Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices

TL;DR

The study investigates how parallel fields affect topological order and confinement in the extended toric code on non-square lattices. By employing continuous-time quantum Monte Carlo and extending percolation-inspired order parameters (POPs) to measure both $e$- and $m$-anyon confinement, the authors reveal that confinement can occur independently for electric and magnetic anyons on honeycomb and triangular lattices, while topological order requires both types to be deconfined. They map detailed phase diagrams across honeycomb, triangular, and cubic lattices, identify multi-critical points, and compare POPs with Fredenhagen-Marcu and SIT order parameters, highlighting the experimental accessibility of POPs. The results clarify the nuanced relationship between topological order and (de)confinement in $\ ext{Z}_2$ lattice gauge theories and provide guidance for quantum simulations of confinement phenomena in realistic lattices.

Abstract

Kitaev's toric code has become one of the most studied models in physics and is highly relevant to the fields of both quantum error correction and condensed matter physics. Most notably, it is the simplest known model hosting an extended, deconfined topological bulk phase. To this day, it remains challenging to reliably and robustly probe topological phases, as many state-of-the-art order parameters are sensitive to specific models and even specific parameter regimes. With the emergence of powerful quantum simulators which are approaching the regimes of topological bulk phases, there is a timely need for experimentally accessible order parameters. Here we study the ground state physics of the parallel field toric code on the honeycomb, triangular and cubic lattices using continuous-time quantum Monte Carlo. By extending the concept of experimentally accessible percolation-inspired order parameters (POPs) we show that electric and magnetic anyons are independently confined on the honeycomb and triangular lattices, unlike on the square lattice. Our work manifestly demonstrates that, even in the ground state, we must make a distinction between topological order and \mbox{(de-)confinement}. Moreover, we report multi-critical points in the aforementioned confinement phase diagrams. Finally, we map out the topological phase diagrams on the honeycomb, triangular and cubic lattices and compare the performance of the POPs with other topological order parameters. Our work paves the way for studies of confinement involving dynamical matter and the associated multi-critical points in contemporary quantum simulation platforms for $\mathbb{Z}_2$ lattice gauge theories.

Figures (6)

• Figure 1: Link- and plaquette percolation as confinement order parameters for e- and m-anyons in the extended toric code (\ref{['eq:eTC']}). (a) On the left, we start with a triangular lattice in the ground state of the bare toric code ($h_x=h_z=0$), i.e. $\hat{A}_v=\hat{B}_p=1$ everywhere. Applying $\hat{\tau}^z$-perturbations ("Z") on a cluster of neighboring links creates a pair of frustrated stars ($\hat{A}_v=-1$) at the open ends of the cluster, which are associated with point-like e-anyons living on lattice sites (red). Applying $\hat{\tau}^x$-perturbations ("X") on a cluster of links connecting neighboring plaquettes creates a pair of frustrated plaquettes ($\hat{B}_p=-1$) at the open ends of the cluster, which are associated with point-like m-anyons living on plaquettes (blue). The problem can be mapped to the dual lattice (here: honeycomb lattice) where the meaning of e- and m-anyons is swapped along with the fields ($h_x \leftrightarrow h_z$). (b) In the e-deconfined phase a percolating cluster of links with $\hat{\tau}^x = -1$ winds around the periodic lattice in at least one spatial dimension (here $\hat{P}^x_x=\hat{P}^x_y=-1$), corresponding to a non-zero percolation probability $\Pi^x > 0$. In the m-deconfined phase a percolating plaquette cluster connected by links with $\hat{\tau}^z = -1$ winds around the periodic lattice in at least one spatial dimension (here $\hat{P}^z_x=1$, $\hat{P}^z_y=-1$), corresponding to a non-zero plaquette percolation probability $\Pi^z > 0$.
• Figure 2: e- and m-confinement phase diagram of the extended toric code (\ref{['eq:eTC']}) on the honeycomb, triangular and cubic lattices. We apply continuous-time QMC at temperature $T=1/L$ to gain insights into the ground state phase diagram and use a crossing-point analysis of Binder cumulants $U$ to extract the critical fields, see inset of (a). On the left (right), we sample in the $\hat{\tau}^x$-basis ($\hat{\tau}^z$-basis) and measure the percolation probability $\Pi^x$ (plaquette percolation probability $\Pi^z$), the FM and the SIT order parameters. (a-b) The honeycomb lattice features an extended e-deconfined region with $\Pi^x>0$ for small $h_x, h_z$ and an e-confined region with $\Pi^x=0$ for larger $h_x, h_z$ which are separated by a continuous phase transition. m-anyons are always deconfined ($\Pi^z>0$) for small $h_z$, giving rise to an e-confined, m-deconfined regime. For large $h_z$, m-anyons feature a confined phase. The phase boundary features two multi-critical points between which the phase transition is of first-order type. For even higher fields, $\Pi^z$ has a percolation transition (dashed line) which is not relevant for topology. (c-d) The toric code on the triangular lattice is dual to the honeycomb lattice, the phase diagram is identical to the honeycomb lattice when exchanging the basis $\hat{\tau}^x \leftrightarrow \hat{\tau}^z$ and $h_x \leftrightarrow h_z$. We identify an m-confined, e-deconfined regime for large $h_z$. (e) On the cubic lattice, the confinement of e-anyons is qualitatively similar to the triangular lattice but features a first-order phase transition (yellow phase boundary, see inset for hysteresis curve) between the e-deconfined (e-confined) region at small (large) $h_x$. The first order line ends at a multi-critical point around $(h_z, h_x) = (1.0(1), 1.8(2))$ after which we find a percolation transition (dashed line). (a-e) We identify the topological phase with the regime where both e- and m-anyons are deconfined. The FM and SIT order parameters can generally probe the topological phase transition in the $\hat{\tau}^z$-basis ($\hat{\tau}^x$-basis) for $h_z$-scans ($h_x$-scans), see insets in c, d. In the other two cases, i.e. in the $\hat{\tau}^x$-basis ($\hat{\tau}^z$-basis) for $h_z$-scans ($h_x$-scans) the SIT features crossover behavior and the FM is too noisy to be evaluated. Additional data is presented in Appendix \ref{['app:details']}.
• Figure 3: Topological phase diagram of the extended toric code (\ref{['eq:eTC']}) on the honeycomb, triangular and cubic lattices. We construct the topological phase diagram using ground state QMC calculations for the FM, SIT and POPs which are partially shown in Fig. \ref{['figConfinementPD']}. The general structure of the phase diagram is remarkably similar to the Fradkin-Shenker model Fradkin1979 for all lattices: the system features an extended topological phase that is persistent for finite fields $h_x, h_z > 0$. The critical line $h_{x, \mathrm{c}}(h_z)$ ($h_{z,\mathrm{c}}(h_x)$) - extracted using a finite-size scaling analysis - is convex, i.e. it slightly shifts up for increasing $h_z$ ($h_x$). A first-order line starts at a multi-critical point at the tip of the topological phase and ends at a multi-critical point in the trivial phase. (a) honeycomb lattice: a continuous phase transition separates the topological from the trivial regime ($h_x, h_z \gg 1$). (b) triangular lattice: the honeycomb lattice is dual to the triangular lattice, which is reflected in the topological phase diagram ($h_x \leftrightarrow h_z$). (c) cubic lattice: the system features a first-order phase transition as we increase $h_x$ while fixing $h_z$ (yellow phase boundary). The phase boundary where $h_x$ is kept constant is associated with a continuous phase transition (solid line).
• Figure 4: Classical confinement phase diagram of Hamiltonian (\ref{['eq:can_cl_ham']}) at finite matter density $d$ on the honeycomb and triangular lattices. At zero matter density, the model can be mapped to the two-dimensional classical Ising model on the dual lattice Wegner1971, respectively. The golden lines correspond to Hamiltonian (\ref{['eq:gc_cl_ham']}) at $\mu=0$. We use the percolation probability $\Pi^x$ in the $\hat{\tau}^x$-basis as a confinement order parameter. (a) honeycomb lattice: we find a thermal deconfinement transition at zero matter density. The critical temperature is the triangular lattice Ising critical temperature $(T/h_x)_\mathrm{c} \approx 3.64$Fisher1967. At finite matter densities, the system is always non-percolating. (b) triangular lattice: the critical temperature at zero density is the honeycomb lattice Ising critical temperature $(T/h_x)_\mathrm{c} \approx 1.52$Fisher1967. In contrast to the honeycomb lattice, we observe an extended region with $\Pi^x > 0$ for finite matter densities. The cubic lattice phase diagram is qualitatively similar to the triangular lattice Linsel2024.
• Figure 5: Magnetizations in $x$- and $z$-direction of the extended toric code (\ref{['eq:eTC']}). We apply continuous-time QMC at temperature $T=1/L$ for $L=20$. We calculate the $x$-magnetization $\langle\hat{\tau}^x\rangle$ and the $z$-magnetization $\langle\hat{\tau}^z\rangle$. (a) For large $h_x$ the system is magnetized in $x$-direction and completely paramagnetic in $z$-direction, here shown for the honeycomb lattice. (b) The opposite is true for large $h_z$, where the system is magnetized in $z$-direction and paramagnetic in $x$-direction, here shown for the triangular lattice.