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arXiv:cond-mat/0603703 · Physical Review Letters

Dynamic Structure Factor of the Calogero-Sutherland Model

Michael Pustilnik

Published Jul 19, 2006• 45 citations
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Abstract

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.

DOI
10.1103/physrevlett.97.036404

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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.
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 qqq 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 qqq.
The dynamic spin structure factor $\mathcal{S}(k,\ensuremath{\omega})$ of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function of $\ensuremath{\omega}$ it is shown to exhibit a power-law singularity at the threshold frequency defined by the energy of a magnon at given kkk. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.
For a one-dimensional model in which the two-body interactions are long-range and strong, the system almost crystallizes. The harmonic modes of such a lattice were used by Krivnov and Ovchinnikov to compute the ground-state wave function and the dynamical density-density correlations. We review this method, and apply it to the Calogero-Sutherland model, whose density-density correlation functions are exactly known for certain values of the coupling constant. We show numerically that the correlations obtained are quite accurate even if the coupling is not very large. Such comparisons have been made earlier by Forrester. The lattice method is considerably simpler than the ones used to derive the exact results, and yields expressions for the correlations- which are easily plotted. The equal-time correlations can be expanded in inverse powers of coupling; we show that the two leading order terms agree with the exact results which are known for integer values of the coupling. The strength-dependent power law fall-off is typical of a Luttinger liquid.In a general one-dimensional model where the two-body interaction decreases as a power of the relative distance, we argue, following Schulz, that at zero temperature the system behaves as a Luttinger liquid if the power exceeds 1, and as a Wigner crystal if it is less than 1.PACS Nos.: 63.20-e, 71.10Pm
The effective extended conformal field theory with symmetry [Formula: see text] that describes the thermodynamic limit of the Calogero–Sutherland (CS) model is considered. The dynamic structure factor of the chiral component in the repulsive regime is determined and compared with the corresponding one of the free bosonic theory, given that both share isomorphic Hilbert spaces but differ in the time evolution of quantum states. In either case, a sharp response function peaked at only one resonant frequency is found, and the physical implications of this outcome are addressed. Furthermore, a detailed comparison between this result and the corresponding one obtained in the first quantized formulation of the CS model is provided. Complete agreement in the parameter region in which both results overlap is found. This outcome provides further support to the equivalence between the first and second quantized treatments of the CS model, in which the computational advantages from integrability in the first formulation are paralleled by those from the algebraic structure of the second.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.