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arXiv:1808.08965 · Physical review. B./Physical review. B

Higher-order topological phases: A general principle of construction

Dumitru Călugăru, Vladimir Juričić, Bitan Roy

Published Jan 4, 2019• 215 citations
arXivHTMLPDF

Abstract

The surface states of a $d$-dimensional conventional or first-order topological state of matter reside on a ($d$-1) dimensional boundary. However, its $n$th order incarnation, commonly known as a higher-order topological (HOT) phase, accommodates ($d$-$n$) dimensional boundary states, with corner and hinge modes standing as their representatives. Here, the authors introduce a general principle of systematically constructing the hierarchy of HOT phases by exploiting the symmetry of the system and the corresponding Clifford algebra. This procedure is applicable for insulating as well as gapless phases, and paves the route toward realizing a fourth-order nodal-loop semimetal, devoid of any surface bound state.

DOI
10.1103/physrevb.99.041301

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The surface states of a ddd-dimensional conventional or first-order topological state of matter reside on a (ddd-1) dimensional boundary. However, its nnnth order incarnation, commonly known as a higher-order topological (HOT) phase, accommodates (ddd-nnn) dimensional boundary states, with corner and hinge modes standing as their representatives. Here, the authors introduce a general principle of systematically constructing the hierarchy of HOT phases by exploiting the symmetry of the system and the corresponding Clifford algebra. This procedure is applicable for insulating as well as gapless phases, and paves the route toward realizing a fourth-order nodal-loop semimetal, devoid of any surface bound state.
A wide variety of higher-order symmetry protected topological phase(HOSPT) with gapless corners or hinges had been proposed as a descendant of topological crystalline insulator protected by spatial symmetry. In this work, we address a new class of higher-order topological state which does not require crystalline symmetries but instead relies on subsystem symmetry for protection. We propose several strong interacting models with gapless hinge or corner based on a `decorated hinge-wall condensate' picture. The hinge-wall, which appears as the defect configuration of Z2Z_2Z2​ paramagnet is decorated with lower-dim SPT state. Such unique hinge-wall decoration structure leads to gapped surfaces separated by gapless hinges. The non-trivial nature of the hinge modes can be captured by a 1+11+11+1D conformal field theory with a Wess-Zumino-Witten term. Besides, we establish a no-go theorem to demonstrate the ungappable nature of the hinge by making a connection between generalized Lieb-Schultz-Mattis theorem and the boundary anomaly of HOSPT state. This universal correspondence engenders a comprehensive criterion to determine the existence of HOSPT under certain symmetry regardless of the microscopic Hamiltonian.
Abstract A wide variety of higher-order symmetry-protected topological phases (HOSPT) with gapless corners or hinges have been proposed as descendants of topological crystalline insulators protected by spatial symmetry. In this work, we address a new class of higher-order topological states that do not require crystalline symmetries but instead rely on subsystem symmetry for protection. We propose several strongly interacting models with gapless hinges or corners based on a decorated hinge-wall condensate picture. The hinge-wall, which appears as the defect configuration of a Z 2 paramagnet, is decorated with a lower-dimensional SPT state. Such a unique hinge-wall decoration structure leads to gapped surfaces separated by gapless hinges. The non-trivial nature of the hinge modes can be captured by a 1 + 1 D conformal field theory with a Wess–Zumino–Witten term. Moreover, we establish a no-go theorem to demonstrate the ungappable nature of the hinges by making a connection between the generalized Lieb–Schultz–Mattis theorem and the boundary anomaly of the HOSPT state. This universal correspondence engenders a comprehensive criterion to determine the existence of HOSPT under certain symmetries, regardless of the microscopic Hamiltonian.
Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in 2+12+12+1 dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a dthd^{th}dth-order SPT phase in d+1d+1d+1 dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in 3+13+13+1 dimensions exhibiting hinge modes protected by a noninvertible symmetry.
We devise a generic recipe for constructing DDD-dimensional lattice models whose ddd-dimensional boundary states, located on surfaces, hinges, corners, and so forth, can be obtained exactly. The solvability is rooted in the underlying lattice structure and as such does not depend on fine tuning, allowing us to track their evolution throughout various phases and across phase transitions. Most saliently, our models provide ``boundary solvable'' examples of the recently introduced higher-order topological phases. We apply our general approach to breathing and anisotropic kagome and pyrochlore lattices for which we obtain exact corner eigenstates, and to periodically driven two-dimensional models as well as to three-dimensional lattices where we present exact solutions corresponding to one-dimensional chiral states at the hinges of the lattice. We relate the higher-order topological nature of these models to reflection symmetries in combination with their provenance from lower-dimensional conventional topological phases.
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.