We present exact results for the susceptibility of the interacting resonant level model in equilibrium. Detailed simulations using both the Numerical Renormalization Group and Density Matrix Renormalization Group were performed in order to compare with closed analytical expressions. By first bosonizing the model and then utilizing the integrability of the resulting boundary sine-Gordon model, one finds an analytic expression for the relevant energy scale TK with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level mode does not correctly reproduce TK -however if the bare parameters in the model are renormalised, then quantities obtained via the direct Bethe ansatz such as the occupation of the resonant level as a function of the local chemical potential do match the numerical results. The case of one lead is studied in the most detail, with many results also extending to multiple leads, although there still remain open questions in this case. arXiv:1810.08246v2 [cond-mat.str-el]

We recapitulate recent developments of the functional renormalization group (FRG) approach to the steady state of systems out of thermal equilibrium. In particular, we discuss second-order truncation schemes which account for the frequency-dependence of the two particle vertex and which incorporate inelastic processes. Our focus is on two different types of one-dimensional fermion chains: (i) infinite, open systems which feature a translation symmetry, and (ii) finite systems coupled to left and right reservoirs. In addition to giving a detailed and unified review of the technical derivation of the FRG schemes, we briefly summarize some of the key physical results. In particular, we compute the non-equilibrium phase diagram and analyze the fate of the Berezinskii–Kosterlitz–Thouless transition in the infinite, open system. Graphic abstract

We study the effects of disorder in a one-dimensional model of Z 3 Fock parafermions which can be viewed as a generalization of the prototypical Kitaev chain. Exact diagonalization is employed to determine level statistics, participation ratios, and the dynamics of domain walls. This allows us to identify ergodic as well as finitesize localized phases. In order to distinguish Anderson from many-body localization, we calculate the time evolution of the entanglement entropy in random initial states using tensor networks. We demonstrate that a purely quadratic parafermion model does not feature Anderson but many-body localization due to the nontrivial statistics of the particles.

Abstract. In this work, we will find a family of small functions η y in the Kernel of an operator defined in the intersection of the Sobolev space H 2,q (S n ) with the orthogonal complement in H 1,2 (S n ) of the first eigenspace of the laplacian on S n , parameterized with a variable y belonging to a small ball contained in B n+1 . We will find L q estimates of these functions and we will use those estimates to find a subcritical solution to the scalar curvature problem on S n , and a solution•F y1 of a nonlinear elliptical problem related to that problem, whereKeywords: Sobolev spaces, conformal deformations, elliptic equations.Estimativos L q de funciones en el núcleo de un operador elíptico y aplicacionesResumen. En este trabajo, vamos a encontrar una familia de pequeñas funciones η y en el kernel de un operador definido en la intersección del espacio de Sóbolev H 2,q (S n ) con el complemento ortogonal en H 1,2 (S n ) del primer espacio propio del laplaciano sobre S n , parametrizado con una variable y que pertenece a una pequeña bola contenida en B n+1 . Encontraremos estimativos L q de estas funciones, las cuales utilizaremos para encontrar una solución subcrítica al problema de curvatura escalar sobre S n y una soluciónde un problema elíptico no lineal relacionado con este problema, donde F y1 : S n → S n es una dilatación centrada. Palabras clave: Espacios de Sóbolev, deformaciones conformes, ecuaciones elípticas.