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arXiv:cond-mat/0603458 · Physical Review Letters

Dynamic Response of One-Dimensional Interacting Fermions

Michael Pustilnik, Maxim Khodas, Alex Kamenev, L. I. Glazman

Published May 1, 2006• 177 citations
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Abstract

We evaluate the dynamic structure factor $S(q,\ensuremath{\omega})$ of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of $S(q,\ensuremath{\omega})$. The sharp peak $S(q,\ensuremath{\omega})\ensuremath{\propto}q\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}uq)$, characteristic for the Tomonaga-Luttinger model, broadens up; $S(q,\ensuremath{\omega})$ for a fixed $q$ becomes finite at arbitrarily large $\ensuremath{\omega}$. The main spectral weight, however, is confined to a narrow frequency interval of the width $\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}{q}^{2}/m$. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number $q$.

DOI
10.1103/physrevlett.96.196405

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We evaluate the dynamic structure factor S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω). The sharp peak S(q,\ensuremathω)\ensuremath∝q\ensuremathδ(\ensuremathω\ensuremath−uq)S(q,\ensuremath{\omega})\ensuremath{\propto}q\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}uq)S(q,\ensuremathω)\ensuremath∝q\ensuremathδ(\ensuremathω\ensuremath−uq), characteristic for the Tomonaga-Luttinger model, broadens up; S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) for a fixed qqq becomes finite at arbitrarily large \ensuremathω\ensuremath{\omega}\ensuremathω. The main spectral weight, however, is confined to a narrow frequency interval of the width \ensuremathδ\ensuremathω\ensuremath∼q2/m\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}{q}^{2}/m\ensuremathδ\ensuremathω\ensuremath∼q2/m. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number qqq.
We evaluate the dynamic structure factor S(q, omega) of a one-dimensional quantum Hamiltonian with the inverse-square interaction (Calogero-Sutherland model). For a fixed small q, the structure factor differs from zero in a finite interval of frequencies of the width deltaomega proportional to q2/m. At the borders of this interval S(q, omega) exhibits power-law singularities with exponents depending on the interaction strength. The singularities are similar in origin to the well-known Fermi-edge singularity in the x-ray absorption spectra of metals.
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We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion k2/2m{k}^{2}/2mk2/2m and long-range density-density interaction whose Fourier transform fq{f}_{q}fq​ is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here q0{q}_{0}q0​ is a momentum-transfer cutoff and kF{k}_{F}kF​ is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter q0/kF{q}_{0}/{k}_{F}q0​/kF​. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different qqq dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For qc⪡q⪡kF{q}_{c}⪡q⪡{k}_{F}qc​⪡q⪡kF​ we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where vvv is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as q2/m{q}^{2}/mq2/m for $q\ensuremath{\rightarrow}0$.
We present a class of one-dimensional, generic, spinless fermion lattice Hamiltonians that express quasi-Fermi liquid physics, manifesting both Luttinger and Fermi liquid features solely due to irrelevant interactions. Using infinite matrix product state techniques, we unveil its universal structure by calculating static and dynamic responses. Key features include a finite discontinuity in the momentum distribution at the Fermi level, despite power-law singularities in the spectral function protected by particle-hole symmetry. Away from half filling, Landau quasiparticles emerge. Charge dynamics show either high-energy bound states or a concentration of spectral weight within the continuum for attractive or repulsive interactions, respectively. These universal features are realized across multiple models and energy scales, thus reifying the quasi-Fermi liquid as a unique paradigm for one-dimensional fermions.
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.