Theoretical Design for a Novel Type of Dirac Fermion in Solids

The relativistic equation for fermions (the Dirac equation) serves as the fundamental equation in the context of high energy physics. Recently, it has been widely recognized that the band structures of electrons in solids often demonstrate energy dispersion that is similar to that of Dirac fermions, which are protected by space-group symmetries. Dirac fermions are characterized by the geometry and topology of the electronic states represented by the Berry phase of Bloch wavefunctions [1]. Interestingly, they also feature a variety of modifications and consequent novel physical attributes that can be directly measured by conducting table-top experiments. Hence, the theoretical design and experimental quest for Dirac fermions in solids are currently very popular research topics in the field of condensed matter physics. Representative examples include two-dimensional (2D) graphene with Dirac fermions at the corner of the first Brillouin zone [2] and the surface state of a three-dimensional topological insulator [3].

Depending on the tilting of the dispersion, there are three types of Dirac fermions in solids, as depicted in Fig. 1(a) for 2D systems [4]. When the tilting is smaller than the velocity, the Fermi surface becomes a point (a-1, type-I). However, when the tilting is greater than the velocity, the Fermi surface becomes lines (a-2, type-II). Type-III corresponds to the critical point between type-I and type-II, wherein the Fermi surface becomes a line with the vanishing Fermi volume (a-3, type-III). Since it is the critical point, usually certain parameters have to be fine-tuned for realizing type-III Dirac fermions in solids. Although such realization of type-III Dirac fermions can be difficult, it may reveal some interesting characteristics such as an enhanced superconducting gap [5].

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Fig. 1. (a) Three types of Dirac fermions. a-1: type-I, a-2: type-II, a-3: type-III. The lower panels depict the corresponding Fermi surfaces. This figure has been taken from Fig. 1 of Ref. 4. (b) Method schematic for realizing type-III Dirac fermions. (c) Application of the method for the 2D SSH model. (b) and (c) are taken from Figs. 1 and 4 of Ref. 6, respectively.

Recently, Mizoguchi and Hatsugai [6] proposed a method for designing type-III Dirac fermions in solids in a systematic manner. A remarkable feature of this method is that the associated parameters do not need to be fine-tuned, and it guarantees the existence of type-III Dirac fermions over a continuous range of the parameters. The corresponding underlying concept is schematically illustrated in Fig. 1(b). First, they constructed the degenerate directionally flat bands (left panel). Here, they employed the method of molecular orbital representation for reducing the rank of the Hamiltonian matrix, which corresponds to the existence of zero-valued eigenvalues. Next, they introduced the additional Hamiltonian, which provides the dispersion to one of the bands to obtain the type-III Dirac fermions. An example of this method, the 2D Su–Schrieffer–Heeger (SSH) model, is shown in Fig. 1(c). Panel c-1 represents the square lattice structure of the model, and the different colors on the bonds correspond to different transfer integrals. The consequent band structure is depicted in Panel c-2, wherein the transfer integrals represented by solid purple lines (t_2x), and the broken purple line (t_2y) are zero. One can see the doubly degenerate flat band along some k-direction. Panel c-3 represents the band structure with added attributes, namely t_2x = 0.2 and t_2y = 0.1, which gives the dispersion to one of the flat bands, whereby type-III Dirac fermions are realized.

Zn₂In₂S₅ is considered as a candidate material for realizing type-III Dirac fermions [7]. However, details of Zn₂In₂S₅ electronic structure should be carefully considered, and it may be difficult to realize the perfectly flat dispersion. Furthermore, artificial systems such as mechanical systems, electric circuits, photonic crystals, and phononic crystals are cleaner, and the proposed design principle will thereby open a new path for topological physics in association with condensed matter physics.

References

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