SciPost Thesis Link
| Title: | Adjacent spin operator correlations in the Heisenberg spin chain | |
| Author: | A.M. Klauser | |
| As Contributor: | (not claimed) | |
| Type: | Ph.D. | |
| Field: | Physics | |
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| Approach: | Theoretical | |
| URL: | http://hdl.handle.net/11245/1.369026 | |
| Degree granting institution: | University of Amsterdam | |
| Supervisor(s): | J.-S. Caux, J. van den Brink | |
| Defense date: | 2012-04-05 |
Abstract:
Contents of the thesis We start the thesis by describing how to solve the Heisenberg spin chain with the coordinate Bethe Ansatz and we create a catalog of the principal excitations. We give then a short introduction to the algebraic Bethe Ansatz formalism and the local representation of spin operators. With these tools, we present our original results: • We show that the response function of the K-edge Resonant Inelastic X- ray Scattering experiment on a Heisenberg spin chain is the spin-exchange correlation function. • In the finite spin chain, a determinant expression for the correlation function of the two adjacent spin operators, $S^−_j S^−_{j+1}$ and $S^z_j S^z_{j+1}$ , is given and the corresponding sum rules and f-sum rules are calculated. We evaluate them numerically and we analyze the contribution of each type of excitations. • The spin-exchange correlation function is computed with the determinant expressions and the response function is compared with a Inelastic Neutron Scattering measurement in the same spin chain. In a more detailed description, the thesis contains the following points. In chapter 2, we follow the method introduced by Bethe to solve the isotropic Heisenberg spin chain model and we give the Bethe equations and Bethe-Takahashi equations which allow one to compute eigenstates. We then give a catalog of the basic excitations above the ground state and we show their densities of states. We introduce in chapter 3 the algebraic Bethe Ansatz. Within this formalism, we explain how to construct the algebraic forms for the Bethe states and the local spin operators. Afterwards we give the formula for the norm of a Bethe state and the explicit reduction of the formula for states containing complex string configurations. Eventually, we recall the determinant expression for the scalar product between two states. In chapter 4, the two experimental setups: Inelastic Neutron Scattering and Resonant Inelastic X-ray Scattering are presented. We first enumerate the necessary ingredients for an inelastic neutron scattering measurement and we recall that the response signal is proportional to the single spin operator dynamical structure function. For the Resonant Inelastic X-ray Scattering, we describe the K-edge scattering process in detail and we show how to determine the response function. Being more specific, we give the result for the spin chain model which is the spin exchange dynamical structure factor. Using the formalism of chapter 3, we construct in chapter 5 first the determinant formula for the form factor of the operators $S^−_jS^−_{j+1}$ and $S^z_j S^z_{j+1}$ . With these results, we give an expression for the two corresponding correlation functions and the sum rules: the integrated intensities and the first frequency moments. Af- terwards, we compute, the map in momentum and energy of the two dynamical structure factors at different magnetization and we analyze the contribution of each type of excitations. The chapter 5 is based on the content of the preprint paper cond-mat/1201.0867 (2012) (see the list of publications). In chapter 6, we use the previous results for the $S^z_j S^z_{j+1}$ form factor in order to compute the spin-exchange dynamical structure factor and we compute also the corresponding sum rules. With a numerical evaluation, we compare then the response functions of the Resonant Inelastic X-ray Scattering and of the Inelastic Neutron Scattering. The results of this chapter have been published previously in the paper Phys. Rev. Lett. 106, 157205 (2011) (see the list of publications).