Continuous systems in one-dimensional statistical quantum mechanics with an infinite range interaction

David Carfì

Masahiko Hyuga, Sho Sugiura, Kazumitsu Sakai +1 more

We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1D) chain and on a two-dimensional (2D) triangular lattice. For the 1D chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential, and the on-site interaction. For the 2D triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.

Kazuya Okamura

J. C. Garrison, Jack Wong, Harry L. Morrison

Thin films are described as idealized systems having finite extent in one direction but infinite extent in the other two. For systems of particles interacting through smooth potentials (e.g., no hard cores), it is shown that an equilibrium state for a homogeneous thin film is necessarily invariant under any continuous internal symmetry group generated by a conserved density. For short-range interactions it is also shown that equilibrium states are necessarily translation invariant. The absence of long-range order follows from its relation to broken symmetry. The only properties of the state required for the proof are local normality, spatial translation invariance, and the Kubo-Martin-Schwinger boundary condition. The argument employs the Bogoliubov inequality and the techniques of the algebraic approach to statistical mechanics. For inhomogeneous systems, the same argument shows that all order parameters defined by anomalous averages must vanish. Identical results can be obtained for systems with infinite extent in one direction only.

Kurt Langfeld, Amanda Turner

This research investigates a class of one-dimensional theories characterized by a distinctly defined infinite interaction range. We propose that such theories emerge naturally through a mesoscopic feedback mechanism. In this proof-of-concept study, we examine Ising-type models and a model with continuous O(3) symmetry, and demonstrate that the natural emergence of phase transitions, criticality, spontaneous symmetry breaking and previously unidentified universality classes is evident. The framework introduced here holds particular relevance for monolayer spintronics research, where the ultimate goal is to achieve a strong ferromagnetic order at room temperature.

Laurent Bruneau, Alain Joye, Marco Merkli