Quantum impurity in a Tomonaga-Luttinger liquid: Continuous-time quantum Monte Carlo approach

Abstract

We develop a continuous-time quantum Monte Carlo (CTQMC) method for quantum impurities coupled to interacting quantum wires described by a Tomonaga-Luttinger liquid. The method is negative-sign free for any values of the Tomonaga-Luttinger parameter, which is rigorously proved, and thus, efficient low-temperature calculations are possible. Duality between electrons and bosons in one dimensional systems allows us to construct a simple formula for the CTQMC algorithm in these systems. We show that the CTQMC for Tomonaga-Luttinger liquids can be implemented with only minor modifications of previous CTQMC codes developed for impurities coupled to noninteracting fermions. We apply this method to the Kane-Fisher model of a potential scatterer in a spin-less quantum wire and to a single spin coupled with the edge state of a two-dimensional topological insulator assuming an anisotropic XXZ coupling. Various dynamical response functions such as the electron Green's function and spin-spin correlation functions are calculated numerically and their scaling properties are discussed.

Luttinger liquids from a microscopic perspective

“…In practice, however, g 2 and g 4 are never fit to scattering data directly, since in general it is not clear how to do this. Rather, the Luttinger parameter-a constant which depends on g 2 and g 4 , and which determines the macroscopic behaviour-is measured experimentally, or calculated via DMRG or Monte Carlo [43,44]. As a result, the assumption that g 2 is related to interbranch scattering and g 4 to intrabranch scattering has never been tested.…”

Microscopic bosonization of band structures: x-ray processes beyond the Fermi edge

“…Our analysis was based on trial wave functions built on coherent states. In future applications of our bosonization approach, the many-boson systems that results could in principle be studied by other numerical means (quantum Monte Carlo [33][34][35][36] or the Numerical Renormalization Group [32]), as well. An interesting open question concerns whether the microscopic bosonic description also holds advantages over the original fermionic one, as it did in the case we studied, when applied in conjunction with these techniques.…”

“…In its bosonized form, Hamiltonian (28) can in principle be analyzed in a variety of ways. For instance, numerical renormalization group [32] or quantum Monte Carlo [33][34][35][36] calculations have previously been developed to deal with similar Hamiltonians. We choose to focus, in what follows, on wave-function-based methods, because according to equation (23), we only need to find the ground state of H V ,enl in order to calculate the transition rate W e ( ).…”

From Luttinger liquids to Luttinger droplets via higher-order bosonization identities

“…VI. Many results for inhomogeneous Luttinger liquids are of course known, e.g., with barriers, 79,80 impurities, 81,82 boundaries, [83][84][85] leads, 22,86 confinements 87 and so on. Models with (effective) position-dependent Luttinger liquid parameters or interaction potentials have also been investigated.…”