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arXiv:2501.13866 · Physical Review X

Disclinations, Dislocations, and Emanant Flux at Dirac Criticality

Maissam Barkeshli, Christopher Fechisin, Zohar Komargodski, Siwei Zhong

Published Dec 8, 2025• 2 citations
arXivHTMLPDF

Abstract

What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers that specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations—defects of crystalline symmetries—generally lead to a certain “emanant” quantized magnetic flux in the continuum. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are ( 2 + 1 ) D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and antiparticles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects, and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single ( 2 + 1 ) D Dirac cone in symmetry class AII is incompatible with a commuting C M rotational symmetry with ( C M ) M = + 1 .

DOI
10.1103/kfd3-qtk7

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What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers that specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations—defects of crystalline symmetries—generally lead to a certain “emanant” quantized magnetic flux in the continuum. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are ( 2 + 1 ) D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and antiparticles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects, and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single ( 2 + 1 ) D Dirac cone in symmetry class AII is incompatible with a commuting C M rotational symmetry with ( C M ) M = + 1 .
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Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.