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:2107.13556 · Physical review. B./Physical review. B

Noncompact atomic insulators

Frank Schindler, Bogdan Andrei Bernevig

Published Nov 29, 2021• 20 citations
arXivHTMLPDF

Abstract

We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically ``trivial'' obstructed atomic insulators. These insulators admit exponentially localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for ${\mathcal{C}}_{2}$ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for ${\mathcal{C}}_{4}$ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a ${\mathcal{C}}_{3}$-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than 18 lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.

DOI
10.1103/physrevb.104.l201114

Related papers

Wannier representation of Floquet topological states

Masaya Nakagawa, Robert-Jan Slager, Sho Higashikawa +1 more

A universal feature of topological insulators is that they cannot be adiabatically connected to an atomic limit, where individual lattice sites are completely decoupled. This property is intimately related to a topological obstruction to constructing a localized Wannier function from Bloch states of an insulator. Here, we generalize this characterization of topological phases toward periodically driven systems. We show that nontrivial connectivity of hybrid Wannier centers in momentum space and time can characterize various types of topology in periodically driven systems, which include Floquet topological insulators, anomalous Floquet topological insulators with micromotion-induced boundary states, and gapless Floquet states realized with topological Floquet operators. In particular, nontrivial time dependence of hybrid Wannier centers indicates impossibility of continuous deformation of a driven system into an undriven insulator, and a topological Floquet operator implies an obstruction to constructing a generalized Wannier function which is localized in real and frequency spaces. Our results pave a way to a unified understanding of topological states in periodically driven systems as a topological obstruction in Floquet states.

arXiv:1903.12197•Physical review. B./Physical review. B•Feb 10, 2020•53 citations

Compactly supported Wannier functions and strictly local projectors

Pratik Sathe, Fenner Harper, Rahul Roy

Abstract Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1 d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.

arXiv:2008.05528•Journal of Physics A Mathematical and Theoretical•Jul 6, 2021•20 citations

Compactly supported Wannier functions and algebraic K -theory

N. Read

In a tight-binding lattice model with $n$ orbitals (single-particle states) per site, Wannier functions are $n$-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactly supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for noninteracting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians---that is, class A in the tenfold classification of Hamiltonians and band structures---a set of compactly supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension $d$ of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a nontrivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the nontrivial bundles that can arise from compactly supported Wannier-type functions are those that may possess, in each of $d$ directions, the nontrivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic $K$-theory, based on a ring of polynomials in ${e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{x}},\phantom{\rule{0.16em}{0ex}}{e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{y}},...,$ which occur as entries in the Fourier-transformed Wannier functions.

arXiv:1608.04696•Physical review. B./Physical review. B•Mar 20, 2017•96 citations

Non-Hermitian topological Euler insulators

Longwen Zhou

Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the $PT$ or $C_2T$ symmetry in two dimensions, which go beyond the tenfold $K$-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.

cond-mat.mes-hallCross-listedarXiv:2608.25411•Aug 27, 2026

When Wannier centers jump: Critical points between atomic insulating phases

Yunchao Zhang, T. Senthil

We study a class of quantum phase transitions between featureless bosonic atomic insulators in $(2+1)$ dimensions, where each phase exhibits neither topological order nor protected edge modes. Despite their lack of topology, these insulators may be ``obstructed'' in the sense that their Wannier centers are not pinned to the physical atomic sites. These insulators represent distinct phases, as no symmetry-preserving adiabatic path connects them. Surprisingly, we find that for certain lattices, the critical point between these insulators can host a conformally invariant state described by quantum electrodynamics in $(2+1)$ dimensions (QED$_3$). The emergent electrodynamics at the critical point can be stabilized if the embedding of the microscopic lattice symmetries suppresses the proliferation of monopoles, suggesting that even transitions between trivial phases can harbor rich and unexpected physics. We analyze the mechanism behind this phenomenon, discuss its stability against perturbations, and explore the embedding of lattice symmetries into the continuum through anomaly matching. In all the models we analyze, we confirm that the QED$_3$ is indeed emergeable, in the sense that it is realizable from a local lattice Hamiltonian.

arXiv:2510.15111•Physical review. B./Physical review. B•Mar 23, 2026•0 citations

Wannier Permanents and Featureless Bosonic Mott Insulators on the 1/3 filled Kagome Lattice

S. A. Parameswaran, Itamar Kimchi, Ari M. Turner +2 more

We study Bose-Hubbard models on tight-binding, non-Bravais lattices, with a filling of one boson per unit cell -- and thus fractional site filling. At integer filling of a unit cell neither symmetry breaking nor topological order is required, and in principle a trivial and featureless (i.e., symmetry-unbroken) insulator is allowed. We demonstrate by explicit construction of a family of wavefunctions that such a featureless Mott insulating state exists at 1/3 filling on the kagome lattice, and construct Hamiltonians for which these wavefunctions are exact ground states. We briefly comment on the experimental relevance of our results to cold atoms in optical lattices. Such wavefunctions also yield 1/3 magnetization plateau states for spin models in an applied field. The featureless Mott states we discuss can be generalized to any lattice for which symmetric exponentially localized Wannier orbitals can be found at the requisite filling, and their wavefunction is given by the permanent over all Wannier orbitals.

•Jun 5, 2012•0 citations
We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically ``trivial'' obstructed atomic insulators. These insulators admit exponentially localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for C2{\mathcal{C}}_{2}C2​ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for C4{\mathcal{C}}_{4}C4​ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a C3{\mathcal{C}}_{3}C3​-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than 18 lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.
A universal feature of topological insulators is that they cannot be adiabatically connected to an atomic limit, where individual lattice sites are completely decoupled. This property is intimately related to a topological obstruction to constructing a localized Wannier function from Bloch states of an insulator. Here, we generalize this characterization of topological phases toward periodically driven systems. We show that nontrivial connectivity of hybrid Wannier centers in momentum space and time can characterize various types of topology in periodically driven systems, which include Floquet topological insulators, anomalous Floquet topological insulators with micromotion-induced boundary states, and gapless Floquet states realized with topological Floquet operators. In particular, nontrivial time dependence of hybrid Wannier centers indicates impossibility of continuous deformation of a driven system into an undriven insulator, and a topological Floquet operator implies an obstruction to constructing a generalized Wannier function which is localized in real and frequency spaces. Our results pave a way to a unified understanding of topological states in periodically driven systems as a topological obstruction in Floquet states.
Abstract Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1 d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.
In a tight-binding lattice model with nnn orbitals (single-particle states) per site, Wannier functions are nnn-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactly supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for noninteracting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians---that is, class A in the tenfold classification of Hamiltonians and band structures---a set of compactly supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension ddd of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a nontrivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the nontrivial bundles that can arise from compactly supported Wannier-type functions are those that may possess, in each of ddd directions, the nontrivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic KKK-theory, based on a ring of polynomials in ${e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{x}},\phantom{\rule{0.16em}{0ex}}{e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{y}},...,$ which occur as entries in the Fourier-transformed Wannier functions.
Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the PTPTPT or C2TC_2TC2​T symmetry in two dimensions, which go beyond the tenfold KKK-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.
We study a class of quantum phase transitions between featureless bosonic atomic insulators in (2+1)(2+1)(2+1) dimensions, where each phase exhibits neither topological order nor protected edge modes. Despite their lack of topology, these insulators may be ``obstructed'' in the sense that their Wannier centers are not pinned to the physical atomic sites. These insulators represent distinct phases, as no symmetry-preserving adiabatic path connects them. Surprisingly, we find that for certain lattices, the critical point between these insulators can host a conformally invariant state described by quantum electrodynamics in (2+1)(2+1)(2+1) dimensions (QED3_33​). The emergent electrodynamics at the critical point can be stabilized if the embedding of the microscopic lattice symmetries suppresses the proliferation of monopoles, suggesting that even transitions between trivial phases can harbor rich and unexpected physics. We analyze the mechanism behind this phenomenon, discuss its stability against perturbations, and explore the embedding of lattice symmetries into the continuum through anomaly matching. In all the models we analyze, we confirm that the QED3_33​ is indeed emergeable, in the sense that it is realizable from a local lattice Hamiltonian.
We study Bose-Hubbard models on tight-binding, non-Bravais lattices, with a filling of one boson per unit cell -- and thus fractional site filling. At integer filling of a unit cell neither symmetry breaking nor topological order is required, and in principle a trivial and featureless (i.e., symmetry-unbroken) insulator is allowed. We demonstrate by explicit construction of a family of wavefunctions that such a featureless Mott insulating state exists at 1/3 filling on the kagome lattice, and construct Hamiltonians for which these wavefunctions are exact ground states. We briefly comment on the experimental relevance of our results to cold atoms in optical lattices. Such wavefunctions also yield 1/3 magnetization plateau states for spin models in an applied field. The featureless Mott states we discuss can be generalized to any lattice for which symmetric exponentially localized Wannier orbitals can be found at the requisite filling, and their wavefunction is given by the permanent over all Wannier orbitals.