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On the spectral properties of inhomogeneous Luttinger liquids

Manuel Simonato

Published Apr 1, 2019• 0 citations
OpenAlex

Abstract

The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.

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The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered
by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are
well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is
known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant
models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be
non-universal.
The LL concept emerged in the study of homogeneous systems with periodic boundary condition
(PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact
LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest
shifted towards the investigation of LLs with PBC including impurities. The fact that these systems
have been shown to scale to chains with open ends led to the study of several models where open
boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of
the power-law suppression close to the boundary differs from the one in the bulk.
The present work is devoted to the study of the local spectral function of two lattice models of LLs
with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the
1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects
of the spectral properties can be understood within the HF theory. Despite what one usually reads
about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory
is already capable of providing meaningful results. Concurrently, this method also allows us to develop
an understanding of the dip appearing in the local spectral function at an energy different from the
chemical potential.
Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface, we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational-invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both set-ups have the same analytical form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In the second part we approximately compute the local spectral function of a microscopic lattice model-the extended Hubbard model-close to an open boundary using the functional renormalization group. We show that it follows the field theoretical prediction in the low-energy regime as a function of energy and temperature, and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
We calculate one-particle spectra for a variety of models of Luttinger liquids with open boundary conditions. For the repulsive Hubbard model, the spectral weight close to the boundary is enhanced in a large energy range around the chemical potential. A power-law suppression, previously predicted by bosonization, only occurs after a crossover at energies very close to the chemical potential. Our comparison with exact spectra shows that the effects of boundaries can partly be understood within the Hartree-Fock approximation.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.
The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.