Three-fold Weyl points in the Schrödinger operator with periodic potentials
aa r X i v : . [ m a t h - ph ] F e b Three-fold Weyl points in the Schr¨odinger operator withperiodic potentials
Haimo Guo ∗ , Meirong Zhang † , Yi Zhu ‡ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, ChinaFebruary 18, 2021
Contents R . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Λ-periodic, Λ-pseudo-periodic functions and Fourier expansions . . . . . . . . 62.3 Decompositions of periodic and pseudo-periodic functions . . . . . . . . . . . 9 W . . . . . . . . . . . . . . . 144.2 Bifurcation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Conical structure of the spectrum near W . . . . . . . . . . . . . . . . . . . . 18 ∗ E-mail: [email protected]. † E-mail: [email protected]. ‡ E-mail: [email protected]. bstract Weyl points are degenerate points on the spectral bands at which energy bands inter-sect conically. They are the origins of many novel physical phenomena and have attractedmuch attention recently. In this paper, we investigate the existence of such points in thespectrum of the 3-dimensional Schr¨odinger operator H = − ∆ + V ( x ) with V ( x ) being ina large class of periodic potentials. Specifically, we give very general conditions on the po-tentials which ensure the existence of 3-fold Weyl points on the associated energy bands.Different from 2-dimensional honeycomb structures which possess Dirac points where twoadjacent band surfaces touch each other conically, the 3-fold Weyl points are conicallyintersection points of two energy bands with an extra band sandwiched in between. Toensure the 3-fold and 3-dimensional conical structures, more delicate, new symmetriesare required. As a consequence, new techniques combining more symmetries are used tojustify the existence of such conical points under the conditions proposed. This paperprovides comprehensive proof of such 3-fold Weyl points. In particular, the role of eachsymmetry endowed to the potential is carefully analyzed. Our proof extends the analysison the conical spectral points to a higher dimension and higher multiplicities. We alsoprovide some numerical simulations on typical potentials to demonstrate our analysis. Keywords: Schr¨odinger operator, Periodic potentials, Weyl points, Conicalcone, Floquet-Bloch theory.
Weyl points are singular points on the 3-dimensional spectral bands of an operator withperiodic coefficients, at which two distinct bands intersect conically. Much attention hasbeen paid to looking for such fundamental singularities in various physical systems in thepast few decades [2, 4, 10, 28]. They are the hallmark of many novel phenomena. Manymaterials such as graphene exhibit such unusual singular points on their energy bands [10, 28].These singular points carry topological charges and play essential roles on the formation oftopological states, for instance chiral edge states or surface states [7, 18, 19, 30]. In the pastdecade, constructing and engineering the conically degenerate spectral points become oneof the major research subjects in many fields. Accordingly, understanding the existence ofthese points on the energy bands and their connections to interesting physical phenomenaare extremely important in both theoretical and applied fields.How to obtain and justify the existence of such degenerate points become urgent invarious physical systems. For instance, it is well known that honeycomb structures giverise to the existence of Dirac points in 2-D systems. The existence of Dirac points in theperiodic system was first reported by Wallace in the tight-binding model and demonstratedin the continuous systems by numerical and asymptotic approaches [1, 3, 7, 33]. However,the rigorous justification on the existence of Dirac points for 2-D Schr¨odinger equation witha generic honeycomb potential was recently given by Fefferman and Weinstein [17]. Theyused very simple conditions to characterize honeycomb potentials and developed a frameworkto rigorously justify the existence of Dirac points. Their framework paved the way for themathematical analysis on such degenerate points, and their method has been successfullyextended to other 2-D wave systems [11, 12, 21, 26]. There are also other rigorous approachesto demonstrate the existence of Dirac points. Lee treated the case where the potential is asuperposition of delta functions centered on sites of the honeycomb structure [25]. Berkolaikoand Comech used the group representation theory to justify the existence and persistence of2irac points [9]. The low-lying dispersion surfaces of honeycomb Schr¨odinger operators inthe strong binding regime, and its relation to the tight-binding limit, was studied in [16].Ammari et al. applied the layer potential theory to honeycomb-structured Minnaert bubbles[5]. Based on the rigorous justification of the existence of Dirac points, a lot of rigorousexplanations on the related physical phenomena have been extensively investigated. Forexample, the effective dynamics of wave packets associated with Dirac points were studied in[6, 14, 20, 34, 35, 36]. The existence of edge states and associated dynamics are studied in[8, 12, 15].Despite successful applications on the aforementioned analysis of the Dirac points in 2-Dsystems, the advances in applications such as materials sciences, condensed matter physics,placed new theoretical demands that are not entirely met. Just as Kuchment pointed out in arecent overview article on periodic elliptic operators [23], ”the story does not end here”. Oneimportant missing piece is the analysis of 3-dimensional degenerate points which are referredto as Weyl points. Another piece is the conical points with higher order multiplicities. Inthe literature, some special structures are proposed to admit Weyl points [29, 32, 38, 39].However, most constructions and demonstrations are based on either tight-binding models,numerical computations or formal asymptotic expansions. To the best of our knowledge, nosimilar construction and rigorous analysis as aforementioned literature have been given forWeyl points with higher-order multiplicities. Due to the importance and potential applica-tions of Weyl points in quantum mechanics, photonics and mechanics, such generic analysisis highly desired. This is the goal of our current work.This work is concerned with the L ( R )-spectrum of the following 3-dimensional Schr¨odingerequation H = − ∆ + V ( x ) , x ∈ R , (1.1)where the potential V ( x ) is real-valued and periodic. By Floquet-Bloch theory [22, 23, 24],the spectrum of H in L ( R ) is the union of all energy bands E b ( k ) , b ≥ k inthe Brillouin zone. For some specific V ( x ), two energy bands may intersect with each otherconically at some k ∗ . This degenerate point k ∗ in the three dimensional energy bands iscalled a Weyl point. There are different types of Weyl points depending on the multiplicityof degeneracy.In this work, we shall give a simple construction of three-fold Weyl points, i.e., two en-ergy bands intersect conically with an extra band between them. We shall also rigorouslyjustify the existence of such degenerate points by using the strategies developed in [17]. Morespecifically, we first propose a very general class of admissible potentials which are character-ized by several symmetries. Different from honeycomb potentials in which the inversion and2 π/ ∗ together with their fundamental cells Ω and Ω ∗ , and then we preciselydiscuss the existence of high symmetry points k h in Ω ∗ . Section 2 concludes with the Fourieranalysis of Λ-periodic functions. Section 3 contains the definition of the admissible potentialscharacterized by several symmetries. We also review the relevant Floquet-Bloch theory forSchr¨odinger operators H = − ∆ + V ( x ). In Section 4, we first propose required conditionsof eigenstructures at high symmetry point W for some eigenvalue µ ∗ , i.e., H1-H2 and theirconsequences. We then prove the energy bands in the vicinity form a conical structurewith an extra band in the middle. In Section 5, we justify that the required conditions
H1-H2 do hold for nontrivial shallow admissible potentials. Specifically, we clearly showthe significance of the R and T symmetries to preserve the multiplicity of eigenvalues of H ε = − ∆ + εV ( x ) at W while ε is sufficiently small. Moreover, the justification is extendedto generic admissible potentials. Section 6 discusses the instability of the Weyl points andperturbations of dispersion bands of when V ( x ) is violated by an odd potential W ( x ). Section7 provides detailed numerical simulations of the energy bands and Weyl points in differentcases for a special choice of admissible potential. In Appendix A, we present the proofs ofcertain Propositions and Lemmas in Section 4 and Section 5. Without specifications, we use the following notations and definitions. • For z ∈ C , z denotes the complex conjugate of z . • For x , y ∈ C n , h x , y i := x · y = x y + ... + x n y n , and | x | := p h x , x i . • For a matrix or a vector A , A t is its transpose and A ∗ is its conjugate-transpose. • Λ ∈ R denotes the lattice, and Λ ∗ ⊂ ( R ) ∗ = R denotes the dual lattice of Λ.Moreover, v j , j = 1 , , ∗ , while q ℓ , ℓ = 1 , , ∗ , which are chosen to satisfy v j · q ℓ = 2 πδ ℓj . • h f, g i D = R D f g is the L ( D ) inner product. In this work, the region D of integrationis assumed to be the fundamental cell Ω if it is not specified. • ∇ = ( ∂ x , ∂ x , ∂ x ) T . • I denotes the 3 × • For κ = ( κ x , κ y , κ z ) ∈ R , κ arg represents κ x κ y κ z | κ | . Λ and the rotation R Consider the following linearly independent vectors in R v = a √ − − , v = a √ , v = a √ − − . Here a > Z v ⊕ Z v ⊕ Z v := { n v + n v + n v : n , n , n ∈ Z } . The parameter a then gives the distance between nearest neighboring sites. The fundamentalperiod cell of Λ is Ω := { x v + x v + x v : 0 ≤ x i ≤ , i = 1 , , } . (2.1)4et q , q , q ∈ R be the dual vectors of v , v , v , in the sense that q ℓ · v j = 2 πδ ℓj , ℓ, j = 1 , , . Explicitly, q = q − , q = q , q = q − , where q = √ πa . Then the dual lattice of Λ is defined asΛ ∗ = Z q ⊕ Z q ⊕ Z q := { m q + m q + m q : m , m , m ∈ Z } . The fundamental period cell of Λ ∗ is chosen to beΩ ∗ := { c q + c q + c q : c i ∈ ( − / , / , i = 1 , , } . In this work, we are interested in the following rotation transformation R in R R = − − . (2.2)Obviously, R t R = RR t = I . Moreover, R ∗ = R t = R − = − − , and R = I. (2.3)By direct calculations, we can conclude the following proposition. Proposition 1 (1)
The eigenvalues of R is i ℓ , ℓ = 1 , , , with the corresponding eigenvectors ω = 1 √ , − i, t , ω = (0 , , t , ω = 1 √ , + i, t = ω . (2.4)(2) R ∗ and R satisfy R ∗ v = v , R ∗ v = v , R ∗ v = v , R ∗ v = v ,R q = q − q , R q = q − q , R q = − q ,R ∗ q = − q , R ∗ q = q − q , R ∗ q = q − q . (2.5) Thus both R and R ∗ leave Λ and Λ ∗ invariant. Definition 1
A point k h ∈ R is defined to be a high symmetry point with respect to R if R k h − k h ∈ Λ ∗ . Remark 1
By understanding Λ ∗ k h := k h + Λ ∗ as shifted lattices, we know that k h is a highsymmetry point if and only if R leaves Λ ∗ k h invariant, i.e., R (Λ ∗ k h ) = Λ ∗ k h . ∗ , there existprecisely four high symmetry points. Lemma 1
A point k h = c q + c q + c q ∈ Ω ∗ is a high symmetry point with respect to R if and only if the coefficients ( c , c , c ) take the following cases ( c , c , c ) = (0 , , , (1 / , / , / , (1 / , / , / , ( − / , − / , − / . (2.6) Proof
By (2.5), we have R k h = ( − c − c − c ) q + c q + c q . Then R k h − k h = ( − c − c − c ) q + ( c − c ) q + ( c − c ) q ∈ Λ ∗ (2.7)is the same as ( − c − c − c , c − c , c − c ) ∈ Z . Due to the restrictions c i ∈ ( − / , / (cid:3) In this work, we only focus on the following specific high symmetry point W := −
14 ( q + q + q ) = q (cid:18) − , − , (cid:19) . It follows from (2.5) and (2.7) that R W = W , and R ℓ W = W + q ℓ for ℓ = 0 , , , . (2.8)Here q := . Λ -periodic, Λ -pseudo-periodic functions and Fourier expansions We say that a function f ( x ) : R → C is Λ-periodic if f ( x + v ) = f ( x ) ∀ x ∈ R , v ∈ Λ . (2.9)More generally, given a quasi-momentum k ∈ R , we say that a function F ( x ) : R → C isΛ-pseudo-periodic with respect to k if F ( x + v ) = e i k · v F ( x ) ∀ x ∈ R , v ∈ Λ . (2.10)Let us introduce the Hilbert space L k , Λ := (cid:8) F ( x ) ∈ L ( R , C ) : F ( x ) satisfies (2.10) (cid:9) , where the inner product is h F, G i := Z Ω F ( x ) G ( x ) d x for F, G ∈ L k , Λ . Similarly, we define H s k , Λ = { F ( x ) ∈ H s ( R , C ) : F ( x ) satisfies (2.10) . }
6n particular, for k = , L , Λ = L ( R / Λ) := (cid:8) f ( x ) ∈ L ( R , C ) : f ( x ) staisfies (2.9) (cid:9) is the space of square-integrable Λ-periodic functions. Obviously, F ( x ) ∈ L k , Λ if and only if f ( x ) := e − i k · x F ( x ) ∈ L , Λ . That is, the mapping f ( x ) F ( x ) := e i k · x f ( x ) (2.11)gives a one-to-one correspondence between L , Λ and L k , Λ . Moreover, it is easy to see that h F, G i = h f, g i ∀ f, g ∈ L , Λ . That is, the mapping (2.11) is an isometry from L , Λ to L k , Λ .Due to the Λ-periodicity of functions f ( x ) ∈ L , Λ , they can be expanded as Fourier seriesof the form f ( x ) = X q ∈ Λ ∗ ˆ f q e i q · x , (2.12)where n ˆ f q o q ∈ Λ ∗ ⊂ l (Λ) is the sequence of Fourier coefficients, indexed using the discreteindexes q from Λ ∗ . Explicitly, ˆ f q := 1 | Ω | Z Ω f ( x ) e − i q · x d x , (2.13)where | Ω | denotes the volume of the cell Ω. Such a form (2.12) of Fourier expansions isconsistent with Example 1 and is more convenient for later uses. Note that n ˆ f q o q ∈ Λ ∗ ∈ l ∗ , the Hilbert space of square-summable complex sequences over the dual lattice Λ ∗ . Remark 2
Given k ∈ R , pseudo-periodic functions F ( x ) = e i k · x f ( x ) ∈ L k , Λ can be ex-panded as F ( x ) = e i k · x X q ∈ Λ ∗ ˆ f q e i q · x = X q ∈ Λ ∗ ˆ f q e i ( k + q ) · x , (2.14) where { ˆ f q } is as in (2.13) . Rotations R and R ∗ in (2.2) and (2.3) can yield a transformation R for functions F ( x ) ∈ L k , Λ by R [ F ]( x ) := F ( R ∗ x ) for x ∈ R . Lemma 2
Let k h be a high symmetry point w.r.t. R . Then • R maps L k h , Λ to itself as a unitary operator. • Define an affine transformation R k h : Λ ∗ → Λ ∗ by R k h ( q ) := R q + R k h − k h for q ∈ Λ ∗ . (2.15)7 hen, for any ℓ ∈ Z , one has R ℓ k h ( q ) = R ℓ q + R ℓ k h − k h = R ℓ ( q + k h ) − k h for q ∈ Λ ∗ . (2.16) In particular, R k h ( q ) = q for q ∈ Λ ∗ . (2.17) • The action R on L k h , Λ is given by R X q ∈ Λ ∗ ˆ f q e i ( k h + q ) · x = X q ∈ Λ ∗ ˆ f q e iR ( k h + q ) · x = X q ∈ Λ ∗ ˆ f q e i ( k h + R k h ( q )) · x . (2.18) Proof • For F ( x ) = e i W · x f ( x ) ∈ L k h , Λ , we can use expansion (2.14) to obtain R h e i k h · x f ( x ) i = X q ∈ Λ ∗ ˆ f q e i ( k h + q ) · R ∗ x = X q ∈ Λ ∗ ˆ f q e iR ( k h + q ) · x = e i k h · x X q ∈ Λ ∗ ˆ f q e i ( R q + R k h − k h ) · x . (2.19)As R leaves Λ ∗ invariant and R k h − k h ∈ Λ ∗ , we have R k h ( q ) = R q + R k h − k h ∈ Λ ∗ for all q ∈ Λ ∗ . Thus X q ∈ Λ ∗ ˆ f q e i ( R q + R k h − k h ) · x ∈ L , Λ and R [ F ] ∈ L k h , Λ . Moreover, for F ( x ) = e i k h · x f ( x ) , G ( x ) = e i k h · x g ( x ) ∈ L k h , Λ , one has hR [ F ] , R [ G ] i = Z Ω F ( R ∗ x ) G ( R ∗ x ) d x = Z Ω f ( R ∗ x ) g ( R ∗ x ) d x = Z R ∗ (Ω) f ( y ) g ( y ) d y = h f, g i = h F, G i , because R ∗ is an orthogonal transformation and both f ( y ) and g ( y ) are Λ-periodic in y ∈ R .This shows that R is unitary. • Let us check (2.16) only for ℓ ∈ N . By (2.15), we have for q ∈ Λ ∗ R ℓ k h ( q ) = R ℓ q + ℓ − X j =0 R j ( R k h − k h ) = R ℓ q + R ℓ k h − k h = R ℓ ( q + k h ) − k h , the desired equalities in (2.16).By letting ℓ = 4 in (2.16), we obtain (2.17) because R = I . • Using (2.14) and (2.15), equality (2.19) can be written as (2.18). (cid:3)
Remark 3
For k h = , one has R = R . For k h = W , one has from (2.8) that R W ( q ) ≡ R q + q for q ∈ Λ ∗ . .3 Decompositions of periodic and pseudo-periodic functions In the following discussions we only consider the special high symmetry point W . Noticefrom (2.8) that R W = I on Λ ∗ , and R ℓ W = I on Λ ∗ for ℓ = 1 , , . Each orbit of the action R W on Λ ∗ consists of precisely four points. Let us introduce S ∗ W := Λ ∗ /R W = Λ ∗ / n q ∼ q ′ : q , q ′ ∈ Λ ∗ , q ′ = R ℓ W ( q ) for some ℓ ∈ Z o ֒ → Λ ∗ . Then functions F ( x ) = e i W · x f ( x ) ∈ L W , Λ can be decomposed into F ( x ) = X q ∈ Λ ∗ ˆ f q e i ( W + q ) · x = X q ∈S ∗ W X ℓ =0 ˆ f R ℓ W ( q ) e i ( W + R ℓ W ( q )) · x = X q ∈S ∗ W X ℓ =0 ˆ f R ℓ W ( q ) e iR ℓ ( W + q ) · x = X q ∈S ∗ W ( ˆ f q e i ( W + q ) · x + ˆ f R W ( q ) e iR ( W + q ) · x + ˆ f R W ( q ) e iR ( W + q ) · x + ˆ f R W ( q ) e iR ( W + q ) · x ) . (2.20)Since R = I and R ∗ = I , one has R = I on L , Λ . Hence eigenvalues σ of the unitaryoperator R must satisfy σ = 1. In fact, one has σ = i ℓ , ℓ = 0 , , , . (2.21)Then we have an orthogonal decomposition for L , Λ L , Λ = L , ⊕ L ,i ⊕ L , − ⊕ L , − i , (2.22)where the eigenspaces are L ,i ℓ := n f ∈ L , Λ : R [ f ] = i ℓ f o , ℓ = 0 , , , . Note that (2.22) also yields an orthogonal decomposition for the space L W , Λ L W , Λ = L W , ⊕ L W ,i ⊕ L W , − ⊕ L W , − i , where L W ,i ℓ := n e i W · x f ( x ) : f ( x ) ∈ L ,i ℓ o ≡ n F ∈ L W , Λ : R [ F ] = i ℓ F o , ℓ = 0 , , , . Let σ be as in (2.21) and F ( x ) = X q ∈ Λ ∗ ˆ f q e i ( W + q ) · x ∈ L W ,σ . Then R ℓ [ F ] = σ ℓ F ∀ ℓ ∈ Z . (2.23)9y (2.18), we have R ℓ [ F ]( x ) = X q ∈ Λ ∗ ˆ f q e iR ℓ ( W + q ) · x = X q ∈ Λ ∗ ˆ f q e i ( W + R ℓ W q ) · x ≡ X q ∈ Λ ∗ ˆ f R − ℓ W ( q ) e i ( W + q ) · x . Since σ ℓ F ( x ) = X q ∈ Λ ∗ σ ℓ ˆ f q e i ( W + q ) · x , we deduce from (2.23) that the Fourier coefficients ˆ f q satisfyˆ f R − ℓ W ( q ) = σ ℓ ˆ f q ∀ q ∈ Λ ∗ , i.e., ˆ f R ℓ W ( q ) = σ − ℓ ˆ f q ∀ q ∈ Λ ∗ , ℓ ∈ Z . (2.24)Combining with general decomposition (2.20), we have the following results. Lemma 3
Let σ be as in (2.21) . • F ( x ) ∈ L W ,σ if and only if there exists { ˆ f q } q ∈S ∗ W ∈ l S ∗ W such that F ( x ) = X q ∈S ∗ W ˆ f q X ℓ =0 σ − ℓ e iR ℓ ( W + q ) · x ! (2.25)= X q ∈S ∗ W ˆ f q (cid:16) e i ( W + q ) · x + σe iR ( W + q ) · x + σ e iR ( W + q ) · x + σe iR ( W + q ) · x (cid:17) . (2.26) • If F ( x ) ∈ L W ,σ , then F ( − x ) ∈ L W , ¯ σ . Proof • Note that R W = I and σ satisfies σ = 1. Substituting relations (2.24), ℓ =0 , , ,
3, into (2.20), we obtain equality (2.25).As for equality (2.26), we need only to notice in (2.25) that σ = 1 , σ − = ¯ σ, σ − = σ , σ − = σ . • We use expansion (2.25) for F ( x ) to obtain F ( − x ) = X q ∈S ∗ W ˆ f q X ℓ =0 σ − ℓ e iR ℓ ( W + q ) · ( − x ) ! = X q ∈S ∗ W ˆ f q X ℓ =0 ¯ σ − ℓ e − iR ℓ ( W + q ) · ( − x ) ! = X q ∈S ∗ W ˆ f q X ℓ =0 ¯ σ − ℓ e iR ℓ ( W + q ) · x ! , which is in L W , ¯ σ , following from the characterization (2.25) for the eigenvalue ¯ σ . (cid:3) Eigenvalues of periodic Schr¨odinger operators
In this work, we introduce the following admissible potentials.
Definition 2 (Admissible Potentials)
Let V ( x ) ∈ C ∞ ( R ) be real-valued. We say that V ( x ) is an admissible potential with respect to Λ if V ( x ) satisfies(1) V ( x ) is Λ -periodic, V ( x + v ) = V ( x ) for all x ∈ R and v ∈ Λ .(2) V ( x ) is real-valued and even, i.e., V ( x ) = V ( x ) , V ( − x ) ≡ V ( x ) for x ∈ R .(3) V ( x ) is R -invariant, i.e., R [ V ]( x ) = V ( R ∗ x ) ≡ V ( x ) for x ∈ R . (4) V ( x ) is T -invariant, i.e., T [ V ]( x ) ≡ V ( T ∗ x ) = V ( x ) , where T is the following matrix T = − −
10 1 0 . (3.1)We remark that the requirements (2) in Definition 2 are the so-called PT -symmetry.Moreover, requirement (4) is a novel symmetry for 3-dimensional potentials which will playan important role in the later analysis for Weyl points. Admissible potentials have thefollowing properties. Corollary 1
Let V ( x ) be an admissible potential. Then its Fourier coefficients ˆ V q satisfy ˆ V − q = ˆ V q ∈ R ∀ q ∈ Λ ∗ , and ˆ V R ℓ q = ˆ V q , ˆ V T ℓ q = ˆ V q ∀ q ∈ Λ ∗ , ℓ ∈ Z . Remark 4
Let us consider the orthogonal matrix T in (3 . . It is easy to see that T mapsthe lattice Λ ∗ to itself and T ∗ = T − . Moreover, T acts on Λ ∗ as follows T q = q − q , T q = − q , T q = q − q ,T W = W + q , T W = W + q , T W = W + q . Typical admissible potentials can be constructed using Fourier expansions.
Example 1
Let us define real, even potentials V ( x ) := cos( q · x ) + cos(( q − q ) · x ) + cos(( q − q ) · x ) + cos( q · x ) ,V ( x ) := cos( q · x ) + cos(( q − q ) · x ) . It is easy to see that these V i ( x ) are R -invariant potentials. Thus, for any real coefficients c i , the potential V ( x ) = X i =1 c i V i ( x )11s also R -invariant. However, V ( x ) is, in general, not T -invariant. In fact, by noting that T q = q − q , we know that V ( x ) is T -invariant if and only if c = c . Therefore V ( x ) := c ( V ( x ) + V ( x ))is an admissible potential as in Definition 2 for any nonzero real number c . (cid:3) The role of the R - and T -invariance of admissible potentials V ( x ) can be stated as thefollowing commutativity with the Schr¨odinger operator H of (1.1) we are going to study. Lemma 4 (1)
Transformations R and T are isometric, i.e., hR f ( x ) , R g ( x ) i = h f ( x ) , g ( x ) i , and hT f ( x ) , T g ( x ) i = h f ( x ) , g ( x ) i for all f ( x ) , g ( x ) ∈ L W , Λ . (2) The commutators [ H, R ] := H R − R H and [ H, T ] := H T − T H vanish on H W , Λ . The proofs are direct.
Let Λ be the lattice defined in (2.1) and V : R → R be an admissible potential in the senseof Definition 2. For each quasi-momentum k ∈ R , we consider the Floquet-Bloch eigenvalueproblem H Φ( x , k ) = µ ( k )Φ( x , k ) , x ∈ R , Φ( x + v , k ) = e i k · v Φ( x , k ) , x ∈ R , v ∈ Λ , (3.2)where µ ( k ) is the eigenvalue and the second condition is the pseudo-periodic condition forΦ( x , k ). By setting Φ( x , k ) = e i k · x φ ( x , k ) , or φ ( x , k ) = e − i k · x Φ( x , k ) , we know that problem (3.2) is converted into the following periodic eigenvalue problem H ( k ) φ ( x , k ) = µ ( k ) φ ( x , k ) , x ∈ R ,φ ( x + v , k ) = φ ( x , k ) , x ∈ R , v ∈ Λ . (3.3)Here the shifted Schr¨odinger operator H ( k ) is defined via ∇ k φ ( x ) := e − i k · x ∇ (cid:16) e i k · x φ ( x ) (cid:17) = ∇ φ ( x ) + i k φ ( x ) = ( ∇ + i k ) φ ( x ) ,H ( k ) φ ( x ) := e − i k · x ∆ (cid:16) e i k · x φ ( x ) (cid:17) + V ( x ) φ ( x )= − ( ∇ + i k ) · ( ∇ + i k ) φ ( x ) + V ( x ) φ ( x ) ≡ − ∇ k · ∇ k φ ( x ) + V ( x ) φ ( x ) . The general properties of the Schr¨odinger operator with a periodic potential is given bythe Floquet-Bloch theory. We end this section by listing some most important conclusions ofthis theory without including their proofs. We refer readers to [13, 17, 23, 24, 31] for details.12 roposition 2 (Floquet-Block theory)(1) For any k ∈ Ω ∗ , the Floquet-Bloch eigenvalue problem (3.3) has an ordered discretespectrum µ ( k ) ≤ µ ( k ) ≤ µ ( k ) ≤ . . . such that µ b ( k ) → + ∞ as b → + ∞ . Furthermore, there exist eigenpairs { φ b ( x , k ) , µ b ( k ) } b ∈ N for each k ∈ Ω ∗ such that { φ b ( x , k ) } b ≥ can be taken to be a complete orthonormal basis of L , Λ . Accordingly, problem (3.2) has eigenpairs { Φ b ( x , k ) , µ b ( k ) } b ∈ N , where n Φ b ( x , k ) := e i k · x φ b ( x , k ) o b ∈ N is a complete orthonormal basis of L W , Λ . (2) The eigenvalues µ b ( k ), referred as dispersion bands, are Lipschitz continuous functionsof k ∈ Ω ∗ .(3) For each b ≥ µ b ( k ) sweeps out a closed real interval I b over k ∈ Ω ∗ , and the unionof I b composes of the spectrum of H in L , Λ :spec( H ) = [ b ≥ , k ∈ Ω ∗ I b , where I b = (cid:2) min k ∈ Ω ∗ µ b ( k ) , max k ∈ Ω ∗ µ b ( k ) (cid:3) . (4) Given k ∈ Ω ∗ , Φ b ( x , k ) is smooth in x ∈ Ω. Moreover, the set of eigenfunctions S b ≥ , k ∈ Ω ∗ Φ b ( x , k ) is a complete orthonormal set of L ( R ). Consequently, any f ( x ) ∈ L ( R )can be written in the summation form f ( x ) = 1 | Ω ∗ | X b ≥ Z Ω ∗ e f b ( k )Φ( x , k )d k , (3.4)where e f b ( k ) = h Φ b ( x , k ) , f ( x ) i = Z R Φ b ( x , k ) f ( x )d x . Here the summation (3.4) is convergent in the L -norm. In this section, we are going to prove the existence of Weyl points on the energy bands ofSchr¨odinger operators with admissible potentials that we propose in Definition 2. The strat-egy used in this work is inspired by the framework that Fefferman and Weinstein developedfor Dirac points in 2-D honeycomb structures [17]. More specifically, (1) we first propose re-quired conditions of eigen structure at W for some eigenvalue µ ∗ , i.e., the conditions H1-H2 below; (2) we then prove the energy bands in the vicinity form a conical structure with anextra band in the middle under these conditions; (3) we justify that the required conditions
H1-H2 do hold for nontrivial shallow admissible potentials; (4) we extend the justificationof required conditions to generic admissible potentials.Compared to the study on Dirac points for the 2-D honeycomb case, the main difficultiesof our current work arise from two perspectives: higher dimension and higher multiplicity.To the best of our knowledge, we have not found rigorous analysis on such degenerate pointsin the literature. Higher dimension makes the calculations more cumbersome. On the other13and, the higher multiplicity forces us to deal with a larger bifurcation matrix which hasmore freedoms which we need to reduce, for instance, the relations among the entries of thematrix. Some new symmetry arguments are introduced to conquer these difficulties.
In this section, we are interested in the three-fold degeneracy of the high symmetry point W .So let us consider the W -quasi periodic eigenvalue problem H Φ( x , W ) ≡ [ − ∆ + V ( x )]Φ( x , W ) = µ ∗ Φ( x , W ) , x ∈ R , Φ( x + v , W ) = e i W · v Φ( x , W ) , x ∈ R , v ∈ Λ . (4.1)We first assume that there exists an eigenvalue µ ∗ such that the following assumption isfulfilled. H1 µ ∗ is a three-fold eigenvalue of H in problem (4.1) with the corresponding eigenspace E µ ∗ such that E µ ∗ ⊥ L W , , and dim {E µ ∗ ∩ L W ,i } = 1 . Then the following proposition characterizes the fine structure of the eigenspace E µ ∗ . Proposition 3
Assume that H1 holds. Then there exist functions Φ ℓ ( x ) ∈ L W ,i ℓ , j = 1 , , such that { Φ ( x ) , Φ ( x ) , Φ ( x ) = Φ ( − x ) } form an orthonormal basis of E µ ∗ . A direct consequence of above proposition is that µ ∗ is an L W ,i ℓ -eigenvalue of multiplicity1 for each ℓ = 1 , , Under the assumption H1 , we always can find an orthonormal basis { Φ ( x ) , Φ ( x ) , Φ ( x ) } for E µ ∗ as in Proposition 3. However, the choice is not unique and a gauge freedom for eacheigenfunction Φ ℓ ( x ) is allowed.Giving such a basis, let us define a complex-valued matrix M ( κ ) for κ ∈ R / { } by M ( κ ) := h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i . It is called the bifurcation matrix which appears naturally in the eigenvalue problem. Weshall see in the later section that the leading order structure of the eigenvalues of H ( k ) for k in the vicinity of W is closely related to M ( κ ). In this subsection, the main properties of M ( κ ) and their justifications are provided. We want to remark that M ( κ ) depends on thechoice of the basis set { Φ ( x ) , Φ ( x ) , Φ ( x ) } due to the gauge freedom. It is evident that M ( κ ) is Hermitian since 2 iκ · ∇ is self-adjoint.We consider the admissible potential V ( x ) in the sense of Definition 2. Recall that[ H, T ] = 0 can imply T E µ ∗ = E µ ∗ . In other words, there exists a 3 × Q T such that T Φ T Φ T Φ = Q T Φ Φ Φ = c c c c c c c c c Φ Φ Φ . T : L W , Λ → L W , Λ preserves the inner product, i.e., hT F, T G i = h F, G i for all f, g ∈ L W , Λ . It immediately follows that Q T is unitary, i.e., Q ∗T Q T = I . In otherwords, {T Φ , T Φ , T Φ } is also an orthonormal basis of E µ ∗ which defines a new bifurcationmatrix M T ( κ ). Namely, M T ( κ ) ≡ hT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ ihT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ ihT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ i hT Φ , iκ · ∇T Φ i . Similarly, by using the symmetry R , we can define another bifurcation matrix M R ( κ )and the corresponding unitary transformation Q T . In fact, it is easy to obtain Q R = i − − i . (4.2)However, the explicit form for Q T is unknown to us.One has the following relations for these bifurcation matrices. Proposition 4
For any κ ∈ R / { } , there hold M ( κ ) = M R ( Rκ ) = Q ∗R M ( Rκ ) Q R , (4.3) M ( κ ) = M T ( T κ ) = Q ∗T M ( T κ ) Q T , (4.4) where R and T are the orthogonal matrices in (2.2) and (3.1) . Proof
We only give the proof to (4.4), while the proof of (4.3) is similar.By Lemma 2 in [26], one has for ℓ, m = 1 , , h Φ ℓ , ∇ Φ m i = hT Φ ℓ , T ∇ Φ m i = hT Φ ℓ , T ∗ ∇T Φ m i . Therefore( M ( κ )) ℓm = h Φ ℓ , iκ · ∇ Φ m i = hT Φ ℓ , iκ · T ∗ ∇T Φ m i = hT Φ ℓ , iT κ · ∇T Φ m i = X ℓ =1 3 X m =1 c iℓ c jm h Φ ℓ , i ( T κ ) · ∇ Φ m i = ( M T ( T κ )) ℓm . (4.5)By recalling that Q T = ( c ij ), we know that (4.5) is equality (4.4). (cid:3) By substituting (4.2) into (4.3), we obtain M R ( Rκ ) = Q ∗R h Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ ih Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ ih Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ i h Φ , iRκ · ∇ Φ i Q R = h Φ , iκ · R ∗ ∇ Φ i i h Φ , iκ · R ∗ ∇ Φ i −h Φ , iκ · R ∗ ∇ Φ i− i h Φ , iκ · R ∗ ∇ Φ i h Φ , iκ · R ∗ ∇ Φ i i h Φ , iκ · R ∗ ∇ Φ i−h Φ , iκ · R ∗ ∇ Φ i − i h Φ , iκ · R ∗ ∇ Φ i h Φ , iκ · R ∗ ∇ Φ i . (4.6)Recall the transformation R : C → C has eigenpairs listed in (2.4). We can then obtainthe following structural result for the bifurcation matrix M ( κ ).15 heorem 1 There exist υ , υ , υ ∈ C such that M ( κ ) = κ · υ ω κ · υ ω κ · υ ω κ · υ ω κ · υ ω κ · υ ω , (4.7) where ω j , j = 1 , , are eigenvectors of R listed in (2.4) . Moreover, there have | υ | = | υ | = | υ | , (4.8) υ υ υ + υ υ υ = 0 . (4.9) Proof
The proof is split into several steps.1. Entries of M ( κ ). Note ( M ( κ )) ℓj = ( M R ( Rκ )) ℓj = ( Q ∗R M ( Rκ ) Q R ) ℓj holds for ℓ, j =1 , ,
3. By comparing the elements in M R ( κ ) displayed in (4.6) with M ( κ ), it is easily seenthat for κ ∈ R , one has i j − ℓ h Φ ℓ , iκ · R ∗ ∇ Φ j i = h Φ ℓ , iκ · ∇ Φ j i = ⇒ κ · i j − ℓ h Φ ℓ , iR ∗ ∇ Φ j i = κ · h Φ ℓ , iκ · ∇ Φ j i . Since κ ∈ R is arbitrary, we claim that R h Φ ℓ , i ∇ Φ j i = i j − ℓ h Φ ℓ , i ∇ Φ j i . (4.10)Equalities in (4.10) have shown that, for each pair ( ℓ, j ), h Φ ℓ , i ∇ Φ j i is either the zerovector or an eigenvector of R associated with the eigenvalue i j − ℓ . If ℓ = j ∈ { , , } , weknow that i j − ℓ = 1 is not an eigenvalue of R and therefore h Φ ℓ , i ∇ Φ ℓ i = 0 for ℓ = 1 , , . On the other hand, the other six equalities of (4.10) imply that there exist constants υ ℓ , ˜ υ ℓ ∈ C such that h Φ , i ∇ Φ i = υ ω , h Φ , i ∇ Φ i = ˜ υ ω , h Φ , i ∇ Φ i = υ ω , h Φ , i ∇ Φ i = ˜ υ ω , h Φ , i ∇ Φ i = υ ω , h Φ , i ∇ Φ i = ˜ υ ω . (4.11)Since M ( κ ) = ( M ( κ )) ∗ and ω = ω , we have necessarily ˜ υ ℓ = υ ℓ for ℓ = 1 , , .
2. Proof of | υ | = | υ | . According to the definition of Φ ( x ) ∈ L W , − , we have R [Φ ]( x ) = Φ ( R ∗ x ) = − Φ ( x ) . Thus Φ ( R ∗ ( − x )) = − Φ ( − x ) , Φ ( R ∗ ( − x )) = − Φ ( − x ) , R [Φ ( − x )] = − Φ ( − x ) . The last equality means that Φ ( − x ) ∈ L W , − . Since dim( E µ ∗ T L W , − ) = 1 by H1 andΦ ( − x ) is also L -normalized, thereforeΦ ( x ) ≡ e iθ Φ ( − x ) for some θ ∈ R . ∇ Φ ( x ) ≡ − e iθ ∇ Φ ( − x ) and h Φ , i ∇ Φ i = Z Φ ( x ) · ie − iθ ∇ Φ ( − x ) d x = Z Φ ( − x ) · ie − iθ ∇ Φ ( x ) d x = e iθ h Φ , i ∇ Φ i (by changing x to − x )= e iθ h Φ , i ∇ Φ i . From the definition of υ ℓ in (4.11), we obtain υ ω = e iθ υ ω and | υ | = | υ | .3. Proof of (4.8) and (4.9). The proof of | υ | = | υ | is different. For any κ ∈ R , weconsider the characteristic polynomial of the bifurcation matrix M ( κ ) p ( a, κ ) := det ( aI + M ( κ )) = det a h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i a h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i a . It is a cubic polynomial of a with coefficients depending on κ . Since Q T is unitary, it followsfrom (4.4) that p ( a, κ ) = det ( aI + Q ∗T M ( T κ ) Q T ) = det ( Q ∗T ( aI + M ( T κ )) Q T )= det ( aI + M ( T κ )) . Thus p ( a, κ ) satisfies the following invariance p ( a, κ ) ≡ p ( a, T κ ) . (4.12)In particular, by taking κ = e := (0 , ,
0) in (4.12), one has from (4.7) that p ( a, e ) ≡ a − | υ | a. (4.13)Similarly, one has T e = (0 , ,
1) = e and by using (4.7) again, we have p ( a, T e ) ≡ a − | υ | a + 12 ( υ υ υ + υ υ υ ) . (4.14)By comparing the coefficients of (4.13) and (4.14), we deduce from the invariance (4.12) thatthere hold | υ | = | υ | and equality (4.9). Together with equality | υ | = | υ | in the above step,we have obtained all equalities in (4.8) and (4.9). (cid:3) We have also the following gauge invariance for υ υ υ and | υ ℓ | . Corollary 2 (1)
The quantity υ υ υ is gauge invariant in the sense that it does not dependon the choice of the orthonormal basis of E µ ∗ . (2) The quantity | υ | = | υ | = | υ | is also gauge invariant. Proof
Let { Φ ℓ ( x ) : ℓ = 1 , , } and n ˆΦ ℓ ( x ) : ℓ = 1 , , o be two sets of orthonormal eigen-functions as in Proposition 3. Then there exist τ ℓ ∈ R such that τ = − τ , andˆΦ ℓ ( x ) = e iτ ℓ Φ ℓ ( x ) , ℓ = 1 , , .
17y direct calculations, one hasˆ υ ω = h ˆΦ ( x ) , i ∇ ˆΦ ( x ) i = e − iτ + iτ h Φ ( x ) , i ∇ Φ ( x ) i , ˆ υ ω = h ˆΦ ( x ) , i ∇ ˆΦ ( x ) i = e − iτ − iτ h Φ ( x ) , i ∇ Φ ( x ) i , ˆ υ ω = h ˆΦ ( x ) , i ∇ ˆΦ ( x ) i = e iτ + iτ h Φ ( x ) , i ∇ Φ ( x ) i . Therefore ˆ υ = e − iτ + iτ υ , ˆ υ = e − iτ − iτ υ , ˆ υ = e iτ υ . These yield the invariance ˆ υ ˆ υ ˆ υ = υ υ υ . (4.15)For (2), by taking the norms in (4.15) and using equalities (4.8), we obtain | ˆ υ ℓ | = | υ ℓ | . This leads to the desired invariance of | υ ℓ | . (cid:3) Due to the equalities in Theorem 1 and the invariance in Corollary 2, let us define υ F := | υ ℓ | ∈ [0 , + ∞ ) , ℓ = 1 , , . (4.16)The quantity υ F of (4.16) is referred to as the Fermi velocity in quantum mechanics.Now we introduce another standing assumption in this paper, which can be simply statedas H2 υ F = 0 . With the eigenstructure at W , we are able to obtain the corresponding eigenstructure whenquasi-momentum k is near W . The results are stated as follows. Theorem 2
Suppose that V ( x ) is an admissible potential in the sense of Definition andconsider the Schr¨odinger operator H = − ∆ + V ( x ) . Assume that there exists b > suchthat µ b − = µ b = µ b +1 = µ ∗ is an L W , Λ -eigenvalue of H and the assumptions H1-H2 arefulfilled.Then, for sufficiently small but nonzero ( κ x , κ y , κ z ) ∈ R , eigenvalues of H satisfy µ b +1 ( W + κ ) = µ ∗ + ξ + υ F | κ | + o ( | κ | ) ,µ b ( W + κ ) = µ ∗ + ξ υ F | κ | + o ( | κ | ) ,µ b − ( W + κ ) = µ ∗ + ξ − υ F | κ | + o ( | κ | ) , (4.17)where υ F is the Fermi velocity defined before, and ξ + ≥ ξ ≥ ξ − are the three (real) roots ofthe following cubic equation ξ − ξ + 2 κ arg = 0 , κ arg := κ x κ y κ z | κ | . (4.18)18 roof The proof is based on the Lyapunov-Schmidt reduction. Thanks to the eigenstruc-ture at W and the explicit form of the bifurcation matrix which we established in last section,we now only need to do a perturbation expansion and a rigorous justification. Compared tothe 2-D honeycomb case [17], we encounter more complicated computations on the bifurca-tion. We complete it in several steps.1. Decomposition of spaces. For k = W , we have φ ℓ ( x ) = e − i W · x Φ ℓ ( x , W ) ∈ L ,i ℓ ⊂ L , Λ , ℓ = 1 , , , such that H ( W ) φ ℓ = µ (0) φ ℓ , ℓ = 1 , , , where µ (0) := µ ∗ . These define a space X = X W := span { φ , φ , φ } . Consider perturbation k = W + κ , where κ ∈ R is small enough. From the definingequalities in (3.2), one has H ( W + κ ) = H ( W ) − iκ · ( ∇ + i W ) + κ · κ = H ( W ) − iκ · ∇ W + κ · κ. To study eigenvalue problem (3.3), let us decompose ψ ( x , W + κ ) = ψ (0) ( x ) + ψ (1) ( x ) , ψ (0) ∈ X , ψ (1) ∈ X ⊥ , and write µ ( W + κ ) = µ (0) + µ (1) , µ (1) ∈ R . Here the orthogonal complement X ⊥ is taken from L , Λ . Then H ( W + κ ) ψ ( x , W + κ ) = µ ( W + κ ) ψ ( x , W + κ )can be expanded as (cid:16) H ( W ) − µ (0) I (cid:17) ψ (1) = F (1) = F (1) ( κ, µ (1) , ψ (0) , ψ (1) ):= (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) ψ (1) + (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) ψ (0) . (4.19)2. Splitting of the equation using the Lyapunov-Schmidt strategy. To solve Eq. (4.19)using such a strategy, let us introduce the orthogonal projections Q k : H ( R / Λ) → X = span { φ , φ , φ } and Q ⊥ := I − Q k : H ( R / Λ) → X ⊥ . Applying Q k and Q ⊥ to Eq. (4.19), we obtain an equivalent system( H ( W ) − µ (0) I ) ψ (1) = Q ⊥ F (1) ( κ, µ (1) , ψ (0) , ψ (1) ) , (4.20)0 = Q k F (1) ( κ, µ (1) , ψ (0) , ψ (1) ) , (4.21)because Q k ψ (0) = ψ (0) , Q ⊥ ψ (1) = ψ (1) and Q k ψ (1) = Q ⊥ ψ (0) = 0 . (4.22)19y using (4.19) for F (1) , we have Q ⊥ F (1) = Q ⊥ (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) ψ (1) + Q ⊥ (2 iκ · ∇ W ) ψ (0) , (4.23) Q k F (1) = Q k (2 iκ · ∇ W ) ψ (1) + Q k (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) ψ (0) . (4.24)By the assumptions of the theorem on eigenfunctions of H ( W ), one knows that, whenrestricted to X ⊥ , H ( W ) − µ (0) I has a bounded inverse E = E ( W , µ (0) ) = ( H ( W ) − µ (0) I ) − : X ⊥ → Q ⊥ H ( R / Λ) . By (4.22) and (4.23)-(4.24), equation (4.20) is equivalent to ψ (1) = E Q ⊥ (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) ψ (1) = E Q ⊥ (2 iκ · ∇ W ) ψ (0) , i.e. (cid:16) I − E Q ⊥ (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17)(cid:17) ψ (1) = E Q ⊥ (2 iκ · ∇ W ) ψ (0) . (4.25)Due to the regularity, the mapping f T Q ⊥ (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) f is a bounded operator defined on H s ( R / Λ) for any s .In the following we assume that | κ | + | µ (1) | is sufficiently small. Then the left-hand sideof (4.25) is invertible. Given any ψ (0) ∈ X , Eq. (4.25) has then the unique solution in Q ⊥ H ( R / Λ) : ψ (1) = P ψ (0) := (cid:16) I − T Q ⊥ (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17)(cid:17) − T Q ⊥ (2 iκ · ∇ W ) ψ (0) . (4.26)Here P = P ( µ (1) , κ ) : X → Q ⊥ H ( R / Λ) is a bounded linear operator. Substituting(4.26) into equation (4.21) and making use of (4.24), we obtain an equation for the unknowns( µ (1) , ψ (0) ) M ( µ (1) , κ ) ψ (0) + f M ( µ (1) , κ ) ψ (0) = 0 , (4.27)where M ( µ (1) , κ ) , f M ( µ (1) , κ ) : X → Q ⊥ H ( R / Λ) are M ( µ (1) , κ ) := Q k (2 iκ · ∇ W ) P ( µ (1) , κ ) , f M ( µ (1) , κ ) := Q k (cid:16) iκ · ∇ W − κ · κ + µ (1) (cid:17) . Note that (4.27) is a linear system mapping from ψ (0) ∈ X to X , with an unknown parameter µ (1) ∈ R .Since X is 3-dimensional, we can write ψ (0) in ψ (0) = X ℓ =1 α ℓ φ ℓ , α ℓ ∈ C . (4.28)In order that (4.27) has a nonzero solution ψ (0) ∈ X , it is necessary and sufficient that thecorrections µ (1) = µ ( W + κ ) − µ ( W ) for eigenvalues are determined bydet E ( µ (1) , κ ) = 0 , (4.29)20here E ( µ (1) , κ ) is the 3 × ψ (0) as in (4.28). Precisely, E ( µ (1) , κ ) ≡ M ( µ (1) , κ ) + f M ( µ (1) , κ ) , (4.30)where M ( µ (1) , κ ) = (cid:16) M m,ℓj ( µ (1) , κ ) (cid:17) × := (cid:16)D φ ℓ , M ( µ (1) , κ ) φ j E(cid:17) × , f M ( µ (1) , κ ) = (cid:16) M m,ℓj ( µ (1) , κ ) (cid:17) × := (cid:16)D φ ℓ , f M ( µ (1) , κ ) φ j E(cid:17) × .
3. Explicit computation for nondegeneracy condition. We need to give a more explicitcomputation for equation (4.29).To this end, by using (4.28) for ψ (0) , we have from (4.26) ψ (1) ( x ) = ψ (1) ( x , κ, µ (1) ) ≡ X ℓ =1 α ℓ c ( ℓ ) ( x , κ, µ (1) ) , (4.31)where c ( ℓ ) ( x , κ, µ (1) ), ℓ = 1 , ,
3, are bounded by (cid:13)(cid:13)(cid:13) c ( ℓ ) ( · , κ, µ (1) ) (cid:13)(cid:13)(cid:13) H ≤ C ( | κ | + | µ (1) | ) for | κ | + | µ (1) | ≪ . (4.32)Let us define C ( ℓ ) ( x ) = C ( ℓ ) ( x , κ, µ (1) ) := e i W · x c ( ℓ ) ( x , κ, µ (1) ) . Recall thatΦ ℓ ( x ) = e i W · x φ ℓ ( x ) , ∇ W φ j ( x ) = e − i W · x ∇ Φ j ( x ) , h Φ ℓ , Φ j i = h φ ℓ , φ j i = δ ℓj . Moreover, as Q k ψ (1) = 0, we have from (4.31) that Q k c ( ℓ ) = 0, i.e., c ( ℓ ) ∈ X ⊥ . Thus D Φ ℓ , C ( j ) E = D φ ℓ , c ( j ) E = 0 . The matrix-valued functions M ( µ (1) , κ ) and f M ( µ (1) , κ ) in (4.30) are M ( µ (1) , κ ) = µ (1) + h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i µ (1) + h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i µ (1) + h Φ , iκ · ∇ Φ i = µ (1) I + M ( κ ) , f M ( µ (1) , κ ) = κ · κ + h Φ , iκ · ∇ C (1) i h Φ , iκ · ∇ C (2) i h Φ , iκ · ∇ C (3) ih Φ , iκ · ∇ C (1) i κ · κ + h Φ , iκ · ∇ C (2) i h Φ , iκ · ∇ C (3) ih Φ , iκ · ∇ C (1) i h Φ , iκ · ∇ C (2) i κ · κ + h Φ , iκ · ∇ C (3) i . By noticing (4.32), we know that f M ( µ (1) , κ ) ℓj = O ( | κ | · | µ (1) | + | κ | ) .
21. Bifurcation of eigenvalues. By the results in Theorem 1, M ( µ (1) , κ ) simplifies to µ (1) I + M ( κ ) = µ (1) h Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i µ (1) h Φ , iκ · ∇ Φ ih Φ , iκ · ∇ Φ i h Φ , iκ · ∇ Φ i µ (1) = µ (1) υ ( κ x − iκ y ) υ ( κ z ) υ ( κ x + iκ y ) µ (1) υ ( κ x − iκ y ) υ ( κ z ) υ ( κ x + iκ y ) µ (1) . Thus the bifurcation equation (4.29) is (cid:16) µ (1) (cid:17) − µ (1) (cid:20) | υ | κ z + | υ | + | υ | κ x + κ y ) (cid:21) − h ( κ ) − g ( µ (1) , κ ) = 0 , (4.33)where h ( κ ) = 12 υ υ υ ( κ x − iκ x κ y − κ y ) κ z + υ υ υ ( κ x + 2 iκ x κ y − κ y ) κ z ,g ( µ (1) , κ ) = O ( | κ | α | µ (1) | β ) , α + β ≥ . The definitions and properties of υ i are displayed in Theorem 1. Note that υ υ υ ispurely imaginary, thus we may set arg( υ υ υ ) = π , because the case arg( υ υ υ ) = π issimilar. Hence (4.33) simplifies to( µ (1) ) − µ (1) υ F | κ | = − υ F κ x κ y κ z + g ( µ (1) , κ ) . (4.34)We then follow the arguments as done for Proposition 4.2 in [17]. Setting µ (1) = ξυ F | κ | + o ( | κ | )and substituting into (4.34), we observe that ξ solves the cubic equation (4.18).By the Cauchy-Schwarz inequality, one has | κ arg | ≤ √ . We therefore conclude thatequation (4.18) has precisely three real solutions ξ + ≥ ξ ≥ ξ − by using the discriminant ofcubic equations. Moreover, ξ + + ξ + ξ − = 0. Actually, the Floquet-Bloch eigenvalue problemhas three dispersion hypersurfaces µ b +1 = µ ∗ + ξ + υ F | κ | + o ( | κ | ) ,µ b = µ ∗ + ξ υ F | κ | + o ( | κ | ) ,µ b − = µ ∗ + ξ − υ F | κ | + o ( | κ | ) . Consequently, we have the desired results (4.17) and the proof of the theorem is complete. (cid:3)
From the Theorem 2, we see that the three bands intersect at the degenerate point( W , µ ∗ ).Note that the roots ξ ∗ = ξ ∗ ( κ/ | κ | ) of equation (4.18) depend only on the directions of κ ,not on the sizes | κ | of the quasi-momenta κ .We want to point out that there is a special direction along which two energy bandsadhere to each other to leading order. Specifically, if κ ∈ n ∗ R + with n ∗ = (cid:16) √ , √ , √ (cid:17) , thesolutions of (4.18) take the form ξ + = ξ = √ , ξ − = − √ . W . We remark here that it is not clearwhether the double degeneracy persists by including higher order terms of | κ | . This is aninteresting problem but is beyond the scope of the current work.At the end of this section, we characterize the lower dimensional structure of the threeenergy bands near the Weyl point W . According to the expressions of dispersion bands µ ( κ )in (4.17), we study a special case of dispersion equation (4.18) as follows. If κ arg = 0, orequivalently, either of κ x , κ y , κ z vanishes, the bifurcation equation (4.18) has solutions ξ + = 1 , ξ = 0 , ξ − = − . In the transverse plane which is perpendicular to one axis direction, the three dispersionsurfaces form a standard cone with a flat band in the middle, see Figure 2 in Section 7. Thisis exactly the band structure of the Lieb lattice in the tight binding limit [21, 27]. To thebest of our knowledge, this structure has not been rigorously proved. We demonstrate itsexistence for our potentials in lower reduced planes.Generally speaking, in the reduced plane, the three dispersion bands do not behave thesame as the above case. Note that ( − κ ) arg = − κ arg . Let us fix a direction n . Then ξ n + = − ξ − n − , ξ n = − ξ − n , ξ n − = − ξ − n + , where the superscripts indicate the different choices of bifurcation equations depending onthe directions n or − n . We can actually construct three analytical branches of dispersioncurves and each branch is a straight line to leading order. In fact, let us define E ( λ ) = µ b +1 ( W + λ n ) = µ ∗ + ξ n + λυ F + o ( | λ | ) ,E ( λ ) = µ b ( W + λ n ) = µ ∗ + ξ n λυ F + o ( | λ | ) ,E ( λ ) = µ b − ( W + λ n ) = µ ∗ + ξ n − λυ F + o ( | λ | ) . Then for a fixed direction n , the three branches E j ( λ ) , j = 1 , , λ .Next we allow n to vary in a transverse plane. Namely, let n and n be two orthonormalvectors and consider the dispersion surfaces in the plane spanned by n and n . Then µ ( W + λ n + λ n ) = µ ∗ + λυ F ξ ˆ λi + o ( | λ | ) , where | λ | denotes the length of ( λ , λ ). Note, while λ is fixed, κ arg is a continuous variablewith respect to λ λ , thus ξ ˆ λi depends on λ λ continuously. Consequently, (4.17) exactly admitsa cone (may not be standard and isotropic) adhered by an extra surface in the middle (seeSection 7 for related figures). Theorem 2 states that as long as H1-H2 hold, the Schr¨odinger operator with an admissiblepotential always admits a 3-fold Weyl point at the high symmetry point W . In this section,we shall justify the two assumptions H1-H2 can actually hold generally. We first examineshallow potentials in which case we can treat the small potential as a perturbation to theLaplacian operator. Then we can conduct the perturbation theory. The main difficulty is toprove the 3-fold degeneracy persists at any order of the asymptotic expansion. We remark23hat in the 2-D honeycomb case [17], the 2-fold degeneracy is naturally protected by theinversion symmetry. But that is not enough for higher multiplicity. What are the requiredarguments on the 3-fold degeneracy? We will answer this question in our analysis by imposingnovel symmetry arguments.
We first consider the Floquet-Bloch eigenvalue problem for the operator H ε = − ∆ + εV ( x ),where ε is possibly small and V ( x ) is a nonzero admissible potential. Without loss of general-ity, we consider the case that ε is positive. Then the W -pseudo-periodic eigenvalue problemon the four eigenspaces of L W ,i ℓ , ℓ ∈ { , , , } takes the form H ε Φ ℓ ( x , W ) ≡ [ − ∆ + εV ( x )]Φ εℓ ( x , W ) = µ ε ( W )Φ εℓ ( x , W ) , x ∈ R , Φ εℓ ( x + v , K ) = e i W · v Φ εℓ ( x , K ) , x ∈ R , v ∈ Λ , R [Φ εℓ ( x , W )] = i ℓ Φ εℓ ( x , W ) , ℓ ∈ { , , , } . (5.1)We first study the special case that ε = 0. Note that R is orthogonal and | W | = | R W | = | R W | = | R W | = 34 q . By letting µ (0) = | W | = q , we know that e iR ℓ W · x are eigenfunctions associated with µ (0) .Thus µ (0) is an eigenvalue of H of multiplicity at least 4. To show that the multiplicityof µ (0) is exactly 4, for m = ( m , m , m ) ∈ Z and q = m q + m q + m q ∈ Λ ∗ , theequation | W + q | = | W | will lead to [(2 m + 2 m − + (2 m + 2 m − + (2 m + 2 m − ] q = 3 q . Since m , m , m are integers, it is(2 m + 2 m − = (2 m + 2 m − = (2 m + 2 m − = 1 , with the precisely 4 solutions m = (0 , , , (1 , , , (0 , , , (0 , , . For these m , W + q correspond to R ℓ W = W + q ℓ , ℓ = 1 , , ,
4, cf. (2.8).Summarizing the above calculations, we have
Proposition 5
The Laplacian H ≡ − ∆ admits a real four-fold eigenvalue µ (0) = | W | = q at W , with the eigenspace spanned by n e iR ℓ W · x : ℓ = 1 , , , o . Notice from (2.8) that R ν W = W + q ν . Let us take the following eigenfunctions associatedwith µ (0) Φ ℓ ( x ) = Φ ℓ ( x , W ) := 1 p | Ω | (cid:16) e i W · x + i ℓ e iR W · x + i ℓ e iR W · x + i ℓ e iR W · x (cid:17) = 1 p | Ω | X ν =0 i − ℓν e iR ν W · x ∈ L W ,i ℓ = 1 p | Ω | X ν =0 i − ℓν e i ( W + q ν ) · x , ℓ = 1 , , ,
4, cf. (2.26). It is easily seen that (cid:10) Φ ℓ , Φ j (cid:11) = δ ℓj , ℓ, j = 1 , , , . Based on the results in Proposition 5, we can justify Assumptions H1 and H2 when ε > Theorem 3
Let V ( x ) be an admissible potential. Suppose that the Fourier coefficient V , , > . Then there exists a constat ε > such that for any ε ∈ (0 , ε ) , H ε = − ∆ + εV ( x ) fulfillsthe assumptions H1 and H2 . Moreover, one has µ ∗ = µ εℓ = | W | + ε ( V , , − V , , ) + O ( ε ) , ℓ = 1 , , , (5.2) (cid:12)(cid:12) υ ε F (cid:12)(cid:12) = q + O ( | ε | ) > . (5.3) Hence the lowest three energy bands intersect at the three-fold Weyl point ( W , µ ∗ ) . Remark 5
The requirement V , , > in Theorem can be replaced by V , , < . In thelatter case, one has the second, third and fourth bands intersect at the Weyl point ( W , µ ∗ ) . The proof of Theorem 3 is inspired by the methods in [26], where the 2-fold Dirac pointsin the 2-D honeycomb structure is studied. The main difficulty in the present case is thejustification of the three-fold degeneracy of the perturbed eigenvalue µ ∗ at W . Recall thatthe two-fold degeneracy is protected by the PT -symmetry of V ( x ) in the 2-D honeycomb case.The potential in our work also possesses the PT -symmetry so that a two-fold eigenvalue µ ∗ at W is guaranteed. However, this is not adequate to admit the three-fold degeneracy of µ ∗ .In fact, we need to combine T -symmetry to ensure that another eigenvalue is the same as µ ∗ at W . This is the main difference compared to the analysis of the previous work. In thefollowing proof, we only list the key calculations and point out the new ingredients.We begin to prove Theorem 3.1. Recall that µ (0) = | W | is the eigenvalue of the Laplacian − ∆ of multiplicity 4.Moreover, µ (0) is also a simple L W ,i ℓ -eigenvalue for ℓ ∈ { , , , } , with the correspondingeigenstates Φ ℓ . Let us decompose Φ εℓ ( x , W ) ∈ L W ,i ℓ asΦ εℓ ( x , W ) = Φ (0) ℓ ( x , W ) + ε Φ (1) ℓ ( x , W ) . Similar to [36], by applying Lyapunov-Schmidt reduction to (5.1), we obtain the expressionfor µ ε for sufficiently small εµ ε = µ εℓ ≡ µ (0) + ε h Φ (0) ℓ , V ( x )Φ (0) ℓ i + O ( ε ) , ℓ ∈ { , , , } . (5.4)We now turn to the calculation of h Φ (0) ℓ , V ( x )Φ (0) ℓ i . By using the R -invariance of V ( x ),25t follows that V , , = h e i W · y , V ( y ) e i W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i ,V , , = h e i W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e i W · y i ,V , , = h e i W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e i W · y i = h e iR W · y , V ( y ) e iR W · y i ,V , , = h e i W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e i W · y i = h e iR W · y , V ( y ) e iR W · y i = h e iR W · y , V ( y ) e iR W · y i , (5.5)where V α,β,γ = Z Ω e − i ( α q + β q + γ q ) V ( y ) d y . By inserting the expansion (5 .
2) of Φ (0) ℓ ( x , W ) and the coefficients (5.5) into (5.4), andnoticing that V ( x ) is even, it follows that µ εℓ = µ (0) + ε ( V , , − V , , − ) + O ( ε ) , ℓ = 1 , ,µ (0) + ε ( V , , + V , , − − V , , ) + O ( ε ) , ℓ = 2 ,µ (0) + ε ( V , , + V , , − + 2 V , , ) + O ( ε ) , ℓ = 4 . (5.6)2. Since V ( x ) is T -invariant, we have V , , − = V , , >
0. In particular, (5.6) is simplifiedto µ ε = µ ε = µ (0) + ε ( V , , − V , , ) + O ( ε ) , ℓ = 1 , ,µ ε = µ (0) + ε ( V , , − V , , ) + O ( ε ) , ℓ = 2 ,µ ε = µ (0) + ε ( V , , + 3 V , , ) + O ( ε ) , ℓ = 4 . (5.7)Here one shall notice that the O ( ε ) terms in µ ε , and µ ε are undetermined. This meansthat we could not assert that µ ε , = µ ε . However, it follows from (5.7) that these eigenvaluesare ordered so that µ ε = µ ε ≈ µ ε < µ ε . Let E µ ε denote the eigenspace of H ε Φ ε = µ ε Φ ε . Then the above analysis shows that E µ ε ⊂ L W ,i ⊕ L W , − i ⊕ L W , − , and 2 ≤ dim E µ ε ≤ . (5.8)The next step is to verify that µ ε is really a three-fold eigenvalue, i.e., dim E µ ε = 3, withthe help of the following lemma. Lemma 5
We assert that T Φ ε / ∈ L W ,i ⊕ L W , − i for ε is sufficiently small. The detailed proof of Lemma 5 is displayed in Appendix B.We continue the proof for Theorem 3. Recall that [ H, T ] = 0. Thus T ( − ∆ + εV ( x ))Φ ε = ( − ∆ + εV ( x )) T Φ ε = µ ε T Φ ε . T Φ ε ∈ E µ ε . By Lemma 5, we deduce that T Φ ε / ∈ L W ,i ⊕ L W , − i . Hence { Φ ε ( x ) , Φ ε ( − x ) , T Φ ε ( x ) } are linearly independent eigenfunctions in E µ ε . Thus dim E µ ε ≥
3. By (5.8), we conclude thatdim E µ ε = 3 and µ ε is a three-fold eigenvalue. Moreover, result (5.2) follows from (5.7).3. We then embark on the proof of (5.3). In analogy with the construction of υ ℓ inTheorem 1, we introduce υ (0) ℓ by υ (0)1 ω = h Φ (0)1 , i ∇ Φ (0)2 i υ (0)2 ω = h Φ (0)2 , i ∇ Φ (0)3 i υ (0)3 ω = h Φ (0)3 , i ∇ Φ (0)1 i . Actually in the following we will present the full calculations for each υ (0) ℓ under the abovechoice of Φ (0) ℓ . By discussions given in [17], it is standard to apply the Lyapunov-Schmidtreduction to approximate υ ℓ while 0 < ε ≪
1. The result is h Φ ℓ , i ∇ Φ j i = h Φ (0) ℓ , i ∇ Φ (0) j i + O ( | ε | ) . Here Φ (0) ℓ ( x ) = 1 p | Ω | X ν =0 i − ℓν e i ( W + q ν ) · x , ℓ = 1 , , , . Thus we can directly deduce thatΦ (0) ℓ ( x ) = 1 p | Ω | X ν =0 ( − i ) − ℓν e − i ( W + q ν ) · x , ∇ Φ (0) j ( x ) = − p | Ω | X µ =0 i − jµ e i ( W + q µ ) · x ( W + q µ ) , Φ (0) ℓ ( x )2 i ∇ Φ (0) j ( x ) = − | Ω | X ν =0 3 X µ =0 ( − i ) − ℓν i − jµ e i ( q µ − q ν ) · x ( W + q µ ) . Therefore, by setting d ℓj := ( − i ) − ℓ i − j ≡ i ℓ − j , one has D Φ (0) ℓ , i ∇ Φ (0) j E = − X ν =0 ( d ℓj ) ν ( W + q ν )= − X ν =0 ( d ℓj ) ν ! W − X ν =1 ( d ℓj ) ν q ν ≡ − X ν =1 ( d ℓj ) ν q ν , where ( ℓ, j ) = (1 , , (2 , , (3 , d = d = − i and d = − , one has D Φ (0)1 , i ∇ Φ (0)2 E = D Φ (0)2 , i ∇ Φ (0)3 E = − X ν =1 ( − i ) ν q ν = 12 ( i q + q − i q ) = √
22 (1 + i ) qω , D Φ (0)3 , i ∇ Φ (0)1 E = − X ν =1 ( − ν q ν = 12 ( q − q + q ) = − qω . (5.9)27rom these we directly obtain | υ (0)1 | = | υ (0)2 | = | υ (0)3 | = q . This completes the proof of (5.3). (cid:3) Remark 6
Note from (5.9) that υ (0)1 = υ (0)2 = e i π q and υ (0)3 = − q . Thus υ υ υ | υ υ υ | = υ (0)1 υ (0)2 υ (0)3 + O ( | ε | ) | υ υ υ | = − q + O ( | ε | ) q + O ( | ε | ) = − O ( | ε | ) . (5.10) Recall that υ υ υ is gauge invariant and arg( υ υ υ ) is either π or π by Theorem . Thuswe assert from (5.10) that arg( υ υ υ ) = 3 π when ε is sufficiently small. Theorem 3 studies the 3-fold Weyl points for the Schr¨odinger operator with shallow admissiblepotentials: H ε = − ∆ + εV ( x ) for ε = 0 and small. In this subsection we make some remarkson the extension of these results to generic potentials, i.e., ε = O (1). Following the argumentsestablished by Fefferman and Weinstein for the existence of Dirac points in 2-D honeycombpotentials, see [17, 14], we claim that the assumptions H1 and H2 hold for some ( W , µ ∗ )except for ε in a discrete set C of R . Consequently, the conclusions of Theorem 3 also hold,i.e., there always exists a 3-fold Weyl point, for the Schr¨odinger operator H ε = − ∆ + εV ( x )if ε is not in the discrete set C .The main idea is based on an analytical characterization of L W ,λ -eigenvalue of H ε . Bya similar argument on the analytic operator theory and complex function theory strategy[17, 14], it is possible to establish the analogous result. Due to the length of this work, weomit the details and refer interesting readers to [14, 17]. In the preceding sections, we have demonstrated that the admissible potentials genericallyadmit a 3-fold Weyl point at W . The admissible potentials are characterized by the in-version symmetry, the R -symmetry and the T -symmetry. Actually we have seen the 3-folddegeneracy at W and conical structure in its vicinity are consequence of combined actionsof these symmetries. In this section, we shall discuss the instability of the 3-fold Weyl point( W , µ ∗ ) if some symmetry is broken. More specifically, we only show the case where theinversion symmetry is broken which can be compared with the results to the 2-fold Diracpoints in 2-D honeycomb case. The calculation of the case where T -symmetry is broken isvery cumbersome and we shall not give detailed discussion and only give numerical examplesin Section 7.Consider the perturbed eigenvalue problem( H + δV p ( x ))Ψ δ ( x , W ) = µ δ Ψ δ ( x , W ) , (6.1)where V p ( x ) is real and odd, and δ is the perturbation parameter which is assumed to besmall. 28e expand µ δ and Ψ δ ( x ) near the 3-fold Weyl point ( W , µ ∗ ) asΨ δ ( x ) = Ψ (0) ( x ) + Ψ (1) ( x ) , and µ δ = µ ∗ + µ (1) , where Ψ (0) is the unperturbed eigenfunction corresponding to the the unperturbed eigenvalue µ ∗ . We have stated in Theorem 2 thatΨ (0) ( x ) = X i =1 α ℓ Φ ℓ ( x ) . Calculations analogous to those in the proof of Theorem 2 can lead to a system of homoge-neous linear equations for α , α , α ( µ (1) I − M − M ) α α α = 0 , where M = h Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) ih Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) ih Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) i h Φ ( x ) , δV p ( x )Φ ( x ) i , and M includes higher order terms.Therefore µ δ is the solution for the perturbed eigenvalue problem (6.1) if and only if µ (1) solves det( µ (1) I − M − M ) = 0 . (6.2)Following a standard perturbation theory for Floquet-Bloch eigenvalue problems, we ob-tain that the solutions of (6.2) satisfy µ (1) = e µ + o ( δ ) , where e µ is the leading order effect of the perturbation which solves the equationdet( e µI − M ) = 0 . (6.3)To understand the problem, it is key to compute the explicit form of M . Note that h Φ ( x ) , V p ( x )Φ ( x ) i = −h Φ ( − y ) , V p ( y )Φ ( − y ) i = − h Φ ( y ) , V p ( y )Φ ( y ) i = −h Φ ( y ) V p ( y ) , Φ ( y ) i , h Φ ( x ) , V p ( x )Φ ( x ) i = −h Φ ( − y ) , V p ( y )Φ ( − y ) i = − h Φ ( y ) , V p ( y )Φ ( y ) i = −h Φ ( y ) V p ( y ) , Φ ( y ) i . (6.4)Therefore h Φ ( x ) , V p ( x )Φ ( x ) i = h Φ ( x ) , V p ( x )Φ ( x ) i = h Φ ( x ) , V p ( x )Φ ( x ) i = 0 . Similarly, h Φ ( x ) , V p ( x )Φ ( x ) i = −h Φ ( − y ) , V p ( y )Φ ( − y ) i = − h Φ ( y ) , V p ( y )Φ ( y ) i = −h Φ ( y ) V p ( y ) , Φ ( y ) i , h Φ ( x ) , V p ( x )Φ ( x ) i = −h Φ ( − y ) , V p ( y )Φ ( − y ) i = − h Φ ( y ) , V p ( y )Φ ( y ) i = −h Φ ( y ) , V p ( y )Φ ( y ) i . (6.5)29ombining (6.4) and (6.5), we obtain M = δ − υ ♯ υ ♯ υ ♯ − υ ♯ − υ ♯ + υ ♯ , (6.6)where υ ♯ and υ ♯ represent h Φ ( x ) , V p ( x )Φ ( x ) i and h Φ ( x ) , V p ( x )Φ ( x ) i respectively. Ob-versely, υ ♯ is real.Let us assume that both υ ♯ and υ ♯ are nonzero. Substituting (6.6) into (6.3), we obtain e µ ( e µ − δ ( υ ♯ ) ) = 2 δ | υ ♯ | e µ . (6.7)Then we can conclude from (6.7) that the 3-fold degenerate point ( W , µ ∗ ) splits into 3 simpleeigenvalues under an inversion-symmetry-broken perturbation. More precisely, µ δ = µ ∗ + δ q ( υ ♯ ) + 2 | υ ♯ | + o ( δ ) ,µ δ = µ ∗ + o ( δ ) ,µ δ = µ ∗ − δ q ( υ ♯ ) + 2 | υ ♯ | + o ( δ ) . The above analysis implies that the 3-fold Weyl point does not persist if the inversionsymmetry of the system is broken. We also include the numerical simulations for a typicaladmissible potential with an inversion-symmetry-broken perturbation in Section 7, see Figure2. It is seen that the 3 bands do not intersect at W and there exist two local gaps.We remark that if T -symmetry is broken and inversion-symmetry persists, the 3-folddegenerate point split into a 2-fold eigenvalue and a simple eigenvalue, see Figure 3 in Section7. The reason is that the inversion symmetry naturally protects the 2-fold degeneracy whichis similar to the 2-D honeycomb case. Due to the length of this work, we shall not includethe detailed calculations while some of main ingredients can be found in our analysis to thebifurcation matrix M ( κ ) in Section 4. In this section, we use numerical simulations to demonstrate our analysis. The numericalmethod that we use is the Fourier Collocation Method [37]. The potential that we choose is V ( x ) = 5( cos( q · x ) + cos(( q − q ) · x ) + cos(( q − q ) · x ) + cos( q · x )+ cos( q · x ) + cos(( q − q ) · x )) . (7.1)It is evident that V ( x ) is an admissible potential in the sense of Definition 2.According to our analysis–Theorem 2 and Theorem 3, the first three energy bands inter-sect conically at W . In the following illustrations, we plot the figures of first three energybands in vicinity of W . Since the full energy bands are defined in R , it is not easy tovisualize such high dimensional structure. We just show the figures in the reduced parameterspace, i.e., energy curves with the quasi-momentum being along certain specific directionsand energy surfaces with the quasi-momentum being in a plane.30e plot dispersion bands µ ( k ) near W along a certain direction n , i.e., µ ( λ ) = µ ( W + λ n ) . (7.2)The dispersion curves µ ( λ ) along three different directions are displayed on the top panel ofFigure 1 where we choose three different directions n = (1 , , , (cid:18) , , (cid:19) , √ , √ , √ ! . In the first two cases, we see that the three straight lines intersect at λ = 0, i.e., at the Weylpoint. In the last example, we only see two straight lines intersect since one straight line istwo-fold degenerate to leading order, see discussions in Section 5. The numerical simulationsagree with our analysis given in Theorem 2.We also plot the energy surfaces with the quasi-momentum varying in along two directions,i.e., µ ( λ , λ ) = µ ( W + λ n + λ n ) . The dispersion surfaces µ ( λ , λ ) are displayed on the bottom panel of Figure 1 where inall cases n = (1 , ,
0) and n = (0 , , , , √ , √ ! , (cid:18) , , (cid:19) respectively. From the figure, we see that the three dispersion surfaces intersect at the Weylpoint. The first and third bands conically intersect each other with the second band in themiddle. This result also agrees well with our analysis.We next verify the instability of conical singularity under certain symmetry-breakingperturbations. A perturbation is added to the above admissible potential (7.1). In otherwords, we consider the Schr¨odinger operator H δi = − ∆+ V ( x )+ δV p i ( x ), where V p i ( x ) , i = 1 , δ a small parameter. In our simulations, we choose δ = 0 . • We first examine the role of PT -symmetry. The perturbation that we choose is V p ( x ) = sin( q · x ) + sin(( q − q ) · x ) + sin(( q − q ) · x ) + sin( q · x )+ sin( q · x ) + sin(( q − q ) · x ) . (7.3)Obviously, V p is odd and thus breaks PT -invariance of V ( x ). We plot the same energy bandfunctions of H δ as shown in Figure 2. We see that the three energy band functions separatewith each other and two gaps open. • To see the significance of T -invariance in the formation of three-fold conical structures,we consider the perturbation V p ( x ) which breaks T -invariance. In our simulation, we choose V p ( x ) = cos( q · x ) + cos(( q − q ) · x ) + cos(( q − q ) · x ) + cos( q · x ) . (7.4)Obviously, the perturbed potential (7.4) possess R -invariance and PT -invariance, but doesnot have the T -symmetry since T q = q − q .As before, we display the energy curves and surfaces near W in Figure 3. It is shownthat the original three-fold degenerate cone structure disappears and breaks into one simpleand one double eigenvalue. The nearby structure near the double eigenvalue is not naturallyconical. It may correspond to other interesting phenomena but is beyond the scope of thispaper. 31 (a) -0.1 0 0.1 (b) -0.1 0 0.1 (c)(d) (e) (f) Figure 1: The first three energy bands of H = − ∆ + V ( x ) with V ( x ) in (7.1) . TopPanel:
Energy curves µ ( W + λ n ) in (7.2) with the quasi-momentum along a fixed direction n being (a)(1 , , (cid:0) , , (cid:1) ;(c) (cid:16) √ , √ , √ (cid:17) respectively. Bottom Panel:
Energy surfaces µ ( λ , λ ) with the quasi-momentum along two directions n , n , where n is chosen tobe (1 , ,
0) and n equals (d)(0 , , (cid:16) , √ , √ (cid:17) ; (f) (cid:0) , , (cid:1) . The three energy bandsintersect conically at the origin, i.e., at the Weyl point. -0.1 0.1 (a) -0.1 0.1 (b) -0.1 0.1 (c)(d) (e) (f) Figure 2: The first three energy bands of H δ = − ∆ + V ( x ) + δV p ( x ) with the inversion-symmetry-breaking potential V p ( x ) in (7.3). The setup is the same as that in Fig. 1. The3-fold degenerate point disappears and two local gaps open.32 (a) -0.1 0.1 (b) -0.1 0.1 (c)(d) (e) (f) Figure 3: The first three energy bands of H δ = − ∆ + V ( x ) + δV p ( x ) with the T -symmetry-breaking potential V p ( x ) in (7.4). The setup is the same as that in Fig. 1. The 3-folddegenerate point splits into a two-fold and a simple eigenvalues. The two-fold degeneracycomes from the inversion-symmetry of the system which we keep. There is no general conicalstructure near the perturbed two-fold degenerate point. A Proof of Proposition 3
The purpose of this appendix is to give a detailed proof to Proposition 3. We first prove thefollowing lemma.
Lemma 6
Let µ ∗ be an eigenvalue of H ( W ) of eigenvalue problem (4 . with the correspond-ing eigenspace E µ ∗ . If E µ ∗ ⊂ L W ,i ⊕ L W , − i , then dim E µ ∗ is even. Proof
Let Φ ∈ E µ ⊂ L W ,i ⊕ L W , − i . Then Φ( x ) = c Φ ( x ) + c Φ ( x ) for some constants c , c , where Φ ∈ L W ,i and Φ ∈ L W , − i . We distinguish the following two cases. • c · c = 0, say c = 0 for instance. Then Φ( x ) = c Φ ( x ). Note that Φ( − x ) = c Φ ( − x ) ∈ L W , − i is linearly independent of Φ( x ). Recall that { Φ( − x ) , µ ( W ) } is also aneigenpair of eigenvalue problem (3.2). We directly obtain Φ ( − x ) ∈ E µ ∗ . • c · c = 0. Applying R to Φ( x ), one has R [Φ]( x ) = ic Φ ( x ) − ic Φ ( x ) ∈ E µ ∗ . In thepresent case, it is easy to see that R [Φ]( x ) is linearly dependent of Φ( x ).By the above analysis, we conclude that dim E µ is even. (cid:3) Now we are ready to prove Proposition 3. Since Φ ( x ) ∈ L W ,i solves the Floquet-Blochproblem (4.1), then Φ ( x ) := Φ ( − x ) ∈ L W , − i is also an eigenfunction.As dim E µ ∗ = 3, there exists Φ ′ ( x ) ∈ E µ ∗ such that Φ ′ ( x ) / ∈ L W ,i ⊕ L W , − i . HenceΦ ′ ( x ) = c Φ ′′ ( x ) + c Φ ′′ ( x ) + c Φ ′′ ( x ) ∈ E µ , (A.1)where c = 0 and Φ ′′ ℓ ∈ L W ,i ℓ satisfy R [Φ ′′ ℓ ] = i ℓ Φ ′′ ℓ for ℓ = 1 , , ′′ ( x ) ∈ E µ ∗ .33 Case 1: c = c = 0. In this case the assertion is trivial from (A.1). • Case 2: One of c and c is nonzero and another is zero, say c = 0 and c = 0. Thenwe have equalities Φ ′ ( x ) = c Φ ′′ ( x ) + c Φ ′′ ( x ) ∈ E µ , R [Φ ′ ]( x ) = ic Φ ′′ ( x ) − c Φ ′′ ( x ) ∈ E µ , R [Φ ′ ]( x ) + Φ ′ ( x ) = ( i + 1) c Φ ′′ ( x ) ∈ L W ,i ∩ E µ . Since the multiplicity in L W ,i is one, we conclude from the last equality that Φ ′′ ( x ) = α Φ ( x )for some α . Consequently, we conclude from the first equality that Φ ′′ ( x ) ∈ E µ . • Case 3: Both c and c are nonzero. Basing on the decomposition (A.1), one has R [Φ ′ ]( x ) = ic Φ ′′ ( x ) − c Φ ′′ ( x ) − ic Φ ′′ ( x ) . Then Φ ′′ ( x ) := R [Φ ′ ]( x ) + Φ ′ ( x ) = k Φ ′′ ( x ) + k Φ ′′ ( x ) ∈ L W ,i ⊕ L W , − i (A.2)where k = c ( i + 1) and k = c (1 − i ).Assume that Φ ′′ span { Φ , Φ } . Without loss of generality, we assume that Φ ′′ ( x ) islinearly independent of Φ ( x ). Then R [Φ ′′ ]( x ) + i Φ ′′ ( x ) = 2 ik Φ ′′ ( x ) ∈ L W ,i , which wouldimply that µ ( W ) is not a three-fold eigenvalue. Thus we have shown that Φ ′′ ∈ span { Φ , Φ } is a linear combination of Φ and Φ . It then follows from (A.2) that Φ ′′ = α Φ and Φ ′′ = β Φ for some constants α, β . Going back to (A.1), we obtain Φ ′ ( x ) = c ′ Φ ( x )+ c Φ ′′ ( x )+ c ′ Φ ( x ).This leads to the assertion Φ ′′ ( x ) ∈ E µ ∗ .The proof is complete. (cid:3) B Proof of Lemma In this appendix, we actually give a proof of a stronger conclusion. Assume that Φ c ( x ) ∈ L W ,i ⊕ L W , − i has the form Φ c ( x ) = Φ ( x ) + Φ ( x ) , where Φ ( x ) ∈ L W ,i and Φ ( x ) ∈ L W , − i are of the form (2.26). That is,Φ ( x ) = c Φ (0)1 ( x ) + Φ h ( x ) = c Φ (0)1 ( x ) + X q ∈S ∗ W \{ (0 , , } Φ q ( e i ( W + q ) · x − ie iR ( W + q ) · x − e iR ( W + q ) · x + ie iR ( W + q ) · x ) , Φ ( x ) = c Φ (0)1 ( x ) + Φ h ( x ) = c Φ (0)3 ( x ) + X q ∈S ∗ W \{ (0 , , } Φ q ( e i ( W + q ) · x + ie iR ( W + q ) · x − e iR ( W + q ) · x + ie iR ( W + q ) · x ) . By the symmetry, we have the following conclusion.
Lemma 7 If | c | + | c | > , then T Φ c ( x ) / ∈ L W ,i ⊕ L W , − i . Note that Lemma 5 is just a direct consequence of above conclusion. Indeed, let us recallthat Φ ε ( x ) = (1 + O ( ε ))Φ (0)1 ( x ) + Φ h ( x ). Thus for sufficiently small ε , Φ ε ( x ) satisfies theconditions of Lemma 7, i.e., c = 1 + O ( ε ) = 0. So T Φ ε ( x ) / ∈ L W ,i ⊕ L W , − i . (cid:3)
34t remains to prove Lemma 7. We begin the proof by considering the action T on Φ (0) ℓ ( x ).By employing T on e iR ℓ W · x accordingly, we obtain T Φ (0)1 = 1 p | Ω | ( − e i W · x + e iR W · x + ie iR W · x − ie iR W · x ) , T Φ (0)2 = 1 p | Ω | ( e i W · x + e iR W · x − e iR W · x − e iR W · x ) , T Φ (0)3 = 1 p | Ω | ( − e i W · x + e iR W · x − ie iR W · x + ie iR W · x ) , T Φ (0)4 = 1 p | Ω | ( e i W · x + e iR W · x + e iR W · x + e iR W · x ) . Obviously T Φ (0)4 ( x ) = Φ (0)4 ( x ). By direct calculations, we have the following relationsbetween {T Φ (0) ℓ ( x ) } ℓ =1 and { Φ (0) ℓ ( x ) } ℓ =1 T Φ (0)1 T Φ (0)2 T Φ (0)3 = Q T Φ (0)1 Φ (0)2 Φ (0)3 , (B.1)where Q T = − − + i − i + i − i i − − i − . (B.2)Assume that T Φ c ∈ L W ,i ⊕ L W , − i . Then T Φ c = T Φ + T Φ = c T Φ (0)1 + c T Φ (0)3 + T Φ h + T Φ h . By the relations in (B.1) and (B.2), we have c ( −
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