Crystalline topological phases as defect networks

TL;DR

This work develops a universal defect-network framework for crystalline topological phases with spatial symmetry $G$ and internal symmetry $G_{ ext{int}}$, unifying symmetry-protected and symmetry-enriched theories across bosons and fermions. It connects a physically intuitive defect-network picture with a rigorous mathematical construction grounded in maps $f: X//G \to \Theta_d$ and generalized cohomology, proving an equivalence with the Thorngren–Else classification and the Crystalline Equivalence Principle (up to twists). The authors introduce smooth states as a dual viewpoint and show how defects emerge by sharpening smooth states, with a Thom isomorphism underpinning the relation between anomalies and higher-dimensional boundaries. A structured framework via spectral sequences is proposed to compute crystalline classifications, including invertible and non-invertible phases, and illustrated in two dimensions with translation symmetry. The results provide a unifying, calculable language for crystalline topological phases and open directions toward fracton-like non-smoothable defect networks and broader generalized cohomology approaches.

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.

Figures (10)

• Figure 1: A defect network in 2-D consists of a bulk topological phase, 1-D defects, and 0-D defect junctions (of course, one can also consider the case where the 1-D defects are trivial, in which case a "0-D defect junction" is just a point defect). In higher dimensions, one can also have higher-order junctions.
• Figure 2: A (portion of) a cell decomposition of a 2-dimensional manifold $X$. In the defect network picture, the 2-cells will carry a 2-D topological phase, the 1-cells will carry 1-D defects, and the 0-cells will carry junctions between 1-D defects.
• Figure 3: A 1-D defect (in this case, a 1-D SPT) inducing an anomaly on a 0-cell
• Figure 4: A $C_2$ symmetric $p+ip$ superconductor in 2-D (with $R^2 = 1$) induces an "anomaly", i.e. a Majorana zero mode, two dimensions lower.
• Figure 5: A $k$-cLU can be written as the product of a strict $k$-cLU and a $k-1$-cLU

Theorems & Definitions (12)

• Conjecture 1
• Conjecture 2
• Corollary 1: Crystalline Equivalence Principle
• Conjecture 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof