PHYSICS REPOR1’S (Review Section of Physics Letters) 67, No. 1 (1980) 103—107. North-Holland Publishing Company Phase Structure of the Z(2) Gauge and Matter Theory* D. HORN Department of Physics & Astronomy, Tel Aviv University, Ramat Aviv, Israel Abstract: We discuss the Z(2) theory using a hamiltonian formulation and emphasize the roles of gauge invariance and duality. Whereas the phases of the pure gauge theory can be characterized as electric— or magnetic — confining, one finds that in the presence of matter the two resulting phases can be characterized as matter — or gauge — screening. We investigate their properties by considering the exact vacua at the limiting points of the parameter space. Using such vacua in a mean-field approach we display the existence of a finite line of first-order phase transitions in the matter-screening phase and discuss its physical meaning. The Z(2) lattice gauge theory with a matter field is a model which exhibits dynamical confinement. It allows us to explore the interplay of the confining gauge field and the dynamical matter source. Using a quantum-mechanical formulation on a cubical lattice in D space dimensions we define the hamiltonian [1,2] as H = HG + HM —H 0 = t ~ ~ [o~a-~a-~a-~](p) (1) — HM = X ~ [‘r3a-3’r3](l) + ~ Ti(i). HG is the pure gauge Z(2) theory [3]. The gauge field is defined on the lattice links and represented by Pauli matrices O~i and ~ The interaction is given by the product over the four links of a plaquette (the Wilson action). The matter field is presented by Pauli matrices r1 and r3 on the lattice sites i with the interaction between nearest neighbours mediated by the gauge field a-3 on the link in between. This hamiltonian is locally gauge invariant because there exist the site operators G(i) = Ti(s) fl a~(l) (2) lmi all of which commute with H. The eigenvalues of G(i) can be +1 or —1 and are invariants of the motion. We will interpret G(i) = —1(+1) as representing the presence (absence) of an external charge. This interpretation is based on the fact that the N -+ limit of Z(N) theories [4] turns into periodic electrodynamics on the lattice. Borrowing the electrodynamic terminology a-i(l) = exp{iirE(1)} measures the electric field, a-3(l) = exp{iA(1)} creates an electric field and [cr3o3a-3o3](,p) = exp{iB(,p)} measures the * Work supported in part by the Israel Commission for Basic Research & U.S.-Israel Bi-National Science Foundation (BSF).