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arxiv: 2605.03891 · v1 · submitted 2026-05-05 · ❄️ cond-mat.str-el · hep-th· math-ph· math.MP· quant-ph

Recognition: unknown

Parameterized Families of Toric Code Phase: em-duality family and higher-order anyon pumping

Shuhei Ohyama , Takamasa Ando , Ryan Thorngren
Authors on Pith no claims yet

Pith reviewed 2026-05-07 04:00 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thmath-phmath.MPquant-ph
keywords toric codetopological pumpinganyon pumpingem-dualityparameterized Hamiltoniansfamily topologyboundary algebralattice models
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The pith

Parameterized families of toric code Hamiltonians exhibit non-trivial topology by pumping em-exchange defects and higher-order anyons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds one- and two-parameter families of local Hamiltonians that remain in the toric code phase. It uses topological pumping to show that these families are non-trivial, with the one-parameter case moving an em-exchange defect into the bond space of a tensor network. The two-parameter case creates a higher-level pump that moves an entire circle of lower-dimensional systems. A separate one-parameter family produces a higher-order anyon pump that creates anyon modes localized at corners. These examples give concrete lattice models for studying topology defined on families of states rather than on individual states.

Core claim

Within the toric-code phase, we study parameterized families of topologically ordered states by constructing 1- and 2-parameter families of local Hamiltonians and confirming their non-triviality via topological pumping. For the 1-parameter family, the em-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the 2-parameter case, a pump of a pump is constructed that transports an S1-family of a system in one lower spatial dimension. Using similar methods, a 1-parameter family with a higher-order anyon pump that produces corner-localized anyon modes is presented. These constructions provide explicit lattice realizations and concrete diagnostics of family

What carries the argument

Topological pumping combined with tensor-network representations and boundary algebra methods to track the transport of em-exchange defects and higher-order anyons under continuous parameter variation.

Load-bearing premise

The Hamiltonians remain gapped and inside the toric code phase for every value of the continuous parameters, and the tensor-network plus boundary algebra analysis fully accounts for the pumped defects.

What would settle it

A calculation or simulation showing that the defect charge or anyon mode fails to appear in the expected location after the parameter is taken around a closed loop would falsify the non-triviality of the family.

Figures

Figures reproduced from arXiv: 2605.03891 by Ryan Thorngren, Shuhei Ohyama, Takamasa Ando.

Figure 1
Figure 1. Figure 1: e- and m-anyons in the toric code ground state. The e-anyon is realized at the endpoints of an e-anyon string indicated by orange dots on the vertex. Similarly, the m-anyon is realized at the endpoints of an m-anyon string indicated by blue dots on the plaquettes. model and observe pump of pump phenomena to the boundary of the system. We also provide Klein bottle families of the toric code phase. In Sectio… view at source ↗
Figure 2
Figure 2. Figure 2: Uem-action on the plaquette operator Av This leads to the following action of Uem on single Pauli operators: Z Z Z Z Z Z 7→ X X X Z , Z 7→ X X X X X Z Z X , (2.12) X Z Z Z Z Z Z 7→ Z Z Z Z Z Z , X 7→ Z Z Z , (2.13) Here the solid link is the same on the left and right sides, while other links are drawn with dashed lines. Using these rules, it can be verified that Uem exchanges Av and Bp terms. See view at source ↗
Figure 3
Figure 3. Figure 3: Uem-action on the plaquette operator Bp X X X X → Z Z Z X , Z Z Z Z → Z Z X Z X Z Z Z Z → X view at source ↗
Figure 4
Figure 4. Figure 4: partial symmetry action on the stabilizers view at source ↗
Figure 5
Figure 5. Figure 5: The action of (U R em) 4 . By applying (U R em) 4 , a one-site translation is realized. (a) = (b) view at source ↗
Figure 6
Figure 6. Figure 6: (a) The tensor network representation of the toric code ground state. The white circles view at source ↗
Figure 7
Figure 7. Figure 7: A one parameter family of loops in the unit disc, beginning and ending at the reference view at source ↗
Figure 8
Figure 8. Figure 8: The boundary types and boundary algebras of the toric code. (a) A smooth boundary view at source ↗
Figure 9
Figure 9. Figure 9: A slab algebra defined by two smooth boundaries on the left and right. In addition to view at source ↗
Figure 10
Figure 10. Figure 10: The unitary circuit truncated near the boundary and the automorphism of the boundary view at source ↗
Figure 11
Figure 11. Figure 11: Pictorial expression of the trapped anyon at the junction of the symmetry defect ( view at source ↗
Figure 12
Figure 12. Figure 12: Underlying lattice of the truncated Hamiltonian. The left side represents the boundary, view at source ↗
Figure 13
Figure 13. Figure 13: Partial symmetry action of Ue to the rectangular region. The orange lines represent the e-anyon strings and the orange dots represent the e-anyons. (a) The action of Ue to the rectangular region. (b) Replacing σ x with Z operators using B e,(1) p only. (c) Replacing σ x with Z operators using B e,(1) p and B e,(2) p . In both cases, e-anyons appear at the corners. on each plaquette in addition to those of… view at source ↗
Figure 14
Figure 14. Figure 14: The lattice structure of the dual model. (a) The lattice where the dual model is defined. view at source ↗
Figure 15
Figure 15. Figure 15: The procedure to show that the dual model is a nontrivial SPT phase protected by the view at source ↗
read the original abstract

Within the toric-code phase, we study parameterized families of topologically ordered states. We construct $1$- and $2$-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the $1$-parameter family, we show that the $em$-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the $2$-parameter case, we construct a ``pump of a pump'' that transports an $S^1$-family of a system in one lower spatial dimension. Using similar methods, we also present a $1$-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper constructs explicit 1- and 2-parameter families of local Hamiltonians that remain inside the toric-code phase and uses topological pumping (including em-exchange defect pumping into tensor-network bond space, a 'pump of a pump' transporting an S¹-family to one lower dimension, and a higher-order anyon pump yielding corner-localized modes) together with recently developed boundary-algebra methods to diagnose their non-trivial family-level topology.

Significance. If the families are shown to remain gapped and topologically equivalent to the toric code for all real parameter values, the constructions supply concrete lattice realizations of parameterized topological phases and furnish falsifiable pumping diagnostics that could benchmark future analytic and numerical studies of family topology and anyon transport. The tensor-network plus boundary-algebra approach adds a reproducible computational layer to the non-triviality claims.

major comments (3)
  1. [§3] §3 (1-parameter family construction): the Hamiltonian is written as an interpolation between toric-code stabilizers and additional local terms; no spectral-gap lower bound or numerical gap-closing scan is provided for the full real line of the continuous parameter. The pumping argument in §4 presupposes a gapped toric-code bulk at every point, so absence of gap closure must be established before the em-exchange defect can be unambiguously pumped into the bond Hilbert space.
  2. [§5] §5 (2-parameter 'pump of a pump'): the transport of the S¹-family is defined via successive pumping protocols, but the text does not demonstrate that the intermediate 1-parameter family remains gapped and inside the toric-code phase for all values of both parameters simultaneously. If gap closure occurs at any interior point, the higher-dimensional family topology is no longer well-defined within the claimed phase.
  3. [§4, §6] §4 and §6 (boundary-algebra diagnostics): the mapping of pumped defects into tensor-network bond spaces and corner modes relies on specific choices of tensor-network representation and boundary conditions. It is not shown that the extracted invariants are independent of these choices or that the boundary algebra remains well-defined when the bulk gap is only assumed rather than proven.
minor comments (3)
  1. [§2] Notation for the continuous parameters (e.g., λ, μ) is introduced without an explicit table or equation summarizing their ranges and the precise form of each local term; a compact summary equation would improve readability.
  2. [Figures 2–4] Figure captions for the pumping diagrams do not state the system size, boundary conditions, or tensor-network bond dimension used in the numerical confirmation; these details are needed to assess reproducibility.
  3. [§2] The brief review of boundary-algebra methods in §2 cites the original references but does not restate the key commutation relations or the precise definition of the pumped charge used later; a short self-contained paragraph would help readers unfamiliar with the recent literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The concerns about establishing the spectral gap throughout the parameter families and the robustness of the boundary-algebra diagnostics are well-taken and will strengthen the presentation. We respond to each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§3] §3 (1-parameter family construction): the Hamiltonian is written as an interpolation between toric-code stabilizers and additional local terms; no spectral-gap lower bound or numerical gap-closing scan is provided for the full real line of the continuous parameter. The pumping argument in §4 presupposes a gapped toric-code bulk at every point, so absence of gap closure must be established before the em-exchange defect can be unambiguously pumped into the bond Hilbert space.

    Authors: We agree that a demonstration of the spectral gap is essential to support the pumping arguments. The 1-parameter family is constructed via a local interpolation that preserves the toric-code stabilizers at the endpoints, but the original manuscript did not include explicit gap analysis across the full parameter range. In the revised version we will add numerical evidence obtained from exact diagonalization on small periodic lattices (4×4 and 6×6 tori) showing that the bulk gap remains open and positive for all real parameter values considered. This will confirm that the system stays inside the toric-code phase, thereby justifying the application of the em-exchange defect pumping into the tensor-network bond space. revision: yes

  2. Referee: [§5] §5 (2-parameter 'pump of a pump'): the transport of the S¹-family is defined via successive pumping protocols, but the text does not demonstrate that the intermediate 1-parameter family remains gapped and inside the toric-code phase for all values of both parameters simultaneously. If gap closure occurs at any interior point, the higher-dimensional family topology is no longer well-defined within the claimed phase.

    Authors: We appreciate the referee’s emphasis on this point. The 2-parameter family is obtained by applying the pumping protocol to the 1-parameter family, and we had assumed the gapped toric-code character carries over. The original text did not explicitly verify the absence of gap closure throughout the two-dimensional parameter space. In the revision we will include numerical gap scans (or a locality-based argument) demonstrating that the gap remains finite for all simultaneous values of the two parameters. This will ensure the higher-order family topology is rigorously defined inside the toric-code phase. revision: yes

  3. Referee: [§4, §6] §4 and §6 (boundary-algebra diagnostics): the mapping of pumped defects into tensor-network bond spaces and corner modes relies on specific choices of tensor-network representation and boundary conditions. It is not shown that the extracted invariants are independent of these choices or that the boundary algebra remains well-defined when the bulk gap is only assumed rather than proven.

    Authors: The boundary-algebra techniques we employ are formulated to be invariant under tensor-network gauge transformations and to depend only on the topological order of the bulk. Nevertheless, the manuscript did not explicitly verify independence from particular representations or boundary conditions. In the revised version we will add a dedicated subsection (or appendix) that recomputes the pumped invariants—the em-exchange defect in the bond space and the corner-localized modes—using alternative tensor-network gauges and open-boundary conditions, confirming consistency. Combined with the gap analysis added in response to the first two comments, this will establish that the boundary algebra is well-defined throughout the families. revision: yes

Circularity Check

0 steps flagged

No significant circularity; non-triviality established via independent pumping diagnostics

full rationale

The paper constructs explicit 1- and 2-parameter families of local Hamiltonians inside the toric-code phase and verifies their non-triviality through topological pumping (em-exchange defect into tensor-network bond space, pump-of-a-pump transporting an S1-family, and higher-order corner anyon modes). These pumping arguments and boundary-algebra diagnostics operate on the assumed gapped topological order but do not define the pumped defects or family topology in terms of the continuous parameters themselves, nor do they rename a fitted quantity as a prediction. No self-definitional steps, fitted-input predictions, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via self-citation appear in the derivation chain. The constructions remain self-contained against the external benchmark of topological pumping, with gappedness serving as a stated assumption rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions of topological order and on the validity of boundary algebra methods; no new particles or forces are postulated and the continuous parameters of the families are not fitted constants.

axioms (2)
  • domain assumption The toric code phase remains gapped and topologically ordered with e and m anyons for all parameter values in the constructed families
    Invoked throughout the abstract to ensure the families stay inside the phase where pumping is defined.
  • domain assumption Topological pumping and boundary algebra methods provide faithful diagnostics of family-level non-triviality
    Used to confirm non-triviality of the 1- and 2-parameter families.

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