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arxiv: 2603.14995 · v2 · submitted 2026-03-16 · 🧮 math-ph · cond-mat.mes-hall· math.MP

The Zak phase in topologically insulating chains: invariants and limitations

Federico Manzoni , Domenico Monaco , Gabriele Peluso This is my paper

Pith reviewed 2026-05-15 10:35 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mes-hallmath.MP
keywords Zak phasetopological invariants1D insulatorsAZC symmetry classesquaternionic structureZ2 invariantKitaev chainsymmetry-protected topology
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The pith

Zak phase defines Z2 invariant for 1D insulators but vanishes in classes with quaternionic symmetry

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

This paper examines how the Zak phase captures the topology of one-dimensional translation-invariant insulators that have time-reversal, particle-hole or chiral symmetries. The authors build symmetric Bloch bases from fibered Hamiltonians and spectral projections, then extract a Z2-valued invariant directly from the abelian Zak phase. They show this invariant is well-defined across Altland-Zirnbauer-Cartan classes yet is forced to zero whenever the system admits anti-unitary symmetries that square to minus the identity. The result is illustrated on generalized Kitaev chains with finite-range hopping and varying numbers of chiral channels, where the Zak phase distinguishes only some of the distinct topological phases.

Core claim

We define a Z2-valued topological invariant I^(AZC-class)(H) obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the Z2 invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states, as well as its limitations in being an effective marker for topological phases of insulating chains.

What carries the argument

Symmetric Bloch bases adapted to discrete symmetries of fibered Hamiltonians and spectral projections, from which the abelian Zak phase yields the Z2 invariant I^(AZC-class)(H)

If this is right

  • In non-quaternionic AZC classes the Z2 invariant extracted from the Zak phase classifies topological phases of the chain.
  • In generalized Kitaev models the Zak phase captures only partial information about phases with arbitrary finite-range hopping or multiple chiral channels.
  • Whenever an anti-unitary symmetry squares to minus the identity the Z2 invariant is identically zero.
  • The Zak phase therefore cannot serve as a complete topological marker in all 1D symmetry-protected insulating phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric constraint may force other phase-based invariants to vanish in higher-dimensional systems that possess quaternionic structure.
  • Complete classification of 1D chains may require supplementing the Zak phase with winding numbers or other invariants that survive the quaternionic case.
  • The result raises the question of whether similar vanishing occurs for the Zak phase in disordered or interacting versions of these chains.

Load-bearing premise

Symmetric Bloch bases can be constructed that are adapted to all discrete symmetries of the Hamiltonian.

What would settle it

An explicit 1D Hamiltonian in a quaternionic symmetry class whose computed Zak phase produces a non-vanishing Z2 invariant would disprove the vanishing claim.

read the original abstract

In this work we investigate the topological content of the Zak phase in one-dimensional translation-invariant topological insulators endowed with time-reversal, particle-hole and/or chiral symmetries, extending results from \cite{Monaco_2023}. We analyze the extent to which the Zak phase captures the topology of all Altland--Zirnbauer--Cartan (AZC) symmetry classes in $1$D. Building on the framework of fibered Hamiltonians and spectral projections, we construct symmetric Bloch bases adapted to the discrete symmetries of the system and define a $\mathbb{Z}_2$-valued topological invariant $\mathrm{I}^{(\mathrm{AZC-class})}(H)$ obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the $\mathbb{Z}_2$ invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states, as well as its limitations in being an effective marker for topological phases of insulating chains. As an example, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels, and show how the Zak phase only retains partial information about their different topological phases.

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

2 major / 2 minor

Summary. The manuscript extends prior work on fibered Hamiltonians to 1D translation-invariant insulators in AZC symmetry classes. It constructs symmetric Bloch bases adapted to time-reversal, particle-hole and chiral symmetries, defines a Z2-valued invariant I^(AZC-class)(H) extracted from the abelian Zak phase, proves that this invariant vanishes in classes admitting a quaternionic structure, and illustrates the partial topological information retained by the Zak phase via generalized Kitaev chains with arbitrary-range hopping.

Significance. If the global construction of symmetry-adapted bases is valid, the work supplies a concrete Z2 marker derived from the Zak phase and demonstrates its forced vanishing under quaternionic constraints, thereby clarifying geometric limitations of the Zak phase as a topological diagnostic in 1D. The Kitaev-chain example further shows how the invariant distinguishes only a subset of phases, which is a useful negative result for the field.

major comments (2)
  1. [Construction of symmetric Bloch bases (fibered-Hamiltonian framework)] The global existence of continuous symmetric Bloch bases over the full Brillouin zone S^1 is asserted without proof when the occupied-state bundle is non-trivial. For a non-trivial line bundle (non-zero winding or Z2 index), no globally continuous section exists; the transition functions on overlaps may alter the accumulated Zak phase, rendering I^(AZC-class)(H) either ill-defined or basis-dependent. This assumption is load-bearing for both the definition of the invariant and the vanishing claim in quaternionic classes.
  2. [Vanishing result for quaternionic classes] The vanishing of I^(AZC-class)(H) in classes with anti-unitary symmetries squaring to -1 is stated to follow from the quaternionic structure, yet the argument appears to presuppose that the adapted bases can be chosen globally precisely when the topology is non-trivial. An explicit computation of the transition functions or a homotopy argument showing that the Zak phase is forced to zero independently of local choices is required.
minor comments (2)
  1. [Kitaev-chain example] In the generalized Kitaev-chain example, specify the precise relation between the number of chiral channels and the possible values of the Zak phase; the statement that it 'retains only partial information' would benefit from an explicit table comparing the Z2 invariant against the full topological classification.
  2. [Preliminaries] Notation for the fibered Hamiltonian and spectral projections should be cross-referenced to the earlier Monaco_2023 framework to avoid ambiguity in the definition of the abelian Zak phase.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying these key points that require clarification. We address each major comment below and will incorporate the suggested improvements in a revised version.

read point-by-point responses

  1. Referee: [Construction of symmetric Bloch bases (fibered-Hamiltonian framework)] The global existence of continuous symmetric Bloch bases over the full Brillouin zone S^1 is asserted without proof when the occupied-state bundle is non-trivial. For a non-trivial line bundle (non-zero winding or Z2 index), no globally continuous section exists; the transition functions on overlaps may alter the accumulated Zak phase, rendering I^(AZC-class)(H) either ill-defined or basis-dependent. This assumption is load-bearing for both the definition of the invariant and the vanishing claim in quaternionic classes.

    Authors: We agree that the global construction requires a more explicit justification. Building on the fibered-Hamiltonian framework of our prior work, the symmetric Bloch bases are defined locally on an open cover of the Brillouin zone S^1, with transition functions fixed by the action of the discrete symmetries (time-reversal, particle-hole, and chiral). The abelian Zak phase is extracted from the determinant line bundle of the occupied states, and the Z2 invariant I^(AZC-class)(H) is defined via the parity of this phase; changes induced by transition functions contribute multiples of 2π that preserve the Z2 value. To strengthen the manuscript we will add a dedicated subsection that (i) recalls the local construction, (ii) computes the effect of symmetry-constrained transition functions on the integrated Berry connection, and (iii) proves that the resulting Z2 invariant is independent of the choice of local bases even when the underlying bundle is topologically non-trivial. revision: yes

  2. Referee: [Vanishing result for quaternionic classes] The vanishing of I^(AZC-class)(H) in classes with anti-unitary symmetries squaring to -1 is stated to follow from the quaternionic structure, yet the argument appears to presuppose that the adapted bases can be chosen globally precisely when the topology is non-trivial. An explicit computation of the transition functions or a homotopy argument showing that the Zak phase is forced to zero independently of local choices is required.

    Authors: The vanishing is a direct consequence of the quaternionic structure induced by any anti-unitary symmetry that squares to -1. This structure equips the occupied bundle with a compatible quaternionic multiplication that forces the Berry connection to satisfy an additional reality condition; integrating this connection over S^1 then yields a Zak phase whose parity is necessarily even. We will supply the missing explicit computation by evaluating the transition functions under the quaternionic action and showing that they contribute an even multiple to the total phase. As an alternative perspective we will also sketch a homotopy argument: any Hamiltonian in a quaternionic class can be continuously deformed, within the symmetry class, to a trivial reference Hamiltonian while preserving the quaternionic structure, and the invariant remains zero throughout the deformation. These additions will make the independence from local choices fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained beyond prior framework

full rationale

The paper extends the fibered-Hamiltonian setting from the authors' prior work but defines the new Z2 invariant I^(AZC-class)(H) directly from the abelian Zak phase after constructing symmetry-adapted Bloch bases; the vanishing result in quaternionic classes follows from the anti-unitary symmetry assumptions (squaring to -1) rather than reducing to a fit, self-definition, or unverified self-citation chain. No equation or step equates the output invariant to its input by construction, and the 1D Zak phase computation remains independently falsifiable via explicit models such as the generalized Kitaev chains. The cited framework supplies the technical language but does not force the specific Z2 claim or its vanishing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard mathematical structures of topological band theory plus the assumption that symmetric Bloch bases can be constructed for the given discrete symmetries.

axioms (2)
  • domain assumption Existence of symmetric Bloch bases adapted to time-reversal, particle-hole and chiral symmetries
    Invoked to define the Z2 invariant from the Zak phase
  • domain assumption The Zak phase remains abelian under the symmetry constraints considered
    Used to obtain the Z2-valued invariant
invented entities (1)
  • Z2-valued topological invariant I^(AZC-class)(H) no independent evidence
    purpose: To serve as a topological marker extracted from the Zak phase
    Newly defined in the paper for each AZC class

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