Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals

Abstract

We determine conditions on the filling of electrons in a crystalline lattice to obtain the equivalent of a band insulator-a gapped insulator with neither symmetry breaking nor fractionalized excitations. We allow for strong interactions, which precludes a free particle description. Previous approaches that extend the LiebSchultz-Mattis argument invoked spin conservation in an essential way and cannot be applied to the physically interesting case of spin-orbit coupled systems. Here we introduce two approaches: The first one is an entanglement-based scheme, and the second one studies the system on an appropriate flat "Bieberbach" manifold to obtain the filling conditions for all 230 space groups. These approaches assume only time reversal rather than spin rotation invariance. The results depend crucially on whether the crystal symmetry is symmorphic. Our results clarify when one may infer the existence of an exotic ground state based on the absence of order, and we point out applications to experimentally realized materials. Extensions to new situations involving purely spin models are also mentioned. (1) in 1D, later extended to higher dimensions by Oshikawa (2), Hastings (3), and Affleck (4), holds that if all symmetries remain unbroken, an insulating ground state must be "exotic"-for example, a Luttinger liquid in 1D or a quantum spin liquid in higher dimensions, both of which have fractional "spinon" excitations. These exotic states cannot be represented as simple product states as a consequence of their long-ranged entanglement.This constraint, which we collectively refer to as HastingsOshikawa-Lieb-Schultz-Mattis (HOLSM), has experimental consequences. Indeed, no sign of magnetic or spatial symmetry breaking is observed down to temperatures orders of magnitude below the intrinsic energy scales in certain materials (5), including the quasi-2D Mott insulators κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , β′ Pd (dmit) 2 , and Herbertsmithite ZnCu 3 (OH) 6 Cl 2 , as well as the 3D Mott insulator Na 4 Ir 3 O 8 . Hence if we can apply HOLSM to these systems, a strong case is made for an exotic ground state (assuming that the effects of disorder can be ignored). However, HOLSM invokes spin rotation invariance in an essential way, which is typically broken in real materials due to spin-orbit coupling (SOC). These effects are not small: Herbertsmithite has Dzyaloshinskii-Moriya terms thought to be on the order of 10% of the Heisenberg coupling (6, 7). In the antiferromagnetic hyperkagome compound Na 4 Ir 3 O 8 , the physics are even dominated by SOC effects and charge fluctuation is significant (5). Physically, the only exact symmetries are time reversal (TR) and the crystal symmetries and charge conservation that allow us to define the electron filling. Can HOLSM be extended to this physically more realistic situation?In this work we show that it indeed can, although entirely different theoretical approaches are needed. We introduce two methods that, like the flux-threading arguments of HOLSM, are nonperturbative, but differ from th...

Filling-enforced nonsymmorphic Kondo semimetals in two dimensions

“…This extends previous results for purely electronic systems [24][25][26][27][28] to Kondo lattices. At intermediate J K , glide symmetry is spontaneously broken, leading to a partially Kondo screened insulator (PKSI), where the hybridization between local moments and conduction electrons is modulated within the unit cell while preserving the translational symmetry of the SSL.…”

“…For ν = 4p + 4 [21], both the Fermi volume and the generalized invariant vanish and these constraints do not apply. Although spin-orbit coupling (SOC) is challenging to treat using flux insertion, if we apply existing results [24,26,27] on filling-enforced semimetals to the hybridized bands at these fillings our results remain unchanged if timereversal symmetry is present. There is then also the additional interesting possibility that the PKSI may be a topological Kondo insulator, as topological insulators can emerge naturally from filling-enforced SOC semimetals…”

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