Strong coupling gauge theory, quantum spin systems and the spontaneous breaking of chiral symmetry

Abstract

We demonstrate equivalence between a large class of generalized quantum antiferromagnets and the strong coupling limit of lattice gauge theories with dynamical fermions. We prove that an SU(N c) lattice gauge theory with N L lattice flavors (=2 [ d/2]−1 N L continuum flavors in d spacetime dimensions) of staggered fermions is equivalent to a quantum U( N L) antiferromagnet for N c ⩾3 and to a Sp(2 N L) antiferromagnet for N c=2 (for N L=1 the latter is identical with a spin 1 2 Heisenberg antiferromagnet). We also show that U( N c ) gauge theory with N L lattice flavors and N L even is equivalent to an SU( N L) antiferromagnet. We use this correspondence and results about Heisenberg antiferromagnets to obtain a rigorous proof that, when N L=2, the strong coupling limit of U( N c ) gauge theory breaks chiral symmetry when d⩾4 and for all N c and in d=3 for N c ⩾2, and similarly for SU(2) gauge theories with N L=1 in d⩾3. (The corresponding result is trivial for SU( N c ) gauge theories with N L=1 and N c ⩾3.) The extension of this proof to other gauge theories would require a better understanding of SU( N L), U( N L>1) and Sp(2 N L>2) antiferromagnets.