Effects of quenched disorder in three-dimensional lattice ${\mathbb Z}_2$ gauge Higgs models

Abstract

We study the effects of uncorrelated quenched disorder to the phase diagram and continuous transitions of three-dimensional lattice ${\mathbb Z}_2$ gauge Higgs models. For this purpose, we consider two types of quenched disorder, associated with the sites and plaquettes of the cubic lattice. In both cases, for sufficiently weak disorder, the phase diagram remains similar to that of the pure system, showing two different phases (one of them being a topologically ordered phase), separated by two different continuous transition lines. However, the quenched disorder changes the universality classes of the critical behaviors along some of the transition lines. The random-plaquette disorder turns out to be relevant along the topological ${\mathbb Z}_2$ gauge transition line, so the critical behaviors belong to the different random-plaquette $\mathbb{Z}_2$ gauge (RP${\mathbb Z}_2$G) universality class with length-scale exponent $ν=ν_{\rm rp}\approx 0.82$; on the other hand, it turns out to be irrelevant along the other Ising$^\times$ transition line (a variant of the Ising transitions with a gauge-dependent order parameter), leaving unchanged its asymptotic critical behaviors with $ν=ν_{\cal I}\approx 0.63$. The random-site disorder leads to a substantially different scenario: it destabilizes the Ising$^\times$ critical behaviors of the pure model, changing them into those of the randomly-dilute Ising$^{\times}$ (RDI$^{\times}$) universality class with $ν=ν_{\rm rdi}\approx 0.68$, while the critical behaviors along the other ${\mathbb Z}_2$ gauge topological transition line remains stable, with $ν=ν_{\cal I}\approx 0.63$.

Figures (6)

• Figure 1: Sketch of the $K$-$J$ phase diagram of the 3D lattice ${\mathbb Z}_2$ gauge Higgs model (\ref{['HiggsH']}). The dashed line represents the self-dual line. The thick line corresponds to first-order transitions on the self-dual line, extending for a finite interval. The two lines labelled "${\mathbb Z}_2$" are related by duality, and correspond to Ising-like continuous transitions, whose critical behaviors can be classified as Ising$^\times$ with a local gauge-dependent order parameter (this is the line ending at $[K=\infty,J_{\cal I} = 0.221654626(5)]$) and topological transitions without local order parameters (the line ending at $[K_{{\mathbb Z}_2g}=0.761413292(11),J=0]$). The first-order transition line should give rise to a critical endpoint belonging to the 3D Ising universality class, at $[K_{\rm cep}\approx 0.688,J_{\rm cep} \approx 0.258]$. The three transition lines meet at a multicritical point (MCP) on the self-dual line, at $[K_\star = 0.7525(1),J_\star = 0.22578(5)]$. At the MCP, the system turns out to develop a multicritical behavior characterized by the critical exponents of the XY universality class.
• Figure 2: Scaling of the third and of the fourth cumulants for the model with $J=0.1$ and $q=0.015$. The critical exponent $\nu_{\rm rp}=0.82$ of the RP${\mathbb Z}_2$G universality class has been used, together with the optimal estimate $K_c=0.894$ of the critical point. For comparison we also report in the upper panel data for the third cumulant computed in the RP${\mathbb Z}_2$G model obtained in Ref. BV-26, fixing the nonuniversal normalizations of the scaling function ${\cal B}_3$ by multiplying by 1.0 and 18.8 along the horizontal and vertical directions respectively.
• Figure 3: Some results for the lattice ${\mathbb Z}_2$ gauge Higgs model without disorder (i.e., $q=0$) and $K=1$, along the Ising$^{\times}$ line. Data for the cumulants $K_3=\frac{1}{V}\left(\frac{\partial}{\partial K}\right)^3\log Z$ (plaquette variables related) and $J_3=\frac{1}{V}\left(\frac{\partial}{\partial J}\right)^3\log Z$ (site variables related).
• Figure 4: Scaling of the third and of the fourth order cumulants for the model with $K=1$ and $q=0.010$. The critical exponent $\nu_{\mathcal{I}}=0.629971$ of the Ising universality class has been used, together with $J_c=0.22266$. For comparison we also report in the lower panel data for the fourth cumulant computed without disorder at $K=1$, rescaled using $\nu_{\cal I}$ and $J_c=0.22185$ (see BPV-24-unco), multiplied by 0.98 and 87 along the horizontal and vertical directions, respectively.
• Figure 5: Scaling of the third and of the fourth order cumulants for the model with $K=1$ and $q=0.015$. The critical exponent $\nu_{\mathcal{I}}=0.629971$ of the Ising universality class has been used, together with $J_c=0.22336$.