Weyl Semimetal and Topological Phase Transition in Five Dimensions
Biao Lian Shou-Cheng Zhang




Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA

Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA



We study two Weyl semimetal generalizations in five dimensions (5d) which have Yang monopoles and linked Weyl surfaces in the Brillouin zone, respectively, and carry the second Chern number as a topological number. In particular, we show a Yang monopole naturally reduces to a Hopf link of two Weyl surfaces when the 𝐓𝐏 (time-reversal combined with space-inversion) symmetry is broken. We then examine the phase transition between insulators with different topological numbers in 5d. In analogy to the 3d case, 5d Weyl semimetals emerge as intermediate phases during the topological phase transition.





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Weyl Semimetal and Topological Phase Transition in Five Dimensions
    Shou-Cheng Zhang
    December 30, 2023
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§ INTRODUCTION


The discovery of topological states of matter has greatly enriched the variety of condensed matter in nature <cit.>. These states usually undergo phase transitions involving a change of topology of the ground state wave function, which are called topological phase transitions (TPTs).
In three dimensions (3d), an significant topological state is the Weyl semimetal <cit.>, which plays a key role in TPTs of 3d insulators.
An example is the time-reversal invariant (TRI) transition between a noncentrosymmetric topological insulator (TI) <cit.> and a normal insulator (NI) in 3d, during which an intermediate TRI Weyl semimetal phase inevitably occurs <cit.>. Another example is the TPT between different 3d Chern insulators (CI) <cit.>, where an intermediate Weyl semimetal phase is also required <cit.>. In both examples, the topological numbers of the insulators are transferred via Weyl points of the Weyl semimetal phase, which behave as "Dirac monopoles" of the Berry curvature in the Brillouin zone (BZ). The electrons around each Weyl point obey the Weyl equation, with a chirality equal to the first Chern number C_1=±1 of the Berry curvature around the Weyl point.

Recently, there has been a revival of interest in gapless topological phases in higher dimensions, aimed at understanding roles of higher-dimensional topological numbers <cit.>. In particular, the Weyl semimetal can be generalized to 5d in two ways: the first is to promote Weyl fermions in 3d to chiral fermions in 5d, which are described by a 4-component spinor and have a 2-fold degenerate linear energy spectrum. The Dirac monopoles associated with the Weyl points in 3d become the Yang monopoles in 5d <cit.>, which carry a non-Abelian second Chern number C_2^NA=±1 of the SU(2) Berry curvature of the 2-fold degenerate valence (conduction) band <cit.>. The Yang monopole was first introduced into condensed matter physics in the construction of the four dimensional quantum Hall effect <cit.>. The second way is to keep the energy spectrum non-degenerate, while promoting the Weyl points to linked 2d Weyl surfaces in the 5d BZ <cit.>. In this case, each Weyl surface carries an Abelian second Chern number C_2^A∈ℤ of the U(1) Berry curvature, which is equal to the sum of its linking number with all the other Weyl surfaces <cit.>. Two natural questions are then whether the two 5d Weyl semimetal generalizations are related, and whether they play the role of intermediate phases during the TPT of certain gapped topological states of matter in 5d.

In this letter, we show the two 5d Weyl semimetal generalizations, namely, the Yang monopole and the linked Weyl surfaces in 5d, are closely related via the 𝐓𝐏 symmetry breaking, where 𝐓 and 𝐏 stand for time-reversal and space-inversion, respectively. We then demonstrate they also arise as intermediate phases in the TPT between 5d CI and NI, and between 5D TI and NI with particle-hole symmetry 𝐂 that satisfies 𝐂^2=-1 <cit.>. In analogy to 3d cases, the Weyl arcs on the boundary of the 5d Weyl semimetal <cit.> naturally interpolate between the surface states of different gapped topological phases.





§ YANG MONOPOLES AND LINKED WEYL SURFACES


In 3d, a Weyl semimetal is known as a semimetal which is gapless at several points in the BZ, i.e., Weyl points. The low energy bands near a Weyl point is generically given by a 2×2 Weyl fermion Hamiltonian H_W(𝐤)=∑_i=1^3v_i(k_i-k^W_i)σ^i up to an identity term, where 𝐤 is the momentum, and σ^i (i=1,2,3) are the Pauli matrices. The Weyl point is located at 𝐤^W, while the velocities v_i≠0 (i=1,2,3) play the role of light speed. By defining the U(1) Berry connection a_i(𝐤)=i⟨ u_𝐤|∂_k_i|u_𝐤⟩ of the valence (conduction) band wavefunction |u_𝐤⟩, one can show the first Chern number of the Berry curvature f_ij=∂_k_ia_j-∂_k_ja_i on a 2d sphere enclosing 𝐤^W is C_1=sgn(v_1v_2v_3)=±1, where sgn(x) is the sign of x. Therefore, the Weyl point 𝐤^W can be viewed as a Dirac monopole of the Berry connection.

The first way of generalizing the Weyl semimetal to 5d is to replace the Weyl fermions above by the chiral Dirac fermions in 5d:

    H_Y(𝐤)=∑_i=1^5v_i(k_i-k^Y_i)γ^i ,

where 𝐤 is now the 5d momentum, and γ^i (1≤ i≤5) are the 4×4 Gamma matrices satisfying the anticommutation relation {γ^i,γ^j}=2δ^ij. The band structure of such a Hamiltonian is 4-fold degenerate at 𝐤^Y, and is 2-fold degenerate everywhere else with a linear dispersion. The 2-fold degeneracy enables us to define a U(2) Berry connection a^αβ_i(𝐤)=i⟨ u^α_𝐤|∂_k_i|u^β_𝐤⟩, where |u^α_𝐤⟩ (α=1,2) denote the two degenerate wavefunctions of the valence bands <cit.>. One can then show the non-Abelian second Chern number C_2^NA on a 4d sphere enclosing 𝐤^Y is

    C_2^NA=∮_S^4^4𝐤ϵ^ijkl[(f_ijf_kl)-(f_ij)(f_kl)]/32π^2=±1 ,

where f_ij=∂_k_ia_j-∂_k_ja_i-i[a_i,a_j] is the non-Abelian U(2) Berry curvature. In this calculation, only the traceless SU(2) part of f_ij contributes. Therefore, 𝐤^Y can be viewed as a Yang monopole in the BZ, which is the source of SU(2) magnetic field in 5d <cit.>.

However, the generic 2-fold degeneracy of Hamiltonian H_Y(𝐤) requires the system to have certain symmetries.
A common symmetry of this kind is the combined 𝐓𝐏 symmetry of time-reversal and inversion, which is anti-unitary and satisfies (𝐓𝐏)^2=-1 for fermions. Therefore, the Yang monopole 5d generalization is not in the same symmetry class as that of the generic 3d Weyl semimetal.


We remark here that the above 5d Yang monopole, together with the 3d Weyl point and the 2d Dirac point (e.g., in graphene), correspond exactly to the quaternion (pseudoreal), complex and real classes of the Wigner-Dyson threefold way <cit.>, and the anti-unitary 𝐓𝐏 symmetry plays a key role in the classification. Basically, a matrix Hamiltonian H(𝐤) falls into these three classes if (𝐓𝐏)^2=-1,0,+1, respectively (0 stands for no 𝐓𝐏 symmetry), and one can show d=5,3,2 are the corresponding spacial dimensions where point-like gapless manifold in the BZ are stable. The minimal Hamiltonians of the three classes are listed in Tab. <ref>. In particular, (𝐓𝐏)^2=+1 is possible for systems with a negligible spin-orbital coupling such as graphene, where the electrons can be regarded as spinless.




The second 5d Weyl semimetal generalization requires no symmetry (other than the translational symmetry), thus is in the same symmetry class with the 3d Weyl semimetal. Its band structure is non-degenerate except for a few closed submanifolds ℳ_j called Weyl surfaces where two bands cross each other<cit.>. The effective Hamiltonian near each ℳ_j involves only the two crossing bands and takes the 2×2 form H_W(𝐤)=ξ_0(𝐤)+∑_i=1^3ξ_i(𝐤)σ^i. Therefore, ℳ_j is locally determined by 3 conditions ξ_i(𝐤)=0 (i=1,2,3). In one band α of the two associated with ℳ_j, one can define a U(1) Berry connection a^(α)_i(𝐤)=i⟨ u^α_𝐤|∂_k_i|u^α_𝐤⟩ with its wavefunction |u^α_𝐤⟩, and define the U(1) second Chern number of ℳ_j in band α on a 4d closed manifold 𝒱 that only encloses ℳ_j as

    C_2^A(ℳ_j,α)=∮_𝒱^4𝐤ϵ^ijklf^(α)_ijf^(α)_kl/32π^2∈ℤ ,

where f^(α)_ij is the Berry curvature of a^(α)_i. Remarkably, we showed in an earlier paper that <cit.>

    C_2^A(ℳ_j,α)=∑_ℓ∈α,ℓ≠ jΦ(ℳ_j,ℳ_ℓ) ,

where Φ(ℳ_j,ℳ_ℓ) is the linking number between ℳ_j and ℳ_ℓ in the 5d BZ, and ℳ_ℓ runs over all the Weyl surfaces associated with band α.




The relation between the above two 5d generalizations can be most easily seen in the following 4-band model with Hamiltonian

    H_Y'(𝐤)=∑_i=1^5(k_i-k^Y_i)γ^i+bi[γ^4,γ^5]/2 ,

where b is a real parameter that breaks the 𝐓𝐏 symmetry. When b=0, the Hamiltonian reduces to the Yang monopole Hamiltonian H_Y(𝐤) in Eq. (<ref>), where we have set all the velocities to v_i=1. When b≠0, the 𝐓𝐏 symmetry is broken, and the Yang monopole necessarily evolves into linked Weyl surfaces. This can be seen explicitly by deriving the energy spectrum ϵ^α_𝐤=±[((k̃_1^2+k̃_2^2+k̃_3^2)^1/2± b)^2+k̃_4^2+k̃_5^2]^1/2, where we have defined k̃_i=k_i-k_i^Y (1≤ i≤5). Here 1≤α≤4 denotes the α-th band in energies.
Fig. <ref>(a) and Fig. <ref>(b) show the band structures for b=0 and b≠0, respectively, where k̃_2,k̃_3,k̃_4,k̃_5 are assumed zero. In the b≠0 case, one can readily identify three Weyl surfaces: ℳ_1 between bands ϵ^2_𝐤 and ϵ^3_𝐤, ℳ_2 between bands ϵ^1_𝐤 and ϵ^2_𝐤 and ℳ_2' between bands ϵ^3_𝐤 and ϵ^4_𝐤 (see Fig. <ref>(b)). ℳ_1 is a 2d sphere given by k̃_1^2+k̃_2^2+k̃_3^2=b^2 and k̃_4=k̃_5=0, while ℳ_2 and ℳ_2' coincide and are a 2d plane given by k̃_1=k̃_2=k̃_3=0. In particular, the second band ϵ_𝐤^2 (thick red line in Fig. <ref>(b)) is associated with ℳ_1 and ℳ_2, which form a Hopf link in 5d as can be seen in the 3d subspace k_3=k_5=0 plotted in Fig. <ref>(d). In the limit b→0, the radius of ℳ_1 contracts to zero, so ℳ_1 collapses onto ℳ_2 (and ℳ_2') and becomes the 4-fold degenerate Yang monopole in Fig. <ref>(c).
One can add other small 𝐓𝐏 breaking terms to Eq. (<ref>), and the above picture remains topologically unchanged.



Due to the 𝐓𝐏 symmetry breaking, the U(2) gauge field a_i(𝐤) is broken down to two U(1) gauge fields a_i^(1)(𝐤) and a_i^(2)(𝐤) in bands ϵ_𝐤^1 and ϵ_𝐤^2. One can easily check the Abelian second Chern number of ℳ_1 calculated from a_i^(2)(𝐤) is C_2^A(ℳ_1,2)=1, which is defined on 4d manifold 𝒱 with topology S^2× S^2 as shown in Fig. <ref>(d) <cit.>. This is closely related to the non-Abelian second Chern number C_2^NA=1 of the Yang monopole before symmetry breaking. In fact, ignoring the gauge invariance, we can still define the U(2) gauge field a_i^αβ(𝐤) using the two valence bands of Hamiltonian H_Y'(𝐤), which is singular on ℳ_1 but not on ℳ_2 (since ℳ_2 is between the two bands defining the U(2) Berry connection), and still satisfies C_2^NA=1 on a sphere S^4 enclosing ℳ_1. The sphere S^4 can be deformed adiabatically into 𝒱 in Fig. <ref>(d), so we also have C_2^NA=1 on 𝒱.



To see C_2^NA is equal to C_2^A(ℳ_1,2), we can take the limit 𝒱 is a thin "torus" S^2× S^2, i.e., its smaller radius (distance to ℳ_1) tends to zero. In this limit, one will find ∫_𝒱^4𝐤ϵ^ijklf^12_ijf^21_kl=0, namely, the off-diagonal elements of field strength f_ij do not contribute (see Appendix <ref>). So C_2^NA is solely given by the diagonal field strengths f^11_ij and f^22_ij, which can be roughly identified with U(1) Berry curvatures of band 1 and 2. By calculations, one can show ϵ^ijklf^11_ijf^11_kl=ϵ^ijklf_ijf_kl=0. A heuristic understanding of this is the Berry curvature f^11_ij of band 1 sees only ℳ_2, while the U(1) trace Berry curvature f_ij sees only ℳ_1, so both of them do not see linked Weyl surfaces and have zero contribution to the second Chern number. One can then readily show
C_2^NA=∫_𝒱^4𝐤ϵ^ijklf^22_ijf^22_kl/32π^2 =∫_𝒱^4𝐤ϵ^ijklf^(2)_ijf^(2)_kl/32π^2=C_2^A(2,ℳ_1). We note that in this limit where 𝒱 is closely attached to ℳ_1, only the diagonal elements of f_ij contributes, while in the Yang monopole case which is spherically symmetric, the diagonal and off-diagonal elements are equally important <cit.>.









In high energy physics, a U(2) gauge symmetry can be spontaneously broken down to U(1)×U(1) via the Georgi-Glashow mechanism <cit.> with an isospin 1 Higgs field. In 5d space, SU(2) gauge fields are associated with point-like Yang monopoles, while U(1) gauge fields are associated with monopole 2-branes (codimension 3 objects). We conjecture that a gauge symmetry breaking from U(2) to U(1)×U(1) in 5d will always break an SU(2) Yang monopole into two linked U(1) monopole 2-branes ℳ_1 and ℳ_2, where ℳ_1 is coupled to one of the two U(1) gauge fields, while ℳ_2 is coupled to both U(1) gauge fields with opposite monopole charges.




§ TOPOLOGICAL PHASE TRANSITIONS IN 5D


It is known that 3d Weyl semimetals play an important role in 3d TPTs. An example is the TPT of 3d Chern insulator (CI) with no symmetry, which is characterized by three integers (n_1,n_2,n_3), with n_i being the first Chern number in the plane orthogonal to k_i in the BZ <cit.>. The CI becomes a normal insulator (NI) when all n_i=0. The TPT from a 3d NI to a (0,0,1) CI involves an intermediate Weyl semimetal phase as shown in Fig. <ref>(a)-(c). By creating a pair of Weyl points with opposite monopole charges and annihilating them after winding along a closed cycle in k_3 direction, one creates a Berry flux quanta in the k_1-k_2 plane, and n_3 increases by one <cit.>. At the same time, a fermi arc arises on the real space boundary connecting the projections of the two Weyl points <cit.>, which finally becomes a closed fermi loop along k_3.

Another example is the TPT from TI to NI, which are the two phases in the ℤ_2 classification of 3d TRI insulators <cit.>. When the inversion symmetry is broken, an intermediate TRI Weyl semimetal arises <cit.>, which contains (multiples of) 4 Weyl points as shown in Fig. <ref>(d)-(f). The TPT is done by creating two pairs of Weyl points with opposite charges, winding them along a loop that encloses a TRI point (e.g., Γ point), then annihilating them in pairs with their partners exchanged. Meanwhile, the fermi surface loop of the Dirac surface states of TI breaks into two fermi arcs connecting the 4 Weyl points, which vanish when all the Weyl points are gone.




Similarly, the 5d TPTs involve creation of 5d Weyl semimetal phases. We first examine the TPT of 5d CIs with no symmetry, which are characterized by 5 second Chern numbers n_i in the 4d hyperplanes of the BZ orthogonal to k_i (1≤ i≤5), and 10 first Chern numbers n_ij in the 2d planes parallel to k_i and k_j (1≤ i<j≤5). The second Chern numbers n_i are even under both 𝐓 and 𝐏 transformations, while the first Chern numbers n_ij are odd under 𝐓 and even under 𝐏. Here we shall show that changes of the five n_i will involve creation and annihilation of linked Weyl surfaces in the BZ. A simple example without 𝐓𝐏 symmetry is the following 4-band Hamiltonian

    H_QH(𝐤)=∑_i=1^5ξ_i(𝐤)γ^i+bi[γ^3,γ^4]/2 ,

where ξ_i(𝐤)=sin k_i for 1≤ i≤ 4, and ξ_5(𝐤)=m+∑_i=1^4(1-cos k_i)+η(1-cos k_5). Here m is a tuning parameter, while 0≤ b<η<1-b.

We shall label each band by its order in energies, and assume the lower two bands are occupied. Through an analysis similar to we did below Eq. (<ref>), the Weyl surfaces between bands 2 and 3 are given by ξ_1^2+ξ_2^2+ξ_5^2=b^2 and ξ_3=ξ_4=0, while those between bands 1 and 2 (also 3 and 4) are given by ξ_1=ξ_2=ξ_5=0. These two kinds of Weyl surfaces are drawn as blue and red in the 3d subspace k_2=k_4=0 of the BZ shown in Fig. <ref>, respectively, where they appear as 1d loops.

When m>b, the system is a 5d NI with all n_i and n_ij zero and no Weyl surfaces (Fig. <ref>(a)). The TPT to a 5d CI with n_5=1 is driven by decreasing m, which experiences the following stages: when -b<m<b, a Weyl surface between bands 2 and 3 arise around the origin, which is topologically a 2d sphere in the k_3=k_4=0 hyperplane (Fig. <ref>(b), (c)). When b-2η<m<-b, as shown in Fig. <ref>(d), the 2d sphere between band 2 and 3 splits into two smaller spheres ℳ_1 and ℳ_3 (blue) in the k_3=k_4=0 hyperplane, while another 2d sphere Weyl surface ℳ_2 (red) between bands 1 and 2 is created in k_1=k_2=0 plane, which is linked to both ℳ_1 and ℳ_3. As m is further decreased, ℳ_1 and ℳ_3 will move along ± k_5, respectively, and finally merge into a single Weyl surface when -b-2η<m<b-2η (Fig. <ref>(e)). This Weyl surface then shrinks to zero, and the system becomes a 5d CI with n_5=1 for b-2<m<-b-2η, leaving a cylindrical Weyl surface ℳ_2 between bands 1 and 2 (also one between bands 3 and 4, Fig. <ref>(f)).
We note that if b=0, the 𝐓𝐏 symmetry is restored, and the two blue Weyl surfaces ℳ_1 and ℳ_3 will collapse into two Yang monopoles of opposite monopole charges C_2^NA. The TPT process then becomes the creation, winding and annihilation of two Yang monopoles.



This TPT is also accompanied with a surface state evolution from trivial to nontrivial. It has been shown <cit.> that a 5d Weyl semimetal with linked Weyl surfaces contain protected Weyl arcs in the 4d momentum space of surface states, which have linear dispersions in the other 3 directions perpendicular to the arc. By taking an open boundary condition along k_3 direction, one can obtain a Weyl arc on the 4d boundary connecting the projections of the two Weyl surface hopf links (Fig. <ref>(d), (e)). When the system becomes a CI, the Weyl arc develops into a non-contractible Weyl loop along k_5 as expected.

The second example is the TPT between 𝐓𝐏 breaking 5d insulators with particle-hole symmetry 𝐂 satisfying 𝐂^2=-1, which are shown to be classified by ℤ_2 into 5d TIs and NIs <cit.>. Here we consider an eight-band model Hamiltonian of a 5d TI as follows:

    H_TI(𝐤)
    
    =∑_i=1^6ζ_i(𝐤)Γ^i+H_A ,

where Γ^i (1≤ i≤7) are the 8×8 Gamma matrices so chosen that Γ^1, Γ^2, Γ^3, Γ^7 are real and Γ^4, Γ^5, Γ^6 are imaginary, ζ_i(𝐤)=sin k_i for 1≤ i≤5, ζ_6(𝐤)=m+∑_i=1^5t_i(1-cos k_i) with t_i>0, and

    H_A=iη_0Γ^1Γ^2Γ^7+η_1Γ^7sin k_5+iη_2Γ^3Γ^4Γ^5+iη_3Γ^3Γ^4

is a symmetry breaking perturbation. The

𝐓, 𝐏 and 𝐂 transformation matrices are given by 𝒯=Γ^4Γ^5Γ^7, 𝒫=iΓ^6 and 𝒞=Γ^4Γ^5, and a Hamiltonian H(𝐤) will have these symmetries if 𝒯^† H(-𝐤)𝒯=H^*(𝐤), 𝒫^† H(-𝐤)𝒫=H(𝐤) and 𝒞^† H(-𝐤)𝒞=-H^*(𝐤), respectively. It is then easy to see that H_A respects the 𝐂 symmetry but breaks 𝐓, 𝐏 and 𝐓𝐏 symmetries. In particular, only η_1 and η_2 breaks the 𝐓𝐏 symmetry.



In the absence of H_A, the system is a 5d NI if m>0, and is a 5D TI if m<0. With the symmetry breaking term H_A, the calculation of the band structure of H_TI(𝐤) becomes more complicated.
For simplicity, we shall only examine the limiting case where (t_1/t_2)^2+η_0^2<1, |η_0|≫|η_1|, |η_0|≫|η_2| and |η_0η_1|≫|η_3| (with t_1 and t_2 as defined in the expression of ζ_6(𝐤)). We shall label each band by its order in energies, and keep the Fermi energy at zero, i.e., between band 4 and 5, as required by the 𝐂 symmetry. To a good approximation, the Weyl surfaces between bands 4 and 5 are given by ζ_1^2+ζ_2^2=η_0^2, ζ_3^2+ζ_4^2+ζ_5^2=η_2^2 and ζ_6=0, while those between bands 5 and 6 (also between 3 and 4) are given by ζ_3=ζ_4=ζ_5=0.
The TPT can be driven by tuning m from negative (TI) to positive (NI), and the evolution of these low energy Weyl surfaces are illustrated in Fig. <ref> (a)-(f) in the 3d subspace k_1^2+k_2^2=η_0^2 and k_4=0. The small blue loops are the images of Weyl surfaces between bands 4 and 5, while the red loop at k_3=k_5=0 is that between bands 5 and 6. At first, two pairs of blue Weyl surfaces arise unlinked (panel (b)). As m increases, they merge into four new Weyl surfaces linked with the red Weyl surface, which then wind around the red Weyl surface and merge into unlinked pairs again with their partners exchanged (panels (c)-(e)). Finally, the four unlinked blue Weyl surfaces contract to zero, and the system becomes a 5d NI.
Similar to the CI case, if η_1=η_2=0, the 𝐓𝐏 symmetry is recovered, and the 4 blue Weyl surfaces will collapse into 4 Yang monopoles. The TPT process then involves winding of Yang monopoles instead of linked Weyl surfaces. The topological surface states of the system also involves a topological transition during the TPT. The topological surface states of a noncentrosymmetric 5d TI is generically a "Weyl ring" as shown in Fig.<ref>(g). The TPT then breaks into two Weyl arcs (panel (h)), which finally vanish when entering the NI phase.


In conclusion, we show that 5d Weyl semimetals with Yang monopoles are protected by the 𝐓𝐏 symmetry, and generically reduce to 5d Weyl semimetals with linked Weyl surfaces in the presence of 𝐓𝐏 symmetry breaking. We therefore expect that Yang monopoles generically break into linked U(1) monopole 2-branes in 5d theories of gauge symmetry breaking from U(2) to U(1)×U(1). As gapless states carrying the second Chern number, they emerge as intermediate phases in the TPTs between CIs and NIs or between TIs and NIs in 5d space, generalizing the connection between gapless and gapped topological phases in 3d <cit.>.


Acknowledgments. We would like to thank Jing-Yuan Chen for helpful discussions during the research. This work is supported by the NSF under grant number DMR-1305677.










§ DERIVATION OF C_2^NA=C_2^A(ℳ_1,2) ON MANIFOLD 𝒱

It is sufficient to do the calculation in the limit 𝒱 is thin, i.e., close to Weyl surface ℳ_1. For the model given in Eq. (<ref>), such a 4d manifold 𝒱 can be given by (κ-b)^2+k̃_4^2+k̃_5^2=ϵ^2, where κ=√(k̃_1^2+k̃_2^2+k̃_3^2), and ϵ≪ b. Using Gamma matrices defined in <cit.>, the wave function |u^2_𝐤⟩ is given by

    |u^2_𝐤⟩=(cosθ/2cosα/2,sinθ/2cosα/2e^iϕ, 
           cosθ/2sinα/2e^iψ,sinθ/2cosα/2e^iϕ+iψ) ,

while |u^1_𝐤⟩ is well-approximated by

    |u^1_𝐤⟩=(sinθ/2,-cosθ/2e^iϕ,0,0) ,

where we have defined the angles α, ψ, θ,ϕ by sinα e^iψ=(k_4+ik_5)/ϵ, and sinθ e^iϕ=(k_1+ik_2)/κ. This approximation basically ignores the dependence of |u^1_𝐤⟩ on k_4 and k_5, which is valid since |k̃_4,5|<ϵ≪ b, and |u^1_𝐤⟩ is nonsingular at ℳ_1. The nonzero components of U(2) Berry connection a^αβ_𝐤 can then be shown to be

    a^11_ϕ=-1+cosθ/2,a^22_ϕ=-1-cosθ/2,  a^22_ψ=-1-cosα/2,
       a^21_θ=i/2cosθcosα/2 , a^21_ϕ=1/2sinθcosα/2 ,




with a^12_i=a^21*_i. It is then straightforward to calculate the non-Abelian field strengths f^αβ_ij. In particular, one can prove that ϵ^ijklf^12_ijf^21_kl=[sin2θ(1-cosα)sinα]/8, which gives 0 when integrated over the four angles. Therefore, the off-diagonal components of f_ij have no contribution to the second Chern number C_2^NA. Further, one can show f_θϕ^22=-f_θϕ^11=f_θϕ^(2)+(sin 2θcosα)/4 and f_αψ^22=f_αψ^(2)=(sinα)/2 are the only rest nonzero terms, where f_ij^(1) and f_ij^(2) are the U(1) Berry connection in band 1 and 2, respectively. Therefore, we have ϵ^ijklf^11_ijf^11_kl=ϵ^ijklf_ijf_kl=0, and ϵ^ijklf^22_ijf^22_kl=ϵ^ijklf^(2)_ijf^(2)_kl+sin 2θsin2α. The non-Abelian second Chern number is then

    C_2^NA   =∫_0^πθ∫_0^2πϕ∫_0^πα∫_0^2πψϵ^ijklf^22_ijf^22_kl/32π^2
       =∮_𝒱^4𝐤ϵ^ijklf^(2)_ijf^(2)_kl/32π^2=C_2^A(2,ℳ_1)=1 .

If we rewrite the non-Abelian field strength as f_ij=f_ij^at^a where t^a=(1,σ^1,σ^2,σ^3)/2 (a=0,1,2,3) are the generator of U(2), the non-Abelian second Chern number on 𝒱 can be expressed as

    C_2^NA=-c^0+c^1+c^2+c^3 ,

where we have defined c^a=∫_𝒱^4𝐤ϵ^ijklf^a_ijf^a_kl/64π^2. In the limit 𝒱 is close to ℳ_1, the above calculations tell us that c_0=c_1=c_2=0, and c_3=C_2(2,ℳ_1)=1. In contrast, in the Yang monopole case which is rotationally symmetric, one can show c_0=0, and c_1=c_2=c_3=1/3. Therefore, the TP symmetry breaking also breaks the symmetry between c_1, c_2 and c_3.



§ WEYL SURFACES OF MODEL HAMILTONIAN EQ. (7)

Compared to the four-band model in Eq. (6) which can be easily diagonalized, the eight-band model H_TI(𝐤) in Eq. (7) has a band structure more difficult to calculate. Here we present an easier way to examine the band structure with the assumptions (t_1/t_2)^2+η_0^2<1, |η_0|≫|η_1|, |η_0|≫|η_2| and |η_0η_1|≫|η_3|.

For the moment we shall assume η_3=0. To solve the Schrödinger equation H_TI|ψ⟩=E|ψ⟩, one can first rewrite it into H_TI^2|ψ⟩=E^2|ψ⟩, which reduces to

    (E^2-∑_i=1^6ξ_i^2-η_0^2-η_1^2sin^2 k_5-η_2^2)|ψ⟩=(Λ_0+Λ_2)|ψ⟩

after making use of the properties of Gamma matrices, where we have defined

    Λ_0=2η_0(ζ_1Γ^2Γ^7-ζ_2Γ^1Γ^7+η_1sin k_5Γ^1Γ^2) ,


    Λ_2=2η_2(ζ_3Γ^4Γ^5-ζ_4Γ^3Γ^5+ζ_5Γ^3Γ^4) .

One can easily show that Λ_0^2=4η_0^2(ζ_1^2+ζ_2^2+η_1^2sin^2 k_5)=η_0^2χ_0^2, Λ_2^2=4η_2^2(ζ_3^2+ζ_4^2+ζ_5^2)=η_2^2χ_2^2, and [Λ_0,Λ_2]=0. Therefore, they can be simultaneously diagonalized, i.e., Λ_0=±η_0χ_0, Λ_2=±η_2χ_2. One then obtain the energy spectrum of the eight bands as

    E=±√((η_0±χ_0)^2+(η_2±χ_2)^2+ζ_6^2) .

One can then see the Weyl surfaces between bands 4 and 5 are given by χ_0^2=ζ_1^2+ζ_2^2+η_1^2sin^2 k_5=η_0^2, χ_2^2=ζ_3^2+ζ_4^2+ζ_5^2=η_2^2 and ζ_6=0. Since |η_0|≫|η_1|, one can approximately ignore the η_1^2sin^2 k_5 term.

It is also easy to see that the Weyl surfaces between bands 3 and 4 are given by χ_2=0, i.e., ζ_3=ζ_4=ζ_5=0, which is exactly the k_1-k_2 plane. However, another set of Weyl surfaces are given by χ_0=0, i.e., ζ_1=ζ_2=sin k_5=0, which gives the k_3-k_4 plane in touch with the above Weyl surface (the k_1-k_2 plane). Such a configuration is unstable against perturbations in 5d.

This touching of Weyl surfaces is removed when one adds the η_3 term. Via a perturbation analysis, one can show the η_3 term splits the above two kinds of Weyl surfaces in the k_5 direction for about a distance of order |η_3E/η_0η_1| away.

The Weyl surfaces between bands 4 and 5 and between bands 5 and 6 can then be plotted according to the expression of the functions ζ_i, which are as illustrated in Fig. <ref>. In particular, the condition (t_1/t_2)^2+η_0^2<1 limits the number of Weyl surfaces between bands 4 and 5 to only four.



26
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