Applications of the density matrix renormalization group to mesoscopic phenomena

Dominique Gobert, Ulrich Schollwöck

Published Jan 1, 2005• 1 citation

Abstract

In this thesis, I analyze properties of mesoscopic systems that exhibit strong quantum correlations. To this purpose, I adapt and use the density-matrix renormalization group (DMRG), a numerical method that has been developed specifically for analyzing such systems. The following systems are discussed: - Josephson effect between superconducting nanograins: I investigate the Josephson effect, i.e. the dependence of the energy of two weakly coupled superconductors on the difference between their superconducting phase, in the regime that the level spacing d is comparable to the bulk superconducting gap Delta. Because the BCS solution is inapplicable in this regime, I use the DMRG to calculate the ground state of the two coupled superconductors and extract the Josephson energy E_J. The standard BCS result for E_J is reproduced only in the continuum limit of vanishing d. As d is increased, however, E_J turns out to display a nonmonotonic behaviour. A tight-binding approximation for weak Josephson coupling explains the physical mechanism underlying this reentrance in a transparent way. - Well-defined quasiparticles in small metallic grains: I analyze zero-temperature spectral functions of mesoscopic systems such as quantum dots and metallic grains, in the limit of large conductance, where they can be described by a "universal Hamiltonian" model. I show that within this model, an important class of spectral functions is dominated by one single energy eigenstate only. For an interacting system this is a very peculiar property, which implies an infinite lifetime of the quasiparticles. Moreover, I show that the dominating eigenstate contains only a limited subclass of all excitations, which I characterize as the "No-Gaudino" excitations. Hence, these are sufficient to explain many of the properties of the systems under consideration. Besides its own physical significance, the dominance of the "No-Gaudino" states has also high practical value, because it permits the calculation of zero-temperature spectral functions with high accuracy using the DMRG. I illustrate the use of this method by calculating the tunneling density of states of metallic grains and the magnetic response of mesoscopic rings. - Real-time dynamics in spin-1/2 chains I study the nonequilibrium transport properties in spin-1/2 chains by solving the time evolution of a non-stationary initial state, and investigate the influence of different interaction strength and dimerization on the magnetization transport. To this purpose, I use the "adaptive time-dependent DMRG", a DMRG variant that was recently developed to solve the many-body Schroedinger equation with high accuracy. I find that the magnetization possesses a well-defined long-time limit, whose nature does not depend on the dimerization, but only on the strength J_z of the S^z-S^z-interaction: For |J_z| < 1 I find ballistic magnetization transport, and for J_z >1 almost no transport, with a sharp crossover at |J^z|=1. I explain this crossover as a subtle consequence of a quantum phase transition which occurs at the precise value J_z| = 1. - Many-body scattering states: I present a general method for calculating many-body scattering states that does not rely on the assumptions of perturbation theory or near-equilibrium. Due to the complexity of the problem, I limit the discussion to a description of the algorithm itself and a few proof-of-principle calculations only. The strategy is to calculate the many-body scattering state that results when a scatterer (e.g. a quantum dot) is connected to two leads, to which a voltage bias V is applied. This state is obtained by solving the many-body Lippmann-Schwinger equation using the DMRG.

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