Integrable spin systems with long-range interactions

Kazuhiro Hikami, Miki Wadati

Filippo Caleca, Fabio Mezzacapo, Tommaso Roscilde

Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, $\sim r^{-α}$ with $α< d$, where $d$ is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.

Kazuhiro Hikami, P. P. Kulish, Miki Wadati

Presented is a general method to construct quantum spin Hamiltonians with long-range pairwise interactions. The Hamiltonian is given explicitly in terms of the R -matrices on an inhomogeneous lattice. Applications to spin-1/2 chains, higher spin chains and Z n models are shown.

Robin Steinigeweg, Heinz–Jürgen Schmidt

Manas Kulkarni, Fabio Franchini, Alexander G. Abanov

Rob Klabbers, Jules Lamers

We consider the construction of quantum-integrable spin chains with q -deformed long-range interactions by ‘freezing’ integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars–Schneider system. We focus on the elliptic case. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbor to long-range interacting spins. We first define an action of the modular group on the spinless elliptic Ruijsenaars–Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantization to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium while preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane–Shastry chains along with their xxz -like q -deformations (face type), or the antiperiodic Haldane–Shastry chain of Fukui–Kawakami, its elliptic generalization of Sechin–Zotov, and their completely anisotropic q -deformations due to Matushko–Zotov (vertex type). Finally, we show how freezing fits in the setting of ‘hybrid’ integrable systems.