Frustrated Phase in the Z 2 Gauge Model ( * )

Our purpose in th is l e t te r is to discuss t he f rus t r a t ed phase in t he Z 2 gauge mode l in 2 + 1 dimensions. Th is s tudy is based on a p rev ious ly p roposed real -space renorma l i za t ion group rea l iza t ion (1) and takes profi t of t he dual proper t ies of t he I s ingl ike models . I n par t icu la r , we present resul ts for the r e l evan t cr i t ical func t ions and exponen t s ob ta ined ana l i t iea l ly and s imu l t aneous ly w i t h t he cor responding ferromagne t i c ones. The (2 + 1)-dimensional q u a n t u m Is ing-gauge mode l on a square l a t t i ce is defined b y the H a m i l t o n i a n

Our purpose in this letter is to discuss the frustrated phase in the Z 2 gauge model in 2 + 1 dimensions.This study is based on a previously proposed real-space renormalization group realization (1) and takes profit of the dual properties of the Ising-like models.In particular, we present results for the relevant critical functions and exponents obtained analitieally and simultaneously with the corresponding ferromagnetic ones.
The (2 + 1)-dimensional quantum Ising-gauge model on a square lattice is defined by the Hamiltonian , for all i links emerging from i defined at each site i of the lattice.
The Hamiltonian (1) allows, in the limit A -+-0% a fully frustrated configuration of spins (that one could call (~ antiferromagnetic )) due to the sign of A) that is shown in fig. 1.The linear combination of this configuration with all its gauge transformed under the operator (2) builds up the fully frustrated fundamental state.This situation distinguishes itself from the corresponding <~ ferromagnetic )) one by the minus sign to the expectation value of the so-called frustration operator <~z~(%>A~_~o.A'= AB(rl)ri = A (l + rt2) 2" These equations, implying only a functional dependence on rl 2, preserve the mentioned symmetry ~-+-V.Then, we obtain simultaneously the ferromagnetic and the antiferromagnetic properties of the systcm (~) without including higher-order couplings in the calculation.We should remark that this is the important property of the group realization, that shows the possibility of studying the frustrated phase of the gauge model (1) via the standard duality transformation.
From eqs. (4) we can easily compute the relevant function in the critical region, defined by the fixed points of the RG that are obtained by searching for the zeroes of the beta-function: where a is the lattice constant and whose nontrivial zeroes are (6) ~lo = -4---: :J: 1.95, corresponding to the ferromagnetic and antiferromagnetic critical points, respectively.Each of these critical points defines two phases characterized by a different behaviour of the ground state of the Hamiltonian (5).Correspondingly we found two phases, for each critical point, in the gauge model (1).In other words, for the frustrated ease, the model shows a free phase for ~ > Vr x and a confinement one for V ~ ~7~.Notice that ~ : 1.95 should be compared with the value 1.55 obtained in a 2 d-1 Ising modelperturbative calculation (a).
The critical exponent of the free phase, or monopole excitation, is given by (~) where fl~ is the slope of the beta-function ( 5) at the critical point.
On the other hand, the excitation of the confinement phase, called boxiton, is described by the exponent It results Um = %, as it should be, however the value obtained is well below the hightemperature standard ferromagnetic value of 0.625.From the standard behaviour of the correlation length near the fixed point we immediately obtain the corresponding critical exponent ~ = 0.66.This value of ~ is in very good agreement with the ferromagnetic high-temperature expansion value (0.639).We believe we found in general good results for space-like parameters, because our RG realization preserves duality.As was pointed out in ref.
(4) vb = Vm should be equal to r~ due to the space-time symmetry of the classical Hamiltonian.Then, the discrepancy found should be traced back to the breaking of this symmetry in the RG calculation.
Finally, the string tension exponent, related to the renormalization of A, is given by ~ = 1.04.This value is not too far from the expected Widom sealing relation v, = 2~ b.The difference, as was stated above is due to the asymmetry introduced for the space-time different scales.
We have discussed the frustrated phase in a lattice gauge model, in particular the Ising gauge, or Z2, model in (2 -4-1)-dimensions, where via the duality transformation we can apply renormalization group techniques.It is thank to our previously proposed realization of the RG that we can take profit of duality in order to study the mentioned phase.This phase manifests itself without introducing other couplings in the model that would destroy the duality properties.
a local gauge invariance, since H commutes with the operator (2) Fig. 1. -A fully frustrated configuration of spins contributing to the fundamentM state.