Spin systems with dimerized ground states

TL;DR

This paper develops a comprehensive theory for Heisenberg spin systems with dimerized ground states, identifying key conditions and classes, and visualizing the set of such systems geometrically.

Contribution

It provides a unified framework for understanding DGS systems, including classical and quantum cases, and introduces a geometric visualization of their parameter space.

Findings

Classical DGS systems are fully characterized.

Quantum DGS systems require necessary or sufficient conditions.

New DGS examples can be generated through convex combinations.

Abstract

In view of the numerous examples in the literature it is attempted to outline a theory of Heisenberg spin systems possessing dimerized ground states (``DGS systems") which comprises all known examples. Whereas classical DGS systems can be completely characterized, it was only possible to provide necessary or sufficient conditions for the quantum case. First, for all DGS systems the interaction between the dimers must be balanced in a certain sense. Moreover, one can identify four special classes of DGS systems: (i) Uniform pyramids, (ii) systems close to isolated dimer systems, (iii) classical DGS systems, and (iv), in the case of $s=1/2$, systems of two dimers satisfying four inequalities. Geometrically, the set of all DGS systems may be visualized as a convex cone in the linear space of all exchange constants. Hence one can generate new examples of DGS systems by positive linear combinations of examples from the above four classes.