various universality classes are reviewed and compared with experimental results. A discussion of pertinent physical ideas and phenomenological approaches to the metal-insulator transition not contained in the sigma-model approach is given, and phase-transition aspects of related problems, like disordered superconductors and the quantum Hall effect, are discussed. The review concludes with a list of open problems.
We studied the correlated quasi-one-dimensional systems by one-loop renormalization group techniques in weak coupling. In contrast to conventional g-ology approach, we formulate the theory in terms of bilinear currents and obtain all possible interaction vertices. Furthermore, the one-loop renormalization group equations are derived by operator product expansions of these currents at short length scale. It is rather remarkable that these coupled non-linear equations, after appropriate rescaling, can be casted into potential flows. The existence of what we nicknamed “RG potential ” provides a natural explanation of the emergent symmetry enhancement in ladder systems. Further implications arisen from the RG potential are also discussed at the end. Quasi-one-dimensional (Q1D) systems have attracted extensive attentions from both experimental and theoretical aspects. 1),2) Due to the low dimensionality, strong quantum fluctuations often give rise to surprising behaviors, which are rather different from our intuitions built in higher dimensions. It is then exciting to explore