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arXiv:1710.06373 · Physical review. B./Physical review. B

Second-order functional renormalization group approach to one-dimensional systems in real and momentum space

Björn Sbierski, Christoph Karrasch

Published Dec 14, 2017• 5 citations
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Abstract

We devise a functional renormalization group treatment for a chain of interacting spinless fermions which is correct up to second order in interaction strength. We treat both inhomogeneous systems in real space as well as the translationally invariant case in a $k$-space formalism. The strengths and shortcomings of the different schemes as well as technical details of their implementation are discussed. We use the method to study two proof-of-principle problems in the realm of Luttinger liquid physics, namely, reflection at interfaces and power laws in the occupation number as a function of crystal momentum.

DOI
10.1103/physrevb.96.235122

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We devise a functional renormalization group treatment for a chain of interacting spinless fermions which is correct up to second order in interaction strength. We treat both inhomogeneous systems in real space as well as the translationally invariant case in a kkk-space formalism. The strengths and shortcomings of the different schemes as well as technical details of their implementation are discussed. We use the method to study two proof-of-principle problems in the realm of Luttinger liquid physics, namely, reflection at interfaces and power laws in the occupation number as a function of crystal momentum.
We compare two fermionic renormalization group (RG) methods which have been used to investigate the electronic transport properties of one-dimensional metals with two-particle interaction (Luttinger liquids) and local inhomogeneities. The first one is a poor man's method set-up to resum 'leading-log' divergences of the effective transmission at the Fermi momentum. Generically the resulting equations can be solved analytically. The second approach is based on the functional RG (fRG) method and leads to a set of differential equations which can only for certain set-ups and in limiting cases be solved analytically, while in general it must be integrated numerically. Both methods are claimed to be applicable for inhomogeneities of arbitrary strength and to capture effects of the two-particle interaction, such as interaction dependent exponents, up to leading order. We critically review this for the simplest case of a single impurity. While on first glance the poor man's approach seems to describe the crossover from the 'perfect' to the 'open chain fixed point' we collect evidence that difficulties may arise close to the 'perfect chain fixed point'. Due to a subtle relation between the scaling dimensions of the two fixed points this becomes apparent only in a detailed analysis. In the fRG method the coupling of the different scattering channels is kept which leads to a better description of the underlying physics.
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.
We study one-dimensional spinless fermions at zero and finite temperature TTT using the density-matrix renormalization group. We consider nearest- as well as next-nearest-neighbor interactions; the latter render the system inaccessible by a Bethe ansatz treatment. Using an infinite-system algorithm we demonstrate the emergence of Luttinger liquid physics at low energies for a variety of static correlation functions as well as for thermodynamic properties. The characteristic power-law suppression of the momentum distribution n(k)n(k)n(k) function at T=0T=0T=0 can be directly observed over several orders of magnitude. At finite temperature, we show that n(k)n(k)n(k) obeys a scaling relation. The Luttinger liquid parameter and the renormalized Fermi velocity can be extracted from the density response function, the specific heat, or the susceptibility without the need to carry out any finite-size analysis. We illustrate that the energy scale below which Luttinger liquid power laws manifest vanishes as the half-filled system is driven into a gapped phase by large interactions.
The functional renormalization group (fRG) approach has the property that, in general, the flow equation for the two-particle vertex generates O(N4)\mathcal{O}({N}^{4})O(N4) independent variables, where NNN is the number of interacting states (e.g., sites of a real-space discretization). In order to include the flow equation for the two-particle vertex, one needs to make further approximations if NNN becomes too large. We present such an approximation scheme, called the coupled-ladder approximation, for the special case of an onsite interaction. Like the generic third-order-truncated fRG, the coupled-ladder approximation is exact to second order and is closely related to a simultaneous treatment of the random phase approximation in all channels, i.e., summing up parquet-type diagrams. The scheme is applied to a one-dimensional model describing a quantum point contact.
We introduce approximate, functional renormalization group based schemes to obtain correlation functions in pure excited eigenstates of large fermionic many-body systems at arbitrary energies. The algorithms are thoroughly benchmarked and their strengths and shortcomings are documented using a one-dimensional interacting tight-binding chain as a prototypical testbed. We study two ``toy applications'' from the world of Luttinger liquid physics: the survival of power laws in lowly excited states as well as the spectral function of high-energy ``block'' excitations, which feature several single-particle Fermi edges.
We recapitulate recent developments of the functional renormalization group (FRG) approach to the steady state of systems out of thermal equilibrium. In particular, we discuss second-order truncation schemes which account for the frequency-dependence of the two particle vertex and which incorporate inelastic processes. Our focus is on two different types of one-dimensional fermion chains: i) infinite, open systems which feature a translation symmetry, and ii) finite systems coupled to left and right reservoirs. In addition to giving a detailed and unified review of the technical derivation of the FRG schemes, we briefly summarize some of the key physical results. In particular, we compute the non-equilibrium phase diagram and analyze the fate of the Berezinskii-Kosterlitz-Thouless transition in the infinite, open system.