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arXiv:0709.0960 · Physical Review Letters

Exact Edge Singularities and Dynamical Correlations in Spin- 1 / 2 Chains

Rodrigo G. Pereira, Steven R. White, Ian K. Affleck

Published Jan 17, 2008• 133 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.100.027206

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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.
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.
Transition rates and dynamic spin structure factor at zero temperature for the spin-1/2 XXZ chain at critical regime in a magnetic field are numerically evaluated in terms of the exact determinant representations for the form factors and norms of the Bethe eigenstates. We have seen that the transition rates converges toward the constant function with the value 1 in the limit delta -> 0. The observed critical exponent of the singularity at the lower boundary is compared with the one predicted from the comformal field theory. We confirm that they are in good agreement. Further we have discovered that a small peak emerges near the upper boundary in the line shape of S(q,omega) for 0
We combine Bethe ansatz and field theory methods to study the longitudinal dynamical structure factor S zz ( q ,ω) for the anisotropic spin-1/2 chain in the gapless regime. Using bosonization, we derive a low-energy effective model, including the leading irrelevant operators (band curvature terms) which account for boson decay processes. The coupling constants of the effective model for finite anisotropy and finite magnetic field are determined exactly by comparison with corrections to thermodynamic quantities calculated by Bethe ansatz. We show that a good approximation for the shape of the on-shell peak of S zz ( q ,ω) in the interacting case is obtained by rescaling the result for free fermions by certain coefficients extracted from the effective Hamiltonian. In particular, the width of the on-shell peak is argued to scale like δω q ∼ q 2 and this prediction is shown to agree with the width of the two-particle continuum at finite fields calculated from the Bethe ansatz equations. An exception to the q 2 scaling is found at finite field and large anisotropy parameter (near the isotropic point). We also present the calculation of the high-frequency tail of S zz ( q ,ω) in the region using finite-order perturbation theory in the band curvature terms. Both the width of the on-shell peak and the high-frequency tail are compared with S zz ( q ,ω) calculated by Bethe ansatz for finite chains using determinant expressions for the form factors and excellent agreement is obtained. Finally, the accuracy of the form factors is checked against the exact first moment sum rule and the static structure factor calculated by density matrix renormalization group (DMRG).
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.
The longitudinal spin structure factor for the XXZ-chain at small wave vector q is obtained using Bethe ansatz, field theory methods, and the density matrix renormalization group. It consists of a peak with a peculiar, non-Lorentzian shape and a high-frequency tail. We show that the width of the peak is proportional to q2 for finite magnetic field compared to q3 for a zero field. For the tail we derive an analytic formula without any adjustable parameters and demonstrate that the integrability of the model directly affects the line shape.
We compute all dynamical spin–spin correlation functions for the spin-1/2 XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulae for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.