Path integrals for dimerized quantum spin systems

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 Néel state as a function of a suitable coupling constant. We present here two path-integral formulations appropriate for spin S=1/2S=1/2 dimerized systems. The first one deals with a description of the dimers degrees of freedom in an SO(4)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 an SO(4)SO(4) formulation and bond-operators.