Interacting Heisenberg dimer Gibbs model

In the ordered basis $e_0{e}_0,e_0{e}_1,e_1{e}_0,e_1{e}_1$, the field operator ${\sigma }_z\otimes I+I\otimes {\sigma }_z$ has eigenvalues $2,0,0,-2$. The middle two basis vectors combine into the singlet and the $m=0$ triplet of [F2]; both have field eigenvalue $0$. The outer vectors are the $m=+1$ and $m=-1$ triplets, with field eigenvalues $2$ and $-2$. Thus the singlet and triplet basis from [F2] is a common orthonormal eigenbasis for $C$ and the field operator. The displayed real eigenvalues make $H_{J,h}$ self-adjoint on its full domain ${\mathbb{C}}^2\otimes {\mathbb{C}}^2$; the singlet energy is $-3J$ and the triplet energies are $J-2h,J,J+2h$.

The canonical trace is the sum of the four positive weights associated with step 1.1. It is therefore finite and positive, and the adopted preparation [A1] gives the displayed partition function. At $h=0$ and $J>0$, setting $x=\beta J$ identifies its normalized state with ${\rho }_x$ in [F3], so that theorem gives the exact separability and entanglement alternatives, including the equality case.

Fix $J,h$ and let $E_{\min }$ be the least of the four energies from step 1.1, with multiplicity $g$. Write the other energies as $E_{\min }+{\mathrm{\Delta }}_i$ with ${\mathrm{\Delta }}_i>0$. After factoring out $e^{-\beta E_{\min }}$, the denominator is $D_{\beta }=g+{\sum }_i{e}^{-\beta {\mathrm{\Delta }}_i}\to g$ and $1\le D_{\beta }\le 4$. The Gibbs probability of each ground vector is $1\mathrm{/}{D}_{\beta }$, so the total ground entropy tends to $-k_{B}g(1\mathrm{/}g)\log (1\mathrm{/}g)=k_B\log g$. For an excited vector, $p_i=e^{-\beta {\mathrm{\Delta }}_i}\mathrm{/}{D}_{\beta }$ and $-p_i\log p_i=p_i(\beta {\mathrm{\Delta }}_i+\log D_{\beta })\le e^{-\beta {\mathrm{\Delta }}_i}(\beta {\mathrm{\Delta }}_i+\log 4)$; this tends to zero by [F9], so the finite sum of excited entropy terms vanishes. At $h=0$, the triplet is the three-dimensional ground space for $J<0$, all four levels coincide for $J=0$, and the singlet is the unique ground state for $J>0$, giving the stated limits.

Fix $J,h$ and any ${\beta }_0>0$. For each of the four energies $E$ from step 1.1, the addition formula [F4] gives $e^{-\beta E}=e^{-{\beta }_{0}E}{\sum }_{n\ge 0}[-E(\beta -{\beta }_0){]}^n\mathrm{/}n!$, a convergent power series about ${\beta }_0$. The four-term sum $Z(\beta )$ is therefore real analytic near ${\beta }_0$ by [F5]–[F6], and $Z({\beta }_0)>0$ by step 2.1. Put $y_0=\log Z({\beta }_0)$. By [F8], $\exp (y_0)=Z({\beta }_0)$; by [F7], ${\exp }^{\mathrm{\prime }}(y_0)=\exp (y_0)=Z({\beta }_0)\mathrm{\ne }0$. The real analytic inverse theorem gives a local analytic inverse to $\exp$ near $Z({\beta }_0)$, and [F8] identifies that inverse with $\log$. The composition clause of [F6] now makes $\log Z$ real analytic near ${\beta }_0$. Since ${\beta }_0$ was arbitrary, $-{\beta }^{-1}\log Z$ is real analytic for every $\beta >0$: the identity and constant functions have local polynomial series, so the quotient clause of [F6] makes $\beta \mapsto {\beta }^{-1}$ analytic on this domain. Thus the finite-volume free energy has no singularity at the entanglement threshold.

For $J>0$, the lowest triplet energy is $J-2\mathrm{\mid }h\mathrm{\mid }$. It equals the singlet energy $-3J$ exactly at $\mathrm{\mid }h\mathrm{\mid }=2J$. The two energy levels can cross as parameters vary, but step 2.1 is a strictly positive finite sum of exponentials at every $\beta >0$, so this zero-temperature ground-space crossing creates no finite-temperature singularity.