Invariant states for time dynamics of one-dimensional lattice quantum fermi systems

A study is made of the problem of describing the set of invariant states for the time dynamics corresponding to a (formal) Hamiltonian $H_0$ of a one-dimensional lattice quantum Fermi system. Assuming that the invariant state $\varphi$ is a KMS state for some “Hamiltonian” $H$, we prove that $H$ is proportional to $H_0$, i.e., that $\varphi$ is a KMS state for $\beta H_0$. As a consequence, in the considered situation every “natural” invariant state is an equilibrium Gibbs state. Use is made here of the condition that $H_0$ is not a quadratic form in the creation and annihilation operators. In such a case the time dynamics admits a much richer set of invariant states. If all terms in $H_0$ except the quadratic ones are diagonal, it can be shown that $H=\beta H_0+N$. Here, $N$ is an arbitrary diagonal quadratic form.