Dynamic Response of One-Dimensional Interacting Fermions

Abstract

We evaluate the dynamic structure factor S(q, ω) 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, ω). The sharp peak S(q, ω) ∝ qδ(ω − uq), characteristic for the Tomonaga-Luttinger model, broadens up; S(q, ω) for a fixed q becomes finite at arbitrarily large ω. The main spectral weight, however, is confined to a narrow frequency interval of the width δω ∼ 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.PACS numbers: 71.10.Pm, 72.15.Nj Low-energy properties of fermionic systems are sensitive to interactions between fermions. The effect of interactions is the strongest in one dimension (1D), where single-particle correlation functions exhibit power-law singularities, in a striking departure from the behavior in higher dimensions. Much of our current understanding of 1D fermions is based on the Tomonaga-Luttinger (TL) model [1]. The crucial ingredient of the model is the assumption of a strictly linear fermionic dispersion relation. The TL model, often used in conjunction with a powerful bosonization technique [2], allows one to evaluate various correlation functions, such as momentum-resolved [3,4] and local [5] single-particle densities of states.Unlike the single-particle correlation functions, the two-particle correlation functions of the TL model exhibit behavior rather compatible with that expected for a Fermi liquid with the linear spectrum of quasiparticles. For example, the dynamic structure factor (the densitydensity correlation function)at small q takes the form [3] S TL (q, ω) ∝ qδ(ω − uq). It means that the quanta of density waves propagating with plasma velocity u are true eigenstates of the TL model; these bosonic excitations have an infinite lifetime. Below we show that such a simple behavior is an artefact of the linear spectrum approximation. In reality, the spectrum of fermions always has some nonlinearity,where the upper/lower sign corresponds to the right/left movers (R/L), and k = p ∓ p F are momenta measured from the Fermi points ± p F . (For Galilean-invariant systems the expansion (2) terminates at k 2 .) The finite curvature (1/m = 0) affects drastically the functional form of S(q, ω). In a clear deviation from the results of TL model, power-law singularities now arise not only in the single-particle correlation functions, but in the structure factor as well. We show that the singularities in these two very different objects have a common origin, proliferation of low-energy particle-hole pairs, and evaluate the corresponding exponents.Because of the success of the bosonization technique [2], it is tempting to treat the spectrum nonlinearity as a weak interaction between the TL bosons. Indeed, the nonlinearity gives rise to a three-boson interaction with the coupling constant ∝ 1/m [6]. However, attempts to treat this ...

Charge dynamics in half-filled Hubbard chains with finite on-site interaction

“…[4][5][6][7][8] In addition, ultracold atoms trapped in optical lattices have emerged as a new means to study coherent dynamics of 1D models, including integrable ones which are not realizable in condensed matter systems. 9 At the same time, significant progress has been achieved in developing analytical [10][11][12][13][14][15][16][17][18][19][20][21][22] and numerical [23][24][25] techniques to study dynamical correlation functions in the high energy regime where conventional Luttinger liquid theory 26,27 does not apply. Analytically, it is possible to compute exponents of power-law singularities that develop near thresholds of the spectrum of dynamical correlation functions at arbitrarily high energies.…”

Current density functional theory for one‐dimensional fermions

“…For example, a continuum of collective excitations is found in [27,28] for a certain frequency range, ω − < ω < ω + , where ω ± are q- and interaction-dependent. In addition, spectral weight is shifted (for a repulsive interaction) to the lower end of the continuum, leading to a power-law divergence near ω − .…”

“…In addition, in the exact data more and more spectral weight is shifted down to the left sub-peaks of the low energy doublets, a feature which is not obtained within CDFT-LDA. The transfer of spectral weight to lower frequencies eventually leads to the formation of the power-law divergence at the lower end of the continuum in the infinite system [27,28].…”

Fermi gas with attractive potential and spinS=32in a one-dimensional trap: Response functions for superfluidity and FFLO signatures

“…In recent papers [56][57][58][59] it was shown that neglecting irrelevant operators perturbing the Luttinger liquid Hamiltonian can lead to incorrect results for on-shell singularities in correlation functions. Introducing a coupling to a mobile impurity and taking the leading irrelevant operators into account nonperturbatively it is possible to recover the exact singularity threshold and critical exponent.…”