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arXiv:2403.13435 · Physical review. B./Physical review. B

Measurement-only dynamical phase transition of topological and boundary order in toric code and gauge Higgs models

Takahiro Orito, Yoshihito Kuno, Ikuo Ichinose

Published Jun 13, 2024• 4 citations
arXivHTMLPDF

Abstract

We extensively study long-time dynamics and the fate of a topologically ordered state in the toric code model evolving through a projective measurement-only circuit. The circuit is composed of several measurement operators corresponding to each term of the toric code Hamiltonian with magnetic-field perturbations, which is a gauge-fixed version of a ($2+1$)-dimensional gauge Higgs model. We employ a cylinder geometry with distinct upper and lower boundaries to classify stationary states after long-time measurement dynamics. The appearing stationary states depend on measurement probabilities for each measurement operator. The Higgs, confined, and deconfined phases emerge in the time evolution by the circuit. We find that both Higgs and confined phases are separated from the deconfined phase by topological entanglement entropy. We numerically clarify that both Higgs and confined phases are characterized by a long-range order on the boundaries accompanying edge modes, and they shift with each other by a crossover reflecting properties in the bulk phase diagram.

DOI
10.1103/physrevb.109.224306

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We extensively study long-time dynamics and the fate of a topologically ordered state in the toric code model evolving through a projective measurement-only circuit. The circuit is composed of several measurement operators corresponding to each term of the toric code Hamiltonian with magnetic-field perturbations, which is a gauge-fixed version of a (2+12+12+1)-dimensional gauge Higgs model. We employ a cylinder geometry with distinct upper and lower boundaries to classify stationary states after long-time measurement dynamics. The appearing stationary states depend on measurement probabilities for each measurement operator. The Higgs, confined, and deconfined phases emerge in the time evolution by the circuit. We find that both Higgs and confined phases are separated from the deconfined phase by topological entanglement entropy. We numerically clarify that both Higgs and confined phases are characterized by a long-range order on the boundaries accompanying edge modes, and they shift with each other by a crossover reflecting properties in the bulk phase diagram.
Study of boundary phase transition in toric code under cylinder geometry via bulk projective measurement is reported. As the frequency of local measurement for bulk qubits is increased, spin-glass type long-range order on the boundaries emerges indicating spontaneous-symmetry breaking (SSB) of Z2{Z}_{2}Z2​ symmetry. From the lattice gauge theory viewpoint, this SSB is a signal of a transition to Higgs phase with symmetry-protected topological order. We numerically elucidate the properties of this phase transition in detail, especially its criticality, and give a physical picture using nonlocal gauge-invariant symmetry operators. Phase transition in the bulk is also studied and its relationship to the boundary transition is discussed.
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.
Gauge-Higgs theory in flat spacetime provides a prime theoretical model of multiple phases of matter with distinct entanglement structures. More recently, there has been interest in hyperbolic space and the influence its extensive boundaries have on these structures. Entanglement transitions between non-equilibrium phases, such as topological phases, can be realized by the competition of non-commutating measurements. Here we study the entanglement phases of gauge-Higgs models on hyperbolic lattices described by measurement-only dynamics. We chart 2D phase diagrams by employing topological entanglement entropy and introducing a generalized measurement-induced entanglement tailored for hyperbolic Higgs phases. Utilizing duality, we find a phase absent in Euclidean space, wherein the bulk lattice hosts separated clusters of symmetry-protected topological order. Transitions between this "archipelago" phase and neighboring topological and trivial phases feature a critical exponent δδδ belonging to a mean-field universality class. Our essential tool is a 3D resource state inspired by measurement-based quantum computation that simulates the spacetime evolution of the 2D quantum state with a third direction of entanglement. This allows us to analyze spacetime "sponges" of entanglement, which map to classical percolation theory and allow us to infer values of critical points on our phase diagrams' boundaries.
In quantum many-body spin systems, the interplay between the entangling effect of multiqubit Pauli measurements and the disentangling effect of single-qubit Pauli measurements may give rise to two competing effects. By introducing a randomized measurement pattern with such bases, a phase transition can be induced by altering the ratio between them. In this work, we numerically investigate a measurement-based model associated with the (2+1)DZ2(2+1)\mathrm{D}\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}(2+1)DZ2​ Fradkin-Shenker Hamiltonian that encompasses the deconfining, confining, and Higgs phases. We determine the phase diagram in our measurement-only model by employing entanglement measures. For the bulk topological order, we use the topological entanglement entropy. We also use the mutual information between separated boundary regions to diagnose the boundary phase transition associated with the Higgs or the bulk symmetry-protected topological (SPT) phase. We observe the structural similarity between our phase diagram and the one in the standard quantum Hamiltonian formulation of the Fradkin-Shenker model with the open rough boundary. First, a deconfining phase is detected by nonzero and constant topological entanglement entropy. Second, we find a (boundary) phase transition curve separating the Higgs=SPT phase from the rest. In certain limits, the topological phase transitions reside at the critical point of the formation of giant homological cycles in the bulk three-dimensional (3D) space-time lattice, as well as the bond percolation threshold of the boundary 2D space-time lattice when it is effectively decoupled from the bulk. Additionally, there are analogous mixed-phase properties at a certain region of the phase diagram, emerging from how we terminate the measurement-based procedure. Our findings pave an alternative pathway to study the physics of Higgs=SPT phases on quantum devices in the near future.
By imaginary-time evolution with the Hamiltonian, an arbitrary state arrives in the system's ground state. In this paper, we conjecture that this dynamics can be simulated by a measurement-only circuit (MOC), where each projective measurement is set in a suitable way. Based on terms in the Hamiltonian and ratios of their parameters (coefficients), we propose a guiding principle for the choice of the measured operators called ``stabilizers'' and also the probability of projective measurement in the MOC. In order to examine and verify this conjecture of the parameter ratio and probability ratio correspondence in a practical way, we study a generalized (1+1)(1+1)(1+1)-dimensional Z2{Z}_{2}Z2​ lattice gauge Higgs model, whose phase diagram is very rich, including the symmetry-protected topological phase, deconfinement phase, etc. We find that the MOC constructed by the guiding principle reproduces the phase diagram in a very similar way to that of the ground state of the gauge Higgs Hamiltonian. The present paper indicates that the MOC can be broadly used to produce interesting phases of matter, which are difficult to be simulated by ordinary Hamiltonian systems composed of stabilizer-type terms.
It is now widely recognized that the toric code is a pure gauge-theory model governed by a projective Hamiltonian with topological orders. In this paper, we extend the interplay between quantum information system and gauge-theory model from the viewpoint of subsystem code, which is suitable for gauge systems including matter fields. As an example, we show that Z2{Z}_{2}Z2​ lattice gauge-Higgs model in (2+12+12+1)-dimensions with specific open boundary conditions is nothing but a kind of subsystem code. In the system, Gauss-law constraints are stabilizers, and order parameters identifying Higgs and confinement phases exist and they are nothing but logical operators in subsystem codes residing on the boundaries. Mixed anomaly of them dictates the existence of boundary-zero modes, which is a direct consequence of symmetry-protected topological order in Higgs and confinement phases. After identifying phase diagram, subsystem codes are embedded in the Higgs and confinement phases. As our main findings, we give an explicit description of the code (encoded qubit) in the Higgs and confinement phases, which clarifies duality between Higgs and confinement phases. The degenerate structure of subsystem code in the Higgs and confinement phases remains even in very high-energy levels, which is analogous to notion of strong-zero modes observed in some interesting condensed-matter systems. Numerical methods are used to corroborate analytically-obtained results and the obtained spectrum structure supports the analytical description of various subsystem codes in the gauge theory phases.