Band Theory Without Any Hamiltonians or “The Way Band Theory Should Be Taught”

Abstract

In this chapter, we introduce the theory of Topological Quantum Chemistry. Within this formalism, we can predict the presence or absence of topological phases by studying the behavior of orbitals lying in some special positions of the crystal. Throughout the chapter, we analyze and study the main concepts of the theory following a well-known example, graphene.

Appendices

Appendix A: Definitions

Definition A.1

(Bravais lattice) A Bravais lattice is an infinite set of translations $\mathbf {t}$ generated by d linearly independent vectors $\mathbf {a}_i$, where d is the dimension of the crystal

$$\begin{aligned} \mathbf {t}=n_1 \mathbf {a}_1 +\cdots +n_d \mathbf {a}_d\;\;, \;\ n_i\in Z \end{aligned}$$

(A.1)

The Bravais lattice is thus isomorphic to $Z^d$.

Definition A.2

(Crystal) A crystal is a Bravais lattice arrangement of atoms, invariant under a space group G.

Definition A.3

(Group of the crystal) The group of the crystal is the space group G under which the crystal remains invariant. G is always an infinite group, as it includes all integer translations along the Bravais lattice. In Seitz notation, the elements of a space group G are denoted as

$$\begin{aligned} g=\{R|\mathbf {r}\} \end{aligned}$$

(A.2)

where R is a point group element and $\mathbf {r}$ is a translation, which may or may not belong to the Bravais lattice. The action of $g\in G$ on a real space point $\mathbf {q}$ is given by

$$\begin{aligned} g \mathbf {q}=\{R|r\}\mathbf {q} = R \mathbf {q} +\mathbf {r} \end{aligned}$$

(A.3)

The Bravais lattice is always a subgroup of the space group G. Its elements are of the form $\{E|\mathbf {t}\}$, where E is the identity operation.

Definition A.4

(Stabilizer group/Site-symmetry group) The stabilizer group or site-symmetry group of a position q is the set of symmetry operations $g\in G$ that leave q fixed. It is denoted by $G_q=\{g|gq=q\}\subset G$. There are a couple of things to remark:

• $g\in G_q$ may include a translation, $g=\{R|\mathbf {r}\}$, with $\mathbf {r}\ne 0$

• However, since any site-symmetry group leaves a point invariant, $G_q$ is necessarily isomorphic to one of the 32 crystallographic point groups.

Definition A.5

(Wyckoff position) A general Wyckoff position is a position q in the unit cell of the crystal with a trivial site-symmetry group, i.e., the only element in $G_q$ is the identity operation. A special Wyckoff position is a position q in the unit cell of the crystal with a non-trivial site-symmetry group; i.e., q is invariant under some symmetry operations, such as mirror planes and rotation axis.

Definition A.6

(Orbit of q) The orbit of q is the set of all positions which are related to q by elements of the symmetry group G; i.e., $Orb_q=\{gq|g\in G\}$ and belong to the same unit cell.

Definition A.7

(Coset representatives) The coset representatives of a site-symmetry group can be defined as the set of elements that generate the orbit of a Wyckoff position. Then each element $q_\alpha$ in the orbit of q may be written as $q_\alpha =g_\alpha q$.

Definition A.8

(Coset decomposition) The coset decomposition of the full space group is defined by

$$\begin{aligned} G=\bigcup _\alpha g_\alpha (G_q\ltimes Z^d) \end{aligned}$$

(A.4)

where $G_q$ is the site-symmetry group and $g_\alpha$ are the coset representatives. The piece multiplying the coset representatives is obtained as the semi-direct product of $G_q$ and the translation group, that in d dimensions is isomorphic to $Z^d$. Each term $g_\alpha (G_q\ltimes Z^d)$ in (A.4) is a (left) coset.

This can be understood as follows. Let us take a position q with site-symmetry group $G_q$. Then $G_q$ plus the translations in the Bravais lattice creates a replica of q at every primitive cell in the crystal. Acting with each coset representative $g_\alpha$ creates, throughout the crystal, replicas of every position in the orbit of q.

Definition A.9

(Multiplicity of a Wyckoff position) The multiplicity of a Wyckoff position is defined as the number of elements (positions) in the orbit of some Wyckoff position. It is obviously equal to the number of coset representatives.

This is what motivates the names for the different maximal Wyckoff positions 1a, 2b, 3c, etc. The number tells you the multiplicity of the position, while the letter labels the positions, from more to less symmetric.

Definition A.10

(Maximal Wyckoff position) A Wyckoff position q is said to be non-maximal if there exists a group H such that $G_q\subset H\subset G$. A Wyckoff position that is not non-maximal is maximal.

A sufficient (although not necessary) condition for a position q to be maximal is that q is the unique point fixed by every operation in $G_q$. As a particular case, in 2D, any site-symmetry group that contains rotations is maximal.

Definition A.11

(Little group) Two reciprocal space vectors $\mathbf {k}_1$ and $\mathbf {k}_2$ are said to be equivalent, $\mathbf {k}_1\equiv \mathbf {k}_2$, if there exists a reciprocal lattice vector $\mathbf {K}$ such that $\mathbf {k}_2=\mathbf {k}_1 +\mathbf {K}$. Then the little group $G_{\mathbf {k}}$ of a vector $\mathbf {k}$ in reciprocal space is the set of elements $g\in G$ such that $g \mathbf {k}\equiv \mathbf {k}$. Note that the action of space group elements on reciprocal space vectors is defined by

$$\begin{aligned} g\mathbf {k}=\{R|\mathbf {t}\}\mathbf {k}=R\mathbf {k} \end{aligned}$$

(A.5)

Definition A.12

(Small representation) A small representation is a representation of the little group.

Appendix B: Proof That the Site-Symmetry Groups for the 3c Wyckoff Positions are Isomorphic to $C_{3v}$

In this appendix, we will prove two statements: First, that the site-symmetry group for the position $q=\left( \frac{1}{3},\frac{1}{3}\right)$ is isomorphic to $C_{3v}$, and second, that the site-symmetry groups for positions in the same orbit are isomorphic to each other.

B.1 Site-Symmetry Group of $q=\left( \frac{1}{3},\frac{1}{3}\right)$

First, we introduce the set of relations that define the group $C_{3v}$:

$$\begin{aligned} \begin{aligned} C_3^3=1:&\\&(x,y)\rightarrow C_3\rightarrow (y,-x-y)\rightarrow C_3\rightarrow (-x-y,x)\rightarrow C_3\rightarrow (x,y)\\ C_3m_{1\bar{1}}=&\;m_{1\bar{1}}C_3^{-1}:\\&(x,y)\rightarrow m_{1\bar{1}}\rightarrow (y,x)\rightarrow C_3\rightarrow (x,-x-y)\\&(x,y)\rightarrow C_3^2\rightarrow (-x-y,x)\rightarrow m_{1\bar{1}}\rightarrow (x,-x-y) \end{aligned} \end{aligned}$$

(B.1)

Now, let’s see if the generators of the site-symmetry group follow the same relations:

$$\begin{aligned} \begin{aligned} \{C_3|01\}^3&=1:\\&(x,y)\rightarrow \{C_3|01\}\rightarrow (y,-x-y+1)\rightarrow \{C_3|01\}\\&\rightarrow (-x-y+1,x)\rightarrow \{C_3|01\}\rightarrow (x,y)\\ \{C_3|01\}&\{m_{1\bar{1}}|0\}=\{m_{1\bar{1}}|0\}\{C_3|01\}^{-1}:\\&(x,y)\rightarrow \{m_{1\bar{1}}|0\}\rightarrow (y,x)\rightarrow \{C_3|01\}\rightarrow (x,-x-y+1)\\&(x,y)\rightarrow \{C_3|01\}^2\rightarrow (-x-y+1,x)\rightarrow \{m_{1\bar{1}}|0\}\rightarrow (x,-x-y+1)\\ \end{aligned} \end{aligned}$$

(B.2)

As we see, the group generators satisfy the same relations. Thus the groups are isomorphic.

B.2 Site-Symmetry Group of Positions in the Same Orbit

We know that the positions for the different elements in the same orbit are related to each other by

$$\begin{aligned} q_\alpha =g_\alpha q \end{aligned}$$

(B.3)

for some q in the orbit and $g_\alpha$ a coset representative. Thus, for some $h\in G_q$,

$$\begin{aligned} hq=q\rightarrow g_\alpha hg_\alpha ^{-1}q_\alpha =q_\alpha \end{aligned}$$

(B.4)

and we see that $g_\alpha hg_\alpha ^{-1}\in G_{q_\alpha }$. This is the definition of conjugate group. As two conjugate groups are isomorphic, it is enough to compute the site-symmetry group for one point in each orbit.

Appendix C: Wannier Function Transformation Properties

We will denote our Wannier functions on the unit cell by two indices: the orbital (latin) and site (greek). In the case of spinful $p_z$ orbitals on 2b Wyckoff positions (graphene), the Wannier functions will be denoted as $W_{i\alpha }$, where i denotes spin up or down, and $\alpha$ denotes the site of the orbit. Wannier functions transform around each site as orbitals:

$$\begin{aligned} gW_{i1}=[\rho (g)]_{ji}W_{j1} \end{aligned}$$

(C.1)

This follows from the Hamiltonian. If the Hamiltonian commutes with the symmetry operations, then its eigenstates²⁸ transform under representations of the symmetry group. In a unit cell, we have $\alpha$ positions in the orbit. The Wannier functions at those points are given, in terms of the functions around one position:

$$\begin{aligned} W_{i\alpha }(r)=g_\alpha W_{i1}(r)=W_{i1}(g_\alpha ^{-1}r) \end{aligned}$$

(C.2)

Let’s see under which representation these transform:

$$\begin{aligned} \begin{aligned} hW_{i\alpha }=g_\alpha gg_\alpha ^{-1}g_\alpha W_{i1}=g_\alpha gW_{i1}=g_\alpha [\rho (g)]_{ji}W_{j1}=[\rho (g_\alpha ^{-1}hg_\alpha )]_{ji}W_{j\alpha } \end{aligned} \end{aligned}$$

(C.3)

where $h\in G_{q_\alpha }$ and $g\in G_{q_1}$.

Now, we can construct all Wannier functions on the full lattice by translating these functions along the lattice. $\{E|t_\mu \}W_{i\alpha }(r)=W_{i\alpha }(r-t_\mu )$, so we have a total of $n\times n_q\times N$ Wannier functions, where $n_q$ is the number of orbitals per position in the orbit, n the multiplicity of the Wyckoff position and N the number of cells of our crystal. These functions form a basis for the representation of the space group induced from the representation of the site-symmetry group. Let the representation of the spatial group be $\rho _G$. Then, $\rho _G\equiv \rho \uparrow G$. This procedure is called induction. Let’s proceed to see how Wannier states transform under an element $h=\{R|\mathbf {t}\}$:

$$\begin{aligned} \begin{aligned} hW_{i\alpha }(\mathbf {r-t_\mu })&=h\{E|\mathbf {t}_\mu \}W_{i\alpha }(\mathbf {r})\\&=\{E|R\mathbf {t}_\mu \}hW_{i\alpha }(\mathbf {r})\\&=\{E|R\mathbf {t}_\mu +\mathbf {t}_{\beta \alpha }\}g_\beta gg_\alpha ^{-1}W_{i\alpha }(\mathbf {r})\\&=\{E|R\mathbf {t}_\mu +\mathbf {t}_{\beta \alpha }\}g_\beta gW_{i1}(\mathbf {r})\\&=\{E|R\mathbf {t}_\mu +\mathbf {t}_{\beta \alpha }\}g_\beta [\rho (g)]_{ji}W_{j1}(\mathbf {r})\\&=\{E|R\mathbf {t}_\mu +\mathbf {t}_{\beta \alpha }\}[\rho (g)]_{ji}W_{j\beta }(\mathbf {r})\\&=[\rho (g)]_{ji}W_{j\beta }(\mathbf {r}-R\mathbf {t}_\mu - t_{\beta \alpha }) \end{aligned} \end{aligned}$$

(C.4)

where in the third line we have used that the action of an element h on a Wyckoff position $q_\alpha$ is given by

$$\begin{aligned} hq_\alpha =\{E|\mathbf {t}_{\beta \alpha }\}q_\beta ,\quad g_\beta ^{-1}\{E|-\mathbf {t}_{\beta \alpha }\}hg_\alpha q_1=q_1\equiv gq_1=q_1 \end{aligned}$$

(C.5)

where the vector $\mathbf {t}_{\beta \alpha }$ represents the possibility of an element to take some Wyckoff away to another cell.²⁹ We see here that we can know how any Wannier in any position in any cell transform just by knowing how they transform around one of the positions of the orbit under an element $g\equiv g_\beta ^{-1}\{E|-\mathbf {t}_{\beta \alpha }\}hg_\alpha \in G_{q_1}$. We can obtain from (1.19) that:

$$\begin{aligned} \mathbf {t}_{\beta \alpha }=hq_\alpha -q_\beta \end{aligned}$$

(C.6)

Appendix D: Elementary Band Representation

In the main text, we have worked out an example of elementary band representation. We will give here some more general results about them. First, let’s state some facts.

We say that two band representations $\rho _G$ and $\sigma _G$ are equivalent if and only if there exists a unitary matrix-valued function $S(\mathbf {k},t,g)$ smooth in $\mathbf {k}$ and continuous in t such that, for all $g\in G$

• $S(\mathbf {k},t,g)$ defines a band representation according to (1.14) for all $t\in [0,1]$

• $S(\mathbf {k},0,g)=\rho ^{\mathbf {k}}_G(g)$

• $S(\mathbf {k},1,g)=\sigma ^{\mathbf {k}}_G(g)$

In the analyzed case of graphene, t would be the parameter of the line that connects two points.

A necessary condition is that both $\rho ^{\mathbf {k}}_G(g)$ and $\sigma ^{\mathbf {k}}_G(g)$ restrict to the same little group representations at all points in the Brillouin Zone. However, it is not sufficient: It may happen that both representations satisfy this condition but $S(\mathbf {k},t,g)$ is not a band representation for all t. We need a sufficient condition for equivalence:

Given two sites $q,\,q'$ (not necessarily in the same Wyckoff position) and representations of their site-symmetry groups ($\rho$ of $G_q$ and $\rho '$ of $G_{q'}$), the band representations $\rho \uparrow G$ and $\rho '\uparrow G$ are equivalent if and only if there exists a site $q_0$ and representation $\sigma$ of $G_{q_0}$ such that $\rho =\sigma \uparrow G_q$ and $\rho '=\sigma \uparrow G_{q'}$.

Now let’s discuss the compositeness of a band representation; i.e., if it is elementary or composite. We say that a band representation is composite if it can obtained as a sum of other band representations. A band representation that is not composite is called elementary.

Now that we know when a band is elementary, we will see what conditions must be met for these to exist.

All band representations admit a description in terms of localized Wannier functions. They are induced from the representations of some site-symmetry group with local orbitals. Notice that if we induce a band representation from a reducible representation of the site-symmetry group:

$$\begin{aligned} \left( \rho _1\oplus \rho _2\right) \uparrow G=\left( \rho _1\uparrow G\right) \oplus \left( \rho _2\uparrow G\right) \end{aligned}$$

(C.7)

where we have used the distributive property of the direct sum. So, if we are interested in elementary band representations, we only need to take care of irreducible representations of the site-symmetry group. Moreover, since $(\rho \uparrow H) \uparrow G=\rho \uparrow G$, we only need to consider maximal subgroups of the space group.

We have determined that all elementary band representations can be induced from irreducible representations of the maximal site-symmetry groups. But this condition is not true in the opposite way; not all irreducible representation of the maximal site-symmetry groups induce an elementary band representation. These last cases, when what is induced is not an elementary band representation, are called exceptions. This may seem annoying, but they have already been tabulated in Topological Quantum Chemistry [10].

Hence (with some exceptions), band representations induced from irreducible representations of maximal site-symmetry groups give elementary band representations, whose bands are connected in the first BZ (they have no gap).

Band representations describe systems in the atomic limit, as they can be described by maximally localized Wannier orbitals. A trivial insulator is one whose bands can be obtained from maximally localized Wannier orbitals, so it does not have edge states.

So, a set of bands that is not a band representation cannot be described in terms of localized Wannier orbitals and is, hence, topological. We call this bands, that are a solution to compatibility relations, a quasi-band representation.

Let’s analyze the following example, alike the graphene case. Suppose we have a Hamiltonian constructed from localized orbitals, whose EBR $\rho _G=\rho \uparrow G$, and that the energy bands of this system can be divided into two disconnected sets of bands overall $\mathbf {k}$ in the first BZ, separated by a spectral gap. This means that the action of every element in the symmetry group on one of the states of one the bands does not take it out of it. Formally, let $P_i$ be the projector into the band i. Then:

$$\begin{aligned}{}[P_i,H]=0,\quad [P_i,g]=0 \end{aligned}$$

(C.8)

for all $g\in G$. Now suppose that the bands of projector $P_i$ transform under a band representation $\rho _G^i$. Then, the full $\rho _G$ representation could be constructed as a direct sum of the band representations of the different bands. We reached a contradiction: Starting with an elementary band representation, we got a composite band representation. So, all bands that transform according to an elementary band representation must be connected along the first BZ, otherwise they are not a band representation and, thus, they are topological, in the sense that there is at least one of them that is topological.

Going back to the graphene case, we saw that the EBR we induced can be connected or disconnected. If it is connected, it describes a trivial insulator while, if disconnected, it describes a topological material.