Low-temperature Bethe root pattern conjecture for the antiferromagnetic massive regime

Let $T$ be the temperature, $N$ the Trotter number, and $h$ satisfy $0<h<h_\ell$. Let $\mathcal X$ and $\mathcal Y$ be the sets of holes and particles, respectively, consisting of complex parameters in the strip $|\operatorname{Im}\lambda|<\gamma/2$, and let $\mathfrak e\in\mathbb Z/2\mathbb Z$. For pseudo-spin-zero states, assume $\operatorname{card}\mathcal X=\operatorname{card}\mathcal Y=\ell$. Define the shift function by

$$F(\lambda|\mathcal X,\mathcal Y)=\frac{1}{2\pi\mathrm i}\sum_{\mu\in\mathcal Y\ominus\mathcal X}\phi(\lambda,\mu).$$

Low-$T$ Bethe root pattern conjecture. All excitations of the quantum transfer matrix at low $T$ and sufficiently large $N$ can be parametrised in this way, and, up to corrections of the form $T^\infty+a_N(T)$ with $\lim_{N\to\infty}a_N(T)=0$ for every fixed $T>0$, the particles and holes satisfy

$$\varepsilon(y|h)=2\pi\mathrm iT\bigl(\ell_y+\mathfrak e/2+F(y|\mathcal X,\mathcal Y)\bigr),\qquad y\in\mathcal Y,$$ $$\varepsilon(x|h)=2\pi\mathrm iT\bigl(m_x+\mathfrak e/2+F(x|\mathcal X,\mathcal Y)\bigr),\qquad x\in\mathcal X,$$

where $\ell_y,m_x\in\mathbb Z$ are mutually distinct. For a solution $(\mathcal X_n,\mathcal Y_n)$ with index $\mathfrak e_n$, writing $F(\lambda|\mathcal X_n,\mathcal Y_n)=F_n(\lambda)$, the corresponding auxiliary function is

$$\mathfrak a_n(\lambda|h)=(-1)^{\mathfrak e_n}\exp\!\left(-\frac{\varepsilon(\lambda-\mathrm i\gamma/2|h)}{T}+2\pi\mathrm iF_n(\lambda-\mathrm i\gamma/2)\right)\left(1+\mathcal O\bigl(T^\infty+a_N(T)\bigr)\right),$$

uniformly for $0<|\operatorname{Im}\lambda|<\gamma$, away from $0$ and $\mathrm i\gamma$. This describes the low-temperature excitations of the quantum transfer matrix in the antiferromagnetic massive regime; the statement is presented as an assumption underlying the analysis and its resolution is not established in the supplied text.