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arXiv:cond-mat/0702612

Scattering approach to impurity thermodynamics

Pankaj Mehta, Natan Andrei

Published Feb 26, 2007• 0 citations
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Abstract

Recently the authors developed a scattering approach that allows for a complete description of the steady-state physics of quantum-impurities in and out of equilibrium. Quantum impurities are described using scattering eigenstates defined ab initio on the open, infinite line with asymptotic boundary conditions imposed by the leads. The scattering states on the open line are constructed for integrable quantum-impurity models by means of a significant generalization of the Bethe-Ansatz which we call the Scattering Bethe-Ansatz (SBA). The purpose of the paper is to present in detail the scattering approach to quantum-impurity models and the SBA and show that they reproduce well-known thermodynamic results for several widely studied models: the Resonance Level model, Interacting Resonance Level model and the Kondo model. Though the SBA is more complex than the traditional Thermodynamic Bethe Ansatz (TBA) when applied to thermodynamical questions, the scattering approach (SBA) allows access to an array of new questions that cannot be addressed otherwise, ranging from scattering of electrons off magnetic impurities to nonequilibrium dynamics.

DOI
10.48550/arxiv.cond-mat/0702612

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Recently the authors developed a scattering approach that allows for a complete description of the steady-state physics of quantum-impurities in and out of equilibrium. Quantum impurities are described using scattering eigenstates defined ab initio on the open, infinite line with asymptotic boundary conditions imposed by the leads. The scattering states on the open line are constructed for integrable quantum-impurity models by means of a significant generalization of the Bethe-Ansatz which we call the Scattering Bethe-Ansatz (SBA). The purpose of the paper is to present in detail the scattering approach to quantum-impurity models and the SBA and show that they reproduce well-known thermodynamic results for several widely studied models: the Resonance Level model, Interacting Resonance Level model and the Kondo model. Though the SBA is more complex than the traditional Thermodynamic Bethe Ansatz (TBA) when applied to thermodynamical questions, the scattering approach (SBA) allows access to an array of new questions that cannot be addressed otherwise, ranging from scattering of electrons off magnetic impurities to nonequilibrium dynamics.
We derive the transport properties of a quantum impurity system, often realized as a quantum dot, subject to a source-drain bias voltage at zero temperature and magnetic field. Using the Scattering Bethe Anstaz(SBA), a generalization of the traditional Thermodynamic Bethe Ansatz(TBA) to open systems out of equilibrium, we derive some transport results for the small quantum dot, described by Anderson impurity Hamiltonian. Exact dot occupation out of equilibrium and the nonlinear conductance are obtained by introducing phenomenological spin- and charge-fluctuation distribution functions in the computation of the current. The current and dot occupation as a function of voltage are evaluated numerically. We also vary the gate voltage and study the transition from the mixed valence to the Kondo regime in the presence of a non-equilibrium current. For the larger quantum dot the Interacting Resonance Level Model is used to describe the strog correlation problem in this system. We show some current vs voltage results and compare with other theoretical results. We conclude with the difficulties we encountered by using this method and suggest some future extensions related to this SBA approach.
We develop an exact nonperturbative framework to compute steady-state properties of quantum impurities subject to a finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates which satisfy an appropriate Lippman-Schwinger equation. Introducing a generalization of the equilibrium Bethe ansatz--the nonequilibrium Bethe ansatz--we explicitly construct the scattering eigenstates for the interacting resonance level model and derive exact, nonperturbative results for the steady-state properties of the system.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.
By mapping steady-state nonequilibrium to an effective equilibrium, we formulate nonequilibrium problems within an equilibrium picture. The Hamiltonian in the open system is rewritten in terms of scattering states with appropriate boundary condition. We first study the analytic properties of many-body scattering states, impose the boundary-condition operator in a statistical operator and prove that this mapping is equivalent to the linear-response theory in the low-bias limit. In an example of infinite-UUU Anderson impurity model, we approximately solve the scattering state creation operators, based on which we derive the bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ to construct the nonequilibrium ensemble in the form of the Boltzmann factor ${e}^{\ensuremath{-}\ensuremath{\beta}(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y})}$. The resulting effective Hamiltonian is solved by noncrossing approximation. We obtain the $I\text{\ensuremath{-}}V$ features of Kondo anomaly conductance at zero bias, inelastic transport via the charge excitation on the quantum dot, and significant inelastic current background over a wide range of bias.