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arXiv:1811.08173 · Physica Scripta

Friedel oscillations and dynamical density of states of an inhomogeneous Luttinger liquid

Joy Prakash Das, Chandramouli Chowdhury, Girish S. Setlur

Published May 21, 2020• 5 citations
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Abstract

Abstract In this work, the four-point Green functions relevant to the study of Friedel oscillations are calculated for a Luttinger liquid (LL) with a cluster of impurities around an origin using the recently developed non-chiral bosonization technique (NCBT). Analytical expressions are obtained for the most singular part of the envelope of the oscillatory part of the local density in the form of power laws, providing closed expressions for the governing exponents. A comparison is done with the work of Egger and Grabert and an exact match of the exponent is obtained both for points near and far from the impurity. The two-point functions (most singular part) obtained using the same method are used to calculate the dynamical density of states (DDOS), which exhibits a power law in energy and closed analytical expressions for the DDOS exponent is calculated. These results interpolates between the weak barrier and weak link cases which are typically studied in the literature. The dependence of the DDOS on the nature of interactions and the strength of the impurity clusters are highlighted. Finally, the results of DDOS are favorably compared with existing literature such as density of states far away and near the impurity. We also compare with the often studied special case when the Luttinger parameter g = 1/2 and find agreement with our results for strong barriers.

DOI
10.1088/1402-4896/ab957f

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Abstract In this work, the four-point Green functions relevant to the study of Friedel oscillations are calculated for a Luttinger liquid (LL) with a cluster of impurities around an origin using the recently developed non-chiral bosonization technique (NCBT). Analytical expressions are obtained for the most singular part of the envelope of the oscillatory part of the local density in the form of power laws, providing closed expressions for the governing exponents. A comparison is done with the work of Egger and Grabert and an exact match of the exponent is obtained both for points near and far from the impurity. The two-point functions (most singular part) obtained using the same method are used to calculate the dynamical density of states (DDOS), which exhibits a power law in energy and closed analytical expressions for the DDOS exponent is calculated. These results interpolates between the weak barrier and weak link cases which are typically studied in the literature. The dependence of the DDOS on the nature of interactions and the strength of the impurity clusters are highlighted. Finally, the results of DDOS are favorably compared with existing literature such as density of states far away and near the impurity. We also compare with the often studied special case when the Luttinger parameter g = 1/2 and find agreement with our results for strong barriers.
The Unified Field Bosonization Technique (UFBT) is employed to investigate the single-particle properties of inhomogeneous Luttinger liquids containing localized clusters of impurities in the presence of forward-scattering interactions. Within this framework, fermionic two-point correlation functions are used to derive the leading-order singular contributions to the dynamical density of states (DDOS) and the cross-barrier tunneling conductance. The results reveal distinct power-law behaviours of the DDOS near the impurity cluster and in the far-field regime. The tunneling conductance exhibits a power-law temperature dependence, decreasing with decreasing temperature for repulsive interactions and increasing for attractive interactions, while remaining temperature-independent in the noninteracting limit. The impurity cluster modifies the amplitudes and spatial structure of these observables, whereas their characteristic power-law exponents are determined solely by the interaction parameters. The formulation accommodates arbitrary impurity strengths without requiring perturbative expansions in the impurity or interaction strengths. The characteristic scaling behaviour emerges directly from the fermionic correlation functions, without an explicit renormalization-group (RG) treatment of the impurity strength, while the weak-interaction limit recovers the leading perturbative correction to the tunneling response. These findings establish UFBT as a unified framework for describing local spectral properties and cross-barrier transport in inhomogeneous Luttinger liquids.
Abstract In this work, we show in pedagogical detail that the most singular contributions to the slow part of the asymptotic density-density correlation function of Luttinger liquids with fermions interacting mutually with only short-range forward scattering and also with localised scalar static impurities (where backward scattering takes place) has a compact analytical expression in terms of simple functions that have second order poles and involve only the scale-independent bare transmission and reflection coefficients. This proof uses conventional fermionic perturbation theory resummed to all orders, together with the idea that for such systems, the (connected) moments of the density operator all vanish beyond the second order—the odd ones vanish identically and the higher order even moments are less singular than the second order moment which is the only one included. This important result is the crucial input to the recently introduced ‘Non-Chiral Bosonization Technique’ (NCBT) to study such systems. The results of NCBT cannot be easily compared with the results obtained using conventional bosonization as the former only extracts the most singular parts of the correlation functions albeit for arbitrary impurity strengths and mutual interactions. The latter, ambitiously attempts to study all the parts of the asymptotic correlation functions and is thereby unable to find simple analytical expressions and is forced to operate in the vicinity of the homogeneous system or the half line (the opposite extreme). For a fully homogeneous system or its antithesis viz. the half-line, all the higher order connected moments of the density vanish identically which means the results of chiral bosonization and NCBT ought to be the same and indeed they are.
In this work, analytical expressions for the Green function of a Luttinger liquid are derived with one and two mobile impurities (heavy particles) using a combination of bosonization and perturbative approaches. The calculations are done in the random phase approximation (RPA) limit using the powerful non-chiral bosonization technique (NCBT) which is nothing but the resummation of the most singular parts of the RPA terms of the Green function expanded out in powers of the forward scattering between fermions with the source of inhomogeneities treated exactly . The force acting on the heavy particle(s) is studied as a function of its terminal velocity, both in the linear and non-linear regime. Linear mobility (which is valid for impurities moving much slower than a certain crossover speed) has a power law temperature dependence whose exponent has a closed algebraic expression in terms of the various parameters in the problem. This expression interpolates between the ballistic regime of no-coupling with the fermions and the no-tunneling regime. When the speed of the impurity is much larger than this crossover speed, the applied force depends non-linearly on the speed and this too is a power law with a closely related exponent. The case of two mobile impurities is also studied whose mobility exhibits peculiar resonances when their mutual separation is appropriately chosen.
We introduce the Unified Field Bosonization Technique (UFBT), a direct bosonization framework for strongly inhomogeneous one-dimensional Luttinger liquids (LLs) containing static impurities. UFBT incorporates impurity scattering through a symmetrized combination of bosonic phase fields and yields closed-form expressions for arbitrary N-point correlation functions for a broad class of static impurity potentials, including delta barriers, finite barriers and finite wells. The formalism requires neither renormalization-group analysis nor perturbative expansions, providing an analytical description of the inhomogeneous system at the bosonized level. The resulting correlation functions capture the leading singular contributions relevant to the present analysis. The technique is validated by recovering known limiting cases, showing agreement with the first-order perturbative expansion in the interaction strength, and demonstrating consistency with the Schwinger-Dyson equations. A key result is that the correlation function exponents remain independent of the impurity strength, while the impurity dependence is captured by the spatial structure and amplitudes of the correlation functions. The resulting correlation functions establish a foundation for analytical studies of transport, Friedel oscillations, and the local and dynamical density of states in strongly inhomogeneous Luttinger liquids.
We present an analytical study of impurity-induced density correlations in interacting one-dimensional Luttinger liquids using the Unified Field Bosonization Technique (UFBT). Unlike conventional approaches that treat impurities perturbatively, UFBT incorporates localized impurities exactly at the free-fermion level while treating interactions non-perturbatively. Closed analytical expressions for fermionic Green functions and density-density correlation functions are obtained for strongly inhomogeneous systems, enabling a unified description of the impurity-induced 2kF2k_F2kF​ Friedel oscillations and the 4kF4k_F4kF​ Wigner-crystal like correlations. The corresponding scaling behaviour is derived analytically, revealing a crossover in the dominant long-distance correlations at g=1/3g=1/3g=1/3: Friedel correlations dominate for g>1/3g>1/3g>1/3, whereas Wigner-crystal like correlations become dominant for g<1/3g<1/3g<1/3, corresponding to the regime of sufficiently strong repulsive interactions. A reflection-coupled harmonic analysis is developed to account for the left-right mixing induced by impurity backscattering and to consistently reproduce the resulting density correlations. The spinless limit is also obtained as a direct reduction of the spinful formulation. These results provide an analytically controlled framework for describing impurity-induced and interaction-driven density correlations in strongly inhomogeneous one-dimensional quantum systems.
We use a functional integral formalism developed earlier for the pure Luttinger liquid (LL) to find an exact representation for the electron Green function of the LL in the presence of a single backscattering impurity in the low-temperature limit. This allows us to reproduce results (well known from the bosonization techniques) for the suppression of the electron local density of states (LDOS) at the position of the impurity and for the Friedel oscillations at finite temperature. In addition, we have extracted from the exact representation an analytic dependence of LDOS on the distance from the impurity and shown how it crosses over to that for the pure LL.