Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits

TL;DR

This study analyzes a measurement-only circuit (MoC) that drives a toric code (TC) on a triangular lattice by alternating stabilizer measurements with competitive local Pauli terms. Using stabilizer methods, it maps the phase diagram via topological entanglement entropy ($TEE$), 1-form-symmetry disorder parameters (string/loop operators), and the emergence of logical qubits, uncovering distinct Higgs, confinement, and TC regions. Finite-size scaling yields critical points and exponents (e.g., $p_{gc}^x \\approx 0.846$, $\\nu^x \\approx 1.69$; $p_{gc}^z \\approx 0.876$, $\\nu^z \\approx 1.74$) showing that the $TEE$ transition does not coincide with 1-form symmetry breaking, and that string/loop observables follow near-2D-percolation exponents, while the $TEE$ exponent differs. The triangular geometry, lacking self-duality, reveals nuanced relationships between topological order, 1-form symmetry breaking, and logical operators, offering insights into TC phase structure and guiding future studies of MoC-generated topological states.

Abstract

Measurement-only (quantum) circuit (MoC) gives possibility to realize the states with rich entanglements, topological orders and quantum memories. This work studies the MoC, in which the projective-measurement operators consist of stabilizers of the toric code and competitive local Pauli operators. The former correspond to terms of the toric code on a triangular lattice and the later to external magnetic and electric fields. We employ efficient numerical stabilizer algorithm to trace evolving states undergoing phase transitions. We elucidate the phase diagram of the MoC system with the observables such as, topological entanglement entropy (TEE), disorder parameters of 1-form symmetries and emergent logical operators. We clarify the locations of the phase transitions through the observation of the above quantities and obtain precise critical exponents to examine if the observables exhibit the critical behavior simultaneously under the MoC and transitions belong to the same universality class. In contrast to the TC Hamiltonian system and toric code MoC on a square lattice, the system on the triangular lattice is not self-dual nor bipartite, and then, coincidence by symmetries, such as critical behaviors across the TC and Higgs/confined phase, does not takes place. Then, the toric code MoC on the triangular lattice provides us a suitable playground to clarify the mutual relationship between the TEE, spontaneous symmetry breaking of the 1-form symmetries, and emergence of logical operators. Obtained results indicate that toric code MoC on the triangular lattice exhibits a few distinct phase transitions with different location and critical exponents, and some of them are closely related with the two-dimensional percolation transition.

Figures (13)

• Figure 1: Toric code on triangular lattice. Left panel: Stabilizers $A_s,B_p$, and loop and string, on which observables are defined. Right panel: Schematic picture of MoC. Operators $X_e$, $Z_e$, $A_s$ and $B_p$ are applied to the state under consideration.
• Figure 2: Left: Subsystems $A,B,C$ for calculation of the topological entanglement entropy (TEE). Right: Phase diagram obtained by the TEE calculated by the MoC. The boundary between Higgs and confinement regimes is determined by behavior of the TEE, string and loop operators, some of which are shown in Appendix D. The dots are locations of the measurement. The label "TC" denotes the toric code phase, a topological ordered phase. Data in Appendix D indicate that a Higgs-confinement phase transition takes place for $p_g=0.6$, whereas for $p_g\leq 0.4$, at most crossover.
• Figure 3: Overview of observables from Higgs-confinement phase to TC. There are three kinds of observables, which we investigate in detail to clarify the phase diagram and critical behavior of the system. These three are; (i) topological entanglement entropy (TEE), (ii) $X$ and $Z$ non-contractible loops ($X/Z$-loop), (iii) $X$ and $Z$ open string ($X/Z$-string). (a) MoC starts with the $X$-product state and observed stabilizers are $Z_e$ (probability $p_z$) and $A_s$ and $B_p$ (probability ${p_g \over 2}$ for each). Under the MoC, $p_z+p_g=1$, whereas $p_x=0$. The system undergoes a phase transition from the Higgs regime to the TC. (b) MoC starts with the $Z$-product state and observed stabilizers are $X_e$ (probability $p_x$) and $A_s$ and $B_p$ (probability ${p_g \over 2}$ for each). Under the MoC, $p_x+p_g=1$, whereas $p_z=0$. The system undergoes a phase transition from the confinement regime to the TC. Detailed study reveals that the three observables exhibit a transition-like behavior at different values of $p_g$.
• Figure 4: Topological entanglement entropy for various system sizes. Top: Transition from confinement regime to TC phase in the parameter space of $p_x+p_g=1$. Bottom: Transition from Higgs regime to TC phase in the parameter space of $p_z+p_g=1$. $A_h$ is area of hexagon complex in unit of triangle. The system size is $(L_x,L_y)=(24,24)$.
• Figure 5: Variance of topological entanglement entropy for various system sizes. Top: Transition from confinement regime to TC phase in the parameter space of $p_x+p_g=1$. Bottom: Transition from Higgs regime to TC phase in the parameter space of $p_z+p_g=1$. $A_h$ is area of hexagon complex in unit of triangle. The insets show the data collapse by the finite-size scaling.