We apply the density matrix renormalization group (DMRG) method to the BCS pairing Hamiltonian which describes ultrasmall superconducting grains. Our version of the DMRG uses the particle (hole) states around the Fermi level as the system block (environment). We observed a smooth logarithmic-like crossover between the few electron regime and the BCS-bulk regime.
We apply the density matrix renormalization group (DMRG) method to the BCS pairing Hamiltonian which describes ultrasmall superconducting grains. Our version of the DMRG uses the particle (hole) states around the Fermi level as the system block (environment). We observed a smooth logarithmic-like crossover between the few electron regime and the BCS-bulk regime.
We study the properties of ultrasmall metallic grains with sizes in the range 20--400 electrons. Using a particle-hole version of the density-matrix renormalization-group (DMRG) method we compute condensation energies, spectroscopic gaps, pairing parameters, and particle-hole probabilities of the ground-state wave function. The results presented in this paper confirm that the bulk superconducting regime (large grains) and the fluctuation dominated regime (small grains) are qualitatively different, but show that the crossover between them is very smooth with no signs of critical level spacings separating them. We compare our DMRG results with the exact ones obtained with the Richardson solution finding complete agreement. We also propose a simplified version of the DMRG wave function, called the particle-hole BCS Ansatz, which agrees qualitatively with the DMRG solution and illustrates what is lacking in the projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.
Abstract. The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of onedimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the realspace DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to Quantum Chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with an outlook for future applications of the method.
The density matrix renormalization group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the real-space DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to quantum chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with a description of the outlook for future applications of the method.
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 thermodynamic behaviour of the enhanced hole-hopping model of high temperature superconductivity is investigated using the numerical Density Renormalization Group (DMRG) technique. The enhanced hole-hopping or Δt model is a Hubbard-like lattice model that has been proposed to account for superconductivity in the high temperature superconducting materials. Extensive results for this model have been obtained by others within the BCS approximation. This thesis does not attempt to motivate the use of the Δt model but rather it is the goal of this work to characterize the accuracy of two techniques used to study this model. In particular, the ground state energy and binding energy for a pair of particles as calculated within the DMRG are compared to similar results obtained from BCS and Exact Diagonalization studies. The DMRG is a relatively new numerical technique and consequently a detailed discussion of its implementation is given. Application of the DMRG necessarily confines investigation to one dimension. Analysis of finite size effects is also presented where warranted.
The particle-hole density matrix renormalization group $(p\ensuremath{-}h$ DMRG) method is discussed as a possible new approach to large-scale nuclear shell-model calculations. Following a general description of the method, we apply it to a class of problems involving many identical nucleons constrained to move in a single large j-shell and to interact via a pairing plus quadrupole interaction. A single-particle term that splits the shell into degenerate doublets is included so as to accommodate the physics of a Fermi surface in the problem. We apply the $p\ensuremath{-}h$ DMRG method to this test problem for two j values, one for which the shell model can be solved exactly and one for which the size of the Hamiltonian is much too large for exact treatment. In the former case, the method is able to reproduce the exact results for the ground state energy, the energies of low-lying excited states, and other observables with extreme precision. In the latter case, the results exhibit rapid exponential convergence, suggesting the great promise of this new methodology even for more realistic nuclear systems. We also compare the results of the test calculations with those from Hartree-Fock-Bogoliubov approximation and address several other questions about the $p\ensuremath{-}h$ DMRG method of relevance to its usefulness when treating more realistic nuclear systems.