Jennifer Cano

Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals

INTRODUCTION Condensed-matter systems have recently become a fertile ground for the discovery of fermionic particles and phenomena predicted in high-energy physics; examples include Majorana fermions, as well as Dirac and Weyl semimetals. However, fermions in condensed-matter systems are not constrained by Poincare symmetry. Instead, they must only respect the crystal symmetry of one of the 230 space groups. Hence, there is the potential to find and classify free fermionic excitations in solid-state systems that have no high-energy counterparts. RATIONALE The guiding principle of our classification is to find irreducible representations of the little group of lattice symmetries at high-symmetry points in the Brillouin zone (BZ) for each of the 230 space groups (SGs), the dimension of which corresponds to the number of bands that meet at the high-symmetry point. Because we are interested in systems with spin-orbit coupling, we considered only the double-valued representations, where a 2π rotation gives a minus sign. Furthermore, we considered systems with time-reversal symmetry that squares to –1. For each unconventional representation, we computed the low-energy k · p Hamiltonian near the band crossings by writing down all terms allowed by the crystal symmetry. This allows us to further differentiate the band crossings by the degeneracy along lines and planes that emanate from the high-symmetry point, and also to compute topological invariants. For point degeneracies, we computed the monopole charge of the band-crossing; for line nodes, we computed the Berry phase of loops encircling the nodes. RESULTS We found that three space groups exhibit symmetry-protected three-band crossings. In two cases, this results in a threefold degenerate point node, whereas the third case results in a line node away from the high-symmetry point. These crossings are required to have a nonzero Chern number and hence display surface Fermi arcs. However, upon applying a magnetic field, they have an unusual Landau level structure, which distinguishes them from single and double Weyl points. Under the action of spatial symmetries, these fermions transform as spin-1 particles, as a consequence of the interplay between nonsymmorphic space group symmetries and spin. Additionally, we found that six space groups can host sixfold degeneracies. Two of these consist of two threefold degeneracies with opposite chirality, forced to be degenerate by the combination of time reversal and inversion symmetry, and can be described as “sixfold Dirac points.” The other four are distinct. Furthermore, seven space groups can host eightfold degeneracies. In two cases, the eightfold degeneracies are required; all bands come in groups of eight that cross at a particular point in the BZ. These two cases also exhibit fourfold degenerate line nodes, from which other semimetals can be derived: By adding strain or a magnetic field, these line nodes split into Weyl, Dirac, or line node semimetals. For all the three-, six- and eight-band crossings, nonsymmorphic symmetries play a crucial role in protecting the band crossing. Last, we found that seven space groups may host fourfold degenerate “spin-3/2” fermions at high symmetry points. Like their spin-1 counterparts, these quasiparticles host Fermi surfaces with nonzero Chern number. Unlike the other cases we considered, however, these fermions can be stabilized by both symmorphic and nonsymmorphic symmetries. Three space groups that host these excitations also host unconventional fermions at other points in the BZ. We propose nearly 40 candidate materials that realize each type of fermion near the Fermi level, as verified with ab initio calculations. Seventeen of these have been previously synthesized in single-crystal form, whereas others have been reported in powder form. CONCLUSION We have analyzed all types of fermions that can occur in spin-orbit coupled crystals with time-reversal symmetry and explored their topological properties. We found that there are several distinct types of such unconventional excitations, which are differentiated by their degeneracies at and along high-symmetry points, lines, and surfaces. We found natural generalizations of Weyl points: three- and four-band crossings described by a simple k · S Hamiltonian, where Si is the set of spin generators in either the spin-1 or spin-3/2 representations. These points carry a Chern number and, consequently, can exhibit Fermi arc surface states. We also found excitations with six- and eightfold degeneracies. These higher-band crossings create a tunable platform to realize topological semimetals by applying an external magnetic field or strain to the fourfold degenerate line nodes. Last, we propose realizations for each species of fermion in known materials, many of which are known to exist in single-crystal form. Fermi arcs from a threefold degeneracy. Shown is the surface density of states as a function of momentum for a crystal in SG 214 with bulk threefold degeneracies that project to (0.25, 0.25) and (–0.25, –0.25). Two Fermi arcs emanate from these points, indicating that their monopole charge is 2. The arcs then merge with the surface projection of bulk states near the origin. In quantum field theory, we learn that fermions come in three varieties: Majorana, Weyl, and Dirac. Here, we show that in solid-state systems this classification is incomplete, and we find several additional types of crystal symmetry–protected free fermionic excitations. We exhaustively classify linear and quadratic three-, six-, and eight-band crossings stabilized by space group symmetries in solid-state systems with spin-orbit coupling and time-reversal symmetry. Several distinct types of fermions arise, differentiated by their degeneracies at and along high-symmetry points, lines, and surfaces. Some notable consequences of these fermions are the presence of Fermi arcs in non-Weyl systems and the existence of Dirac lines. Ab initio calculations identify a number of materials that realize these exotic fermions close to the Fermi level.

Topological quantum chemistry

Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (this http URL). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPATIBILITY RELATIONS provides compatibility relations between the irreducible representations of double space groups at different k-vectors of the Brillouin zone when there is a group-subgroup relation between the corresponding little groups. The program DSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description and the search of topological insulators.

Bulk-Edge Correspondence in 2+1-Dimensional Abelian Topological Phases

The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at

Chiral anomaly factory: Creating Weyl fermions with a magnetic field

\nu=8

Interactions along an entanglement cut in 2+1 D Abelian topological phases

and 12, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at

Majorana Zero Modes in Semiconductor Nanowires in Contact with Higher-Tc Superconductors

\nu=8/7, 12/11, 8/15, 16/5

Unexpected tunneling current from downstream neutral modes

. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers

Microwave absorption by a mesoscopic quantum Hall droplet

p

Graph theory data for topological quantum chemistry

in which every quasiparticle has a topological spin that is a