A real-space renormalization-group calculation for the quantum Z_2 gauge theory on a square lattice

TL;DR

This paper enhances a real-space renormalization-group method to analyze the quantum Z_2 gauge theory on a square lattice, improving accuracy and deriving gauge-invariant operators, but finds discrepancies in critical exponent predictions.

Contribution

It introduces a second-order perturbation expansion into the RG algorithm for the quantum Z_2 gauge theory, improving the treatment of higher energy states and deriving gauge-invariant operators.

Findings

Reaffirmed previous RG fixed point analysis.

Calculated critical exponents with poor agreement to known values.

Estimated critical amplitude ratio near the phase transition.

Abstract

We revisit Fradkin and Raby's real-space renormalization-group method to study the quantum Z_2 gauge theory defined on links forming a two-dimensional square lattice. Following an old suggestion of theirs, a systematic perturbation expansion developed by Hirsch and Mazenko is used to improve the algorithm to second order in an intercell coupling, thereby incorporating the effects of discarded higher energy states. A careful derivation of gauge-invariant effective operators is presented in the Hamiltonian formalism. Renormalization group equations are analyzed near the nontrivial fixed point, reaffirming old work by Hirsch on the dual transverse field Ising model. In addition to recovering Hirsch's previous findings, critical exponents for the scaling of the spatial correlation length and energy gap in the electric free (deconfined) phase are compared. Unfortunately, their agreement is poor. The leading singular behavior of the ground state energy density is examined near the critical point: we compute both a critical exponent and estimate a critical amplitude ratio.