Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1712.07911 · Physical review. B./Physical review. B

Lattice models with exactly solvable topological hinge and corner states

Flore K. Kunst, Guido C. P. van Miert, Emil J. Bergholtz

Published Jun 11, 2018• 147 citations
arXivHTMLPDF

Abstract

We devise a generic recipe for constructing $D$-dimensional lattice models whose $d$-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.

DOI
10.1103/physrevb.97.241405

Related papers

Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension

Flore K. Kunst, Guido C. P. van Miert, Emil J. Bergholtz

We present a generic and systematic approach for constructing $D\ensuremath{-}\text{dimensional}$ lattice models with exactly solvable $d\ensuremath{-}\text{dimensional}$ boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of ``sweet spots'' in the space of possible tight-binding models---the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher order) topological and nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.

arXiv:1809.05293•Physical review. B./Physical review. B•Feb 19, 2019•28 citations

Higher-order topological phases: A general principle of construction

Dumitru Călugăru, Vladimir Juričić, Bitan Roy

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.

arXiv:1808.08965•Physical review. B./Physical review. B•Jan 4, 2019•215 citations

Higher-order topological phase with subsystem symmetries

Yizhi You

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.

New Journal of Physics•Sep 1, 2024•4 citations

Higher-order topological phase without crystalline symmetry

Yizhi You

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 $Z_2$ 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 $1+1$D 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.

arXiv:1908.04299•Aug 12, 2019•8 citations

Higher-order symmetry-protected topological states for interacting bosons and fermions

Yizhi You, Trithep Devakul, F. J. Burnell +1 more

Here, the authors propose a zoology of exotic materials that are gapped in their interior and on their surfaces but support gapless mode at corners or along hinges. Exemplifying a much richer bulk-boundary correspondence, this phenomenology is now termed a higher-order topological phase. The protected corner or hinge modes are a consequence of spatial symmetry, which traps and pins the low-energy modes localized at the high symmetric point.

arXiv:1807.09788•Physical review. B./Physical review. B•Dec 3, 2018•104 citations

Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders

Yohei Fuji, Akira Furusaki

Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.

arXiv:2112.07926•Physical Review Research•Nov 3, 2023•3 citations
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.
We present a generic and systematic approach for constructing $D\ensuremath{-}\text{dimensional}$ lattice models with exactly solvable $d\ensuremath{-}\text{dimensional}$ boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of ``sweet spots'' in the space of possible tight-binding models---the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher order) topological and nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.
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.
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.
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.
Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.