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Progress of Theoretical Physics Supplement

DMRG Studies of Impurities in Luttinger Liquids

U. Schollwöck, V. Meden, Walter Metzner, Kurt Schönhammer

Published Jan 1, 2002• 3 citations
Publisher

Abstract

Using the Density Matrix Renormalization Group (DMRG) and various analytical techniques (functional renormalization) we consider the effects of local impurities in Luttinger liquids. We find that the universal physics predicted by bosonization sets in only at extremely long, experimentally arguably irrelevant length scales or small energy scales respectively. The explicit construction of the RG flow allows to trace this behaviour to an extremely weak renormalization in real space of the impurities associated to the generation of a very long-ranged oscillating scattering potential.

DOI
10.1143/ptps.145.312

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Using the Density Matrix Renormalization Group (DMRG) and various analytical techniques (functional renormalization) we consider the effects of local impurities in Luttinger liquids. We find that the universal physics predicted by bosonization sets in only at extremely long, experimentally arguably irrelevant length scales or small energy scales respectively. The explicit construction of the RG flow allows to trace this behaviour to an extremely weak renormalization in real space of the impurities associated to the generation of a very long-ranged oscillating scattering potential.
Using a fermionic renormalization group method we present a simple real space picture of the strong influence an impurity has on the electronic properties of a Luttinger liquid. We compute the flow of the renormalized impurity potential for a single impurity over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. We confirm that low energy properties close to the impurity are as if the chain is cut in two pieces with open boundary conditions at the end points, but show that this universal behavior is only reached for extremely large systems. The accuracy of the renormalization group scheme is demonstrated by a direct comparison with data obtained from the density-matrix renormalization group method.
Using a functional renormalization group, we compute the flow of the renormalized impurity potential for a single impurity in a Luttinger liquid over the entire energy range from the microscopic scale of a lattice-fermion model down to the low-energy limit. The nonperturbative method provides a complete real-space picture of the effective impurity potential. We confirm the universality of the open chain fixed point, but it turns out that very large systems (104−−105{(10}^{4}--{10}^{5}(104−−105 sites) are required to reach the fixed point for realistic choices of the impurity and interaction parameters.
We improve the recently developed functional renormalization group (fRG) for impurities and boundaries in Luttinger liquids by including renormalization of the two-particle interaction, in addition to renormalization of the impurity potential. Explicit flow equations are derived for spinless lattice fermions with nearest-neighbor interaction at zero temperature, and a fast algorithm for solving these equations for very large systems is presented. We compute spectral properties of single-particle excitations, and the oscillations in the density profile induced by impurities or boundaries for chains with up to 106{10}^{6}106 lattice sites. The expected asymptotic power laws at low energy or long distance are fully captured by the fRG. Results on the relevant energy scales and crossover phenomena at intermediate scales are also obtained. A comparison with numerical density matrix renormalization results for systems with up to 1000 sites shows that the fRG with the inclusion of vertex renormalization is remarkably accurate even for intermediate interaction strengths.
In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin-1/2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.
Many quantum mechanical problems (such as dissipative phase fluctuations in metallic and superconducting nanocircuits or impurity scattering in Luttinger liquids) involve a continuum of bosonic modes with a marginal spectral density diverging as the inverse of energy. We construct a numerical renormalization group in this singular case, with a manageable violation of scale separation at high energy, capturing reliably the low energy physics. The method is demonstrated by a nonperturbative solution over several energy decades for the dynamical conductance of a Luttinger liquid with a single static defect.
This thesis comprises two parts centered around the functional renormalization-group framework: in the first part, I study the role of symmetries and conservation laws in approximate solutions, while in the second part I analyze Friedel oscillations and transport in Luttinger liquids with impurities. The functional renormalization group (fRG) has been developed as a new computational tool in the theory of interacting Fermi systems. The effective behavior of a given microscopic model is calculated by solving coupled differential flow equations for the Green functions with an energy scale as the flow parameter. The symmetries of the microscopic model imply Ward identities between Green and response functions. It is shown that solutions of truncated flow-equation hierarchies satisfy Ward identities if the cutoff bare action is gauge invariant. However, truncations are generally not self-consistent approximations in the sense of Baym and Kadanoff. The fRG is then applied to study Luttinger liquids. By computing the full spatial effective potential of a single impurity, long-range Friedel oscillations are observed in the density profile with the expected power laws for systems with up to 10^7 lattice sites. For a double barrier enclosing a dot region we find temperature regimes in which the conductance follows power laws with universal exponents, as well as non-universal crossover regimes in intermediate parameter regions.