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Riemann surfaces are geometric constructions in complex analysis that may represent multi-valued holomorphic functions using multiple sheets of the complex plane. We show that the energy dispersion of surface states in topological semimetals can be represented by Riemann surfaces generated by holomorphic functions in the two-dimensional momentum space, whose constant height contours correspond to Fermi arcs. This correspondence is demonstrated in the recently discovered Weyl semimetals and leads us to predict new types of topological semimetals, whose surface states are represented by double- and quad-helicoid Riemann surfaces. The intersection of multiple helicoids, or the branch cut of the generating function, appears on high-symmetry lines in the surface Brillouin zone, where surface states are guaranteed to be doubly degenerate by a glide reflection symmetry. We predict the heterostructure superlattice [(SrIrO$_3$)$_2$(CaIrO$_3$)$_2$] to be a topological semimetal with double-helicoid Riemann surface states.
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Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. In this work, we introduce a model describing a semimetal in four spatial dimensions, whose nodal points act like tensor monopoles in momentum space. This system is shown to exhibit monopole-to-monopole phase transitions, as signaled by a change in the value of the topological Dixmier-Douady invariant as well as by the associated surface states on its boundary. We use this model to reveal an intriguing 4D parity magnetic effect, which stems from a parity-type anomaly. In this effect, topological currents are induced upon time-modulating the separation between the fictitious monopoles in the presence of a magnetic perturbation. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.
The Fermi arcs of topological surface states in the three-dimensional multi-Weyl semimetals on surfaces by a continuum model are investigated systematically. We calculated analytically the energy spectra and wave function for bulk quadratic- and cubic-Weyl semimetal with a single Weyl point. The Fermi arcs of topological surface states in Weyl semimetals with single- and double-pair Weyl points are investigated systematically. The evolution of the Fermi arcs of surface states variating with the boundary parameter is investigated and the topological Lifshitz phase transition of the Fermi arc connection is clearly demonstrated. Besides, the boundary condition for the double parallel flat boundary of Weyl semimetal is deduced with a Lagrangian formalism.
We theoretically demonstrate that the chiral structure of the nodes of nodal semimetals is responsible for the existence and universal local properties of the edge states in the vicinity of the nodes. We perform a general analysis of the edge states for an isolated node of a 2D semimetal, protected by {em chiral symmetry} and characterized by the topological winding number $N$. We derive the asymptotic chiral-symmetric boundary conditions and find that there are $N+1$ universal classes of them. The class determines the numbers of flat-band edge states on either side off the node in the 1D spectrum and the winding number $N$ gives the {em total} number of edge states. We then show that the edge states of chiral nodal semimetals are {em robust}: they persist in a finite-size {em stability region} of parameters of chiral-asymmetric terms. This significantly extends the notion of 2D and 3D topological nodal semimetals. We demonstrate that the Luttinger model with a quadratic node for $j=frac32$ electrons is a 3D topological semimetal in this new sense and predict that $alpha$-Sn, HgTe, possibly Pr$_2$Ir$_2$O$_7$, and many other semimetals described by it are topological and exhibit surface states.
Fermi arc states are features of Weyl semimetal (WSM) surfaces which are robust due to the topological character of the bulk band structure. We demonstrate that Fermi arcs may undergo profound restructurings when surfaces of different systems with a well-defined twist angle are tunnel-coupled. The twisted WSM interface supports a moire pattern which may be approximated as a periodic system with large real-space unit cell. States bound to the interface emerge, with interesting consequences for the magneto-oscillations expected when a magnetic field is applied perpendicular to the system surfaces. As the twist angle passes through special arcless angles, for which open Fermi arc states are absent at the interface, Fermi loops of states confined to the interface may break off, without connecting to bulk states of the WSM. We argue that such states have interesting resonance signatures in the optical conductivity of the system in a magnetic field perpendicular to the interface.
We address the problem of hybridization between topological surface states and a non-topological flat bulk band. Our model, being a mixture of three-dimensional Bernevig-Hughes-Zhang and two-dimensional pseudospin-1 Hamiltonian, allows explicit treatment of the topological surface state evolution by continuously changing the hybridization between the inverted bands and an additional parasitic flat band in the bulk. We show that the hybridization with a flat band lying below the edge of conduction band converts the initial Dirac-like surface states into a branch below and one above the flat band. Our results univocally demonstrate that the upper branch of the topological surface states is formed by Dyakonov-Khaetskii surface states known for HgTe since the 1980s. Additionally we explore an evolution of the surface states and the arising of Fermi arcs in Dirac semimetals when the flat band crosses the conduction band.
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