nLab quantum lattice system

Idea

A quantum lattice system is a quantum mechanical system whose states/observables are arise from basic states/observables assigned to each vertex of some lattice, such as $\mathbb{Z}^d$.

Quantum lattice systems are of relevance in solid state physics, where the lattice typically corresponds to an actual physical lattice of atoms, or as approximations to continuous structures, as in lattice gauge theory.

Description in AQFT

In the context of AQFT the typical discussion of a quantum lattice system proceeds as follows.

For $\mathbf{L}$ a lattice, let $P_f(\mathbf{L})$ be its poset of finite subsets.

Typically there is associated with each lattice site $x \in \mathbf{L}$ a Hilbert space $\mathcal{H}_x$, typically finite dimensional, $\mathcal{H}_x \simeq \mathbb{C}^d$, representing finitely many physical degrees of freedom at that site (for instance spin polarizations of a partical). The Hilbert space associated to some $\Lambda \in P_f(\mathbf{L})$ is then typically the tensor product $\mathcal{H}_\Lambda \coloneqq \otimes_{x \in \Lambda} \mathcal{H}_x$, and hence the corresponding algebra of observables is that of bounded operators $\mathcal{A}_\Lambda \coloneqq \mathcal{B}(\mathcal{H}_\Lambda)$.

The complete algebra of observables is then the colimit/inductive limit

$$\mathcal{A} \coloneqq \underset{\longrightarrow_{\Lambda}}{\lim} \mathcal{A}_\Lambda \,.$$

in the category of C-star algebras, which is the norm-closure of the union of all these algebras. If we call each $\mathcal{A}_{\Lambda}$ a local algebra, then this is the algebra of those observables that can be approximated to arbitrary precision in norm by a local algebra. This is called a uniformly hyperfinite algebra.

Next, there is typically for each $\Lambda \in P_f(\mathbf{L})$ a Hamiltonian $H_\Lambda$ with $\exp(i t H) \in \mathcal{A}_\Lambda$ for all $t \in \mathbb{R}$. Under suitable (subtle) conditions (e.g. Bratteli-Robinson, part I, chapter 6.2) the 1-parameter time evolution $\mathbb{R} \to \mathrm{Aut}(\mathcal{A}_\Lambda)$ induced by this unitary extends to an suitably continuous family of automorphisms of $\mathcal{A}$ (this is no longer given in general by an inner automorphism even though for each finite $\Lambda$ it is).

Related concepts

• harmonic oscillator

• hydrogen atom

References

Detailed discussion of quantum lattice systems in statistical physics formulated with AQFT tools is in

• O. Bratteli, D. Robinson, Operator algebras and quantum statistical mechanics Texts and monographs in physics, Springer, New York (part I 1987, part II 1997)