In this brief note we outline two key concepts in quantum many-body (QMB) theory, which are immensly helpful both theoretically and numerically.
These are the simplest to study, as we see now. Simply stated, a QMB system is non-interacting if it can be diagonalized exactly over single particle terms. The Hamiltonian of a non-interacting, also called free (or sometimes Gaussian if we borrow a bit of field theory jargon) and admits some flavor of the following decomposition,
\begin{equation} H = \sum_k \varepsilon_k a_k^\dagger a_k \end{equation}
here \(k\) labels the different particles, which do not interact with each other. Here \(\{a_k,a_k^\dagger\}_k\) are creation and annhilation operators for each particle \(k\), which could be bosonic or fermionic. Nonetheless, the key is for \(N\) particle, this object is a matrix of linear size \(O(N)\), instead of exponential, and in-fact the eigenstates are super simple. For any \(\mathcal{K} \subseteq \{1,2,\dots,N\}\),
\begin{equation} H \left[\prod_{k \in \mathcal{K}} a_k^\dagger |0\rangle\right] = \left(\sum_{k\in \mathcal{K}} \varepsilon_k\right) \left[\prod_{k \in \mathcal{K}} a_k^\dagger |0\rangle\right] \end{equation}
Well, the particles interact. For instance, Hamiltonians containing terms like \(n_i n_j\) with \(n_k := a_k^\dagger a_k\) cannot be brought to the form above. This induces an exponentially complex Hilbert space, with no analytical solutions in general except for special cases. So the Hamiltonian cannot be written as a sum of decoupled single-particle terms. These effects include scattering and collective phenomenon, and are really the most interesting to study.
A QMB system is said to be integrable if it possesses an extensive set of conserved quantities (operators) \(\{Q_i\}_i\), which are
The Hamiltonian is trivially a conserved quantity, always. These systems are solvable as well, because each eigenstate of the Hamiltonian can be uniquely specified as per it’s charge under the conserved quantitites \(Q_i\), \(\psi \leftrightarrow (q_1,q_2,\dots)\). Such equations can be solved with polynomial complexity as well, as opposed to exponential. Integrable systems exhibit regular, predictable behavior due to their conserved quantities. Integrable systems avoid thermalization owing to the extensive set of conserved quantities.
No such extensive set of conserved quantities. One corner here is that the only conserved quantity is energy, i.e., the Hamiltonian. In random circuits, even that is taken away, making them interesting unstructured objects to study.
Then, one could take a Cartesian product, and have three (non-interacting non-integrable does not really exist) classes,