[Paper Review] The Dynamical Correlation Function of the XXZ Model

Robert Weston, Bougourzi, A. H.|arXiv (Cornell University)|Jul 13, 1994

Theoretical and Computational Physics5 references3 citations

TL;DR

This paper computes the dynamical correlation function of the one-dimensional spin-1/2 XXZ model using spectral decomposition into form factors. By evaluating the two-particle contribution via integral expressions from the Kyoto school and expanding to twelfth order in the anisotropy parameter $ q $, it provides analytical and numerical results for $ S(\omega, k) $ at various $ k $ and $ \Delta $, showing non-trivial behavior near $ k = 0 $ and $ \pi $, with explicit plots for different anisotropy values.

ABSTRACT

We perform a spectral decomposition of the dynamical correlation function of the spin $1/2$ XXZ model into an infinite sum of products of form factors. Beneath the four-particle threshold in momentum space the only non-zero contributions to this sum are the two-particle term and the trivial vacuum term. We calculate the two-particle term by making use of the integral expressions for form factors provided recently by the Kyoto school. We evaluate the necessary integrals by expanding to twelfth order in $q$. We show plots of $S(w,k)$, for $k=0$ and $π$ at various values of the anisotropy parameter, and for fixed anisotropy at various $k$ around $0$ and $π$.

Motivation & Objective

• To compute the dynamical correlation function $ S(\omega, k) $ of the integrable XXZ spin chain beyond the free-fermion point.
• To analyze the spectral decomposition of the correlation function into form factors, focusing on contributions below the four-particle threshold.
• To evaluate the two-particle form factor contribution using integral expressions derived by the Kyoto school.
• To provide high-order analytic expansions (up to 12th order in $ q $) of the correlation function for numerical and qualitative analysis.
• To present plots of $ S(\omega, k) $ at $ k = 0 $, $ \pi $, and intermediate values, across different anisotropy parameters $ \Delta $

Proposed method

• Performing a spectral decomposition of the dynamical correlation function into an infinite sum of form factor products.
• Focusing on the two-particle and vacuum contributions, as higher thresholds are beyond the physical region of interest.
• Using integral representations of form factors recently derived by the Kyoto school for the XXZ model.
• Expanding the resulting integrals in powers of $ q $, up to twelfth order, to achieve high-precision analytical expressions.
• Numerically evaluating and plotting $ S(\omega, k) $ for $ k = 0 $, $ \pi $, and intermediate $ k $ values at fixed anisotropy parameters.
• Utilizing $ q $-expansion to access the behavior of the correlation function in the massive regime of the XXZ model.

Experimental results

Research questions

• RQ1 What is the structure of the dynamical correlation function $ S(\omega, k) $ of the XXZ model below the four-particle threshold?
• RQ2 How do the two-particle and vacuum form factor contributions dominate the spectral decomposition in this regime?
• RQ3 What is the analytical form of the two-particle contribution to $ S(\omega, k) $, and how does it depend on the anisotropy parameter $ \Delta $?
• RQ4 How does $ S(\omega, k) $ vary with momentum $ k $ near $ k = 0 $ and $ \pi $ for different values of $ \Delta $?
• RQ5 What insights do high-order $ q $-expansions (up to 12th order) provide into the behavior of the dynamical structure factor?

Key findings

• The two-particle form factor term is the dominant non-trivial contribution to $ S(\omega, k) $ below the four-particle threshold.
• The vacuum term contributes trivially, while higher particle-number form factors vanish in this energy-momentum region.
• The two-particle contribution is computed analytically via integral expressions from the Kyoto school, expanded to 12th order in $ q $.
• Plots of $ S(\omega, k) $ show distinct peak structures at $ k = 0 $ and $ k = \pi $, with intensity and width sensitive to the anisotropy parameter $ \Delta $.
• The correlation function exhibits non-trivial momentum dependence, with enhanced weight near $ k = \pi $ for certain $ \Delta $ values.
• The high-order $ q $-expansion enables accurate numerical evaluation and visualization of $ S(\omega, k) $ across the Brillouin zone.