Path integrals for dimerized quantum spin systems

TL;DR

This paper develops two path-integral formulations for $S=1/2$ dimerized quantum spin systems, enabling analysis of quantum phase transitions and providing a foundation for deriving effective field theories.

Contribution

It introduces two novel path-integral approaches for dimerized quantum spin systems, utilizing SO(4) and bond-operator formalisms with Faddeev-Jackiw quantization.

Findings

Formulated path-integrals for dimerized systems using SO(4) and bond-operator methods.

Applied the formalism to a spin-Peierls chain to derive its field-theory.

Demonstrated the consistency of measures and constraints from operator algebra.

Abstract

Dimerized quantum spin systems may appear under several circumstances, e.g\ by a modulation of the antiferromagnetic exchange coupling in space, or in frustrated quantum antiferromagnets. In general, such systems display a quantum phase transition to a N\'eel state as a function of a suitable coupling constant. We present here two path-integral formulations appropriate for spin $S=1/2$ dimerized systems. The first one deals with a description of the dimers degrees of freedom in an SO(4) manifold, while the second one provides a path-integral for the bond-operators introduced by Sachdev and Bhatt. The path-integral quantization is performed using the Faddeev-Jackiw symplectic formalism for constrained systems, such that the measures and constraints that result from the algebra of the operators is provided in both cases. As an example we consider a spin-Peierls chain, and show how to arrive at the corresponding field-theory, starting with both a SO(4) formulation and bond-operators.