Pfaffian Formalism for Higher-Order Topological Insulators

Abstract

We generalize the Pfaffian formalism, which has been playing an important role in the study of time-reversal invariant topological insulators (TIs), to 3D chiral higher-order topological insulators (HOTIs) protected by the product of four-fold rotational symmetry C 4 and the time-reversal symmetry T. This Pfaffian description reveals a deep and fundamental link between TIs and HOTIs, and allows important conclusions about TIs to be generalized to HOTIs. As examples, we demonstrate in the Letter how to generalize Fu-Kane's parity criterion for TIs to HOTIs, and also present a general method to efficiently compute the Z 2 index of 3D chiral HOTIs without a global gauge.Introduction.-In comparison to the well-studied topological insulators (TIs), which have a gapped d-dimensional bulk and topologically-protected gapless states on its d − 1 dimensional boundaries [1][2][3][4][5][6], the recently proposed higher-order topological insulators (HOTIs) have a similar gapped bulk, but the gapless states emerge at lower dimensions, e.g. the 1D hinge of a 3D insulator . In this Letter, we focus on second-order topological insulators characterized by nontrivial magneto-electric polarization P 3 , e.g., 3D chiral second-order topological insulators (CSOTIs) with gapless chiral hinge states propagating in alternative directions. These second-order TIs have a strong connection to TIs, and in particular, if the time-reversal symmetry T is enforced, 2P 3 recovers the Z 2 index of a TI [30]. If the time-reversal symmetry is broken, 2P 3 still defines a Z 2 topological index, as long as a space inversion, rotoinversion or C n T symmetry is preserved [7][8][9][10][11][12][13][14][15], where C n represents n-fold rotation with n = 2, 4, 6, and this Z 2 index, in the absence of time-reversal symmetry, characterizes a second-order TI. For systems invariant under space-inversion or some rotoinversion, this topological index is fully dictated by high-symmetry points and can be conveniently obtained using the framework of topological quantum chemistry [31] or symmetry indicators [32,33]. However, in general, calculating P 3 requires more sophisticated techniques like the nested Wilson loops [7,17,18,[34][35][36].

Topological Materials Discovery from Crystal Symmetry

“…However in practice, the Kane-Mele Z 2 invariant is exceedingly difficult to compute in realistic models, as it requires a careful treatment of matrix square roots and Pfaffians, as well as the computationally intensive process of determining, storing, and integrating the occupied states at each k point. Though a handful of calculations based on the Kane-Mele Pfaffian invariant have been employed to diagnose the topology of TI and TCI models [162,163], the numerical prediction of TI and TCI phases in real materials without SIs (see Sec. VI) has primarily been accomplished through the equivalent Wilson-loop method introduced in [32-34] (see Sec.…”

Higher-order topological corner states induced solely by onsite potentials with mirror symmetry

“…Since the discovery of quantized multipole insulators, higher-order (HO) topological phases and materials have attracted great interests because of their novel bulkboundary correspondences [1][2][3][4][5][6]. In contrast to conventional (or first-order) topological phases, the topologically protected boundary states of HO topological phase exhibit lower dimensions.…”

Floquet Second-Order Topological Phases in Momentum Space

“…The presence of these unique topological matter is usually guaranteed by the coexistence of crystal and non-spatial symmetries, and their classifications go beyond the tenfold way of first-order topological insulators and superconductors [ 9 , 10 , 11 , 12 ]. Besides great theoretical efforts in the study of higher-order topological insulators [ 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ], superconductors [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 ] and semimetals [ 49 , 50 , 51 , 52 , 53 , 54 ], HOTPs have also been observed in solid state materials [ 55 , 56 ,…”