Continuous topological phase transition between $\mathbb{Z}_2$ topologically ordered phases

TL;DR

This work demonstrates a direct, continuous topological transition between the toric code and double semion phases by introducing a perturbed $\mathbb{Z}_4$ quantum double model and mapping it to a two-coupled $(2+1)$D quantum Ising model via a nontrivial duality. The analysis identifies the transition as belonging to the XY$^*$ universality class, with deconfinement of certain $\mathbb{Z}_4$ anyons at criticality and a two-parameter pattern of 1-form symmetry breaking. By analyzing both nonlocal order parameters (condensation patterns) and area-law coefficients of Wilson loops, the authors extract critical exponents consistent with an $O(2)$ Wilson-Fisher fixed point in a gauged setting. The results establish a framework for topological phase transitions beyond anyon condensation and suggest emergent fractionalization at critical points between incompatible topological orders.

Abstract

Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct $\mathbb{Z}_2$ topological orders in (2 + 1)D. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a directly continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the $\mathbb{Z}_4$ topological order, we introduce a perturbed $\mathbb{Z}_4$ quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous (2 + 1)D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that $\mathbb{Z}_4$ anyons, fractionalized from the $\mathbb{Z}_2$ topological orders, become deconfined at the transition between $\mathbb{Z}_2$ topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.

Figures (8)

• Figure 1: Schematic illustration of the motivation and main results. (a) Previous works TC_DS_EDTC_DS_QMC_2021Scaffidi_2023 reveal that the transition between the TC and DS phases can be first order or via an intermediate phase. The existence of a direct continuous transition between them is an open question. (b) We confirm a direct continuous TC-DS transition point, where the adjacent TC and DS phases spontaneously break two different 1-form symmetries. We further reveal deconfined $\mathbb{Z}_4$ anyons at their transition point.
• Figure 2: Perturbed $\mathbb{Z}_4$ QD model and duality transformation. (a) The multi-site local Hamiltonian terms of the perturbed $\mathbb{Z}_4$ QD model, $e$ of $W_e^2$ is the edge where the $X^2$ operator is located. (b) The perturbed $\mathbb{Z}_4$ QD model is mapped to a two-coupled Ising model, the blue (red) lattice is the primal (dual) lattice, and we shift the dual lattice to overlap with the primal lattice.
• Figure 3: Phase diagram of the perturbed $\mathbb{Z}_4$ QD model on the $h_x=-1$ plane. The model has three tuning parameters: $h_x$, $h_z$ and $h_w$. The solid blue lines indicate the 3D Ising* transitions. The red star marks the multi-critical point, which belongs to the XY* universality class. Arrows (i) and (ii) denote the lines $h_z=h_w$ and $h_w=2.980-h_z$, respectively, both crossing the multi-critical point.
• Figure 4: Condensation order parameters and area law coefficients near and at the continuous TC-DS transition point $(h_x,h_z,h_w)\approx(-1.000,1.490,1.490)$. Different colors correspond to different order parameters, and the colors from light to dark correspond to the bond dimensions $(D,\chi)=(4,100),(5,100),(6,100),(7,100)$, where $D$ is the iPEPS bond dimension and $\chi$ is the bond dimension of the corner transfer matrix renormalization group method contracting the iPEPS CTMRG_1CTMRG_2Fast_CTMRG. The vertical dashed line marks the location of the multi-critical point ($h_c=1.490$). From the XY* universality class, we have three order parameter critical exponents $\beta_{\phi}$, $\beta_{\phi}'$, $\beta_{-}$ and two correlation length critical exponents $\nu_{+}$ and $\nu_{-}$. (a) Area law coefficients along the line (i) in Fig. \ref{['fig:phasediagram']}. (b) Condensation order parameters along the line (i). (c) Double-log plot of the area law coefficients in (a), where the slope of the straight lines is $2\nu_{+}$. (d) Double-log plot of the condensation order parameters in (b), where the slopes of the straight lines are $\beta_{\phi}$ and $\beta_{-}$, respectively. (e) Area law coefficients along the line (ii) in Fig. \ref{['fig:phasediagram']}. (f) Condensation order parameters along the line (ii). (g) Double-log plot of the area law coefficients in (e), where the slope of the straight lines is $2\nu_{-}$. (h) Double-log plot of the condensation order parameters in (f), where the slope of the straight lines is $\beta_{\phi}'$.
• Figure S1: String and loop operators defining the condensation order parameters and the Wilson loop order parameters. (a) Sting operator creating two $\pmb{e}^2$ at $v$ and $v'$. (b) Sting operator creating $\pmb{m}^2$ at $p$ and $p'$. (c) Sting operator creating $\pmb{e}^2\pmb{m}^2$ at $p$ and $p'$. (d) Wilson loop operator associated to $\pmb{e}$. (e) Wilson loop operator associated to $\pmb{m}$. (f) Wilson loop operator associated to $\pmb{e}\pmb{m}$.