Band rearrangement through the 3D-Dirac equation with boundary conditions, and the corresponding topological change
Rearrangement of energy bands against a parameter is studied through the 3D-Dirac equation on a ball in under the APS and the chiral bag boundary conditions on the boundary two-sphere, where APS is an abbreviation of Atiyah-Patodi-Singer. 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 semi-quantum 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 semi-quantum 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 bulk-edge correspondence.
Toshihiro Iwai and Boris Zhilinskii
Kyoto University, 606-8501 Kyoto, Japan
Université du Littoral Côte d’Opale, Dunkerque, France
E-mail: email@example.com, firstname.lastname@example.org
Keywords: energy band, Dirac equation, APS and chiral bag boundary conditions, mapping degree, bulk-edge correspondence
Mathematics Subject Classification 2010: 35Q41, 53C80, 81Q70, 81V55
A relation between band rearrangement and topological change in quantum states or what is called the bulk-edge 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 , gap node , gapless excitation, contact of the conduction band and the valence band at a single point , and Dirac points , 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 high-energy excitations . 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 rotation-vibrational problem is taken as a reference problem. In this modeling, the rotational variables describing high-density energy levels due to low-energy excitations can be treated as classical ones which range over a two-dimensional sphere on account of the conservation of the angular momentum, but the vibrational variables describing high-energy excitations remain to be treated as quantum ones. Such construction is called a semi-quantum model and can be naturally generalized to quantum systems possessing subsystems of “slow” and “fast” variables of different nature.
Within semi-quantum rotation-vibration 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 semi-quantum Hamiltonians is defined to be Hermitian matrices on the two-sphere 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 semi-quantum Hamiltonians, the Chern number is defined for the complex line bundles associated with each of non-degenerate 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  and  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  and hence one needs to set up classical and quantum mechanics in terms of fiber bundles by taking into account the non-separability 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 rotation-vibration 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 .
At this stage in Introduction, it is significant to compare the idea as to fast- and slow variables in molecular physics with bulk-edge 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 two-band system, and the bulk-state energy levels are of high density, so that the bulk states can be treated in terms of classical variables in a semi-quantum 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 , bulk and boundary invariants for lattice systems are studied by means of K-theory.
Interest in topological study of intersection of potential energy surfaces on the basis of Hermitian matrices dates back to  and led to the Berry phase [11, 12, 62]. Hermitian matrices have been effectively used [55, 63] in semi-quantum molecular models as well. The relevance of Chern numbers to band rearrangements in isolated molecules was initially suggested in  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 two-torus in contrast to the two-sphere for rotation-vibration 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 semi-quantum models [37, 41], the authors have observed that elementary band rearrangement (or level-crossing) 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 delta-Chern. Further, they pointed out, in previous papers [39, 40], that the linearization method works well for the study of band rearrangement both in semi-quantum 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 semi-quantum model. It was shown that the delta-Chern [37, 41] defined for a parametric family of Hamiltonians in the semi-quantum model corresponds to the spectral flow or its extension for the 2D Dirac equation in question, where the spectral flow of a one-parameter family of self-adjoint 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 , and where the delta-Chern is a jump in the Chern number which takes place when the parameter crosses a critical value. The authors have already given non-linear two-dimensional models which exhibits the correspondence between spectral flows and changes in Chern numbers . The present correspondence is, in principle, the same as the bulk-edge correspondence for topological insulators without time-reversal symmetry .
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 Atiyah-Patodi-Singer . The corresponding semi-quantum 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 delta-Chern 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 bulk-edge correspondence holds for a wider class of model Hamiltonians or not. A further way to approach the present question is to consider higher-dimensional cases. If group actions such as a time-reversal transformation with half-integer spin degrees of freedom are taken into account, then new Hamiltonians comes to be defined in dimensions larger than two, in both quantum and semi-quantum 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 semi-quantum Hamiltonian, both of which admit the time-reversal and the chiral symmetries. Since the Chern number is defined for vector bundles over even-dimensional manifolds and since the manifold on which the model Hamiltonians are to be defined is of odd-dimensions, 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 semi-quantum Hamiltonian.
Among several representations of the gamma matrices, the choice made in the present article is
where denotes the unit matrix. Then, the 3D Dirac Hamiltonian takes the form
where is a parameter, which originally denotes the mass but is allowed to take negative values in the present study. In Dirac’s textbook , 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 edge-states eigenvalues, where the edge-state eigenvalues are responsible for band rearrangement but the bulk-state eigenvalues form separate two bands, positive and negative. The spectral flow and its extension are defined for edge-state 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 bulk-state energy levels is inversely proportional to the radius of the ball, so that the bulk-state eigenvalues become of high density for a sufficiently large . Hence, the bulk-state bands are allowed to be treated in terms of classical variables or in momentum variables. Accordingly, the quantum Hamiltonian turns into a semi-quantum 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 semi-quantum 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 semi-quantum 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 particle-hole symmetry of the quantum and semi-quantum Hamiltonians. Thus, the bulk-edge correspondence proves to hold in the present setting.
This paper is organized as follows: In Sec. 2, the Dirac Hamiltonian (3) and the corresponding semi-quantum Hamiltonian are characterized from the viewpoint of the time-reversal 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 two-sphere 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 pseudo-symmetry) can explain the pattern of eigenvalues as functions of the parameter . In the last subsection of this section, the -dependence of the bulk-state eigenvalues is discussed and then it is shown that the semi-quantum Hamiltonian can be viewed as a bulk Hamiltonian in the limit as . In Sec. 4, the corresponding semi-quantum 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 bulk-edge 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
which we call a semi-quantum Dirac Hamiltonian. In this section, we characterize the Hamiltonian (4) after the classification scheme for Hermitian matrices by means of discrete symmetries such as time-reversal, particle-hole, and chiral symmetries [3, 26, 64]. Remarks on the so-called AZ classification will be made in the last section, as far as the present Hamiltonian is concerned.
According to [47, 45], the introduction of time-reversal symmetry along with the spin degrees of freedom for spin- converts a two-level Hamiltonian acting on into a four-level Hamiltonian acting on , which we denote by . We now require that this Hamiltonian admits the time-reversal and the chiral (or sublattice) symmetries by imposing the constraints,
respectively, where the overline on denotes the complex conjugation. The time-reversal symmetry condition (5a) renders the Hamiltonian in the form
then the Hamiltonian (7) is put in the form
According to , the chiral (or sublattice) symmetry is the product of the time-reversal and the particle-hole symmetries. In our present case, the product of the time-reversal and the chiral operators is given by
where denotes the complex conjugation. This operator serves as the particle-hole operator. As is easily seen, the Hamiltonian admits the particle-hole symmetry,
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
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 time-reversal, the particle-hole, and the chiral symmetries, in this order,
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 three-dimensional 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  and in the molecular physics [37, 41], in the latter of which the linearization at a degeneracy point for eigenvalues of a semi-quantum Hamiltonian is rigorously treated to obtain “delta-Cherns,” while the base manifold is the unit two-sphere.
In the corresponding full quantum model, the Dirac Hamiltonian may admit the time-reversal, the particle-hole, 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
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 time-reversal, the particle-hole, and the chiral symmetries,
In closing this section, we have to mention the parity. For a four-component 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
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
Let be a four-component spinor defined on . Then, the actions of on and on are related through the diagram
Here, the space is identified with , the set of trace-less Hermitian matrices, and the is represented as the action such as
where is a skew-symmetric matrix defined through and where denote the standard basis vectors of .
The diagram (22) determines the unitary operator acting on spinors by
For , the generator of the unitary operator is denoted by and expressed as
The operators are called the total angular momentum operators or the spin-orbital angular momentum operators, satisfying the commutation relations
Like (12), the Dirac Hamiltonian admits the symmetry
For and , the above equation is differentiated with respect to at to provide
which implies that the Hamiltonian and the total angular momentum operators commute,
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 two-component spinors. We now provide basis spinors of the representation spaces, viewing the operator as acting on two-spinors. On account of the Clebsch-Gordan 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 ,
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
and in the case of by
where both and are eigenstates of with the eigenvalue . We refer to the two-component spinors and as the spin-up and the spin-down states, respectively. From (31) and (32), it turns out that the the space of four-component spinors as the representation space of the total angular momentum with the eigenvalue is given by
In describing the four-component spinors, we may express them as
Though these four-component 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,
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
On introducing the operator
two types of the four-spinors given in (34) can be distinguished by the property
The operator was introduced in  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
These formulae show that the action of exchanges the spin-up and the spin-down 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
Furthermore, we can show that
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  and written out, on account of (43), as
In order to see a property of , we use the formula
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
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 .
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
In applying the separation of variable method, Eq. (34) shows that there are two ways to express unknown states,
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
In a similar manner, Eq. (47) with reduces to the following radial equations
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 first-order differential equations (49) for are put together to give rise to the uncoupled second-order differential equations
For , these equations are modified spherical Bessel equations. We set
Then, solutions to (51) take the form
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,
we find that
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
It turns out that solutions to the Dirac equation for take the form
We proceed to (50). Carrying out the same procedure, we eventually find that solutions to the Dirac equation for take the form
3.4.2 The case of
The coupled first-order differential equations (49) for are put together to provide the uncoupled second-order differential equations
For , these equations are spherical Bessel equations. On introducing the parameter
solutions to (59) are put in the form
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,
we find from (49) that
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
It then follows that solutions to the Dirac equation for take the form
We turn to the radial equations (50). Following a similar procedure, we verify that solutions to the Dirac equation for take the form