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On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

K. K. Kozlowski

Abstract

This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -- the Fourier transforms of dynamical two-point functions -- in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations: the method of analysis \textit{does not rely, at any stage}, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state). In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations maximising the energy at fixed momentum.

On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

Abstract

This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -- the Fourier transforms of dynamical two-point functions -- in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations: the method of analysis \textit{does not rely, at any stage}, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state). In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations maximising the energy at fixed momentum.

Paper Structure

This paper contains 46 sections, 25 theorems, 643 equations, 6 figures.

Table of Contents

  1. An outline of the problem and main results
  2. The XXZ chain
  3. Singularities of response functions
  4. The main achievement of the work
  5. The principal theorems
  6. Outline of the paper
  7. Some history of the analysis of dynamic response functions
  8. Heuristic approaches
  9. Exact approaches
  10. Numerical and Bethe Ansatz based approaches
  11. The restricted sum approach
  12. Main results
  13. The setting and some generalities on the model
  14. The behaviour of the longitudinal dynamic response function in the two hole excitations in $\mathscr{S}^{(z)}(k,\omega)$
  15. The series representation for the dynamic response functions
  16. ...and 31 more sections

Key Result

Theorem \oldthetheorem

Let $a<b$ be two reals. Let $\mathfrak{z}_{\pm}(\lambda)$ be two real-holomorphic functions in a neighbourhood of the interval $\mathscr{J}= [ a \,; b ]$, such that Let $\Delta_{\upsilon}$ be real analytic on $\mathrm{Int}(\mathscr{J})$ and such that $\Delta_{\upsilon} \geq 0$. Let $\mathscr{G}$ be in the smooth class of $\mathscr{J}$ associated with the functions $\Delta_{\pm}$ and with a const

Figures (6)

  • Figure 1: Singularity curves issued from the sectors involving up to two particles, two-holes and no $r$-strings with $r\geq 2$ for $\Delta=0.57$ and in presence of a magnetic field $h$ which fixes the per site magnetisation $\mathfrak{m}=1-2D$ such that $D=0.21$. Continuous curves correspond to one massive -hole or particle- excitation. Dotted curves correspond to a collective, coordinated, multi-particle-hole excitation. This excitation is such that all particles and holes building it up have the same velocity.
  • Figure 2: Velocity $\mathfrak{v}_1$ plotted for $\Delta=0.57$ and magnetic field $h$ such that the per-site magnetisation $\mathfrak{m}=1-2D$ is parameterised by $D=0.21$ (lhs) and for $\Delta=-0.60$ and $D=0.30$ for the rhs.
  • Figure 3: Deformed contours in the case $| \boldsymbol{ \texttt{u} } |> \boldsymbol{ \texttt{v} }$ and for $\Re(t) \mathfrak{s}_{ \boldsymbol{ \texttt{u} } }<0$
  • Figure 4: Deformed contours in the case $| \boldsymbol{ \texttt{u} } |> \boldsymbol{ \texttt{v} }$ and for $\Re(t) \mathfrak{s}_{ \boldsymbol{ \texttt{u} } } > 0$
  • Figure 5: Deformed contours in the case $| \boldsymbol{ \texttt{u} } |< \boldsymbol{ \texttt{v} }$ and for $\Re(t) < 0$
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 19 more