On a \(C^*\)-algebra approach to phase transition in the two-dimensional Ising model
The phase transition of the two-dimensional Ising model is investigated in terms of the state on the \(C^*\)-algebra of Pauli spins on a one- dimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two- dimensional Ising model (with nearest neighbour interaction) through the transfer matrix method. It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary is derived. It is also shown that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operators \(E_ 1\) and \(E_ 2\) (on the direct sum of two copies of the basic Hilbert space) satisfy the following two conditions: (1) \(E_ 1-E_ 2\) is in the Hilbert-Schmidt class, (2) \(E_ 1\wedge(1-E_ 2)\) has an even dimension, where the even- oddness of \(\dim E_ 1\wedge(1-E_ 2)\) is called \({\mathbb{Z}}_ 2\)-index of \(E_ 1\) and \(E_ 2\) and is continuous in \(E_ 1\) and \(E_ 2\) relative to the norm topology (as long as the condition (1) is satisfied).
phase transition of the two-dimensional Ising model
state on the \(C^*\)- algebra of Pauli spins on a one-dimensional lattice
thermodynamic limit of the Gibbs ensemble
scientific article; zbMATH DE number 3849889