Order parameters in \(XXZ\)-type spin \(1\over2\) quantum models with Gibbsian ground states (Q2495156)

Consider hyper-cubic lattice \(\mathbb{Z}^d\). Denote by \(\sigma^x_i,\sigma^y_i,\sigma^z_i\) Pauli matrices at site \(i\in \mathbb{Z}^d\). For a given \(A\subset \mathbb{Z}^d\) denote \(\sigma^\gamma_{[A]}=\prod_{i\in A}\sigma_i^\gamma\), here \(\gamma\in\{x,y,z\}\). The following is considered Hamiltonian \[ H_{\Lambda}=H_{0\Lambda}+V_{\Lambda},\qquad H_{0\Lambda}=\sum\limits_{A,A'\subseteq \Lambda, A\cap A'=\varnothing} \phi_{A,A'}\sigma^x_{[A]}\sigma^y_{[A']}, \] where \( \phi _{A,A'}\) are real valued coefficients, \(V_{\Lambda}\) depends on \(S^z_{\Lambda}\). The author finds an expression for \(V_{\Lambda}\) which guarantees that \(\Psi_{\Lambda}\) given by \[ \Psi_{\Lambda}=\sum\limits_{s_{\Lambda}}e^{-\frac{\alpha}{2} U_0(s_{\Lambda})}\Psi^0_{\Lambda}(s_{\Lambda}),\qquad \alpha\in \mathbb R^+, \] where the summation is performed over \((\times \{-1,1\}^{| \Lambda| })\), \[ \Psi^0_{\Lambda}(s_{\Lambda})=\bigotimes_{x\in \Lambda}\psi_0(s_x),\qquad \psi_0(1)=(1,0), \qquad \psi_0(-1)=(0,1), \] where \(U_0\) is some classical potential, is the eigen (ground) state. He also finds two order parameters for two spin components \(x\), \(z\) simultaneously for large values of the parameter \(\alpha\) playing the role of the inverse temperature. It is shown that the ferromagnetic order in \(x\) direction exists for all dimensions \(d\geq 1\) for a wide class of models considered. The results generalize the ones obtained in [\textit{T. Dorlas, W. Skrypnik}, J. Phys. A, Math. Gen., 37, 6623--6632 (2004; Zbl 1060.82003)].