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Physical review. B, Condensed matter

Critical behavior of Tomonaga-Luttinger liquids with a mobile impurity

Yasumasa Tsukamoto, Tatsuya Fujii, Norio Kawakami

Published Aug 15, 1998• 36 citations
Publisher

Abstract

The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as $c=1$ shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.

DOI
10.1103/physrevb.58.3633

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The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as c=1c=1c=1 shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.
The power-law dependence of correlation functions at low energies is what clearly distinguishes the Tomonaga-Luttinger (TL) liquid and the Fermi liquid (FL). The correlation function critical exponent for the FL is usually fixed to an integer, while that of the TL liquid varies continuously. We calculate the nature of power-law dependence in the charge, spin and electron correlation functions for the TL liquid at both zero and small magnetic fields. In particular, our approach centres on the Bethe Ansatz and conformal field theory (CFT). This is important since the concept of CFT relates critical exponents of various correlation functions with the finite-size corrections to the energy spectrum. Our findings show that increase in the particle density nc, and external magnetic field B, increases the value of the critical exponents while increase in the interaction strength u, decreases the value of the critical exponents.
Bariev's model for correlated hopping in one dimension is integrable and soluble within the framework of two nested Bethe Ansätze. For repulsive interaction and finite magnetization the low energy excitation spectrum is a two-component Luttinger liquid with conformal charges c=1. In zero magnetic field the spectrum of spin excitations is gapped. For higher energy excitations the single-hole (single-particle) spectral functions display edge singularities that deviate from those expected from a Luttinger liquid. The system can be described as a mobile impurity coupled to the Luttinger liquid yielding an effective x-ray edge type model. Using the Bethe Ansatz solution for repulsive interaction we obtain expressions for the critical exponents for the single-particle and single-hole Green's functions in zero magnetic field.
The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge c=1c=1c=1. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.
Exact critical properties of the one-dimensional SU(N) interacting fermion model with open boundaries are studied by using the Bethe ansatz method. We derive the surface critical exponents of various correlation functions using boundary conformal field theory. They are classified into two types, i.e., the exponents for the chiral SU(N) Tomonaga-Luttinger liquid and those related to the orthogonality catastrophe. We discuss a possible application of the results to the photoemission (absorption) in the edge state of the fractional quantum Hall effect. \textcopyright{} 1996 The American Physical Society.
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.
Applying boundary conformal field theory we study the low-energy critical behavior of the Calogero-Sutherland model of BCNBC_NBCN​-type. The universality class of the model is found to be a {\it chiral} Tomonaga-Luttinger liquid. Various correlation exponents depending on the interaction strength are obtained.