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arXiv:2311.16651 · Physical review. B./Physical review. B

Bulk-measurement-induced boundary phase transition in toric code and gauge Higgs model

Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Published Feb 28, 2024• 9 citations
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Abstract

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 ${Z}_{2}$ 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.

DOI
10.1103/physrevb.109.054432

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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.
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.
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.
Since the discovery of phase transitions driven by topological defects, the classification of phases of matter has been significantly extended beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking (SSB). In particular, intrinsic and symmetry-protected topological (SPT) orders have been discovered in (mostly gapped) quantum many-body ground states. However, these are commonly viewed as zero-temperature phenomena, and their robustness in a gapless ground state or against thermal fluctuations remains challenging to tackle. Here we introduce an explicit construction for SPT-type states with hidden order associated with SSB: They feature (quasi-)long-range correlations along appropriate edges but short-range order in the bulk, ground state degeneracy associated with SSB, and nonlocal string order in the bulk. We apply our construction to predict two types of finite-temperature SPT transitions protected by 1-form symmetries in the Ising and BKT class, respectively, where the usual signs of criticality appear despite the absence of a diverging correlation length in the bulk. While the state featuring hidden Ising order is gapped, the other SPT state associated with the BKT-SPT transition has hidden U ( 1 ) or XY order and constitutes an intrinsically gapless SPT state, associated with a gapless Goldstone mode. Specifically, in this work we discuss spins with global Z 2 or U ( 1 ) symmetry coupled to link variables constituting a loop gas model with a 1-form symmetry. By mapping this system to an Ising-gauge theory, we demonstrate that one of the SPT phases we construct corresponds to the Higgs-SPT phase at T = 0 – which we show here to remain stable at finite temperature. Our work paves the way for a more systematic search for hidden order SPT phases, including in gapless systems, and raises the question if a natural (finite- T ) spin liquid candidate exists that realizes hidden order in the Higgs-SPT class.
Where in the landscape of many-body phases of matter do we place the Higgs condensate of a gauge theory? On the one hand, the Higgs phase is gapped, has no local order parameter, and for fundamental Higgs fields is adiabatically connected to the confined phase. On the other hand, Higgs phases such as superconductors display rich phenomenology. In this work, we propose a minimal description of the Higgs phase as a symmetry-protected topological (SPT) phase, utilizing conventional and higher-form symmetries. In this first part, we focus on 2+1D Z2\mathbb Z_2Z2​ gauge theory and find that the Higgs phase is protected by a higher-form magnetic symmetry and a matter symmetry, whose meaning depends on the physical context. While this proposal captures known properties of Higgs phases, it also predicts that the Higgs phase of the Fradkin-Shenker model has SPT edge modes in the symmetric part of the phase diagram, which we confirm analytically. In addition, we argue that this SPT property is remarkably robust upon explicitly breaking the magnetic symmetry. Although the Higgs and confined phases are then connected without a bulk transition, they are separated by a boundary phase transition, which we confirm with tensor network simulations. More generally, the boundary anomaly of the Higgs SPT phase coincides with the emergent anomaly of symmetry-breaking phases, making precise the relation between Higgs phases and symmetry breaking. The SPT nature of the Higgs phase can also manifest in the bulk, e.g., at transitions between distinct Higgs condensates. Finally, we extract insights which are applicable to general SPT phases, such as a 'bulk-defect correspondence' generalizing discrete gauge group analogs of Superconductor-Insulator-Superconductor (SIS) junctions. The sequel to this work will generalize 'Higgs=SPT' to continuous symmetries, interpreting superconductivity as an SPT property.
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.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.