Higher-order bulk-boundary correspondence for topological crystalline phases (1805.02598v2)

Abstract: We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order topological phases with an additional order-two symmetry or antisymmetry is contained in this work.

• The paper introduces an algebraic framework linking bulk topological classes to higher-order boundary states using novel homomorphisms.
• It classifies crystalline phases by decomposing bulk groups into subgroups that directly correlate with observable boundary phenomena.
• The study guides the search for second-order topological insulators, impacting material science, quantum computing, and condensed matter research.

Higher-order Bulk-boundary Correspondence for Topological Crystalline Phases

The paper "Higher-order bulk-boundary correspondence for topological crystalline phases", authored by Luka Trifunovic and Piet W. Brouwer, addresses the classification of topological crystalline phases that are distinguished by an order-two (anti)symmetry. The research provides a comprehensive framework for understanding the relation between higher-order topological phases and their boundary states in the presence of crystalline symmetries. The focus is on developing a formal bulk-boundary correspondence that extends beyond the conventional first-order phases.

Overview

Central to this paper is the concept of topological insulators and superconductors, which exhibit robust boundary states protected by the topology of their bulk band structure. For phases protected by nonlocal crystalline symmetries, this protection is contingent upon the boundary respecting the symmetry. The research expands on this by including higher-order topological phases, which may manifest with boundary states of codimension less than the system's dimension, denoted by $d - 1$.

Key Results

The authors introduce a subgroup sequence of bulk classifying groups which resolve the phases according to their boundary signature, ranging from those without protected boundary states to those with higher-order boundary states. This classification uses a framework that relates these bulk classifying groups to the configuration of boundary states at various codimensions, harnessing the isomorphism properties of the involved homomorphisms.

Numerically, the formulation of the bulk-boundary correspondence derives from an algebraic approach. The authors detail a homomorphism $ω$ that is employed to relate bulk classification across different dimensions—a novel approach that utilizes dimension-raising isomorphisms $κ$ and $ρ$, and a boundary map $δ$. For a specified lower-order phase $n$, the paper establishes the correspondence $K_i^{(n+1)} = K^{(n)}/K^{(n+1)}$, showcasing how higher-order boundary states correlate with properties of crystallographic symmetries.

Implications and Future Directions

These findings are significant for the theoretical classification of crystalline phases with potentially no straightforward boundary phenomena, elucidating circumstances where nontrivial topologies might lack topologically protected boundary states. The implications of this work can extend into condensed matter systems where symmetry properties play a crucial role.

Practically, enriched classification can guide the search for implementations in materials where higher-order topologies manifest, such as in predicting material candidates for second-order topological insulators. The findings will impact fields exploring quantum computing and materials science by highlighting new pathways to manipulate and utilize topological properties.

Future research scope includes expanding this framework to non-order-two symmetries and scrutinizing the observable signatures of phases with nontrivial bulk topology yet without protected boundary states, potentially integrating answering how weak invariants might synergize with the introduced formalism.

Overall, this paper constructs a robust algebraic foundation that considers both bulk topology and boundary phenomena of topological crystalline phases within an expansive and rigorous mathematical structure, providing a profound contribution to the understanding of higher-order topological phenomena.