Analyticity of ground states of the XY-model
type: Item
instance of: scholarly article
title: Analyticity of ground states of the XY-model
zbMATH Open document ID: 0591.58018
DOI: 10.1007/BF00417469
author: Taku Matsui
author: Huzihiro Araki
published in: Letters in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: We continue to study ground states of the XY-model on a one-dimensional infinite lattice specified by the Hamiltonian \[ -J[\sum \{(1+\gamma)\sigma_ x^{(j)}\sigma_ x^{(j+1)}+(1-\gamma)\sigma_ y^{(j)}\sigma_ y^{(j+1)}\}+2\lambda \sum \sigma_ z^{(j)}] \] with real parameters \(J\neq 0\), \(\lambda\) and \(\gamma\). This model has a unique ground state \(\phi_{\lambda,\gamma}\) if either \(| \lambda | \geq 1\) or \(\gamma =0\) and \(| \lambda | <1\); it has two pure ground states \(\phi^{\pm}_{\lambda,\gamma}\) for \(| \lambda | <1\), \(\gamma\) \(\neq 0\), except for the Ising model case \((\lambda,\gamma)=(0,\pm 1)\) where there are, besides the cyclic representations associated with \(\phi^{\pm}_{\lambda,\gamma}\), two additional ground-state representations with infinitely many vectors giving rise to ground states [Commun. Math. Phys. 101, 213-245 (1985; Zbl 0581.46058); Statistical physics and dynamical systems, Rigorous results, Pap. 2nd Colloq. Workshop, Koeszeg/Hung. 1984, Prog. Phys. 10, 17-39 (1985; Zbl 0565.46036)]. Our main result is as follows. Theorem. For any local observable A, i.e., any polynomial of Pauli spin matrices \(\sigma_ x^{(j)}\), \(\sigma_ y^{(j)}\), and \(\sigma_ z^{(j)}\) at arbitrary lattice sites \(j=0,\pm 1,\pm 2,...\), \(\phi_{\lambda,\gamma}(A)\), \(\phi^+_{\lambda,\gamma}(A)\) and \(\phi^-_{\lambda,\gamma}(A)\) are all real analytic in (\(\lambda\),\(\gamma)\), \(\phi_{\lambda,\gamma}(A)\) in the domain \(D=\{(\lambda,\gamma):\quad | \lambda | >1\}\) and both \(\phi^+_{\lambda,\gamma}(A)\) and \(\phi^-_{\lambda,\gamma}(A)\) in the union of 2 domains \(D_+=\{(\lambda,\gamma):\quad | \lambda | <1,\quad \gamma >0\}\) and \(D_-=\{(\lambda,\gamma):\quad | \lambda | <1,\quad \gamma <0\}.\)
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 3949342
zbMATH Keywords: XY-model on a one-dimensional infinite lattice
zbMATH Keywords: Ising model
zbMATH Keywords: ground states
MaRDI profile type: Publication
cites work: On a \(C^*\)-algebra approach to phase transition in the two-dimensional Ising model
cites work: Ground states of the XY-model
cites work: On a \(C^ *\)-algebra approach to phase transition in the two-dimensional Ising model. II
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description: scientific article; zbMATH DE number 3949342