[Paper Review] Crystalline topological phases as defect networks

Dominic V. Else, Ryan Thorngren|arXiv (Cornell University)|Oct 24, 2018

Topological Materials and Phenomena4 citations

TL;DR

This paper introduces a defect network picture that unifies crystalline topological phases—both SPT and SET phases in bosonic and fermionic systems—by describing them as G-symmetric networks of defects in a G_int-symmetric topological phase. The key contribution is a general framework that establishes equivalence between the defect network picture and the broader classification of Thorngren and Else (2018), proving that crystalline topological phases are classified by the same invariants as internal symmetry phases under the Crystalline Equivalence Principle, even in non-invertible and twisted cases.

ABSTRACT

A crystalline topological phase is a topological phase with spatial symmetries. In this work, we give a very general physical picture of such phases: a topological phase with spatial symmetry $G$ (with internal symmetry $G_{\mathrm{int}} \leq G$) is described by a *defect network*: a $G$-symmetric network of defects in a topological phase with internal symmetry $G_{\mathrm{int}}$. The defect network picture works both for symmetry-protected topological (SPT) and symmetry-enriched topological (SET) phases, in systems of either bosons or fermions. We derive this picture both by physical arguments, and by a mathematical derivation from the general framework of [Thorngren and Else, Phys. Rev. X 8, 011040 (2018)]. In the case of crystalline SPT phases, the defect network picture reduces to a previously studied dimensional reduction picture, thus establishing the equivalence of this picture with the general framework of Thorngren and Else applied to crystalline SPTs.

Motivation & Objective

• To unify competing frameworks for classifying crystalline topological phases, particularly the block state picture and the general framework of Thorngren and Else (2018).
• To extend the block state picture to non-invertible topological phases by introducing the defect network formalism.
• To establish that crystalline topological phases with spatial symmetry G are classified by the same invariants as internal symmetry phases, as per the Crystalline Equivalence Principle.
• To prove the injectivity of the restriction map from G-bundles to G_m-bundles in generalized cohomology, ensuring classification stability under unit cell enlargement.
• To generalize the framework to include internal symmetries, magnetic translations, and twisted cohomology, ensuring broad applicability to fermionic and interacting systems.

Proposed method

• Proposes a defect network picture: a G-symmetric network of defects in a G_int-symmetric topological phase, where G is the full symmetry group including spatial symmetries.
• Derives the defect network picture via physical arguments and a mathematical derivation from the general framework of Thorngren and Else (2018), using spectral sequences and obstruction theory.
• Applies Atiyah-Hirzebruch and Serre spectral sequences to analyze the restriction map between cohomology groups of BG and BG_m, proving injectivity when m is coprime to torsion in cohomology.
• Uses obstruction theory to show that if a map f∘i is homotopic to a constant, then f is also homotopic to a constant, provided m is coprime to torsion in homotopy groups up to dimension l.
• Extends the argument to twisted cohomology and fiber bundles over BG, showing the same injectivity holds under the same coprimality conditions.
• Considers generalizations to non-abelian groups and extensions such as magnetic translations, proving the restriction map remains injective if m is 1 modulo the order of the extension class.

Experimental results

Research questions

• RQ1 Can the block state picture for crystalline topological phases be generalized to non-invertible phases, such as SET phases with anyons?
• RQ2 Is the defect network picture equivalent to the general classification framework of Thorngren and Else (2018) for crystalline SPT and SET phases?
• RQ3 Does the Crystalline Equivalence Principle—that crystalline phases are classified like internal symmetry phases—hold for non-invertible and twisted phases?
• RQ4 Under what conditions is the restriction map from H^n(BG, A) to H^n(BG_m, A) an isomorphism or injection, particularly when m is coprime to torsion in cohomology?
• RQ5 Can the classification framework be extended to include internal symmetries, magnetic translations, and twisted cohomology groups?

Key findings

• The defect network picture provides a unified description of crystalline topological phases, valid for both SPT and SET phases in bosonic and fermionic systems.
• The framework proves that the classification of crystalline topological phases with symmetry G is equivalent to the classification of internal symmetry phases with group G, under the Crystalline Equivalence Principle.
• The restriction map from H^n(BG, A) to H^n(BG_m, A) is an isomorphism when m is coprime to all torsion in A, as shown via Serre spectral sequences and torsion coprimality.
• For twisted cohomology, the same injectivity result holds, provided m is coprime to torsion in the twisted coefficients, extending the framework to time-reversal and orientation-reversing symmetries.
• The argument generalizes to arbitrary generalized cohomology theories, including spin cobordism and TQFTs, with m required to be 1 modulo the product of torsion in π≤n(pt) of the target space.
• The method applies to extensions such as magnetic translations, where enlarging the unit cell by an odd factor preserves the symmetry group, and the restriction map remains injective.