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On form-factor expansions for the XXZ chain in the massive regime

M. Dugave, F. Göhmann, K. K. Kozlowski, J. Suzuki

Abstract

We study the large-volume-$L$ limit of form factors of the longitudinal spin operators for the XXZ spin-$1/2$ chain in the massive regime. We find that the individual form factors decay as $L^{-n}$, $n$ being an even integer counting the number of physical excitations -- the holes -- that constitute the excited state. Our expression allows us to derive the form-factor expansion of two-point spin-spin correlation functions in the thermodynamic limit $L\rightarrow +\infty$. The staggered magnetisation appears naturally as the first term in this expansion. We show that all other contributions to the two-point correlation function are exponentially small in the large-distance regime.

On form-factor expansions for the XXZ chain in the massive regime

Abstract

We study the large-volume- limit of form factors of the longitudinal spin operators for the XXZ spin- chain in the massive regime. We find that the individual form factors decay as , being an even integer counting the number of physical excitations -- the holes -- that constitute the excited state. Our expression allows us to derive the form-factor expansion of two-point spin-spin correlation functions in the thermodynamic limit . The staggered magnetisation appears naturally as the first term in this expansion. We show that all other contributions to the two-point correlation function are exponentially small in the large-distance regime.

Paper Structure

This paper contains 25 sections, 11 theorems, 243 equations, 3 figures.

Table of Contents

  1. The structure of excited states revisited
  2. The Bethe equations of the massive XXZ chain and the counting function
  3. Large-$L$ behaviour of the counting function
  4. The higher-level Bethe Ansatz equations
  5. Energy and momentum of excited states
  6. General parametrisation of the excited states for large $L$
  7. The form factors of the spin operator $\sigma^z$
  8. Formulae at finite $L$
  9. The large-$L$ behaviour of the form factor: the result
  10. Large-$L$ behaviour of the coefficients $\widehat{\mathcal{D}}_k$
  11. The $\widehat{\mathcal{A}}$-coefficients
  12. The regular factor $\widehat{\mathcal{A}}_{\mathrm{reg}}$
  13. The singular factor $\widehat{\mathcal{A}}_{\mathrm{sing}}$
  14. The form-factor series
  15. The form-factor series
  16. ...and 10 more sections

Key Result

Proposition 1.1

Let the set of Bethe roots $\{\mu_a\}_1^N$ satisfy the above stated hypotheses and set Then, the counting function $\widehat{\xi}_{\mu}$ admits the large-$L$ asymptotic expansion In each of the three situations above, the remainder is uniform in $\omega$ provided that it is located at a finite distance to the boundary of the domain of interest.

Figures (3)

  • Figure 1: Contour $\Gamma_{\mu}=\Gamma^{(\uparrow)}_{\mu} \cup \Gamma^{(\downarrow)}_{\mu}$.
  • Figure 2: Comparison of our result $A_2(\nu_1,\nu_2 \, | \, \iota=1)$ (red dots) and the prediction $2 \, | f_-(\nu_1,\nu_2) |^2$ from the vertex-operator approach (black line) as a function of $\nu_1$ for a fixed value of $\nu_2$ (left panel). The right panel shows the same plot with logarithmic coordinate.
  • Figure 3: Comparison of the exact nearest-neighbour correlator KatoShiroishiTakahashiNextNearestCorrelationFctInMassiveXXZ (black line) and the 2-spinon approximation (red line). The relative difference between both curves is of the order $10^{-3}$ for $\Delta>2$. Note that both functions remain finite in the limit $\Delta\rightarrow 1$ with the known ratio of ca. 73% (cf. BougourziFledderjohanKarbachMullerMutterDynamicTwoSpinonStructureFactor)

Theorems & Definitions (11)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem \oldthetheorem
  • Lemma 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • ...and 1 more