Dynamical structure factor at small q for the XXZ spin-1/2 chain

Department of Physics and Astronomy, University of California, Irvine, Irvine, California, United States
Journal of Statistical Mechanics Theory and Experiment (Impact Factor: 2.4). 06/2007; 2007(08). DOI: 10.1088/1742-5468/2007/08/P08022
Source: arXiv


We combine Bethe Ansatz and field theory methods to study the longitudinal dynamical structure factor S^{zz}(q,omega) 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,omega) 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 delta omega_{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,\omega) in the region delta omega_{q}<< omega-vq << J 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,omega) 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). Comment: 67 pages, 25 figures

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Available from: J. Sirker, Feb 22, 2015
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    • "The determinant representation for the scalar product also was used for the calculation of form factors of local operators [17] [18] [19]. These results were used for the analytical [20] [21] and numerical analysis of correlation functions [22] [23] [24] [25]. "
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    ABSTRACT: We study quantum integrable models with $GL(3)$ trigonometric $R$-matrix solvable by the nested algebraic Bethe ansatz. We analyze scalar products of generic Bethe vectors and obtain an explicit representation for them in terms of a sum with respect to partitions of Bethe parameters. This representation generalizes known formula for the scalar products in the models with $GL(3)$-invariant $R$-matrix.
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    • "Furthermore explicit and compact formulas for the scalar products sometimes allow one to study the correlation functions even in such models, for which the solution of the inverse scattering problem is not known [3, 5–7, 10]. This approach was successfully applied for the quantum integrable models with GL(2)-invariant or GL(2) trigonometric R-matrix [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]. In all these works a determinant representation for the scalar products of the Bethe vectors obtained in [23] was essentially used. "
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    ABSTRACT: Using strong-disorder renormalization group, numerical exact diagonalization, and quantum Monte Carlo methods, we revisit the random antiferromagnetic XXZ spin-1/2 chain focusing on the long-length and ground-state behavior of the average time-independent spin-spin correlation function C(l)=\upsilon l^{-\eta}. In addition to the well-known universal (disorder-independent) power-law exponent \eta=2, we find interesting universal features displayed by the prefactor \upsilon=\upsilon_o/3, if l is odd, and \upsilon=\upsilon_e/3, otherwise. Although \upsilon_o and \upsilon_e are nonuniversal (disorder dependent) and distinct in magnitude, the combination \upsilon_o + \upsilon_e = -1/4 is universal if C is computed along the symmetric (longitudinal) axis. The origin of the nonuniversalities of the prefactors is discussed in the renormalization-group framework where a solvable toy model is considered. Moreover, we relate the average correlation function with the average entanglement entropy, whose amplitude has been recently shown to be universal. The nonuniversalities of the prefactors are shown to contribute only to surface terms of the entropy. Finally, we discuss the experimental relevance of our results by computing the structure factor whose scaling properties, interestingly, depend on the correlation prefactors.
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