Fundamental research in condensed matter physics has driven tremendous advances in modern electronic capabilities. Transistors, optical fiber, LEDs, magnetic storage media, plasma displays, semiconductors, superconductors – the list of technologies born of fundamental research in condensed matter physics is staggering. Scientists working in this field continue to explore and discover surprising novel phenomena that hold promise for tomorrow's technological advances.

An important line of inquiry in this field involves topology – a mathematical framework for describing surface states that remain stable even when the material is deformed by stretching or twisting. The inherent stability of topological surface states has implications for a range of applications in electronics and spintronics.

Now, a team of researchers from the US and China has discovered an exotic new form of topological state in a large class of three-dimensional (3D) semi-metallic crystals called Dirac semimetals. The researchers developed extensive mathematical machinery to bridge the gap between theoretical models containing forms of ‘higher-order’ topology (topology that manifests only at the boundary of a boundary) and the physical behavior of electrons in real materials. They report their findings in a paper in *Nature Communications*.

Over the past decade, Dirac and Weyl fermions have been predicted and experimentally confirmed in a number of solid-state materials, most notably in crystalline tantalum arsenide (TaAs), the first topological Weyl fermion semimetal to be discovered. Several researchers observed that TaAs exhibits two-dimensional (2D) topological surface states known as ‘Fermi arcs’. But similar phenomena observed in Dirac fermion semimetals have eluded understanding, until now.

In the context of semimetals, a Fermi arc is a surface state that behaves like one-half of a 2D metal; the other half is found on a different surface.

"This is not something that's possible in a purely 2D system, and can only happen as a function of the topological nature of a crystal," says team member Barry Bradlyn, professor of physics at the University of Illinois at Urbana-Champaign. "In this work, we found that the Fermi arcs are confined to the 1D hinges in Dirac semimetals."

In earlier work, certain members of this research team, including Xi Dai from Hong Kong University of Science and Technology and Andrei Bernevig from Princeton University, experimentally demonstrated that the 2D surfaces of Weyl semimetals must host Fermi arcs, regardless of the details of the surface. This is a topological consequence of the Weyl points (fermions) present deep within the bulk of the crystal.

"Weyl semimetals have layers like onions," notes Dai. "It's remarkable that you can keep peeling the surface of TaAs, but the arcs are always there."

Researchers have also observed arc-like surface states in Dirac semimetals, but attempts to develop a similar mathematical relationship between such surface states and Dirac fermions in the bulk of the material have been unsuccessful. It became clear that the Dirac surface states arise from a different, unrelated mechanism, and it was concluded the Dirac surface states were not topologically protected.

In the current study, the researchers were surprised to encounter Dirac fermions that appeared to exhibit topologically protected surface states, contradicting this conclusion. Working on models of Dirac semimetals derived from topological quadrupole insulators – higher-order topological systems recently discovered by Bernevig in collaboration with Taylor Hughes from the University of Illinois – they found that this new class of materials exhibits robust, conducting electronic states in 1D, or two fewer dimensions than the bulk 3D Dirac points.

Initially confounded by the mechanism through which these ‘hinge’ states appeared, the researchers worked to develop an extensive, exactly solvable model for the bound states of topological quadrupoles and Dirac semimetals. They found that, in Dirac semimetals, Fermi arcs are generated by a different mechanism than the arcs in Weyl semimetals.

"In addition to settling the decades-old problem of whether condensed matter Dirac fermions have topological surface states," says team member Benjamin Wieder, a postdoctoral researcher at Princeton University, "we demonstrated that Dirac semimetals represent one of the first-solid state materials hosting signatures of topological quadrupoles."

"Unlike Weyl semimetals, whose surface states are cousins of the surfaces of topological insulators, we have shown that Dirac semimetals can host surface states that are cousins of the corner states of higher-order topological insulators," says Bradlyn.

The team discovered that almost all condensed matter Dirac semimetals should exhibit hinge states. "Our work provides a physically observable signature of the topological nature of Dirac fermions, which was previously ambiguous," notes team member Jennifer Cano, a professor of physics at the State University of New York at Stony Brook.

"It's clear that numerous previously studied Dirac semimetals actually do have topological boundary states, if one looks in the right place," Bradlyn adds.

Through first-principles calculations, the researchers theoretically demonstrated the existence of overlooked hinge states on the edges of known Dirac semimetals, including the prototypical material, cadmium arsenide (Cd_{3}As_{2}).

"With an amazing team combining skills from theoretical physics, first-principles calculations and chemistry, we were able to demonstrate the connection between higher-order topology in two dimensions and Dirac semimetals in three dimensions, for the first time," says Bernevig.

The team's findings have implications for the development of new technologies, including in spintronics, because the hinge states can be converted into edge states whose direction of propagation is tied to their spin, much like the edge states of a 2D topological insulator. Additionally, nanorods of higher-order topological semimetals could realize topological superconductivity on their surfaces when placed in close proximity to conventional superconductors. This could potentially realize multiple Majorana fermions, which have been proposed as ingredients for achieving fault-tolerant quantum computation.

*This story is adapted from material from the University of Illinois at Urbana-Champaign, with editorial changes made by Materials Today. The views expressed in this article do not necessarily represent those of Elsevier. Link to original source.*