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 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> <a:mi mathvariant="normal">D</a:mi> </a:mrow> </a:math> 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mrow> <f:mo stretchy="false">(</f:mo> <f:mn>2</f:mn> <f:mo>+</f:mo> <f:mn>1</f:mn> <f:mo stretchy="false">)</f:mo> <f:mi mathvariant="normal">D</f:mi> </f:mrow> </f:math> Dirac cone in symmetry class AII is incompatible with a commuting <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>C</k:mi> <k:mi>M</k:mi> </k:msub> </k:math> rotational symmetry with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>M</m:mi> </m:msub> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>M</m:mi> </m:msup> <m:mo>=</m:mo> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:math> .