2008/02/07 by Peyman Pirooznia, Florian Schuetz, Florian Schütz +1
Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Physics of Superconductivity and Magnetism #Quantum, superfluid, helium dynamics #cond-mat.stat-mech #cond-mat.str-el
paper · pdf · doi:10.1103/physrevb.78.075111
published as Phys. Rev. B 78, 075111 (2008) · 33 Revtex pages, 17 figures
arxiv created 2008/02/07 · openalex publication_date 2008/08/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We calculate the dynamic structure factor S(\ensuremathω,q) of spinless fermions in one dimension with quadratic energy dispersion k2/2m and long-range density-density interaction whose Fourier transform fq is dominated by small momentum transfers q\ensuremath\lesssimq0⪡kF. Here q0 is a momentum-transfer cutoff and 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. 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 fq=f0+f0^\ensuremath''q2/2+O(q4) with f0^\ensuremath''\ensuremath≠0, we show that the momentum scale qc=1/|mf0^\ensuremath''| separates two regimes characterized by a different q dependence of the width \ensuremathγq of the collective zero sound mode and other features of S(\ensuremathω,q). For qc⪡q⪡kF we find that the line shape is non-Lorentzian with an overall width \ensuremathγq\ensuremath∝q3/(mqc) and a threshold singularity [(\ensuremathω\ensuremath-\ensuremathωq^\ensuremath-)ln2(\ensuremathω\ensuremath-\ensuremathωq^\ensuremath-)]^\ensuremath-1 at the lower edge \ensuremathω\ensuremath→\ensuremathωq^\ensuremath-=vq\ensuremath-\ensuremathγq, where v 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 \ensuremathμq=1\ensuremath-2\ensuremathηq with \ensuremathηq\ensuremath∝qc2/q2. Although for q\ensuremath\lesssimqc we have not succeeded to explicitly evaluate our functional bosonization result for S(\ensuremathω,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 for q\ensuremath→0.