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arXiv:cond-mat/0703128 · Physical Review B

Indirect forces between impurities in one-dimensional quantum liquids

P. Wächter, V. Meden, Kurt Schönhammer

Published Jul 31, 2007• 26 citations
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Abstract

We investigate the indirect interaction between two isolated impurities in a Luttinger liquid described by a microscopic lattice model. To treat the electron-electron interaction $U$, the functional renormalization group method is used. For comparison, we also study the $U=0$ case. We find that for a wide range of impurity parameters the impurity interaction ${V}_{12}$ as a function of their separation $r$ oscillates with decaying amplitude between being attractive and repulsive. For half-filling of the band and in a crossover regime between weak and strong impurities, the interaction becomes purely attractive. For $U=0$ and independent of the impurity strength, the amplitude of the interaction energy falls off as $1∕r$. For $U>0$, the decay for small separations and weak to intermediate impurities is governed by a $U$ dependent exponent larger than $\ensuremath{-}1$, which crosses over to $\ensuremath{-}1$ for large $r$. The crossover scale depends on the impurity strength and $U$. We present simple pictures which explain our results in the limits of weak and strong impurities. We finally also consider attractive interactions $U<0$.

DOI
10.1103/physrevb.76.045123

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We investigate the indirect interaction between two isolated impurities in a Luttinger liquid described by a microscopic lattice model. To treat the electron-electron interaction UUU, the functional renormalization group method is used. For comparison, we also study the U=0U=0U=0 case. We find that for a wide range of impurity parameters the impurity interaction V12{V}_{12}V12​ as a function of their separation rrr oscillates with decaying amplitude between being attractive and repulsive. For half-filling of the band and in a crossover regime between weak and strong impurities, the interaction becomes purely attractive. For U=0U=0U=0 and independent of the impurity strength, the amplitude of the interaction energy falls off as 1∕r1∕r1∕r. For U>0U>0U>0, the decay for small separations and weak to intermediate impurities is governed by a UUU dependent exponent larger than \ensuremath−1\ensuremath{-}1\ensuremath−1, which crosses over to \ensuremath−1\ensuremath{-}1\ensuremath−1 for large rrr. The crossover scale depends on the impurity strength and UUU. We present simple pictures which explain our results in the limits of weak and strong impurities. We finally also consider attractive interactions U<0U<0U<0.
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.
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.
We study Friedel oscillations in one-dimensional electron liquids for arbitrary electron-electron interaction and arbitrary impurity strength. Explicit results for spinless as well as spin- 12\frac{1}{2}21​ electrons are given. In the case of Luttinger-liquid leads, the Friedel oscillations decay as ${x}^{\ensuremath{-}g}$ far away from the impurity where ggg is the interaction constant. For a weak scatterer, a slower decay is found at small-to-intermediate distances from the impurity, with a crossover to the asymptotic ${x}^{\ensuremath{-}g}$ decay.
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.
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 study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.