Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs

A Weyl semimetal is a new state of matter that hosts Weyl fermions as emergent quasiparticles and admits a topological classification that protects Fermi arc surface states on the boundary of a bulk sample. This unusual electronic structure has deep analogies with particle physics and leads to unique topological properties. We report the experimental discovery of a Weyl semimetal, tantalum arsenide (TaAs). Using photoemission spectroscopy, we directly observe Fermi arcs on the surface, as well as the Weyl fermion cones and Weyl nodes in the bulk of TaAs single crystals. We find that Fermi arcs terminate on the Weyl fermion nodes, consistent with their topological character. Our work opens the field for the experimental study of Weyl fermions in physics and materials science.

https://science.sciencemag.org/content/sci/early/2015/07/15/science.aaa9297.full.pdf

Discovery of a Weyl fermion semimetal and topological Fermi arcs

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

Maximally-localized Wannier Functions: Theory and Applications

The intersections of topology, geometry and strong correlations offer many opportunities for exotic quantum phases to emerge in condensed matter systems. Weyl fermions, in particular, provide an ideal platform for exploring the dynamical instabilities of single-particle physics under interactions. Despite its fundamental role in relativistic field theory, the concept of causality and the associated spacetime light cone and event horizon has not been considered in connection with interacting Weyl fermionic excitations in quantum matter. Here, by using charge-density wave (CDW) as an example, we unveil the behavior of interacting Weyl fermions and show that a Weyl fermion in a system can open a band gap by interacting only with other Weyl fermions that lie within its energy-momentum dispersion cone. In this sense, causal connections or interactions are only possible within overlapping dispersion cones and each dispersion cone thus constitutes a solid-state analogue of the more conventional ‘event horizon’ of high-energy physics. Our study provides a universal framework for considering interacting relativistic quasiparticles in condensed matter by separating them into energy-like and momentum-like relationships in analogy with the time-like and space-like events in high-energy physics. Finally, we consider two different candidate materials for hosting the Weyl CDW phase: (TaSe4)2I and Mo3Al2C. Our study greatly enriches the phenomenology and unveils new connections between condensed matter and high-energy physics.

Relativistic horizon of interacting Weyl fermions in condensed matter systems

A way to represent the band structure that distinguishes between energy-momentum and energy-crystal momentum relationships is proposed upon the band-unfolding concept. This momentum-resolved band structure offers better understanding of the physical processes requiring the information of wave functions in momentum space and provides a direct connection to angle-resolved photoelectron spectroscopy (ARPES) spectra. Following this approach, we demonstrate that Dirac cones in graphene are intrinsically broken in momentum space and can be described by a conceptual unit cell smaller than the primitive unit cell. This hidden degree of freedom can be measured by ARPES experiments as missing weight that is retrievable by investigating the effect of different polarized light. Having the energy-momentum relationship, we provide alternative understanding of the retrieved momentum intensity beyond the periodic-zone scheme, that is, the retrieved momentum intensity is assisted with the properties of final states, not from the Dirac cones directly. The revealed broken Dirac cones and momenta supplied by the lattice give interesting ingredients for designing advanced nanodevices.

https://iopscience.iop.org/article/10.1088/1361-648X/aacac2/pdf

Realization of intrinsically broken Dirac cones in graphene via the momentum-resolved electronic band structure.

: The formation of a charge density wave (CDW) in two-dimensional (2D) materials caused by Peierls instability is a controversial topic. This study investigates the extensively debated role of Fermi surface nesting in causing the CDW state in 2H-NbSe 2 materials. Four NbSe 2 structures (i.e., normal, stripe, filled, and hollow structures) are identified on the basis of the characteristics in scanning tunneling microscopy images and first-principles simulations. The calculations reveal that the filled phase corresponds to Peierls’ description; that is, it exhibits fully opened gaps at the CDW Brillouin zone boundary, resulting in a drop at the Fermi level in the density of states and the scanning tunneling spectroscopy spectra. The electronic susceptibility and phonon instability in the normal phase indicate that the Fermi surface nesting is triggered by two nesting vectors, whereas the involvement of only one nesting vector leads to the stripe phase. This comprehensive study demonstrates that the filled phase of NbSe 2 can be categorized as a Peierls-instability-induced CDW in 2D systems.

Revealing the Charge Density Wave caused by Peierls instability in two-dimensional NbSe$_{2}$

Charge density wave (CDW) resulted from a small distortion in the lattice is able to create new orders beyond the original lattice. In 2H-NbSe2, one of the layered transition metal dichalcogenides (TMD), the 3x3 charge order appears in two-dimensional (2D) layers. Although CDW is usually described by a sine wave, the spatial distribution within a 2D layer has never been systematically visualized. Here by using scanning tunneling microscopy (STM) and density functional theory (DFT), we have monitored the evolution of 3x3 CDW along c-axis and realized a nearly tomography scan of CDW of the topmost layer. The results show that the strength of 3x3 charge order varies while increasing the tunneling current. The 3x3 charge order is relatively strong at the outermost Se level and decreases while probing in between Se and Nb levels. Interestingly, the 3x3 charge order gets strong again as reaching Nb level but along with a phase shift. We further calculated the orbital charge distributions and found that both CDW intensity modulation and phase shift are strongly correlated with the distribution of Se p orbitals and Nb d orbitals.

Tomography Scan of Charge Density Wave in NbSe2

Chi-Cheng Lee (李啟正), 2 Xiaoqian M. Chen (陈小千), Chen-Lin Yeh (葉 承霖), 4 H. C. Hsueh (薛宏中), Peter Abbamonte, and Wei Ku (顧威) 5 Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, IL 61801, USA Department of Physics, Tamkang University, Tamsui, Taipei 25137, Taiwan Physics Department, State University of New York, Stony Brook, New York 11790, USA (Dated: May 21, 2012)

Chi-Cheng Lee

Papers

Relativistic horizon of interacting Weyl fermions in condensed matter systems

Realization of intrinsically broken Dirac cones in graphene via the momentum-resolved electronic band structure.

Revealing the Charge Density Wave caused by Peierls instability in two-dimensional NbSe$_{2}$

Tomography Scan of Charge Density Wave in NbSe2

Propagation of tightly bound excitons: a simple rst-principles method of exciton kinetics