Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation

We analyze topological phase transitions and higher Berry curvature in one-dimensional quantum spin systems, using a framework that explicitly incorporates the symmetry group action on the parameter space. Based on a $G$-compatible discretization of the parameter space, we incorporate both group cochains and parameter-space differentials, enabling the systematic construction of equivariant topological invariants. We derive a fixed-point formula for the higher Berry invariant in the case where the symmetry action has isolated fixed points. This reveals that the phase transition point between Haldane and trivial phases acts as a monopole-like defect where higher Berry curvature emanates. We further discuss hierarchical structures of topological defects in the parameter space, governed by symmetry reductions and compatibility with subgroup structures.

💡 Research Summary

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The paper develops a systematic, discrete matrix‑product‑state (MPS) framework for studying one‑dimensional quantum spin chains whose Hamiltonians depend on a set of external parameters τ that form a space M equipped with a symmetry group G action. The central idea is to triangulate the parameter space in a G‑compatible way (a G‑simplicial complex) so that each vertex, edge, and higher‑dimensional simplex transforms either into itself or into another simplex under the group. At each vertex a pure state (or, more generally, an injective MPS tensor) is assigned.

Equivariant families of pure states
The authors first review the standard discrete Berry connection: for an edge Δ¹=(τ₀,τ₁) the Berry phase is A(Δ¹)=arg⟨ψ(τ₀)|ψ(τ₁)⟩. The Berry flux on a triangle Δ² is the discrete exterior derivative dA(Δ²)=A(∂Δ²). Summing fluxes over a closed 2‑cycle Σ yields an integer Chern number ν(Σ). When a group element g acts on a state, the relation
  ĥg|ψ(τ)⟩ = e^{iα_g(τ)}|ψ(gτ)⟩
introduces a 1‑cochain α_g. The compatibility condition δA = dα (the “descent equation”) encodes how the Berry connection changes under G. Consequently the Berry flux is G‑invariant (δF=0) and the Chern number transforms as ν(gΣ)=ϕ_g ν(Σ), where ϕ_g = ±1 distinguishes unitary (ϕ_g=1) and anti‑unitary (ϕ_g=−1) elements. This leads to general constraints: if an anti‑unitary symmetry maps a surface to its orientation‑reversed image, the Chern number must vanish; similarly, a loop invariant under such a symmetry can only carry a Berry phase of 0 or π.

Discrete MPS formulation
Moving from abstract states to concrete many‑body wavefunctions, the authors adopt injective MPS with translational invariance. An MPS is specified by a set of tensors A^i(g,τ) and a positive diagonal matrix Λ(τ). The fundamental theorem of MPS guarantees that two MPS representing the same physical state differ by a gauge transformation V(g,τ) on the virtual indices. The G‑equivariance condition translates into a relation between the physical symmetry operator ĥg and the virtual gauge V(g,τ), together with a phase α_g(τ) that appears exactly as in the pure‑state case. The overlap matrix Ω(τ,τ′)=⟨Ψ(τ)|Ψ(τ′)⟩ provides a gauge‑invariant quantity whose argument reproduces the discrete Berry connection. Because Ω can be evaluated with standard DMRG or TEBD algorithms, the whole construction is numerically tractable.

Fixed‑point formulas and topological invariants
A major technical achievement is the derivation of fixed‑point formulas for several higher‑dimensional topological invariants when the G‑action has isolated fixed points. Near a fixed point τ₀ the parameter space can be approximated by a small ball B^d. The authors expand the Berry connection and the cochain α_g to leading order, integrate over the boundary sphere S^{d‑1}=∂B^d, and obtain closed expressions for:

• The ordinary Berry phase (d=1) – reproducing the Thouless pump invariant.
• The π‑higher Berry phase (d=2) – a Z₂‑valued invariant that detects a monopole‑like source of curvature at a symmetry‑protected critical point.
• The Dixmier–Douady–Kapustin–Spodyneiko (DDKS) number (d=3) – an integer invariant associated with a gerbe connection, interpreted as the third Chern‑Simons‑type invariant of the family.

These formulas involve only data on the fixed points (the projective representation of the stabilizer group G_{τ₀} and the local MPS tensors), making them analytically transparent and computationally cheap.

Physical examples
The paper applies the formalism to several concrete families:

1. One‑parameter Thouless pump – a family of Hamiltonians periodic in a single angle θ. The fixed‑point formula reproduces the integer charge pumped per cycle, identified with the Chern number of the Berry curvature on the θ‑circle.

2. Two‑parameter π‑higher Berry phase – a family interpolating between the Haldane phase and a trivial phase, protected by either time‑reversal T or Z₂×Z₂ spin‑flip symmetry. The critical point where the gap closes acts as a point‑like defect; integrating the Berry curvature over a small sphere surrounding it yields π, confirming the monopole picture.

3. Three‑parameter DDKS families – models with combined spatial rotations (C_n) and time‑reversal, or with C₂^x×C₂^y symmetry. Different symmetry reductions lead to distinct fixed‑point formulas (labeled (1) and (2) in the text) that compute the DDKS integer, the π‑higher Berry phase, or a conventional pump invariant, depending on the subgroup considered.

These examples illustrate how the same underlying MPS data can encode multiple topological responses, each revealed by choosing a different subgroup of G and a corresponding parameter subspace.

Defect hierarchy
In Section 5 the authors discuss a hierarchy of topological defects in parameter space. Fixed points of the G‑action are interpreted as sources (or sinks) of higher Berry curvature. When a symmetry is reduced (e.g., from G to a subgroup H), some defects may split or disappear, giving rise to new invariants associated with H. This hierarchical viewpoint unifies the description of monopole‑like Berry sources, domain‑wall‑type defects, and higher‑dimensional gerbe defects within a single cohomological framework.

Conclusion and outlook
The work provides a concrete, gauge‑invariant, and numerically implementable method for extracting higher‑order Berry invariants from families of 1D spin chains with symmetry. By marrying simplicial cohomology with the MPS formalism, the authors bridge abstract topological field‑theoretic concepts (gerbes, Dixmier–Douady classes) and practical many‑body calculations. Future directions mentioned include extending the approach to non‑translationally invariant systems, to higher spatial dimensions, and to non‑injective (symmetry‑broken) MPS, as well as exploring connections to anomaly inflow and bulk‑boundary correspondence in higher‑dimensional SPT phases.

Overall, the paper makes a significant contribution to the toolbox for diagnosing symmetry‑protected topological phases and phase transitions in low‑dimensional quantum systems, offering both analytical insight and a clear path toward numerical verification.