Band rearrangement through the 3DDirac equation with boundary conditions, and the corresponding topological change
Abstract
Rearrangement of energy bands against a parameter is studied through the 3DDirac equation on a ball in under the APS and the chiral bag boundary conditions on the boundary twosphere, where APS is an abbreviation of AtiyahPatodiSinger. The notion of spectral flow and its extension is introduced to characterize the energy eigenvalue redistribution against the parameter, which reflects an analytical property of edge eigenstates. It is shown that though band rearrangement takes place the net spectral flow is zero () for both boundary conditions. The corresponding semiquantum Hamiltonian defined on the 3D momentum space is studied in parallel and it is shown that a change against the parameter is observed in the mapping degree defined for the semiquantum model Hamiltonian, which is viewed as reflecting a topological property of the bulk eigenstates of the full quantum system. Specifically, there are two mappings assigned, for which changes in mapping degree take the values . The present correspondence is viewed as a bulkedge correspondence.
Toshihiro Iwai and Boris Zhilinskii

Kyoto University, 6068501 Kyoto, Japan

Université du Littoral Côte d’Opale, Dunkerque, France

Email: iwai.toshihiro.63u@st.kyotou.ac.jp, zhilin@univlittoral.fr
Keywords: energy band, Dirac equation, APS and chiral bag boundary conditions, mapping degree, bulkedge correspondence
Mathematics Subject Classification 2010: 35Q41, 53C80, 81Q70, 81V55
1 Introduction
A relation between band rearrangement and topological change in quantum states or what is called the bulkedge correspondence have been increasingly attractive in condensed matter physics, especially in topological insulators, a review of which is found in [10, 25], for example. It is to be recognized that among different fields of physics similar topological ideas are shared under respective nomenclatures such as band rearrangement or energy level redistribution, gap closing [59], gap node [23], gapless excitation, contact of the conduction band and the valence band at a single point [1], and Dirac points [10], etc. The authors have been interested in this theme from the viewpoint of molecular physics [16, 35, 37, 38, 41].
A setup for pursuing this theme in molecular physics is stated as follows: There is a class of molecular systems such that the whole set of dynamical variables can be split into subsets associated with low and highenergy excitations [19]. Depending on modelings, the dynamical variables in a molecular system are separated into electronic and vibrational variables, orbital and spin ones, or rotational and vibrational ones, to cite a few. Among such modelings, the rotationvibrational problem is taken as a reference problem. In this modeling, the rotational variables describing highdensity energy levels due to lowenergy excitations can be treated as classical ones which range over a twodimensional sphere on account of the conservation of the angular momentum, but the vibrational variables describing highenergy excitations remain to be treated as quantum ones. Such construction is called a semiquantum model and can be naturally generalized to quantum systems possessing subsystems of “slow” and “fast” variables of different nature.
Within semiquantum rotationvibration models, the authors have been studying band rearrangements for isolated molecules in terms of Chern numbers in a series of papers [35, 36, 37, 41, 16]. A family of semiquantum Hamiltonians is defined to be Hermitian matrices on the twosphere along with some control parameters. Symmetry is often taken into account, and the family of the Hamiltonians are supposed to be invariant under a prescribed group which is a continuous or a finite subgroup of the orthogonal group . For this family of semiquantum Hamiltonians, the Chern number is defined for the complex line bundles associated with each of nondegenerate eigenvalues. The parameter space is divided into connected regions to every point of which the same Chern number is assigned but the different regions have respective Chern numbers (see figures in [35] and [37] in the presence of and symmetries, respectively). The boundaries between different regions correspond to degeneracy of eigenvalues, which is responsible for the modification of band structure. The variation in Chern numbers in crossing the boundary between different regions is counted by means of related winding numbers and the orbit of the symmetry group, and is shown to qualitatively explain the band rearrangement for an isolated molecule [37, 41]. A general description of the stratification of the set of eigenvalues and eigenvectors of a family of Hermitian matrices is found in [7, 8].
If the coupling between rotation and vibration is not weak, the simple splitting of quantum energy levels is not available [28, 54]. Strictly speaking, rotational and vibrational motions cannot be separated on the level of classical mechanics [24] and hence one needs to set up classical and quantum mechanics in terms of fiber bundles by taking into account the nonseparability of rotation and vibration [31, 32]. In this setting, a number of articles have been published [33, 34, 57]. Nevertheless, if a finite set of vibrational levels can be chosen, effective model Hamiltonians for rotationvibration systems are of practical use to study a question as to what kinds of rearrangement of band structure are allowed under the variation of control parameters in the presence of symmetry [38].
At this stage in Introduction, it is significant to compare the idea as to fast and slow variables in molecular physics with bulkedge correspondence in topological insulator theory. In the topological insulators, while the edge state is described quantum mechanically, the bulk states are allowed to be described in terms of classical variables, specifically, in the momentum variables [21, 48]. This procedure can be interpreted from the viewpoint of molecular physics as follows: The insulator is viewed as a twoband system, and the bulkstate energy levels are of high density, so that the bulk states can be treated in terms of classical variables in a semiquantum model. Hence, the Chern number comes into sight as a topological invariant which characterizes the bands. Incidentally, a study of topological insulators has been made also under the named of bulk boundary principle [22, 50]. In particular, in [50], bulk and boundary invariants for lattice systems are studied by means of Ktheory.
Interest in topological study of intersection of potential energy surfaces on the basis of Hermitian matrices dates back to [29] and led to the Berry phase [11, 12, 62]. Hermitian matrices have been effectively used [55, 63] in semiquantum molecular models as well. The relevance of Chern numbers to band rearrangements in isolated molecules was initially suggested in [49] and studied further for formation of more concrete relation between Chern numbers and the numbers of states in the band [18, 19, 20]. As is mentioned above, a spectacular appearance of Chern numbers is made in the description of quantum Hall effect [6, 44, 58], though the base manifold of the relevant line bundle is a twotorus in contrast to the twosphere for rotationvibration problems for isolated molecules. Increasing interest has centered on topological phenomena such as the quantum spin Hall effect and topological insulators [42, 52, 25, 9, 10], and further attempt has been made for classifying topological phases [43, 53, 56].
In the course of the study on band rearrangement in semiquantum models [37, 41], the authors have observed that elementary band rearrangement (or levelcrossing) takes place between two adjacent bands and can be characterized by a topological invariant which can be evaluated through the linearization of the Hamiltonian at a critical point, and they have called the elementary topological change a deltaChern. Further, they pointed out, in previous papers [39, 40], that the linearization method works well for the study of band rearrangement both in semiquantum and full quantum models. In [39, 40], 2D Dirac equations with boundary conditions are intensively solved with interest in qualitative modifications of band structures in the full quantum model together with interest in analogous modification in the corresponding semiquantum model. It was shown that the deltaChern [37, 41] defined for a parametric family of Hamiltonians in the semiquantum model corresponds to the spectral flow or its extension for the 2D Dirac equation in question, where the spectral flow of a oneparameter family of selfadjoint operators is originally defined to be the net number of eigenvalues passing through zero in the positive direction as the parameter runs [5, 51] and then extended in [40], and where the deltaChern is a jump in the Chern number which takes place when the parameter crosses a critical value. The authors have already given nonlinear twodimensional models which exhibits the correspondence between spectral flows and changes in Chern numbers [16]. The present correspondence is, in principle, the same as the bulkedge correspondence for topological insulators without timereversal symmetry [21].
The oneparameter family of Dirac operators studied in [39, 40] is given by
(1) 
where is a control parameter and where , are the standard Pauli matrices. The eigenvalue problem for is set on the disk under the APS and the chiral bag boundary conditions on the circle , where APS is an abbreviation of AtiyahPatodiSinger [5]. The corresponding semiquantum Hamiltonian is given by replacing the operators by the real variables , .
A question arises as to whether the correspondence between the spectral flow and the deltaChern is a key, or not, to the understanding of the relation between band rearrangement and topological change, independently of a choice of models. Put in another way, the present question is as to whether the bulkedge correspondence holds for a wider class of model Hamiltonians or not. A further way to approach the present question is to consider higherdimensional cases. If group actions such as a timereversal transformation with halfinteger spin degrees of freedom are taken into account, then new Hamiltonians comes to be defined in dimensions larger than two, in both quantum and semiquantum models. Another view of the extension of Hamiltonians is taken through gamma matrices. If gamma matrices are adopted, corresponding Dirac Hamiltonians are to be defined on or . The present paper deals with the Dirac Hamiltonian defined on and the corresponding semiquantum Hamiltonian, both of which admit the timereversal and the chiral symmetries. Since the Chern number is defined for vector bundles over evendimensional manifolds and since the manifold on which the model Hamiltonians are to be defined is of odddimensions, another topological invariant should be introduced to describe a topological character. In the last section of this paper, remarks will be made on Dirac Hamiltonians defined on together with the corresponding semiquantum Hamiltonian.
Among several representations of the gamma matrices, the choice made in the present article is
(2) 
where denotes the unit matrix. Then, the 3D Dirac Hamiltonian takes the form
(3) 
where is a parameter, which originally denotes the mass but is allowed to take negative values in the present study. In Dirac’s textbook [15], the alpha matrices are used in place of the gamma matrices (2), where while .
In order to obtain discrete energy eigenvalues for , a boundary condition is required to be posed on the boundary of a bounded domain, since no external field is present. In this paper, the eigenvalue problems for are put on the ball of radius under the APS and the chiral bag boundary conditions. It will be shown that the eigenvalues for with the APS and the chiral bag boundary conditions are broken up into bulk and edgestates eigenvalues, where the edgestate eigenvalues are responsible for band rearrangement but the bulkstate eigenvalues form separate two bands, positive and negative. The spectral flow and its extension are defined for edgestate eigenvalues and has the value for both boundary conditions. In spite of the zero value, the band rearrangement takes place indeed.
From the viewpoint of energy level density, it is shown that the difference between adjacent two bulkstate energy levels is inversely proportional to the radius of the ball, so that the bulkstate eigenvalues become of high density for a sufficiently large . Hence, the bulkstate bands are allowed to be treated in terms of classical variables or in momentum variables. Accordingly, the quantum Hamiltonian turns into a semiquantum Hamiltonian by replacing the classical momentum variables for the momentum operators . The zero spectral flow for the 3D Dirac equation is shown to have a counterpart in the corresponding semiquantum model, which is described in terms of the mapping degrees (or winding numbers) of mappings to be defined through the projection operator onto each of eigenspaces for the semiquantum Hamiltonian. To each projection operator, there are assigned two mappings which have the mapping degrees of opposite sign. Each of the mapping degree changes the sign as the parameter passes the critical value (zero), but the net change, the sum of variation in respective mapping degrees, is zero. The zero spectral flow and the zero net change in mapping degrees are in keeping with the particlehole symmetry of the quantum and semiquantum Hamiltonians. Thus, the bulkedge correspondence proves to hold in the present setting.
This paper is organized as follows: In Sec. 2, the Dirac Hamiltonian (3) and the corresponding semiquantum Hamiltonian are characterized from the viewpoint of the timereversal and the chiral symmetries. Section 3 is concerned with the full quantum model. In the beginning of this section, a brief review is made of the total angular momentum operators together with the associated spinor spherical harmonics on . After the 3D Dirac Hamiltonian is described in spherical polar coordinates, a boundary operator is introduced on the twosphere of radius . Then, feasible solutions to the Dirac equation are obtained without referring to the boundary condition, which is the first step toward solutions to the boundary value problem. Depending on ranges of the parameter, the feasible solutions are classified into three classes. After the APS and the chiral bag boundary conditions are specified, eigenvalues for the 3D Dirac equation with respective boundary conditions are found together with associated eigenstates, which are classified into bulk states, edge states, and zero modes or critical states, where the radial functions for bulk and edge states are described in terms of Bessel functions and modified Bessel functions, respectively, and zero modes or critical states take the form of solid harmonics. Discrete symmetry (or pseudosymmetry) can explain the pattern of eigenvalues as functions of the parameter . In the last subsection of this section, the dependence of the bulkstate eigenvalues is discussed and then it is shown that the semiquantum Hamiltonian can be viewed as a bulk Hamiltonian in the limit as . In Sec. 4, the corresponding semiquantum Hamiltonian is studied. Two mappings are defined through the projection operators onto the eigenspaces. A jump in the mapping degree against the parameter is shown to have a topological meaning. Sec. 5 presents an answer to the aforementioned question as to the correspondence between band rearrangement and topological change or the bulkedge correspondence. In Sec. 6, remarks on further study and related fields to the present study are mentioned. Appendix A contains calculations for the derivation of spherical spinor harmonics and for the description of the Hamiltonian in the polar spherical coordinates.
2 Dirac Hamiltonians with discrete symmetry
In the momentum representation, the momentum operators are replaced by the classical variables , and then the Dirac Hamiltonian (3) is brought into
(4) 
which we call a semiquantum Dirac Hamiltonian. In this section, we characterize the Hamiltonian (4) after the classification scheme for Hermitian matrices by means of discrete symmetries such as timereversal, particlehole, and chiral symmetries [3, 26, 64]. Remarks on the socalled AZ classification will be made in the last section, as far as the present Hamiltonian is concerned.
According to [47, 45], the introduction of timereversal symmetry along with the spin degrees of freedom for spin converts a twolevel Hamiltonian acting on into a fourlevel Hamiltonian acting on , which we denote by . We now require that this Hamiltonian admits the timereversal and the chiral (or sublattice) symmetries by imposing the constraints,
(5a)  
(5b) 
respectively, where the overline on denotes the complex conjugation. The timereversal symmetry condition (5a) renders the Hamiltonian in the form
(6) 
Furthermore, the chiral symmetry condition (5b) brings the Hamiltonian (6) into
(7) 
We note that the Hermitian matrix of the form (6) is already found in [47, 45]. If we set with and further rewrite the parameters as
(8) 
then the Hamiltonian (7) is put in the form
(9) 
According to [56], the chiral (or sublattice) symmetry is the product of the timereversal and the particlehole symmetries. In our present case, the product of the timereversal and the chiral operators is given by
(10) 
where denotes the complex conjugation. This operator serves as the particlehole operator. As is easily seen, the Hamiltonian admits the particlehole symmetry,
(11) 
At this stage, and are merely parameters. We now break up the parameters into dynamical variables and a control parameter by taking into account the action on the Hamiltonian . Under the adjoint action of , , the transforms according to
(12) 
where is the rotation group defined through . This shows that the action is accompanied with the transform but the is left invariant. For this reason, we can interpret and as dynamical variables and a control parameter, respectively. With this in mind, we denote by , which is exactly the same as (4). Then, Eq. (12) means that the Hamiltonian defined on is equivariant with respect to the action. In a summary, we list the conditions that the satisfies for the timereversal, the particlehole, and the chiral symmetries, in this order,
(13)  
(14)  
(15) 
We make a further remark on the Hamiltonian from a generic point of view. Though we have set the parameters according to (8), we may set the parameters to take a more generic form. For example, a generic Hamiltonian is expressed as , where and are variables on a threedimensional manifold and a control parameter, respectively. If there exist a critical point and a critical parameter value such that , the Hamiltonian is viewed as a linearization of at . The linearization of a Hamiltonian at a critical point is effectively used both in the topological insulator theory [25] and in the molecular physics [37, 41], in the latter of which the linearization at a degeneracy point for eigenvalues of a semiquantum Hamiltonian is rigorously treated to obtain “deltaCherns,” while the base manifold is the unit twosphere.
In the corresponding full quantum model, the Dirac Hamiltonian may admit the timereversal, the particlehole, and the chiral symmetries as well. To translate the symmetry equations for into those for , we need to take into account the inversion . This is because in the corresponding full quantum model, the momentum operators undergo the inversion when the complex conjugation is applied. For this purpose, we may use the operator , which has the transformation property
(16) 
Then, by the operation with for Eqs. (13) and (14) and without the operation for Eq. (15) and further by replacing by , we find that the Dirac Hamiltonian satisfies, respectively, the equations for the timereversal, the particlehole, and the chiral symmetries,
(17)  
(18)  
(19) 
In closing this section, we have to mention the parity. For a fourcomponent spinor defined on , the parity transformation is defined to be , , where is given in (2) and expressed also as . As is well known, the Dirac Hamiltonian is parity invariant and transforms according to
(20) 
3 Full quantum Dirac models
This section, broken up into several subsections, deals with the eigenvalue problem for the Dirac operator (3) on the ball of radius under both the APS and the chiral bag boundary conditions.
3.1 A review of the total angular momentum operators
In order to solve the Dirac equation by the use of rotational symmetry, we need to review the total angular momentum operators and the spinor spherical harmonics on .
3.1.1 The symmetry
As is already adopted in (12), the action on is expressed by the matrix
(21) 
Let be a fourcomponent spinor defined on . Then, the actions of on and on are related through the diagram
(22) 
Here, the space is identified with , the set of traceless Hermitian matrices, and the is represented as the action such as
(23) 
where is a skewsymmetric matrix defined through and where denote the standard basis vectors of .
The diagram (22) determines the unitary operator acting on spinors by
(24) 
For , the generator of the unitary operator is denoted by and expressed as
(25) 
The operators are called the total angular momentum operators or the spinorbital angular momentum operators, satisfying the commutation relations
(26) 
Like (12), the Dirac Hamiltonian admits the symmetry
(27) 
For and , the above equation is differentiated with respect to at to provide
(28) 
which implies that the Hamiltonian and the total angular momentum operators commute,
(29) 
It then follows that the eigenvalue problem reduces to subproblems on the eigenspaces of or on the representation spaces of .
3.1.2 The representation spaces for
We proceed to write out the basis states of the representation spaces for the total angular momentum operators. In view of the expression (25), we see that the representation space is decomposed into the direct sum of two spaces of twocomponent spinors. We now provide basis spinors of the representation spaces, viewing the operator as acting on twospinors. On account of the ClebschGordan theorem applied to the spin and the orbital angular momentum coupling, for the eigenvalues of , there are two possibilities of constructing the same value of ,
(30) 
where is the parameter for the representation of the orbital angular momentums and are spin eigenvalues.
Basis spinors of the representation spaces for the total angular momentum are composed of spherical harmonics and basis vectors for spin matrices (see Appendix A for the construction of basis spinors). The basis spinors are given, in the case of , by
(31) 
and in the case of by
(32) 
where both and are eigenstates of with the eigenvalue . We refer to the twocomponent spinors and as the spinup and the spindown states, respectively. From (31) and (32), it turns out that the the space of fourcomponent spinors as the representation space of the total angular momentum with the eigenvalue is given by
(33) 
In describing the fourcomponent spinors, we may express them as
(34) 
Though these fourcomponent spinors are related by the chiral operator , we have to distinguish them. In fact, as will be seen later, the APS boundary condition is invariant under the chiral operator, but the chiral bag boundary condition is not so. A way to distinguish them is to refer to parity. By using a formula about spherical harmonics (see Appendix A), we see that the and possess opposite parity,
(35) 
3.1.3 Some formulae
We here give some formulae in advance, which will be used in solving the Dirac equation. We first need to know how and transform under the action of the operators . A straightforward calculation with the formula (229) given in Appendix A provides
(36a)  
(36b) 
On introducing the operator
(37) 
two types of the fourspinors given in (34) can be distinguished by the property
(38) 
The operator was introduced in [15] and shown to commute with the Hamiltonian of the form . Likewise, the Hamiltonian is shown to commute with ; , so that each of the eigenvalues serves as a quantum number.
We proceed to discuss the transformation of the basis states under the action of . A straightforward calculation with the recursion formulae for the associated Legendre functions, which are given in Appendix A, provides
(39) 
These formulae show that the action of exchanges the spinup and the spindown states.
3.2 Dirac Hamiltonian in spherical polar coordinates
Because of the symmetry, the Hamiltonian can be expressed in terms of the polar spherical coordinates as
(40) 
where
(41) 
and where
(42a)  
(42b)  
(42c) 
Furthermore, we can show that
(43) 
where denotes the identity matrix (see Appendix A for the proof).
3.3 A boundary operator on the sphere
In order to pose the APS boundary condition on the sphere of radius , we need a boundary operator. By restricting the to the sphere of radius , we define the operator . The boundary operator is then defined to be after [4] and written out, on account of (43), as
(44) 
In order to see a property of , we use the formula
(45) 
which can be verified in a straightforward manner. On account of the formula (45), we show that the gamma matrix has the exchanging property for the eigenstates of . Let be an eigenstate of associated with a positive eigenvalue , that is, . Operating this equation with and using the relation (45), we obtain
(46) 
If , this equation implies that is an eigenstate associated with a negative eigenvalue of . This fact will play a key role to posing the APS boundary condition. The is in principle the same as that used in [5].
3.4 Feasible solutions to the Dirac equation
We set out to solve the Dirac equation in the spherical polar coordinates. The APS and the chiral bag boundary conditions will be applied in Sec. 3.8 and in Sec. 3.9, respectively, to the feasible solutions obtained in this section. On account of (43) together with , we put the eigenvalue equation in the form
(47) 
In applying the separation of variable method, Eq. (34) shows that there are two ways to express unknown states,
(48) 
where are unknown functions. Writing out Eq. (47) for , the left one of the two unknown states given in (48), and using the formulae about the actions of and on given in Sec. 3.1.3, we obtain the radial equations
(49a)  
(49b) 
In a similar manner, Eq. (47) with reduces to the following radial equations
(50a)  
(50b) 
We are to find solutions to respective radial equations (49) and (50). In what follows, the procedure for solving those equations is divided into three, according to , , and , and each of cases is further broken up into two, according to whether the unknown state is of the form or (see (48)).
3.4.1 The case of
The coupled firstorder differential equations (49) for are put together to give rise to the uncoupled secondorder differential equations
(51a)  
(51b) 
For , these equations are modified spherical Bessel equations. We set
(52) 
Then, solutions to (51) take the form
(53) 
where are constants and the are modified Bessel functions of the first kind, and where modified Bessel functions of the second kind have been deleted on account of the regularity of solutions at the origin . The constants are to be related through (49). By using the recursion formula for the modified spherical Bessel functions,
(54a)  
(54b) 
we find that
(55) 
Since the region defined by in the space consists of two connected components distinguished by the sign of (see Fig. 1), this relation leads to
(56a)  
(56b) 
It turns out that solutions to the Dirac equation for take the form
(57a)  
(57b) 
We proceed to (50). Carrying out the same procedure, we eventually find that solutions to the Dirac equation for take the form
(58a)  
(58b) 
3.4.2 The case of
The coupled firstorder differential equations (49) for are put together to provide the uncoupled secondorder differential equations
(59a)  
(59b) 
For , these equations are spherical Bessel equations. On introducing the parameter
(60) 
solutions to (59) are put in the form
(61) 
where are constants and the are Bessel functions, and where Neumann functions have been deleted on account of the regularity of solutions at the origin . The constants are related through (49). By using the recursion formula for the spherical Bessel functions,
(62a)  
(62b) 
we find from (49) that
(63) 
Since the region defined by in the space is broken up into two, according to the sign of (see Fig. 1), this relation leads to
(64a)  
(64b) 
It then follows that solutions to the Dirac equation for take the form
(65a)  
(65b) 
We turn to the radial equations (50). Following a similar procedure, we verify that solutions to the Dirac equation for take the form
(66a)  
(66b) 