Excited TBA Equations II: Massless Flow from Tricritical to Critical Ising Model

TL;DR

We analyze the massless flow from the tricritical Ising model ${\cal M}(4,5)$ perturbed by the thermal operator to the critical Ising model ${\cal M}(3,4)$ in a cylinder with integrable boundaries. Using the ABF lattice in Regime IV, we derive massless excited TBA equations labeled by $(\boldsymbol{m},\boldsymbol{n})$ strings and show that the string content changes along the flow via three mechanisms A,B,C, which induce a mapping between finitized Virasoro characters through generalized $q$-Vandermonde identities. Specifically, in the vacuum sector the flow maps $\chi^{4}_{1,1}(q)$ to $\chi^{3}_{1,1}(q)$, consistent with the UV/IR operator content, across sectors. Numerical solutions of the excited massless TBA verify smooth UV-to-IR flows of scaling energies and 1-string patterns, recovering the Ising fixed point and illustrating boundary-driven operator flows.

Abstract

We consider the massless tricritical Ising model M(4,5) perturbed by the thermal operator in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A_4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massless renormalization group flow from the tricritical to critical Ising model. As in the massive case of Part I, the excitations are completely classified in terms of (m,n) systems but the string content changes by one of three mechanisms along the flow. Using generalized q-Vandemonde identities, we show that this leads to a flow from tricritical to critical Ising characters. The excited TBA equations are solved numerically to follow the continuous flows from the UV to the IR conformal fixed points.

Figures (7)

• Figure 1: Upper part of the period rectangle $(-\frac{\lambda}{2},2\lambda)\times (-\pi i\varepsilon,\pi i\varepsilon)$ in the complex $u$-plane for the $A_4$ lattice model in Regime IV showing the putative strips 1 and 2. The schematic location of zeros reflects the complex conjugation symmetry and the symmetry (\ref{['xsymm']}). In the scaling limit the imaginary period $\varepsilon\to\infty$ and it is the behaviour of the zeros near the indicated scaling edge at $\hbox{Im}(u)=\pi\varepsilon/2$ that is relevant.
• Figure 2: Schematic representation of the three mechanisms A, B, C by which the string contents change under the flow. In each case the two circled zeros leave the analyticity strip as indicated by the arrows. In mechanism A two 1-strings leave the strip whereas in mechanisms B and C it is a 2-string that leaves the strip. Note that only the location of the 1-strings enter the TBA equations.
• Figure 3: Flow of groundstate energy $c_{\hbox{\small eff}}=-12RE(R)/\pi$ versus $\log(mR)$ for periodic and $(r,s)=(1,1)$ boundary conditions. The difference arises from the boundary term in the TBA equations. The energy is a decreasing function of $mR$ for periodic boundary conditions by Zamolodchikov's $c$-theorem but this theorem does not apply with fixed boundary conditions.
• Figure 4: Flow $\chi_{1,1}^4(q)\mapsto\chi_{1,1}^3(q)$ of scaling energies $-c_{\hbox{\small eff}}/24=RE(R)/2\pi$ versus $\log(mR)$ in the $(r,s)=(1,1)$ sector. The degeneracies of the levels are shown in the margins. The intermediate region of the Mechanism A levels (shown dashed) are schematic and have not been obtained from numerical solution of the TBA equations.
• Figure 5: The scaling energy $-c_{\hbox{\small eff}}/24=RE(R)/2\pi$ versus $\log(mR)$ for the Mechanism A level in the $(r,s)=(1,1)$ sector with string contents $(m_1,m_2)=(4,2)$ and UV quantum numbers $I=(0,0,0,0|0,0)$.