Systematic Construction of tight-binding Hamiltonians for Topological Insulators and Superconductors
Abstract
A remarkable discovery in recent years is that there exist various kinds of topological insulators and superconductors characterized by a periodic table according to the system symmetry and dimensionality. To physically realize these peculiar phases and study their properties, a critical step is to construct experimentally relevant Hamiltonians which support these topological phases. We propose a general and systematic method based on the quaternion algebra to construct the tight binding Hamiltonians for all the three-dimensional topological phases in the periodic table characterized by arbitrary integer topological invariants, which include the spin-singlet and the spin-triplet topological superconductors, the Hopf and the chiral topological insulators as particular examples. For each class, we calculate the corresponding topological invariants through both geometric analysis and numerical simulation.
pacs:
73.43.-f, 74.20.-z, 03.65.Vf, 02.40.ReTopological insulators (TIs) and superconductors (TSCs) are symmetry protected topological phases of non-interacting fermions described by quadratic Hamiltonians Hasan and Kane (2010); ?, which have robust gapless boundary modes protected by the system symmetry Chen et al. (2013). These protected boundary modes have exotic properties and in some cases are characterized as anyons with fractional statistics Nayak et al. (2008), which could be used for the realization of topological quantum computation Kitaev (2003). Notable examples of TIs include the integer quantum Hall states Sarma and Pinczuk (2008) and the recently discovered two-dimensional (2D) quantum spin Hall states Bernevig et al. (2006); ? and the three-dimensional (3D) TIs Fu et al. (2007); ?; ?; ?. Examples of TSCs include the 2D superconductors of spinless fermions Read and Green (2000) and the Helium superfluid B-phase Kitaev (2009); Ryu et al. (2010).
It turns out that the above TI/TSC examples are just a part of a larger scheme: they sit inside a periodic table for TIs/TSCs developed according to symmetry and dimensionality of the system Ryu et al. (2010); Schnyder et al. (2008); Kitaev (2009). The periodic table predicts possible existence of a number of new topological phases, and it is of great interest to search for these new phases in nature. However, the periodic table does not tell where to look for or how to realize these phases. To physically realize these exotic phases and study their properties, it is of critical importance to construct tight-binding Hamiltonians so that they could be realized in real quantum materials such as optical lattice systems Bloch et al. (2012). So far, some clever example Hamiltonians have been found for a few new topological phases Qi et al. (2009); Roy (2008); Schnyder et al. (2008); Moore et al. (2008), typically with the topological invariant , but we lack a systematic method to construct tight-binding Hamiltonians for generic topological phases with arbitrary integer topological invariants.
In this paper, we propose a general and systematic method to construct tight binding Hamiltonians for new topological phases based on the use of quaternion algebra. By this method, we construct the tight-binding Hamiltonians for all the 3D topological phases in the periodic table with arbitrary integer topological invariants, which include the spin-singlet and the spin-triplet TSCs, the chiral and the Hopf TIs as prototypical examples. For each class, the topological invariant is explicitly calculated for the constructed Hamiltonian, using both geometric analysis and numerical simulation, which confirm that we indeed realize all the topologically distinct phases in the corresponding class characterized by a topological invariant of arbitrary integer values. The construction method proposed here should also work for the 2D and 1D cases, and we believe its direct generalization to Clifford algebra should provide a powerful tool to construct tight-binding Hamiltonians for all the integer topological phases in the periodic table.
Before showing the method, let us first briefly introduce the quaternion algebra , which is a generalization of the familiar complex algebra, with the imaginary basis-vectors extended from one () to three (). The basis-vectors () multiply according to the following non-commutative product table Nakahara (2003):
(1) | |||||
Any element of can be expanded as , where () are real numbers. Quaternion has been used recently as a tool to analyze the 3D Landau levels Li and Wu (2013).
For our purpose, it is more convenient to write in the polar-like coordinate with where is the norm of , is the angle, and with is a unity vector denoting the direction in the imaginary space. From the definition, we immediately get
(2) |
To construct tight-binding lattice Hamiltonians for the TIs or TSCs, we typically work in the momentum space. The Hamiltonian coefficients are taken as components of a quaternion , which in general depend on the momentum through the notation . In a -dimensional () space, the momentum takes values from the Brillouin zone (BZ) characterized by a -dimensional torus . The norm of the quaternion characterizes the energy scale (energy gap) of the Hamiltonian, which can be taken as (the energy unit) without loss of generality, and the topological space of is thus characterized by the sphere . The Hamiltonian with as the coefficients can be considered as a map from the space to . If this map is topologically nontrivial characterized by a topological invariant (usually taken as the winding number or Chern number) , geometrically (in the sense of homotopy) it means that the space wraps around one time through the map. Now consider a Hamiltonian where the coefficients are taken as the components of . From the geometric representation of in the polar coordinate in Eq. (2), if the space wraps around one time through the map with , it will wrap times through the map with . So, by this quaternion-power mapping, we can construct Hamiltonians for topologically distinct new phases with arbitrary integer topological invariants. This serves as our physical intuition to construct tight-binding Hamiltonians for new topological phases. In the following, we apply this method to construction of the Hamiltonians for all the 3D topological phases in the periodic table characterized by the integer group , which include the spin-singlet and the spin-triplet TSCs, the chiral and the Hopf TIs.
Spin-singlet TSC (class CI)— The topological phases in the periodic table are classified by three generic symmetries, the time reversal, the particle-hole exchange (charge conjugation), and the chiral symmetry, denoted respectively by with . These three symmetries can be used to classify the system even when weak disorder breaks the lattice translational symmetry. The symmetries and are represented by anti-unitary operators, and can take values either or depending on the effective spin of the system.
Spin singlet TSC is described by a Bogoliubov-de-Gennes (BdG) type of mean-field Hamiltonian and belongs to the symmetry class CI in the periodic table, which means the BdG Hamiltonian has both and symmetries with and . The topological phase is characterized by a topological invariant , which takes values from (even integers). Ref. Schnyder et al. (2009) has proposed a Hamiltonian in a diamond lattice which realizes a special instance of the CI TSC with . Here, we construct tight binding Hamiltonians which can realize all the topologically distinct phases for the CI TSC with arbitrary even integer in a simple cubic lattice. The simplified lattice geometry could be important for an experimental implementation.
To construct the BdG Hamiltonian in the momentum space, first we define a quaternion with the following dependence on the momentum
(3) |
where is a dimensionless parameter. A family of the BdG Hamiltonians can be constructed on the 3D cubic lattice with the form in the momentum space, where denotes the fermionic mode operators with spin and momentum . The Hamiltonian matrix reads
(4) |
where with denoting the th-components of the quaternion , is the identity matrix, and are the Pauli matrices. Expressed in the real space, the Hamiltonian contains spin-singlet -wave pairing described by the quaternion component , and has local hopping and pairing terms up to the th neighboring sites. One can check that indeed has both and symmetries (and thus also the chiral symmetry ) with and (see the supplement for an explicit check Sup ()).
Now we show that the Hamiltonian has topologically distinct phases depending on the parameters and . For this purpose, we need to calculate the topological invariant for . Direct diagonalization of the Hamiltonian leads to the energy spectrum . It is always gapped if and has a twofold degeneracy for each . To calculate the topological index , we first flatten the bands of (which is a continuous transformation that does not change its topological property) by introducing the matrix,
(5) |
where is the projector onto the filled Bloch bands (with energy and wave-vectors from the diagonalization of ). With the chiral symmetry, the matrix can be brought into the block off-diagonal form by a unitary transformation, with
(6) |
for the Hamiltonian . With the matrix , the topological index is defined by the following winding number Ryu et al. (2010):
(7) |
where is the antisymmetric Levi-Civita symbol and . When , the integral in can be calculated analytically and we find sign. In general cases, due to the geometric interpretation of the map , we immediately get
(8) |
This result is confirmed through direct numerical calculations. We integrate Eq. (7) numerically through discretization of the Brillouin zone. The calculation results for different are shown in Fig. 1(a), which quickly converge to the exact results in Eq. (8) as the number of integration grids increases. As one varies , it is evident from Eq. (8) that we can realize all the spin-singlet TSC phases in the CI class through our constructed Hamiltonian with the topological index taking arbitrary even integers.
Spin-triplet TSC (class DIII) — Spin triplet TSCs are described by the BdG Hamiltonians that have both and symmetries with and . It belongs to the symmetry class DIII in the periodic table. The superfluid B phase is a well known example in this class Kitaev (2009); Ryu et al. (2010), but it is not described by a simple lattice model. Tight-binding lattice Hamiltonians have been constructed for the DIII-class spin triplet TSCs with the topological index Fu and Berg (2010); Yan et al. (2010); Sato (2010). Here, we use the quaternion method to construct tight-binding Hamiltonians for the DIII-class TSCs with arbitrary integer topological indices in a simple cubic lattice.
To construct the Hamiltonian, we define a quaternion with the following dependence on
(9) |
with being dimensionless parameters. We will use this form of for all our following examples. We construct a four-band BdG Hamiltonian with the form , with the fermionic mode operators and the Hamiltonian matrix
(10) |
where , , and denote the standard Dirac matrices with the explicit expressions given in the supplement Sup (). This Hamiltonian has spin triplet pairing with the energy spectrum , which is fully gapped when and .
The DIII class TSC has the chiral symmetry, so its matrix for the Hamiltonian can be brought into the block off-diagonal form Sup () and the topological index is also characterized by the winding number in Eq. (7). We find
(11) |
It is evident that the topological index can take arbitrary integer values for our constructed Hamiltonian depending on the parameters . In the particular case with , the Hamiltonian reduces to the model Hamiltonian introduced in Ref. Schnyder et al. (2008); Ryu et al. (2010), which has or .
Chiral TI (class AIII)— Chiral TIs do not have time-reversal or particle-hole symmetry (thus ), but they possess chiral symmetry with and belongs to the symmetry class AIII in the periodic table. Tight-binding Hamiltonians have been constructed for the chiral TIs with the topological index Neupert et al. (2012). Here, we use the quaternion method to construct Hamiltonians with arbitrary integer . We consider a three-band Hamiltonian with the following form , where the fermionic mode operators and the Hamiltonian matrix
(12) |
In , denotes the same quaternion coefficients as defined below Eq. (10) and are the four Gell-Mann matrices with the explicit form given in the supplement Sup (). The Hamiltonian is gapped when and and has a perfectly flat middle band with a macroscopic number of zero-energy modes due to the chiral symmetry Neupert et al. (2012). A topological invariant classifying this family of Hamiltonians can be defined as Neupert et al. (2012)
(13) |
We have calculated this invariant and found that
(14) |
for our constructed . This analytic result is confirmed with direct numerical calculations as shown in Fig. 1(b). In the particular case with , the Hamiltonian reduces to the model Hamiltonian constructed in Ref. Neupert et al. (2012). Through the quaternion power, we extend the model Hamiltonian and realize the chiral TIs with the topological index taking arbitrary integer values.
Hopf insulators (class A)— The Hamiltonians in class A do not have any symmetry (, , or ) except the particle number conservation. Generically, this class of Hamiltonians have no topologically nontrivial phase in 3D, but there is a peculiar exception, called the Hopf insulator, which occurs when the Hamiltonian has just two bands due to existence of the topologically nontrivial Hopf map from to Moore et al. (2008); Deng et al. (2013). The Hopf insulators are characterized by the topological Hopf index , which takes values from the integer set . A model Hamiltonian has been constructed for the Hopf insulator with in Ref. Moore et al. (2008) based on the Hopf map Whitehead (1947). This method was extended in Ref. Deng et al. (2013) to construct Hamiltonians for general Hopf insulators with arbitrary integer based on the generalized Hopf map encountered in mathematics but not in physics literature. Here, with the quaternion algebra, we use only the simple Hopf map but still can construct tight-binding Hamiltonians for the Hopf insulators with arbitrary integer .
To construct the Hamiltonian, we define two complex variables from the quaternion , where is defined by Eq. (9). The Hopf map is defined as , which is a quadratic map from up to normalization. The two-band Hamiltonians can then be constructed as with and
(15) |
The Hopf insulators are characterized by the topological Hopf index, define as
(16) |
where is the Berry curvature with and is the associated Berry connection which satisfies Moore et al. (2008); Deng et al. (2013). From this definition and our geometric interpretation of , we find
(17) |
This analytical expression is also confirmed with direct numerical calculations. Some numerical results for the topological indices are listed in Table (1) for different classes of TIs and TSCs, which agree very well with our analytical expressions.
Before ending the paper, we briefly remark that the quaternion tool proposed here can be extended straightforwardly to the 1D and 2D cases although our focus in this paper is on the 3D topological phases. We can set one (two) of the quaternion components to zero for the 2D (1D) case and observe that the map always preserves the subspace of spanned by ) (). With the power mapping which preserve the symmetry of the Hamiltonian, starting from one particular example of topological Hamiltonians with the topological index , we can always construct a family of Hamiltonians which realize all the topological phases with arbitrary integer .
In summary, we have proposed a powerful tool based on the quaternion algebra to systematically construct tight-binding Hamiltonians for all the topological phases in the periodic table that are characterized by arbitrary integer topological indices. The constructed Hamiltonians make the basis for further studies of properties of these topological phases and phase transitions and provide an important step for future experimental realization.
Acknowledgements.
We thank K. Sun and C. J. Wu for helpful discussions. This work was supported by the NBRPC (973 Program) 2011CBA00300 (2011CBA00302), the IARPA MUSIQC program, the ARO and the AFOSR MURI program.Supplementary Material: Systematic Construction of tight-binding Hamiltonians for Topological Insulators and Superconductors
In this supplementary material, we explicitly check the symmetries for our constructed Hamiltonians. We also give some details for the description of these Hamiltonians and the calculation of their corresponding topological indices.
We first specify the definition of time-reversal (), particle-hole (charge conjugation ), and chiral () symmetries in the momentum () space. A Hamiltonian is represented by a finite matrix in the space (kernel of the Hamiltonian). It has the time reversal symmetry if there exists a unitary matrix such that
(18) |
Similarly, has the particle-hole symmetry if there is a unitary matrix such that
(19) |
The anti-unitary nature of the time reversal and the particle-hole symmetries is manifested in the complex conjugate in Eqs. (1) and (2). Finally, as , the chiral symmetry is unitary and represented by ()
(20) |
Pertaining to the presence/absence of these symmetries, ten classes of single-particle Hamiltonians can be specified, which is intimately related to the classification of random matrices by Altland and Zirnbauer (AZ) Altland and Zirnbauer (1997).
Appendix A Spin-singlet topological superconductor
We first prove that the Hamiltonian for the spin-singlet topological superconductors (TSCs) constructed in Eq. (4) of the main text has the CI-class symmetry with and . Let us denote the parity of a function by , with () for an even (odd) parity under the exchange . From the product table of the quaternion algebra, it is easy to check that with and , we have and for any integer power . The explicit expression of the Hamiltonian in eq. (4) of the main text is
(21) |
From the parity of , the time reversal symmetry can be readily seen as
(22) |
so , the identity matrix. The particle-hole symmetry can be seen as
(23) |
with , where denote the Pauli matrices. Apparently, and (as ), as it is the case for the CI-class symmetry.
To calculate the topological invariant, we note that the system also has the chiral symmetry and the matrix (defined in the main text) can thus be brought into the block off-diagonal form by a unitary transformation Schnyder et al. (2008); Ryu et al. (2010). At the half filling (therefore inside the energy gap) and with a convenient gauge, direct calculation leads to
(24) |
with , as mentioned in the main text.
Appendix B Spin-triplet topological superconductor
The Dirac matrices (also known as the gamma matrices) are a set of matrices, defined as
The fifth gamma matrix is defined by Using the explicit form of these gamma matrices, the Hamiltonian matrix can be written as
(25) |
The matrix can thus be written as
(26) |
with . Note that for the quaternion defined in Eq. (9) of the main text, we have and thus . With the parity properties, one can easily check that
(27) | |||||
(28) |
So the symmetry matrix and with and (as and ), as it is the case for the DIII-class symmetry.
Appendix C Chiral topological insulator
The four Gell-Mann matrices used in the text are defined as
(29) |
The Hamiltonian matrix for the chiral topological insulator has the following explicit form
(30) |
The Hamiltonian does not have time-reversal or particle-hole symmetry, but it has a chiral symmetry with the unitary matrix
(31) |
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