Monopole Charge Density Wave States in Weyl Semimetals

We study a new class of topological charge density wave states exhibiting monopole harmonic symmetries. The density-wave ordering is equivalent to pairing in the particle-hole channel due to Fermi surface nesting under interactions. When electron and hole Fermi surfaces carry different Chern numbers, the particle-hole pairing exhibits a non-trivial Berry phase inherited from band structure topology independent of concrete density-wave ordering mechanism. The associated density-wave gap functions become nodal, and the net nodal vorticity is determined by the monopole charge of the pairing Berry phase. The gap function nodes become zero-energy Weyl nodes of the bulk spectra of quasi-particle excitations. These states can occur in doped Weyl semimetals with nested electron and hole Fermi surfaces enclosing Weyl nodes of the same chirality in the weak coupling regime. Topologically non-trivial low-energy Fermi arc surface states appear in the density-wave ordering state as a consequence of the emergent zero-energy Weyl nodes.

Introduction. -Charge density wave ordering (CDW), the spontaneous ordering of electron density or bond strength, is an important phenomenon in correlated electron systems [1,2]. The broken translational symmetry of CDW ordering often arises from a Peierls instability, which is driven by electron-phonon interactions between nested Fermi surfaces that lead to the softening of phonon modes and accompanying periodic lattice distortions [3]. Novel topological electron excitations can exist at defects in the CDW order, such as half-fermion modes localized around the domain walls of the Peierls distortion in the one-dimensional polyacetylene chain [2,4]. The CDW instability may also be driven by electron-electron interactions, as studied in the context of high-T c cuprates [5][6][7]. Analogously to unconventional superconductivity, CDW order may also possess unconventional symmetries, forming a non-trivial representation of the lattice symmetry group. For example, CDW order with a d-wave form factor was proposed to compete and coexist with superconductivity [5,6].
Study of the Berry phase of Bloch wave states in lattice systems has led to the discovery of a plethora of topological states, such as quantum anomalous Hall insulators [8,9] and topological insulators [10][11][12][13]. Furthermore, the discovery of Weyl semimetals  has opened up a new avenue for studying topological phases in semi-metallic systems. As in quantum anomalous Hall insulators where a Chern number structure arises from quantized Berry flux over the two-dimensional Brillouin zone, in three-dimensional semimetals the Fermi surfaces have a Chern number structure due to the Weyl points acting as sources or sinks of Berry flux.
After doping, magnetic Weyl semimetals can host monopole harmonic superconductivity, a novel class of topological states. As opposed to typical unconventional superconductors, such as d-wave high T c cuprates, and p-wave superfluid 3 He, in monopole harmonic superconductors the gap function ∆(k) cannot be described by spherical harmonics and their lattice counterparts [46]. Instead, these systems carry "pairing monopole charge", a generalization of Berry phase from single-particle states to a two-particle order parameter. When the pairing occurs between two Fermi surfaces with opposite Chern numbers, which can be the case when the enclosed Weyl points have opposite chiralities, Cooper pairs acquire non-trivial Berry phase structure. As a result, the gap function cannot be well defined over the entire Fermi surface. Consequently, the Fermi surface becomes nodal with total vorticity determined by the pairing monopole charge associated with the two-particle Berry phase.
In this article, we study non-trivial Berry phase structure for a class of order parameter gap functions lying in the particle-hole channel. As an example that can be realized in a doped Weyl semimetal, CDW ordering will be considered. When two nested Fermi surfaces, one electron-like and one hole-like, carry different Chern numbers, the CDW order formed between these two Fermi surfaces inherits non-trivial band structure topology that can be seen in its gap function ρ(k). As with the gap function of monopole harmonic superconductivity, ρ(k) cannot be globally well defined in momentum space and becomes nodal. The nontrivial Berry flux enforces a nonzero total vorticity of ρ(k) determined by the difference in Chern number between the two nested Fermi surfaces that is independent of the concrete mechanism for CDW ordering. The nodes of the CDW gap function emerge as new Weyl nodes in the low-energy quasi-particle spectra that are distinct from the original band structure Weyl points. The chiralities of the emergent quasi-particle Weyl nodes are determined by the band structure in which the single-particle Weyl points have been shifted away from the Fermi surfaces after doping.
Gap function Berry flux and nodes for CDW ordering. -We begin with a minimal description of a pair of electron-like and hole-like Fermi surfaces, which carry opposite Chern numbers and are well nested. Such Fermi surfaces can be realized in a 3D Weyl semimetal system arXiv:1810.08715v1 [cond-mat.str-el] 19 Oct 2018 around two Weyl points of the same chirality. Consider two Weyl points of positive chirality located at K + e and K + h with energies −E 0 and E 0 respectively and Fermi energy at µ = 0. A hole-like Fermi surface denoted FS h,C is centered around K + h with Chern number C or, equivalently, monopole charge q = 1 2 C. Similarly, an electronlike Fermi surface denoted FS e,−C is centered around K + e with Chern number −C. An example of a system with this Fermi surface structure is considered in Eq. (6) below.
Since FS h,C and FS e,−C are well nested, they favor a CDW instability, inter-Fermi surface particle-hole pairing, under repulsive interactions. The two-particle CDW order parameter exhibits a non-trivial Berry flux quantization, which can be seen as follows. After projecting to the low-energy Fermi surfaces, the electron creation operators on FS h,C and FS e,−C can be defined as where p is the momentum relative to the Weyl node at K + h(e) , a refers to the spin or pseudospin degrees of freedom, and ξ ± (p) is the spinor eigenfunction carrying monopole charge ±q. Here p lies on the surface S that results from shifting FS e,−C by −K + e towards the origin. We define the particle-hole channel pairing operator which creates an electron on FS e,−C and a hole on FS h,C . The single-particle Berry connection is A ± (p) = a iξ * ±,a (p)∇ p ξ ±,a (p), and the Berry flux penetrating FS ± is given by ‚ S dp · ∇ p × A ± (p) = ±4πq. It can be shown that the pairing Berry connection associated where q CDW = −2q.
The non-zero Berry flux through S leads to a nontrivial vortex structure for the CDW gap function. The CDW interaction Hamiltonian after mean-field decomposition is expressed as where P +− = P † −+ . The gap function ρ −+ is conjugate to the CDW operator P −+ (p), and ρ +− = ρ * −+ . Because of the non-trivial gauge field A −+ , ρ −+ (p) cannot be globally well defined on S. This follows from examining the gauge invariant "velocity" field v −+ (p) = ∇φ −+ (p) − A −+ (p), where φ −+ is the phase of ρ −+ . v −+ is well defined except at the nodes of ρ −+ (p), and each node has an integer-valued vorticity g i = 1 2π¸C i dp · v −+ (p). positive loop direction defined with respect to the local normal vector. The total vorticity of ρ −+ over S is where the sum is over all the nodes on S. As a consequence, the enclosure of a non-zero net monopole charge gives rise to nodes of ρ −+ (p) on S. The non-trivial nodal structure necessitates the use of the monopole harmonic functions [47], as opposed to the usual spherical harmonics to describe the order parameter. Doped Weyl Semimetal with Nested Fermi Surfaces. -To demonstrate the above topological nodal structure, we employ the band Hamiltonian where a, b refers to the pseudospin degree of freedom, typically realized by A and B sublattices; H ρ is the meanfield Hamiltonian for CDW ordering specified below; and we assume the chemical potential µ = 0. The matrix kernel h(k) of the band Hamiltonian is with Pauli matrices τ x,y,z defined in the A, B basis and pseudospin-dependent hopping amplitudes t x,y,z . Here γ = 1/2 controls the location of the Weyl points along k z . For simplicity, we choose t x,y,z = t in this paper. The corresponding lattice model giving rise to h(k) is presented in Supplemental Materials (S.M.) I. The momentumdependent potential V (k) takes the form with I the 2×2 identity matrix. V 0 plays a role similar to a chemical potential by controlling the size of the Fermi surface.
Without loss of generality, we assume V 0 > 0. This model possesses four Weyl points, all located on the k z axis, at where the upper indices ± refer to the chiralities of the Weyl points and the lower indices e and h refer to whether the Fermi surface associated with the Weyl point is electron-like or hole-like. The potential V (k) shifts the points K − h and K + h up in energy, forming the respective hole-like Fermi surfaces FS h,1 and FS h,−1 , as shown in Fig. 1 (a). Similarly, the points K + e and K − e are shifted down in energy, forming the electron pockets FS e,−1 and FS e,+1 . This model is a modification of the models in Refs. [17 and 48] to allow four Weyl points with nesting. The electron and hole Fermi pockets enclosing the Weyl points with the same chirality are nested with the commensurate wavevector Q = (0, 0, π). This nesting condition is satisfied so that portions of the Fermi surface separated by Q have the same shape. Under an open boundary condition along the y-direction and periodic boundary conditions along x and z, the energy spectrum in the absence of the CDW ordering is plotted in Fig. 1 (b) as a function of k z along the k x = 0 cut. The surface Fermi arc states are shown in red. CDW ordering is imposed through the mean-field Hamiltonian where we take ρ(k) = ρτ z and ρ is the magnitude of the CDW ordering. ρ(k) is diagonal in the sublattice A and B basis, which describese two sublattices with different charge densities. Below we will see that this CDW ordering does not open a full gap over the Fermi surface but instead becomes nodal with a non-trivial vorticity. Topological Nodal CDW. -We first consider the CDW gap function connecting FS h,1 and FS e,−1 which enclose the Weyl nodes K + h and K + e , respectively. For small V 0 /t and ρ/t, the Fermi surfaces are close to the Weyl nodes and the single-particle states correspond to the helicity eigenstates satisfyingp · τ ξ ± = ∓ξ ± , where ξ ± corresponds to Berry flux monopole charges of q = ± 1 2 . Explicitly, ξ ± can be represented as ξ + (p) = (− sin which indicates the presence of gap function nodes on the k z -axis, where sin θ p = 0. The angular form of ρ −+ (p) corresponds to the monopole harmonic Y q=−1,l=1,m=0 (θ p , φ p ) [47]. After taking into account the contribution of the Berry connection A −+ (p), the "velocity" field of the gap function is v −+ (p) = − cot θ pφp .
Integrating v around infinitesimal loops near θ p = 0 and π reveals a gap function vorticity of −1 near both poles, hence the total vorticity of −2 on the Fermi surface surrounding FS e,−1 , consistent with q CDW = −1.
The CDW gap function nodes are actually low-energy Weyl points generated by interactions for the mean-field Hamiltonian. Around the nested Fermi surfaces FS e,−1 and FS h,1 , the low-energy two-band Hamiltonian is where ψ(p) = (α −,e (p), α +,h (p)) T and µ = −V (K + e + p). The interaction-induced Weyl node at the north pole (θ p = 0), denoted K + n , has positive chirality as can be shown by expanding H 2band about the north pole, where the helical basis is regular. The south pole (θ p = π), denoted K + s , is the site of a singularity in the helical basis and thus needs to be treated more carefully. Taking into account the 4π-flux from the Dirac string penetrating the south pole, or equivalently changing the gauge choice to place the singularity at the North pole, this Weyl node can also be shown to possess positive chirality as well.
The positive chiralities of K + n,s are in fact determined by the chiralities of the original band structure Weyl nodes K + e,h , which are away from the chemical potential and hence lie in the high energy sector. Nevertheless,

FIG. 2.
The bulk and surface spectra for the topological CDW ordering with emergent Weyl nodes of Eq. (9). The open boundaries are perpendicular to the y-axis, and only surface states located at the y = 0 boundary are shown. Parameter values are ρ/t = 0.1 and V0/t = 0.2. a) The dispersions along the cut at kx = 0 with varying kz in the reduced BZ with 0 ≤ kz ≤ π. K − e and K + h at kz < 0 are folded into the reduced BZ. Surface state spectra are plotted in red. The dispersions for varying kx are shown for constant kz cuts at b) kz = π 4 , c) kz = 3π 8 , d) kz = 5π 8 , and e) kz = 3π 4 . Green and magenta respectively indicate surface states with majority density in the 0 ≤ kz ≤ π and −π ≤ kz ≤ 0 components of the basis. they still determine the chirality of the low-energy Weyl nodes, independent of the details of the mechanism of the CDW ordering. Typically, the low-energy physics is not sensitive to the details at high energy, but the topological structure at low-energy in our case is indeed inherited from the topology at high energy, and thus the emergence of the low-energy Weyl fermions are topologically protected.
Similar analysis can also be performed in parallel for the CDW ordering connecting the nested Fermi surfaces FS h,−1 and FS e,1 surrounding K − h,e , respectively. The two low-energy Weyl nodes denoted K − n,s on the nested FS h,−1 and FS e,1 have negative chirality, which is again determined by the Weyl nodes K − h,e at high energies. In total, the sum of chiralities of all the Weyl nodes, including the original band structure ones and the interactioninduced ones, remain zero as required by the Nielsen-Ninomiya theorem [49,50].
The topologically protected nodes in the CDW gap function as well as the emergence of new surface states can be seen explicitly in the surface spectra, as shown in Fig. 2 for open boundary conditions along the y-direction and periodic boundary conditions along the x-and zdirections, with states on the y = 0 boundary shown in color. The nesting wavevector Q = (0, 0, π) leads to a reduced Brillouin zone (BZ) in which 0 ≤ k z ≤ π. For the cut with constant k x = 0 and varying k z , both of the zero-energy Weyl node pairs K − s,n and K + s,n are connected by Fermi arcs at zero-energy as shown in Fig. 2 (a). The surface states associated with the original Weyl points appear in the central gapped region with two dispersion branches. The number of branches of surface states changes as k z moves across K − s , K − n , K + s and K + n , as shown in Fig. 2 The surface states are shown in magenta and green for clarity, where magenta (green) coloring indicates states in which the momentum component of k + Q carries more (less) weight than that of k, where k lies in the reduced BZ. For the surface states inside the CDW gaps, which component carries more weight changes in crossing zero energy, while in the central gapped region the surface states remain two separate branches with different components carrying majority weight.
A natural question is whether the projection to the low-energy helical Fermi surface remains valid as the CDW ordering strength ρ becomes large. We find that the zero-energy Weyl nodes remain robust until ρ reaches a critical value ρ c . As ρ is increased, the low-energy Weyl nodes K − n and K + s are pushed closer to each other, as are K − s and K + n . As shown in Fig. 3 (a), K − s and K + n first merge at k z = 0, opening a gap, and K − n and K + s remain separated. Further enlarging ρ, K − n and K + s merge next at k z = π 2 after which the gap opens. After both pairs of zero-energy Weyl nodes merge, the system becomes fully gapped around zero energy, as shown in Fig. 3(b). The surface states in this case cross the bulk gap as k x is varied. This process is similar to the transition between weak-and strong-coupling topological superconductivity, in which the strong-coupling state is topologically trivial.
Increasing ρ beyond the point at which the topological behavior is destroyed, there must eventually be an additional gap closing followed by a gap opening. To see this, consider that in the ρ → ∞ limit, the spectrum must be concentrated close to ±ρ, the eigenvalues of H ρ . In this large ρ limit, there can then be no surface states crossing the gap, and the surface states shown in Fig. 3(b) must eventually be destroyed by a second transition as ρ increases beyond ρ = 0.6t.
Conclusions. -We have studied the inter-Fermisurface particle-hole pairing between well-nested Fermi surfaces enclosing Weyl nodes of the same chirality. The CDW ordering in this case gives rise to gap functions possessing topologically protected nodes independent of the details of ordering interactions. These nodes manifest in the bulk single-particle energy spectra as zeroenergy Weyl nodes, which allow the topology of the surface Fermi arcs to change. The nodal structure corresponds to a new, novel type of topological CDW state whose order parameter carries monopole symmetries. As the ordering strength increases to the strong coupling regime, the zero energy Weyl nodes merge and disappear, and the system becomes fully gapped.