A Note on Adiabatic Time Evolution and Quasi-Static Processes in Translation-Invariant Quantum Systems

Abstract

We study the slowly varying, non-autonomous quantum dynamics of a translation-invariant spin or fermion system on the lattice $\mathbb {Z}^d$. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in Jakšić et al. (Approach to equilibrium in translation-invariant quantum systems: some structural results, in the present issue of Ann. H. Poincaré, 2023), we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

A Glossary of Terms

More details can be found in [1].

• $\mathfrak {A}$: the $C^*$-algebra of a spin system on $\mathbb {Z}^d$, or equivalently, the gauge-invariant sector of the fermionic CAR algebra on $\ell ^2(\mathbb {Z}^d)$.

• $\mathscr {F}$: the set of all finite subsets of ${\mathbb {Z}}^d$.

• $\Phi$: an interaction, namely a translation-invariant family $\{\Phi (X)\}_{X\in \mathscr {F}}$ of self-adjoint elements of $\mathfrak {A}$ with $\mathop {\textrm{supp}}\nolimits (\Phi )=X$.

• $\mathscr {B}^r$: the Banach space of interactions satisfying $\Vert \Phi \Vert _r=\sum _{X\ni 0}\textrm{e}^{r(|X|-1)}\Vert \Phi (X)\Vert <\infty$, ($r>0$).

• $\mathbb {Z}^d\ni x\mapsto \varphi ^x$: the group action of $\mathbb {Z}^d$ on $\mathfrak {A}$.

• $\mathscr {S}_\textrm{I}(\mathfrak {A})$: the set of translation-invariant states on $\mathfrak {A}$.

• $s(\nu )$: the specific entropy of a state $\nu \in \mathscr {S}_\textrm{I}(\mathfrak {A})$.

• $s(\nu |\omega )$: the specific relative entropy of two states $\nu ,\omega \in \mathscr {S}_\textrm{I}(\mathfrak {A})$.

• $P(\Phi )$: the pressure of the interaction $\Phi$.

• $E_\Phi = \sum _{X\ni 0} |X|^{-1} \Phi (X)$, so that $\nu (E_\Phi )$ is the expected specific energy of the interaction $\Phi$ in the state $\nu \in \mathscr {S}_\textrm{I}(\mathfrak {A})$.

• $\mathscr {S}_\textrm{eq}(\Phi )=\{\nu \in \mathscr {S}_\textrm{I}(\mathfrak {A})\mid P(\Phi )=s(\nu )-\nu (E_\Phi )\}$: the set of equilibrium states for $\Phi$ (from the Gibbs variational principle).

• $\textrm{WG}(\Phi )$: the set of weak Gibbs states for $\Phi$, see [1, Section 2.2].

• $\alpha _\Phi$: the Heisenberg dynamics generated by the (time-independent) interaction $\Phi$.

• Physical equivalence: two interactions $\Phi , \Psi \in \mathscr {B}^r$ are physically equivalent iff $\alpha _\Phi =\alpha _\Psi$, see [1, Theorem 2.7].