Topological Characterization of Extended Quantum Ising Models

TL;DR

This paper introduces a topological approach to characterize extended quantum Ising models using geometric loops in an auxiliary space, linking quantum phase transitions to changes in topological invariants.

Contribution

It establishes a novel topological framework for quantum phase characterization by associating ground state properties with geometric loops and their winding numbers.

Findings

Ground state energy density exhibits nonanalytic behavior at topological transitions.

Winding number acts as a topological quantum number for quantum phases.

Different models correspond to distinct geometric curves like circles, ellipses, and cardioids.

Abstract

We show that a class of exactly solvable quantum Ising models, including the transverse-field Ising model and anisotropic XY model, can be characterized as the loops in a two-dimensional auxiliary space. The transverse-field Ising model corresponds to a circle and the XY model corresponds to an ellipse, while other models yield cardioid, limacon, hypocycloid, and Lissajous curves etc. It is shown that the variation of the ground state energy density, which is a function of the loop, experiences a nonanalytical point when the winding number of the corresponding loop changes. The winding number can serve as a topological quantum number of the quantum phases in the extended quantum Ising model, which sheds some light upon the relation between quantum phase transition and the geometrical order parameter characterizing the phase diagram.