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arXiv:cond-mat/0404614 · Physical Review Letters

Well-Defined Quasiparticles in Interacting Metallic Grains

Dominique Gobert, Moshe Schechter, Ulrich Schollwöck, Jan von Delft

Published Oct 26, 2004• 5 citations
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Abstract

We analyze spectral functions of mesoscopic systems with large dimensionless conductance, which can be described by a universal Hamiltonian. We show that an important class of spectral functions are dominated by one single state only, which implies the existence of well-defined (i.e., infinite-lifetime) quasiparticles. Furthermore, the dominance of a single state enables us to calculate zero-temperature spectral functions with high accuracy using the density-matrix renormalization group. We illustrate the use of this method by calculating the tunneling density of states of metallic grains, of which we discuss the crossover from the few-electron to the bulk regime.

DOI
10.1103/physrevlett.93.186402

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We analyze spectral functions of mesoscopic systems with large dimensionless conductance, which can be described by a universal Hamiltonian. We show that an important class of spectral functions are dominated by one single state only, which implies the existence of well-defined (i.e., infinite-lifetime) quasiparticles. Furthermore, the dominance of a single state enables us to calculate zero-temperature spectral functions with high accuracy using the density-matrix renormalization group. We illustrate the use of this method by calculating the tunneling density of states of metallic grains, of which we discuss the crossover from the few-electron to the bulk regime.
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.
The physics of the strongly interacting Hubbard chain (with $t/U\ensuremath{\ll}1$) at finite temperatures undergoes a crossover to a spin-incoherent regime when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale. This crossover can be understood by means of Ogata and Shiba's factorized wave function, where charge and spin are totally decoupled, and assuming that the charge remains in the ground state, while the spin is thermally excited and at an effective ``spin temperature.'' We use the time-dependent density matrix renormalization group method to calculate the dynamical contributions of the spin, to reconstruct the single-particle spectral function of the electrons. The crossover is characterized by a redistribution of spectral weight both in frequency and momentum, with an apparent shift by kF{k}_{F}kF​ of the minimum of the dispersion.
In periodic systems of interacting electrons, Fermi and Luttinger surfaces refer to the locations within the Brillouin zone of poles and zeros, respectively, of the single-particle Green's function at zero energy and temperature. Such difference in analytic properties underlies the emergence of well-defined quasiparticles close to a Fermi surface, in contrast to their supposed non-existence close to a Luttinger surface, where the single-particle density-of-states vanishes at zero energy. We here show that, contrary to such common belief, dispersive 'quasiparticles' with infinite lifetime do exist also close to a pseudo-gapped Luttinger surface. Thermodynamic and dynamic properties of such 'quasiparticles' are just those of conventional ones. For instance, they yield well-defined quantum oscillations in Luttinger surface and linear-in-temperature specific heat, which is striking given the vanishing density of states of physical electrons, but actually not uncommon in strongly correlated materials.
The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators.The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions.Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the realfrequency zero-temperature and Keldysh formalisms.Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms.The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels.The former are formalism independent but system specific; the latter are system independent but formalism specific.Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models.We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model.In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles.In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.
The single electron Green's function of the one-dimensional Tomonaga-Luttinger model in the presence of open boundaries is calculated with bosonization methods. We show that the critical exponents of the local spectral density and of the momentum distribution change in the presence of a boundary. The well understood universal bulk behavior always crosses over to a boundary dominated regime for small energies or small momenta. We show this crossover explicitly for the large-U Hubbard model in the low-temperature limit. Consequences for photoemission experiments are discussed.