Disclinations, dislocations, and emanant flux at Dirac criticality

What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers which 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 in the continuum to a certain ``emanant’’ quantized magnetic flux. 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 anti-particles 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 $(C_M)^M = +1$.

💡 Research Summary

The paper develops a comprehensive framework for understanding how crystalline defects—disclinations (rotational defects) and dislocations (translational defects)—affect (2+1)‑dimensional Dirac fermion critical points. The central construct is a group homomorphism ρ: G_UV → G_IR that maps the microscopic lattice symmetry group G_UV (including translations, rotations, reflections) to the symmetry group G_IR of the low‑energy infrared (IR) quantum field theory (QFT). For each Dirac fermion ψ_o, diagonalizing the lattice rotation operator C_M yields an eigenvalue (C_M)_o. Depending on whether (C_M)_o^M = +1 or (−1)^F (F is the fermion number), the associated angular momentum quantum number s_o is quantized as a half‑integer or integer modulo M. These s_o constitute the “topological quantum numbers” that parameterize ρ.

Using ρ, the authors show that a lattice disclination of angle 2π/M is represented in the continuum by a conical geometry of the same angle together with a U(1) holonomy (magnetic flux) Φ = 2π s_o/M for each Dirac fermion. Similarly, a dislocation with Burgers vector b induces a U(1) holonomy Φ = p·b, where p is the momentum of the Dirac point in the basis that diagonalizes the translation operator. Crucially, this flux exists even when no external magnetic field is applied; the authors term it “emanant magnetic flux”.

The emanant flux has observable consequences. In the Dirac theory, adiabatically turning on a localized flux creates electron‑hole pairs from the vacuum, which then circulate around the defect. This leads to a universal azimuthal current j_θ ∝ Φ sin θ and an energy‑density shift ε ∝ Φ², both of which are independent of microscopic details. At half‑flux (Φ = π, i.e., π‑mod‑2π), the defect supports an exactly marginal boundary operator, giving rise to a one‑parameter family of infrared fixed points—a “defect conformal manifold”. Along this manifold the current varies with the marginal coupling while the energy density remains constant. This phenomenon is specific to Dirac criticality and does not appear in free boson theories.

To substantiate the theory, the authors perform extensive numerical simulations on two paradigmatic lattice models: the Qi‑Wu‑Zhang (QWZ) Chern‑insulator model (which hosts two Dirac cones) and the Haldane model (a single Dirac cone). By inserting a disclination into the lattice and threading an auxiliary flux δα at the rotation center, they measure the ground‑state current and energy density as functions of δα. The data show a systematic offset between the clean lattice and the defect lattice, precisely matching the predicted emanant flux Φ = 2π s_o/M. Different on‑site potential patterns (uniform, staggered on one site, staggered on two sites) are explored, confirming that the shift is governed solely by the topological quantum numbers s_o and not by microscopic details. Dislocation simulations similarly reveal flux proportional to the Burgers vector and Dirac momentum.

A notable theoretical result is a no‑go theorem: a single (2+1)‑dimensional Dirac cone in symmetry class AII (time‑reversal with T² = −1) cannot coexist with a commuting M‑fold rotation C_M satisfying (C_M)^M = +1. The proof follows from the incompatibility of the required half‑integer s_o (imposed by the spin‑½ nature of the Dirac fermion) with the constraint (C_M)^M = +1. This extends the fermionic crystalline equivalence principle and imposes new constraints on symmetry‑protected topological classifications.

In summary, the work provides: (i) a clear UV‑IR mapping of crystalline symmetry defects to continuum gauge holonomies; (ii) a universal description of the emanant magnetic flux and its physical signatures (pair creation, current, energy density); (iii) identification of a defect conformal manifold at half‑flux; (iv) numerical verification in concrete lattice models; and (v) a rigorous no‑go constraint on single Dirac cones with certain rotational symmetries. The framework is applicable beyond free fermions, offering a route to analyze interacting critical points enriched by crystalline symmetries.