Excited states by analytic continuation of TBA equations

TL;DR

The paper develops a framework to extract excited-state spectra from integrable model TBA equations by analytic continuation in the coupling, guided by monodromy arguments. Applied to the scaling Lee-Yang model, it derives one-particle and two-particle excited-state TBA equations, including modifications when singularities cross the real axis, and demonstrates striking agreement with TCSA data while analyzing ultraviolet and infrared limits. The authors propose a general n-particle extension and discuss how UV scaling dimensions emerge from Rogers' dilogarithm, establishing consistency with the cylinder spectrum and primary/descendant structures. They argue for broad applicability of the method to purely elastic theories and outline future work to map Riemann surfaces of the finite-volume energy function and extend to more sectors.

Abstract

We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.

Figures (6)

• Figure 1: Two solutions to the basic TBA equation on the negative-$\lambda$ line (points) versus TCSA data (continuous lines)
• Figure 2: Padé extrapolation of $\theta_0$ for the first excited state, from $r=1.5\imath$ (lower right) to $r=4.5$ (upper left), and back.
• Figure 3: Proposed one-particle scaling functions (points) compared with TCSA data (continuous lines)
• Figure 4: A contour plot of ${\rm Im}(F(r))$ as obtained from the basic TBA equations (1.5), (1.9), with $r$ lying in the box $0 \le {\rm Re}(r) \le 1.85$, $0 \le {\rm Im}(r) \le 10.1$, showing the branch points at $r_0$ and $r_1$, and also the segment of the negative-$\lambda$ line, running from $0$ to $r_0$, along which ${\rm Im}(F(r))=0$. The box was scanned from right to left, so it is the lower branch of this segment that is visible.
• Figure 5: ${\rm Im}(F(r))$ in the box $3.8 \le {\rm Re}(r) \le 4.4$, $7.7 \le {\rm Im}(r) \le 8.5$ as obtained from the one-particle TBA equations (2.3), (2.4), showing the branch point at $\tilde{r}_1$.