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Localization of a particle due to dissipation in 1 and 2 dimensional lattices

Matthew Hasselfield

Published Jan 1, 2010• 0 citations
Publisher

Abstract

We study two aspects of the problem of a particle moving on a lattice while subject to dissipation, often called the "Schmid model." First, a correspondence between the Schmid model and boundary sine-Gordon field theory is explored, and a new method is applied to the calculation of the partition function for the theory. Second, a traditional condensed matter formulation of the problem in one spatial dimension is extended to the case of an arbitrary two-dimensional Bravais lattice. A well-known mathematical analogy between one-dimensional dissipative quantum mechanics and string theory provides an equivalence between the Schmid model at the critical point and boundary sine-Gordon theory, which describes a free bosonic field subject to periodic interaction on the boundaries. Using the tools of conformal field theory, the partition function is calculated as a function of the temperature and the renormalized coupling constants of the boundary interaction. The method pursues an established technique of introducing an auxiliary free boson, fermionizing the system, and constructing the boundary state in fermion variables. However, a different way of obtaining the fermionic boundary conditions from the bosonic theory leads to an alternative renormalization for the coupling constants that occurs at a more natural level than in the established approach. Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram for the mobility. We extend a classic one-dimensional, finite temperature calculation to the case of an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tightbinding lattice limits is reproduced in the two-dimensional case, and a perturbation expansion in the potential strength used to verify the change in the critical dependence of the mobility on the strength of the dissipation. With a triangular lattice the possibility of third order contributions arises, and we obtain some preliminary expressions for their contributions to the mobility.

DOI
10.14288/1.0085809

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We study two aspects of the problem of a particle moving on a lattice while subject to dissipation, often called the "Schmid model." First, a correspondence between the Schmid model and boundary sine-Gordon field theory is explored, and a new method is applied to the calculation of the partition function for the theory. Second, a traditional condensed matter formulation of the problem in one spatial dimension is extended to the case of an arbitrary two-dimensional Bravais lattice. A well-known mathematical analogy between one-dimensional dissipative quantum mechanics and string theory provides an equivalence between the Schmid model at the critical point and boundary sine-Gordon theory, which describes a free bosonic field subject to periodic interaction on the boundaries. Using the tools of conformal field theory, the partition function is calculated as a function of the temperature and the renormalized coupling constants of the boundary interaction. The method pursues an established technique of introducing an auxiliary free boson, fermionizing the system, and constructing the boundary state in fermion variables. However, a different way of obtaining the fermionic boundary conditions from the bosonic theory leads to an alternative renormalization for the coupling constants that occurs at a more natural level than in the established approach. Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram for the mobility. We extend a classic one-dimensional, finite temperature calculation to the case of an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tightbinding lattice limits is reproduced in the two-dimensional case, and a perturbation expansion in the potential strength used to verify the change in the critical dependence of the mobility on the strength of the dissipation. With a triangular lattice the possibility of third order contributions arises, and we obtain some preliminary expressions for their contributions to the mobility.
We study quantum Brownian motion (QBM) models for a particle in a dissipative environment coupled to a periodic potential. We review QBM for a particle in a one-dimensional periodic potential and extend the study to that for a particle in two-dimensional (2D) periodic potentials of four Bravais lattice types: square, rectangular, triangular (hexagonal), and centered rectangular. We perform perturbative renormalization group analyses to derive the zero temperature flow diagrams and phase boundaries for a particle in these potentials, and observe localization behavior dependent on the anisotropy of the lattice parameters.
The dissipative Hofstadter model describes quantum particles moving in two dimensions subject to a uniform magnetic field, a periodic potential and a dissipative force. We discuss the dissipative Hofstadter model in the framework of the boundary state formulation in string theory and construct exact boundary states for the model at the magic points by using the fermion representation. The dissipative Hofstadter model at magic points is shown to be equivalent to the critical boundary sine-Gordon model.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.
We study the motion of a particle in a periodic potential with Ohmic dissipation. In D=1D=1D=1 dimension it is well known that there are two phases depending on the dissipation: a localized phase with zero temperature mobility μ=0μ=0μ=0 and a fully coherent phase with μμμ unaffected by the periodic potential. For D>1D>1D>1, we find that this is also the case for a Bravais lattice. However, for non symmorphic lattices, such as the honeycomb lattice and its DDD dimensional generalization, there is a new intermediate phase with a universal mobility μ∗μ^*μ∗. We study this intermediate fixed point in perturbatively accessible regimes. In addition, we relate this model to the Toulouse limit of the D+1D+1D+1 channel Kondo problem. This mapping allows us to compute μ∗μ^*μ∗ exactly using results known from conformal field theory. Experimental implications are discussed for resonant tunneling in strongly coupled Coulomb blockade structures and for multi channel Luttinger liquids.
We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.
It has recently been shown that the dissipative Hofstadter model (dissipative quantum mechanics of an electron subject to uniform magnetic field and periodic potential in two dimensions) exhibits critical behavior on a network of lines in the dissipation/magnetic field plane. Apart from their obvious condensed matter interest, the corresponding critical theories represent non-trivial solutions of open string field theory, and a detailed account of their properties would be interesting from several points of view. A subject of particular interest is the dependence of physical quantities on the magnetic field since it, much like thetarmQCD\\theta_{\\rm QCD}thetarmQCD​, serves only to give relative phases to different sectors of the partition sum. In this paper we report the results of an initial investigation of the free energy, NNN-point functions and boundary state of this type of critical theory. Although our primary goal is the study of the magnetic field dependence of these quantities, we will present some new results which bear on the zero magnetic field case as well.