Dirac equation perspective on higher-order topological insulators

We begin by introducing some basic notions of condensed matter physics, topological phases, and topological crystalline insulators. We illustrate symmetry-protected topological phases with a toy model in zero spatial dimensions. Various aspects of the material in this Tutorial are also covered in Refs. 1–6.

Units are chosen so that the speed of light and Planck’s constant are dimensionless $c=ℏ=1$⁠.

Whereas particle physics aims to decompose the macroscopic material world into its elementary constituents, the goal of condensed matter physics is to understand how the properties of materials emerge from their microscopic degrees of freedom. In principle, this question can be posed not only for materials but also for any many-body system that is comprised of a macroscopic number of particles. However, condensed phases of matter are particularly interesting, because they are not adiabatically (that is, not without a phase transition) connected to the featureless phase at infinite temperature. This characterization implies that their behavior cannot be simply understood as the sum of their parts:⁷ for instance, the elementary particles that make out a condensed matter system often reorganize into collective excitations that behave like particles themselves, the so-called quasi-particles, and yet have markedly different properties, such as a different mass or even different exchange statistics.⁸ One of the basic insights of condensed matter physics is that there is a multitude of condensed phases beyond the rather coarse classification of matter into solids, liquids, and gases, with particularly interesting cases at very low temperatures where quantum mechanical effects come into play.

A crystal is an example of a condensed phase that consists of atomic nuclei and electrons, which for our purposes are its elementary constituents. We assume that the atomic nuclei (often just called atoms) form a crystalline lattice in $D$ spatial dimensions and move so slowly in comparison to the electrons—because of their much larger mass—that we can essentially treat them as a fixed background. Since the atomic lattice breaks spatial rotation and translation symmetry, the crystal realizes a condensed phase that is not adiabatically connected to the infinite-temperature phase (which is perfectly symmetric). From now on, we only focus on the electronic part, which at low temperatures is governed by quantum mechanics⁹ and determines whether the crystal is an insulator or a conductor, among many other measurable properties.

Since electrons are fermionic particles obeying the Pauli exclusion principle, a conglomerate of electrons such as the one populating a crystal cannot be understood as the sum of its parts even when there are no further interactions: in contrast to bosons, electrons cannot all occupy the same single-particle eigenstate. This property already enables a great number of interesting electronic phases of matter, and so we will completely neglect further interactions between the electrons such as the Coulomb interaction. It turns out that this approximation is often surprisingly good and applies to many real materials.

Quantum properties dominate over thermal fluctuations at low temperatures, and therefore we will consider systems at zero temperature for simplicity, allowing us to focus on ground state properties only. $N$ non-interacting electrons then fill up the $N$-lowest energy eigenstates of the single-particle Hamiltonian due to the Pauli exclusion principle. To be precise, let $cα†$ be the creation operator of an electron in the single-particle eigenstate labeled by $α$⁠. Fermionic exchange statistics are implemented by the anti-commutation relations

We can write a generic non-interacting many-body Hamiltonian that preserves electronic charge as

where $ϵα$ are the eigenvalues of the single-particle Hamiltonian $H$⁠, and $μ$ is the chemical potential. The $N$-particle ground state is then given by

where $N$ is a normalization factor, $|0⟩$ denotes the electronic vacuum, and each $α$ appears exactly once as long as $ϵα<μ$⁠. $|Ω⟩$ is called a Slater-determinant state, because the fermionic anti-commutation relations imply that its coefficients in a given basis are given by a determinant.¹⁰ 

The total number of single-particle states scales linearly with the system volume $V$⁠. An insulator is defined by an energy gap between the ground state and the lowest excited state that survives in the thermodynamic limit, where we take $N→∞$⁠, $V→∞$ while keeping the particle density $N/V$ constant. The condition for a gap in the many-body spectrum of $H$ translates to the absence of single-particle eigenstates with energies at the chemical potential $μ$⁠. A gap implies that a small external electric field¹¹ cannot lift electrons to excited single-particle states with nonzero net momentum and, therefore, cannot create a current. In contrast, a conductor (a metal) is characterized by the absence of a gap. A (second-order) phase transition, which features a diverging correlation length and a discontinuous change in some ground state properties, implies a gap closing.

A condensed matter system realizes a gapped topological phase when its ground state cannot be continuously deformed to a trivial reference state adiabatically, that is, without closing the energy gap to the first excited state. In the context of a crystal, a trivial insulator can be defined as an atomic limit where all electrons are tightly bound to the atoms, that is, as a state that is adiabatically connected to a Slater product over single-electron wavefunctions whose weight is exponentially localized around the atomic sites in position space.¹² For this definition, it is important to assume periodic boundary conditions so that boundary effects do not play a role and only bulk crystal properties are probed. We refer to the gap of a crystal with periodic boundary conditions as the bulk gap and adiabatic manipulations as those preserving it. There may be multiple trivial phases: for instance, in a crystal containing multiple types of atoms (such as rock salt and $NaCl$⁠), the states formed by electrons localized around each atom are equally valid atomic limits (in real materials, however, at most one atomic limit is energetically favored).

The concept of topological phases can be refined by demanding that symmetry preservation is necessary to prevent an adiabatic path to a trivial phase, one then speaks of a symmetry-protected topological phase (a pedagogical review is provided in Ref. 13) (see Fig. 1). For example, it may be possible to adiabatically deform the ground state of all electrons in a crystal to an atomic limit that breaks the crystalline inversion symmetry (which maps a coordinate $x$ in the crystal lattice to $−x$⁠) but to no other atomic limit. The state then realizes a topological crystalline insulator¹⁴ protected by inversion symmetry. The adjective “crystalline” here just means that a crystalline symmetry, in addition to the bulk gap, is necessary to stabilize the topological phase. When the symmetry is not crystalline, such as time reversal, the convention is to simply call the system a topological insulator. Historically, topological insulators protected by time-reversal symmetry came before topological crystalline insulators, here we nevertheless focus on the latter class of systems because they are conceptually much simpler. There is only one known example of a non-interacting topological phase that requires no symmetries at all for its protection: the Chern insulator or the integer quantum Hall effect.¹⁵ 

To illustrate these concepts, consider a zero-dimensional toy model of a spinless electron occupying one of the two orbitals in an atom, an $s$ and a $p$ orbital, with single-particle Hamiltonian

where $σa$⁠, $a=x,y,z$⁠, are the $2×2$ Pauli matrices, and there are only two energy levels labeled by their sign $α=±$⁠. Let inversion symmetry be represented by $I=σz$ so that the $s$ orbital corresponds to the state $|s⟩=(1,0)T$⁠, while the $p$ orbital corresponds to $|p⟩=(0,1)T$ (⁠$T$ takes the transpose). For $|γ|≫|α|,|β|$⁠, for instance, the spectrum of $H$ is gapped, and therefore also that of the corresponding many-body Hamiltonian $H$ in Eq. (2) at chemical potential $μ=0$⁠. Let us now define $|s⟩$ as the trivial reference state, it is the ground state of $H$ for $γ=−∞$⁠.¹⁶ Then, the system realizes an inversion symmetry-protected topological phase for large positive $γ$⁠: as long as inversion symmetry is not enforced, it is possible to tune $γ$ from positive to negative values without a gap closing, which happens only when $α2+β2+γ2=0$⁠, as long as $α$ and $β$ are kept non-zero. On the other hand, enforcing inversion symmetry via the commutation relation $[H,I]=0$ implies that $α$ and $β$ vanish, which in turn ensures a gap closing along the interpolation $γ(t)=1−2t$⁠, $t∈[0,1]$ that exchanges $|p⟩$ with $|s⟩$ as the lowest energy eigenstate of $H$⁠. We therefore find that there are no topological insulators in zero spatial dimensions that are protected by the bulk gap alone, but that there do exist topological crystalline insulators protected by inversion symmetry.

Since the vacuum can be viewed as a trivial phase,¹⁷ the boundary of a topological phase generically hosts a phase transition (a phase transition must occur to go between topological and trivial) and thereby a gap closing, as long as it respects the protecting symmetries.¹⁸ The most experimentally relevant kinds of topological insulators are therefore characterized by an insulating (gapped) bulk and a conducting (gapless) boundary. If the protecting symmetry is not crystalline, all boundaries preserve it by default and are therefore gapless, implying that the $D$-dimensional insulating bulk is surrounded by a $(D−1)$ dimensional conducting surface—this is the celebrated bulk-boundary correspondence. Topological crystalline insulators, on the other hand, generically have surfaces that do not preserve the relevant crystalline symmetry on their own and can therefore be gapped. The crystalline symmetry may still enforce the presence of $(D−k)$-dimensional gapless states with $k>1$⁠, thereby giving rise to higher-order topological insulators, where $k$ is referred to as the order.

In this Tutorial, our aim is to give a pedagogical introduction to first- and higher-order topological crystalline insulators from the point of view of their effective low-energy description. Our main tool will be the Dirac equation, which describes elementary electrons in particle physics. In a particularly tractable realization of the principle of emergence discussed at the beginning of this introduction, we will see how variants of the Dirac Hamiltonian arise in the low-energy description of topological (crystalline) insulators (see also Chap. V.5 of Ref. 19). We will then explain how they can be used to argue for the stability of (higher-order) topological phases and, in particular, for the presence of protected boundary states.

In this section, we introduce the Dirac equation in the context of the tight-binding description of electron movement in crystals. We also fix some basic notation and conventions that will be extensively used later on. For simplicity, we only treat one-dimensional (1D) systems in this section, the generalization to higher dimensions being straightforward (it mostly involves making all position and momentum coordinates vector-valued and replacing products between them by dot products).

Consider a 1D crystal.²⁰ We may model it as an infinitely long chain of repeated atom arrangements, the unit cells. In the tight-binding approximation, each unit cell hosts a finite number of sites that can be occupied by electrons. Denote by $cxi†$ the operator that creates an electron at site $i$⁠, $i=1,…,M$⁠, in the unit cell at position $x$⁠, $x=1,…,L$ (we take $x$ to be dimensionless so that $L$ is just the total number of unit cells in the sample). Assuming only nearest-neighbor hoppings between the sites (in a microscopic treatment, these hoppings would be derived from transition matrix elements of the microscopic electronic Hamiltonian between states that are well-localized on the individual sites), the Hamiltonian takes the form

(Here and in the following, we will set the chemical potential $μ$ to zero for simplicity.) By construction, $H$ preserves a global $U(1)$ symmetry that is represented by

with $ϕ∈(0,2π]$ an arbitrary phase. The infinitesimal generator of this symmetry is the total charge,

so that we have $eiϕQcxi†e−iϕQ=eiϕcxi†$⁠. Let $|Ω⟩$ be the many-body ground state of $H$ and perform a $U(1)$ transformation by the angle $2π$⁠. Evidently, such a transformation must be equal to the identity transformation, and so the ground state, which transforms as $|Ω⟩→ei2πQ|Ω⟩$⁠, must have integer charge $Q∈Z$⁠. We will see in Sec. III D how the boundary of a topological phase can break this quantization constraint and host fractional charges at the expense of a topologically nontrivial bulk.

Let us for now return to exploring the general properties of the Hamiltonian $H$⁠. Assuming periodic boundary conditions, we can exploit its discrete translational symmetry and partially diagonalize it by introducing the crystal momenta $k=(1,…,L)2πL$⁠, and the correspondingly Fourier-transformed creation operators

Later on, we will often implicitly take $L→∞$⁠, in which case all sums over $k$ should be interpreted as integrals over the first Brillouin zone $k∈(0,2π]$⁠. Equation (5) becomes

where we introduced the Bloch Hamiltonian

The eigenvalues of $H(k)$ give the single-particle energy spectrum as a function of crystal momentum $k$⁠, the so-called electronic band structure. That is, in the case of a periodic one-dimensional crystal, the eigenstate label $α$ that appears in Eqs. (2) and (3) is a composite label $α=(k,n)$ that specifies the crystal momentum $k$ together with the index of the $n$th eigenstate of $H(k)$⁠, where $n=1,…,M$ (there are $M$ bands). [Equation (4) can be viewed as a Bloch Hamiltonian in zero dimensions, where there is no momentum $k$⁠, and with $M=2$ sites in the unit cell.]

In the remainder of this section, we consider the simplest nontrivial case, where the unit cell has two atomic sites, $M=2$⁠. We can then expand the Bloch Hamiltonian $H(k)$ in terms of the identity matrix and the Pauli matrices (something that is possible for any Hermitian $2×2$ matrix),

where we used scalar product notation and introduced the real-valued vector $d(k)$ with components $da(k)$⁠, $a=x,y,z$⁠. The spectrum of $H(k)$ can be easily inferred by calculating $[H(k)−ϵ0(k)1]2$⁠, which turns out to be proportional to the identity matrix $1$⁠. It is given by

We see that the condition for an energy gap between the two bands $ϵ+(k)$ and $ϵ−(k)$ translates to the inequality $da(k)≠0$⁠, $a=x,y,z$⁠. Moreover, we can always adiabatically tune $ϵ0(k)→0$ without closing the gap between the two energy levels, and so all two-band insulators that only differ by the choice of $ϵ0(k)$ are topologically equivalent. We set $ϵ0(k)=0$ in the following.

We now come to topology. Due to the above criterion for an energy gap, the space of all one-dimensional two-band insulators is isomorphic to the space of all vector-valued functions $d(k)$ that do not map any momentum $k∈(0,2π]$ to the zero vector $0$⁠, that is, $d(k)∈R3∖{0}$⁠. Since the Brillouin zone has the topology of a circle, or one-sphere $S1$ (after all, the replacement $k→k+2π$ has no effect), a Bloch Hamiltonian is characterized by a collection of vectors $d(k)$ that satisfies $d(2π)=d(0)$⁠. It is easy to see that all such loops in $R3$ can be contracted to a point $d(k)=d∗$ without ever touching the origin $0$⁠. Comparing with Eq. (10), this implies that any constant $d∗$ gives rise to a corresponding atomic limit Hamiltonian $H(k)=H∗$ that is independent of $k$ and does not feature any hoppings between unit cells.²¹ We deduce that all one-dimensional two-band insulators are topologically trivial absent symmetries.

Here and in the following, we will specify only a single kind of trivial reference phase: the atomic limits with a Bloch Hamiltonian independent of $k$⁠. There may be other atomic limits (remember from Sec. I B that atomic limits are simply insulators whose ground state can be decomposed into exponentially localized electronic wavefunctions). For instance, the Hamiltonian belonging to an atomic limit where all electrons are exponentially localized around atomic sites at the boundary of the unit cell necessarily has to be $k$-dependent. From our point of view, we will treat such phases as topological as long as they cannot be adiabatically transformed into a trivial reference phase, similar to the more canonical examples of topological phases that do not have a representation in terms of exponentially localized electrons at all.²² 

In order to stabilize nontrivial 1D insulators, we need to incorporate symmetries. As mentioned in Sec. I B, the simplest example of a crystalline symmetry is given by spatial inversion that takes $x$ to $−x$⁠. When each unit cell contains two atoms that are mapped onto each other by inversion, it is represented by $I=σx$⁠. At the same time, inversion symmetry flips a crystal momentum $k$ to $−k$⁠. The condition for inversion symmetry in 1D is therefore that the Bloch Hamiltonian satisfies

[This condition is equivalent to the requirement that the many-body inversion operator commutes with the many-body Hamiltonian in Eq. (2).] In our two-band example, this constraint enforces that $dy(k)=−dy(−k)$ and $dz(k)=−dz(−k)$ are odd functions of $k$⁠, while $dx(k)=dx(−k)$ is even. Moreover, we find that the crystal momenta $k¯=0,π$⁠, which satisfy $k¯=−k¯mod2π$⁠, are special in that, at these momenta, we have the commutation relation $[H(k¯),I]=0$⁠.

These constraints are sufficient to protect topological crystalline insulators: Consider the many-body ground state at zero chemical potential. In our example [Eq. (11)], this situation corresponds to occupying a single site per unit cell with an electron (half-filling), and therefore, a single occupied energy band (each band has $L$ available crystal momenta that may be occupied, and there are $L$ unit cells with two sites each).²³ The occupied band is composed of Bloch states $|u(k)⟩$ that satisfy

At the high-symmetry momenta $k¯$⁠, these states can furthermore be chosen as eigenstates of inversion symmetry,

Due to $I2=1$⁠, we have $λ(k¯)=±1$—the inversion eigenvalues are quantized to only two allowed values (this quantization is a key ingredient for topology). Now, an insulating state where the two occupied inversion eigenvalues differ, $λ(0)=−λ(π)$⁠, is clearly not adiabatically connected to any trivial atomic limit: all constant Bloch Hamiltonians $H(k)=H∗$ have equal occupied inversion eigenvalues, and these cannot be changed without either closing the gap to the unoccupied band with energy $ϵ+(k)$ or breaking inversion symmetry. As an example, consider the inversion-symmetric Bloch Hamiltonian for $[dx(k),dy(k),dz(k)]=[Δ+cos⁡k,sin⁡k,0]$⁠, which reads

This Hamiltonian is in the topological phase for $|Δ|<1$⁠, and in the trivial phase otherwise. [Comparing with Eq. (12), the energy dispersion reads

and so there is a gap closing at $k=π$ for $Δ=1$ and one at $k=0$ for $Δ=−1$⁠.] In fact, $HSSH(k)$ is closely related to the famous Su–Schrieffer–Heeger (SSH) model of polyacetylene²⁴ (for a derivation, see Sec. III). We will show in Sec. III B that the topological phase is characterized by the presence of end states in an open chain geometry. The model in Eq. (16) provides the archetypical model of topological insulators, from which many more phases can be constructed by dimensional and symmetry enhancement.

We have found that there are 1D topological crystalline insulators protected by inversion symmetry. To explore their physical properties with respect to boundaries, we would like to solve for the spectrum and eigenstates of Eq. (16) in geometries with open rather than periodic boundary conditions. This can be done numerically via exact diagonalization, but it is more illuminating to find an approximate analytical solution, in particular, in order to determine which properties are specific to the model at hand, and which are general features of the topological phase.²⁵ 

In order to make the problem tractable, we first need to determine the salient feature distinguishing the topological phase from the trivial phase. Fixing $Δ<1$ for the trivial reference phase (that is, the atomic limit where all electrons are tightly bound in the center of each unit cell as a symmetric, even-parity superposition), this feature is clearly the inversion eigenvalue at $k=0$⁠: in the trivial phase, we have $λ(0)=λ(π)=1$⁠, while in the topological phase we have $λ(0)=−λ(π)=−1$ [remember that $λ(k¯)$ is the eigenvalue of inversion symmetry $I$ in the occupied, lower-energy eigenstate of $HSSH(k)$]. To capture the topological phase, it is therefore sufficient to perform a first-order Taylor expansion of $HSSH(k)$ around $k=0$⁠, arriving at the Hamiltonian

where we identified $m=Δ+1$⁠. This is the Hamiltonian of a Dirac fermion with mass $m$ that propagates in $1+1$ spacetime dimensions.²⁶ (See Fig. 2 for a comparison of the tight-binding and Dirac energy band spectrum.) It is somewhat curious that the relativistic Dirac equation appears in the context of an evidently non-relativistic (non-Lorentz-invariant) crystal. Nevertheless, the Dirac approximation (also called $k⋅p$ approximation) can be studied in its own right for any tight-binding model and is particularly useful in topological band theory: topological properties are insensitive to the non-universal details of the Bloch Hamiltonian that we discarded by approximating around $k=0$⁠.²⁷ 

Note that by virtue of the Taylor expansion, the momentum variable $k$ loses the Brillouin zone periodicity property it previously enjoyed. It should therefore be properly seen as taking values in $k∈(−∞,∞)$ (additionally, we may assume the thermodynamic limit $L→∞$ so that $k$ becomes a continuous variable). However, large values of $k$ are unphysical (the Taylor expansion breaks down), we will therefore only use the predictions of the Dirac approximation in a small window around $k=0$ and deduce the behavior at the remaining momenta by arguments of adiabatic continuity. The advantage that the Dirac approximation has over the original tight-binding model is that it effectively “forgets” the microscopic structure of the crystalline lattice and lives in continuous space:²⁸ in order to go to real space (which we need to study finite sample geometries and boundary effects), we can just make the canonical replacement $k→−i∂/∂x$⁠.

In the following, we will use the Dirac equation approach to capture the boundary physics of the one-dimensional SSH model as well as that of a higher-order topological phase in two dimensions. The general strategy will be the same,

1. Write down the tight-binding model.

2. Identify the momentum at which the gap closes between trivial and topological parameter regimes.

3. Perform a Taylor expansion around this momentum to first order, thus arriving at a Dirac-type Hamiltonian.

4. Use the Dirac framework to derive the boundary physics of the respective topological phase.

In this section, we introduce the inversion-symmetric Su–Schrieffer–Heeger Hamiltonian as the elementary topological insulator in one spatial dimension and use its Dirac approximation (derived in Sec. II D) to study its anomalous boundary physics.

While previous tutorials have discussed the SSH model as a topological phase protected by chiral symmetry,^2,4,29 here we instead focus its role as the simplest topological crystalline insulator, protected by inversion symmetry alone.^5,30

Consider polyacetylene, a polymer chain of $C2H2$ molecules. To minimize potential energy, the structure exhibits an alternation of weak and strong bonds between the two carbon atoms of the unit cell, $A$ and $B$⁠, which effectively each provide one atomic site that electrons can occupy [see Fig. 3(a)]. At half-filling, there is therefore one electron per unit cell. The Hamiltonian contains two terms: the weak, intra-unit cell hopping $t$ that connects $A$ with $B$⁠, and the strong, inter-unit cell hopping $t′$ that connects $B$ with the $A$ atom in the next unit cell. Inversion symmetry maps one carbon atom to another and should therefore again be represented by $σx$⁠. Equation (10) becomes

where in the last line we identified $Δ=t$ and set $t′=1$ in order to recover Eq. (16). We have seen in Sec. II that the spectrum of $HSSH(k)$ is fully gapped when $|Δ|≠1$ or equivalently when $|t|≠|t′|$ (when there is a mismatch in bond strength). Furthermore, we identified the regime $|Δ|<1$ as topological and derived the effective Dirac Hamiltonian, Eq. (18), that captures the sign flip of inversion eigenvalues at $k=0$ as $Δ$ is tuned. For convenience, we reproduce the Dirac Hamiltonian here,

Keeping in mind $m=Δ+1$ and that we fixed $Δ<−1$ as the trivial reference phase, we can identify the parameter range $m<0$ as trivial and $m>0$ as topological in the Dirac approximation.

The Dirac model $HDirac(k)$⁠, just like the original tight-binding Hamiltonian $HSSH(k)$⁠, is gapped at all values of $k$⁠, and therefore bulk-insulating. To study the effect of boundaries in the topological phase, we need to model an interface between the trivial and topological regimes. This is most easily achieved by assigning a position-dependence to the mass $m→m(x)=mx$⁠, with the convention $m>0$⁠, so that $m(x)$ is negative (trivial) in the left half of the system and positive (topological) in the right half. The particular form of the real space dependence beyond such a sign flip is arbitrary—we might as well have chosen $m(x)=msign(x)$⁠, for instance. The reason is that we are only interested in topological properties that survive continuous deformations on either side of the gap-closing point at $x=0$⁠. The only notable effect of our particular choice $m(x)=mx$ is that it is differentiable and so gives smoother results than, for instance, $m(x)=msign(x)$⁠.

We conclude that the following real-space Hamiltonian effectively describes the interface between two SSH chains, one trivial and one topological,

where we expressed $k=−i∂/∂x$ as an operator acting on real space, because its eigenvalue $k$ is not anymore a good quantum number [any non-constant $m(x)$ breaks translation symmetry].

Recall the heuristic argument from Sec. I B that posited that the boundary between a topological and a trivial phase hosts a phase transition, characterized by a gapless energy spectrum. [This argument can be made mathematically rigorous,³¹ here we will find it more illuminating to derive it in the Dirac approximation context.] Guided by this intuition, we will attempt to solve for zero-energy states at the interface, which are the two-component wavefunctions $|Ψ(x)⟩=[ψ(x),ϕ(x)]T$ that satisfy the equation

There is only one normalizable solution to this equation

where $N=(π/m)14$ is a normalization factor. [The solution that multiplies the vector $(0,1)T$ comes with a plus sign in the exponential and is therefore not normalizable.] In conclusion, we find a single zero-energy state that is exponentially localized to the boundary of the topological phase. Its wavefunction decay is shown in Fig. 3(b).

What remains is to interpret the zero-energy end state from the point of view of topology. At this point, it is illuminating to first recount the historical point of view, according to which the SSH model realizes a topological phase protected by a local rather than a crystalline symmetry.⁴ [These conceptually distinct interpretations of topology are both consistent with the Hamiltonian $HSSH(k)$⁠, that is, there is more than one symmetry that prevents adiabatic transformations of the SSH ground state to the trivial reference state.] This local symmetry is a sublattice symmetry $C=σz$ (also known as chiral symmetry) of $HSSH(k)$⁠, and it is realized via

This relation enforces that the Hamiltonian only couples the two inequivalent sites of the unit cell (the two sublattices) to each other but not to themselves (as would be the case for local potentials or longer-range hoppings). [Somewhat confusingly, on the level of the Bloch Hamiltonian, sublattice symmetry implies anti-commutation rather than commutation with $C$⁠. It is nevertheless a valid symmetry of the physical system, because it commutes with the many-body Hamiltonian, Eq. (2)—the two minus signs, one deriving from the anti-commutation, and another from the fermionic exchange statistics, cancel.]

Sublattice symmetry is preserved by the Dirac approximation and the interface geometry in Eq. (21). Crucially, it implies that for every eigenstates of $Hbdry$ at energy $E$⁠, with $|E|>0$⁠, there exists a partner eigenstate at energy $−E$⁠. Only a state at $E=0$ may be unpaired. But this constraint ensures that the single zero-energy mode that we derived in Eq. (23) cannot be moved away from $E=0$ by any local perturbation: a single eigenstate cannot continuously split into two. The edge of a nontrivial SSH insulator therefore hosts a zero-energy bound state that is topologically protected by the bulk gap and sublattice symmetry.³² 

Let us now return to crystalline symmetries. Whereas it is difficult to guarantee sublattice symmetry in realistic systems, inversion symmetry is a property of many naturally arising crystal structures. We understood in Sec. II that the SSH model also realizes an inversion symmetry-protected topological insulator and derived the presence of an end state in Eq. (23). It is then natural to ask what properties of the end state are universal and guaranteed by inversion symmetry alone, that is, not specific to the particular choice of Hamiltonian $HSSH(k)$⁠. Clearly, absent sublattice symmetry there is no preferred notion of “zero energy,”³³ so that lying at zero-energy cannot be a universal property of the domain wall bound state.

At this point, it is important to note that in order to connect bulk crystalline topology to boundary states, we must study a boundary termination that preserves inversion symmetry. Our choice of the mass profile $m(x)$ from before explicitly broke inversion symmetry—we only modeled one (the left) end of the topological phase. In order to restore inversion symmetry, we must also take into account the second zero-energy state that is located at the opposite (the right) end of the sample. The inversion-symmetric and finite system will then have two zero-energy states, but as explained above, nothing prevents the addition of a local edge potential to $HSSH(k)$ that shifts them to finite energies (at least as long as the local potential is the same on the left and right end of the sample, such a manipulation respects inversion symmetry). The property that remains invariant is therefore not the energy of the end states, but their filling: imagine that we gradually tune a finite system from trivial to topological in an $I$-symmetric manner. This process can, for instance, be modeled via the Hamiltonian

where $T∈[0,1]$⁠,

and $m(x,T)$ interpolates continuously between $m(x,0)$ and $m(x,1)$ in an inversion-symmetric fashion: $m(x,T)=m(−x,T)$⁠. $Hbdry,I(0)$ then describes a fully trivial system without boundaries, while $Hbdry,I(1)$ describes a topological system of length $L$ that is embedded in a trivial environment. For large $L$⁠, we may take over the result we obtained for a single edge, Eq. (23), and deduce that $Hbdry,I(1)$ hosts two inversion-related end states as discussed above. The only important feature of the interpolating Hamiltonian $Hbdry,I(T)$ is that it also respects sublattice symmetry at all values of $T$⁠, ensuring that the two end states that are present at $T=1$ but not at $T=0$ must come in with opposite energies $E1(T)=−E2(T)$ as $T$ is tuned from $0$ to $1$⁠, with $E1(1)=E2(1)=0$ [recall Eq. (24) and the surrounding discussion]. Correspondingly, only one state is filled (occupied by an electron in the system’s ground state), namely, the one deriving from the negative-energy manifold of states.³⁴ We conclude that the system hosts two end states that together have only one electron’s worth of electric charge [recall how the total charge $Q$ is defined in Eq. (7) as the number of occupied electrons]. The only inversion-symmetric way to distribute this charge is to assign half a charge to either end of the system, essentially fractionalizing the electronic charge.

The trade-off between the global $U(1)$ symmetry, which naively implies charge quantization for a Hamiltonian of the form of Eq. (5), and the crystalline inversion symmetry, constitutes the so-called filling anomaly.³⁵ It is the only universal boundary characteristic of inversion-protected topological insulators in one dimension [see Fig. 4(a)]. The spectrum of Eq. (19) with open boundary conditions, shown in Fig. 4(b), confirms these results.

While we have used sublattice symmetry to show that the edge states of $HSSH(k)$ are (filling) anomalous, it should be emphasized that sublattice symmetry is not required to stabilize the anomaly: we used sublattice symmetry simply to shortcut an otherwise lengthy calculation of the domain wall charge. It can be broken without destroying either the crystalline bulk topology or the edge state filling anomaly, which are both protected by inversion symmetry and $U(1)$ charge conservation alone. Note also that the filling anomaly, even though it derives from the zero-dimensional edge states, crucially relies on the topologically nontrivial one-dimensional bulk: as we have shown in Sec. II A, there is no well-defined (regularizable) Hamiltonian that preserves $U(1)$ symmetry and has a ground state with fractional charge.

We next leverage what we have learned from the SSH model to understand the simplest possible example of a higher-order topological insulator: the inversion-symmetric case in two spatial dimensions. Again, the Dirac approximation provides the most direct route to a crystalline bulk-boundary correspondence and filling anomaly.

Historically, higher-order phases were discovered much later^37–41 than their first-order counterparts such as the Chern insulator⁴² or the time-reversal symmetric topological insulator.⁴³ Ironically however, at least in 2D, the higher-order case is conceptually simpler. This allows us to bypass the discussion of first-order phases, which have been extensively reviewed elsewhere.^1–4,29

In Sec. III, we have seen how zero-dimensional states can arise at the ends of a one-dimensional insulator that preserves inversion symmetry. This kind of topological bulk-boundary correspondence is called first order, because the gapless boundaries have a dimension that is one lower than that of the bulk. Similarly, higher-order topological insulators in $D$ spatial dimensions are defined by the presence of gapless states along boundary segments with dimension $D−K$⁠, where $K>1$ is the order. The simplest example is corner states in two dimensions, which realize a corner (zero-dimensional) filling anomaly similar to what was discussed in Sec. III D, while the bulk (2D) and edges (1D) are gapped. The filling anomaly gives rise to fractional corner charge, similar to how the SSH filling anomaly gave rise to fractional end charge. There also exist 2D first-order topological insulators, most notably the Chern insulator (the integer quantum Hall effect) and the time-reversal symmetric topological insulator (the quantum spin Hall effect). An overview of first- and higher-order topological phases is given in Fig. 5.

First-order phases realize insulators (gapped band structures) in the bulk and anomalous metals [gapless band structures that cannot be obtained from well-defined, regularizable $(D−1)$-dimensional Hamiltonians] on the boundary. Second-order phases, on the other hand, realize insulators in the bulk and anomalous insulators [gapped band structures that cannot be obtained from well-defined, regularizable $(D−1)$-dimensional Hamiltonians] on the boundary. These anomalous boundaries can then themselves be viewed as first-order topological phases that host their own gapless end states of dimension $D−2$⁠.

Crucially, we cannot simply view the $(D−1)$-dimensional boundaries as first-order topological insulators that have been glued to the boundary of an otherwise trivial $D$-dimensional system:⁴⁴ even though they are gapped, they are anomalous and can only be realized in the presence of a topologically nontrivial bulk. These qualitative considerations will be made concrete in the following.

Unlike in Sec. III A, where we made at least some attempt at motivating the SSH Hamiltonian from the microscopic chemistry of polyacetylene, we will from now on study the properties of toy model Hamiltonians without further physical legitimization. The reason is that these capture the essential physics without distracting us too much with the non-universal properties of any given material. Moreover, higher-order topological phases were only discovered recently, and as of yet only very few “nice” material realizations are known.^45–47 For our purposes, it will therefore be sufficient to study these novel phases of matter on a conceptual level—it is one of the great luxuries of condensed matter theory that very often initially theoretical concepts eventually do turn out to be realized in nature.

Following up on our general characterization of two-dimensional second-order topological insulators in Sec. IV A, let us therefore ask the question: how do we construct a two-dimensional tight-binding that hosts corner states? Evidently, we will again need a crystalline symmetry to ensure topological protection: with only local symmetries in place, any corner states could be brought together along the sample boundary and annihilated with each other—this process would be adiabatic with respect to the bulk gap. We note that the requirement of crystalline symmetries (symmetries that do not leave all spatial positions $x$ invariant) is particular to higher-order phases. First-order phases can be protected by crystalline symmetry operations, but do not have to—we have seen this already for the SSH insulator that can be stabilized by either sublattice or inversion symmetry. For simplicity, we will again consider inversion symmetry here, which in 2D maps both coordinates $x=(x,y)→(−x,−y)$ to their opposite values.

In particular, inversion symmetry maps the one-dimensional boundary of a two-dimensional, inversion-symmetric sample to itself. A natural starting point to construct a second-order topological insulator is then to engineer a phase that on its boundary hosts a phase akin to the one-dimensional, inversion-symmetric SSH model—domain walls in the SSH parameter $m$ [Eq. (20)] then bind zero-dimensional “corner” states. We will show in the following that such a dimensional hierarchy is achieved by the four-band tight-binding Hamiltonian

where $τi$ and $σi$⁠, $i=0,x,y,z$ are two sets of Pauli matrices, and we abbreviated the Kronecker product by $τi⊗σj≡τiσj$⁠. Inversion symmetry is given by $I=τxσ0$⁠, and the topological parameter regime is delimited by $|Δ|<2$⁠. The small constant $δ≪1$ multiplies a collection of symmetry-allowed perturbation terms $O$ that will be essential in gapping out the one-dimensional boundary.⁴⁸ We may again define inversion eigenvalues at the four high-symmetry momenta $k¯=(0,0),(0,π),(π,0),(π,π)$⁠, just as was done in Sec. II C. Declaring $Δ<−2$ as the trivial reference phase,⁴⁹ the topological phase is then characterized by a sign flip of two occupied inversion eigenvalues at $k¯=(0,0)$ [see Fig. 6(a)], prompting us to expand around that momentum in a Dirac approximation:

This Dirac Hamiltonian captures the trivial-to-topological gap-closing transition of $HSOTI(k)$ when $m=Δ+2$ is tuned from negative to positive values.

As for the SSH model, we will first derive the band structure of $HSOTI(k)$ in the presence of a single edge before arguing for the topological signature of a full inversion-symmetric boundary. Without loss of generality, we keep periodic boundary conditions in the $y$ direction so that the edge under consideration has its normal along the (negative) $x$ direction. A mass profile $m(x)=mx$⁠, $m>0$ then locates the left edge of the topological phase at $x=0$⁠. Similar to Eq. (21), any boundary-localized zero-energy states are determined by the Hamiltonian

There are two normalizable solutions to the zero-energy condition $Hx(0)|Ψ(x)⟩=0$ at $ky=δ=0$⁠, given by

where $N=(4π/m)14$ is a normalization factor.

We may now deduce the form of the edge Hamiltonian (the Hamiltonian governing the edge degrees of freedom) by degenerate first-order perturbation theory in $ky$ and $δ$⁠. Let us for the moment fix $O=τ0σx$—this choice preserves inversion symmetry. The edge Hamiltonian is given by the matrix elements of $Hx(ky)$ in the basis ${|Ψ1(x)⟩,|Ψ2(x)⟩}$ spanned by the zero-energy edge states,

where we introduced another set of Pauli matrices $ξi$⁠, $i=0,x,y,z$⁠. We thus find that the edge of $HSOTI(k)$ is governed by the SSH Dirac Hamiltonian, Eq. (18), with the identification $m=δ$⁠. In particular, any finite $|δ|>0$ implies that the edge is gapped out, and so there is no protected filling anomaly stemming from the one-dimensional edge states. The edges of a second-order topological insulator are therefore gapped and seemingly trivial. While some choices of the perturbation $O$ do not lead to a gap (for instance, $τ0σz$⁠), the existence of a single symmetry-allowed gapping term is already enough to trivialize the edge from a topological point of view (also, such a term will be generically present in real materials). The Dirac results are confirmed by a full diagonalization of $HSOTI(k)$ in a slab geometry, see Figs. 6(b) and 6(c).

We have shown that the Hamiltonian $HSOTI(k)$ is gapped in bulk and on edges with normal in the $x$ direction. Similarly, one can show that the states along edges with normal in the $y$ direction are gapped and described by a Hamiltonian just like Eq. (31). We might thus conclude that the entire system is gapped and featureless. There is, however, one catch: the edge termination we considered is not inversion symmetric. Recall how we first derived the SSH end state for a single edge in Sec. III B, before deducing that an inversion-symmetric pair of edges hosts a pair of edge states of which only one is filled. To determine the anomalous boundary physics of second-order topological insulators, we likewise have to consider an inversion-symmetric edge termination, for instance, that of a rectangular sample. A rectangle has four edges (call them left and right, top and bottom) that all come with their respective version of Eq. (29)—in Sec. IV C, we have explicitly treated the left edge only. It does, however, not suffice to repeat the calculation for the remaining three edges separately: due to the gauge freedom in defining the edge states (we may multiply either state with a phase factor, or take arbitrary superpositions, to arrive at an equally valid set of edge states), there would be no meaningful way of comparing the resultant edge Hamiltonians. Instead, we must find the bound states of an arbitrary edge, labeled by an angle $ϕ$—these then continuously evolve with $ϕ$ as we interpolate from one edge to another, and thereby provide a consistent choice of gauge.

For an arbitrary edge that has its normal at an angle $ϕ$ to the $x$ axis, it makes sense to decompose the vector of crystal momenta as

where $k⊥$ is the momentum along the edge normal and $k∥$ is the momentum along the edge itself (note that we are orienting $k∥$ at a positive right angle from $k⊥$⁠). The Dirac Hamiltonian then reads

where $σ1(ϕ)=σxcos⁡ϕ+σysin⁡ϕ$ and $σ2(ϕ)=σycos⁡ϕ−σxsin⁡ϕ$⁠. After decomposing the position coordinate $x$ into $x⊥$ and $x∥$ similar to Eq. (32), we again implement the edge via a domain wall mass profile $m(x⊥)=mx⊥$⁠, $m>0$ so that the topological phase lies at positive values of $x⊥$ (the edge normal points into the topological region). This setup retains $k∥$ as a good momentum quantum number, while we replace $k⊥$ by $−i∂/∂x⊥$⁠,

Edge states at $k∥=δ=0$ then satisfy

Noting that $[σ1(ϕ)]2=1$⁠, the eigenvalues of $σ1(ϕ)$ are constant and given by $−1$ and $+1$⁠. The equation can then be solved to give

where $N=(4π/m)14$ is a normalization factor. The dispersion of these states with edge momentum $k∥$⁠, and their gapping, can again be derived in degenerate first-order perturbation theory by evaluating the matrix elements

where

We will in fact not use any explicit form of the perturbation terms $O$ in the following—it will be sufficient to note that these terms will generically gap the two edge states as long as they project into a nonzero $ζ(ϕ)$ (we have already seen this for the particular choice of $O=τ0σx$ and $ϕ=0$⁠). Importantly, however, we now show that inversion symmetry implies stable zeroes of $ζ(ϕ)$⁠. First, note that it acts on the edge states as follows:

The constraint of inversion symmetry then implies

(Recall that the momentum $k∥$ is defined at a right angle to the edge normal, it rotates with $ϕ$ and so is not separately flipped by inversion.) Now, any perturbation in Eq. (37) that anticommutes with the kinetic term (other terms do not lead to a gap and can be absorbed into a redefinition of $k∥$⁠) can be decomposed as

so that the eigenvalues of $Hedgeϕ(k∥)$ are given by $ϵ±ϕ(k∥)=±k∥2+a(ϕ)2+b(ϕ)2$⁠. Equation (40) yields the constraints

which together imply

This relation necessitates the presence of domain walls in the Dirac mass, $ζ(ϕ)=0$⁠, at some angles $ϕ¯$ and $ϕ¯+π$ where $ζ(ϕ)$ switches sign [see Fig. 7(a)]. The domain walls are movable yet irremovable: perturbations may change the precise location $ϕ¯$⁠, but domain walls can only be annihilated in pairs. Since the partner of the domain wall at $ϕ¯$ is fixed to the location $ϕ¯+π$ by inversion symmetry, the two can never come together and cancel.

We conclude that, in a rectangular geometry, the edge states on the left and right edges, as well as on the top and bottom edges, have to come with the opposite signs of their Dirac mass. Viewing the full rectangular boundary as a 1D system governed by a SSH type Hamiltonian [Eq. (37)], it then hosts Dirac mass domain walls at two oppositely related corners [see Figs. 7(b) and 7(c)].⁵⁰ The situation is therefore just that of Eq. (25) at $t=1$⁠, where we considered a SSH model with two inversion-related domain walls and showed that these host end states with a filling anomaly. Equivalently, the corners of $HSOTI(k)$ bind gapless states that lead to a filling anomaly of second-order: while the infinite slab geometry of Fig. 6(b) is gapped, a filling anomaly arises when introducing open boundary conditions in two directions. Just like for the SSH model, the presence of the filling anomaly is guaranteed by the topologically nontrivial bulk gap and inversion symmetry alone, and neither relies on the particular structure of $HSOTI(k)$⁠, nor on the rectangular sample geometry we chose to derive it. The Dirac result is confirmed by a full diagonalization of the tight-binding Hamiltonian, shown in Fig. 7(d).

The corner states discussed here have been predicted to naturally occur in 2D monolayers of antimony and arsenic at finite buckling.⁵¹ 

I thank Titus Neupert for teaching me most of what is covered in this Tutorial. I am supported by a fellowship at the Princeton Center for Theoretical Science.

The data that support the findings of this study are available from the corresponding author upon reasonable request.