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arXiv:0905.0598 · Physical Review B

Edge exponent in the dynamic spin structure factor of the Yang-Gaudin model

Mikhail B. Zvonarev, Vadim Cheianov, Thierry Giamarchi

Published Nov 10, 2009• 42 citations
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Abstract

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 $k$. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.

DOI
10.1103/physrevb.80.201102

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The dynamic spin structure factor S(k,\ensuremathω)\mathcal{S}(k,\ensuremath{\omega})S(k,\ensuremathω) of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function of \ensuremathω\ensuremath{\omega}\ensuremathω 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.
Exact formulas for the singularities of the dynamical structure factor, Szz(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts which are known exactly from the Bethe ansatz solution. We also study the long-time asymptotics of the self-correlation function 0|Sjz(t)Sjz(0)|0. Utilizing these results to supplement very accurate time-dependent density matrix renormalization group, for short to moderate times, we calculate Szz(q,omega) to very high precision.
We study the dynamic structure factor $S(k,\ensuremath{\omega})$ of the spin-1/2 chain with long-range, power-law decaying unfrustrated (sign-alternating) Heisenberg interactions ${J}_{r}\ensuremath{\sim}{(\ensuremath{-}1)}^{r\ensuremath{-}1}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ by means of stochastic analytic continuation (SAC) of imaginary-time correlations computed by quantum Monte Carlo (QMC) calculations. We do so in both the long-range antiferromagnetic (AFM, for $\ensuremath{\alpha}\ensuremath{\lesssim}2.23$) and quasi-long-range-ordered (QLRO, for $\ensuremath{\alpha}\ensuremath{\gtrsim}2.23$) ground-state phases, employing different SAC parametrizations of $S(k,\ensuremath{\omega})$ to resolve sharp edges characteristic of fractional quasiparticles and sharp peaks expected with conventional quasiparticles. In order to identify the most statistically accurate parametrization, we apply a newly developed cross-validation method as a ``model selection'' tool. We confirm that the spectral function contains a power-law divergent edge in the QLRO phase and a very sharp (likely $\ensuremath{\delta}$-function) magnon peak in the AFM phase. From our SAC results, we extract the dispersion relation in the different regimes of the model, and in the AFM phase we extract the weight of the magnon pole. In the limit where the model reduces to the conventional Heisenberg chain with nearest-neighbor interactions, our $S(k,\ensuremath{\omega})$ agrees well with known Bethe-ansatz results. In the AFM phase the low-energy dispersion relation is known to be nonlinear, ${\ensuremath{\omega}}_{k}\ensuremath{\sim}{k}^{z}$, and we extract the corresponding dynamic exponent $z(\ensuremath{\alpha})$, which in general is somewhat above the form obtained in linear spin-wave theory. We also find a significant continuum above the magnon peak. This study serves as a benchmark for SAC and QMC studies of systems with a transition from conventional to fractionalized quasiparticles.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
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 study the longitudinal and transverse spin dynamical structure factors of the spin-1/2 XXX chain at finite magnetic field h, focusing in particular on the singularities at excitation energies in the vicinity of the lower thresholds. While the static properties of the model can be studied within a Fermi-liquid like description in terms of pseudoparticles, our derivation of the dynamical properties relies on the introduction of a form of the 'pseudofermion dynamical theory' (PDT) of the 1D Hubbard model suitably modified for the spin-only XXX chain and other models with two pseudoparticle Fermi points. Specifically, we derive the exact momentum and spin-density dependences of the exponents ζ(τ)(k) controlling the singularities for both the longitudinal (τ = l) and transverse (τ = t) dynamical structure factors for the whole momentum range k ∈ ]0,π[, in the thermodynamic limit. This requires the numerical solution of the integral equations that define the phase shifts in these exponents expressions. We discuss the relation to neutron scattering and suggest new experiments on spin-chain compounds using a carefully oriented crystal to test our predictions.