Low-Energy Effective Theories of Quantum Link and Quantum Spin Models

TL;DR

This paper demonstrates that quantum spin and link models, when analyzed through effective field theories, reproduce known continuum theories like the O(3) model, principal chiral model, and Yang-Mills theory, confirming their validity as regularizations.

Contribution

The authors derive low-energy effective Lagrangians for D-theory models, establishing their equivalence to conventional continuum theories using coherent state path integral techniques.

Findings

Quantum spin models reproduce the 2D O(3) model in the continuum limit.

Quantum spin models with specific symmetries are equivalent to the principal chiral model.

Quantum link models reduce to 4D Yang-Mills theory in the continuum limit.

Abstract

Quantum spin and quantum link models provide an unconventional regularization of field theory in which classical fields arise via dimensional reduction of discrete variables. This D-theory regularization leads to the same continuum theories as the conventional approach. We show this by deriving the low-energy effective Lagrangians of D-theory models using coherent state path integral techniques. We illustrate our method for the $(2+1)$-d Heisenberg quantum spin model which is the D-theory regularization of the 2-d O(3) model. Similarly, we prove that in the continuum limit a $(2+1)$-d quantum spin model with $SU(N)_L\times SU(N)_R\times U(1)_{L=R}$ symmetry is equivalent to the 2-d principal chiral model. Finally, we show that $(4+1)$-d SU(N) quantum link models reduce to ordinary 4-d Yang-Mills theory.