Edge states in rationally terminated honeycomb structures

Significance

Edge electronic transport in graphene depends on the edge’s shape. We present the first rigorous analytical results—a complete classification of flat bands and explicit analytic expressions for the corresponding edge states—for general rational edges in graphene, that is, sharp terminations of the graphene bulk which are translation invariant with respect to an arbitrary triangular lattice basis vector. Our analysis potentially impacts the study of questions of current mathematical and physical interest. It is a step toward understanding transport along non-translation-invariant (irrational) edge terminations. Further, our approach to transport along structures whose boundaries have arbitrarily large finite periods may shed light on the mathematical analysis of other physical problems having this feature, such as twisted bilayer graphene.

Abstract

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type,” generalizing the classical zigzag and armchair edges. We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l.

A phenomenon of great interest in materials science is the propagation of electrons along line defects or edges in a crystalline material. It was first recognized by refs. 1 and 2 that edge transport depends on the shape of the edge. We study the spectrum arising from the tight-binding model of graphene, sharply terminated along a rational edge, that is, an edge l parallel to a direction of translational symmetry of the underlying period lattice. Unlike standard “zigzag” (ZZ) and “armchair” (AC) edges, a general rational edge gives rise to edge states with nonzero energies and nonflat dispersion curves. We report numerical evidence for such states, as well as a rigorous analysis that determines precisely when zero-energy (“flat band”) edge states arise. We set up basic definitions and notation and then state our results. Detailed proofs of our rigorous results, and explanations of our numerical investigations, are presented in ref. 3.

The Honeycomb Structure, $\mathbb{H}$

The honeycomb structure, $\mathbb{H}$, consists of all the vertices of all the tiles in a tiling of the plane by regular hexagons. $\mathbb{H}$ is invariant under translations by vectors in the equilateral triangular lattice $\mathrm{\Lambda }=\mathbb{Z}\stackrel{°}{{\mathbf{v}}_1}\oplus \mathbb{Z}\stackrel{°}{{\mathbf{v}}_2}$, where we may take $\stackrel{°}{{\mathbf{v}}_1}=(\sqrt{3}/2,1/2)$ and $\stackrel{°}{{\mathbf{v}}_2}=(\sqrt{3}/2,-1/2)$. More precisely, we may express $\mathbb{H}$ as the union of two interpenetrating triangular lattices: $\mathbb{H}=\left(\stackrel{°}{{\mathbf{v}}_A}+\mathrm{\Lambda }\right)\cup \left(\stackrel{°}{{\mathbf{v}}_B}+\mathrm{\Lambda }\right)\equiv {\mathbb{H}}_A\cup {\mathbb{H}}_B$, where $\stackrel{°}{{\mathbf{v}}_A}=0$ and $\stackrel{°}{{\mathbf{v}}_B}=(1/3)\left(\stackrel{°}{{\mathbf{v}}_1}+\stackrel{°}{{\mathbf{v}}_2}\right)$. The points of ${\mathbb{H}}_A$ and ${\mathbb{H}}_B$ are called, respectively, “A sites” and “B sites.” Each site $\omega \in \mathbb{H}$ has three nearest neighbors in $\mathbb{H}$; we denote the set of those three nearest neighbors $\text{NN}(\omega )$.

The Terminated Honeycomb Structure, ${\mathbb{H}}_{♯}$

Any straight line l separates the plane into two half-planes ${\mathrm{\Omega }}^{\pm }$. We pick one of these half-spaces, ${\mathrm{\Omega }}^{+}$, and the terminated structure, ${\mathbb{H}}_{♯}$ arising from l is defined as $\mathbb{H}\cap {\mathrm{\Omega }}^{+}$. We restrict our attention to rational edges; that is, we assume that the line l is parallel to a nonzero vector in the lattice Λ. Other classes of terminated structures were considered by refs. 4 and 5; see the section below on the relation of the current work to previous work. Given a rational edge, we may pick an alternative triangular lattice basis for Λ, one of whose elements is parallel to l; see Fig. 1. To do so, first, choose relatively prime integers a₁₁ and a₁₂ such that the vector

$${\mathbf{v}}_1=a_{11}\stackrel{°}{{\mathbf{v}}_1}+a_{12}\stackrel{°}{{\mathbf{v}}_2}$$ [1]

Fig. 1.

Notation for the honeycomb structure.

is parallel to l. Further, we may choose integers a₂₁ and a₂₂ such that $\det (a_{ij})=1$, and the vector

$${\mathbf{v}}_2=a_{21}\stackrel{°}{{\mathbf{v}}_1}+a_{22}\stackrel{°}{{\mathbf{v}}_2}$$ [2]

points into ${\mathrm{\Omega }}^{+}$. The integers a₂₁ and a₂₂ are not uniquely determined, but our results are independent of the choice of these integers.

To describe ${\mathbb{H}}_{♯}$ in terms of the basis $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$, we introduce the integers $s_1,s_2\in \{-1,0,1\}$ as follows:

$$a_{22}-a_{21}\equiv s_1,\hspace{0.17em}\text{modulo}\hspace{0.17em}3$$ [3a]

$$a_{11}-a_{12}\equiv s_2,\hspace{0.17em}\text{modulo}\hspace{0.17em}3.$$ [3b]

The plane can be tiled by parallelograms whose sides are translates of ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$. Each such parallelogram contains exactly one A site and one B site. For one such parallelogram, those sites are, respectively,

$${\mathbf{v}}_A=0,\hspace{1em}{\mathbf{v}}_B=\frac{1}{3}(s_1{\mathbf{v}}_1+s_2{\mathbf{v}}_2).$$ [3c]

For integers $n_{\text{min}}^A,\hspace{0.17em}{n}_{\text{min}}^B$, our terminated honeycomb ${\mathbb{H}}_{♯}$ is then given by

$$\begin{array}{l}{\mathbb{H}}_{♯}=\{{\mathbf{v}}_A+m{\mathbf{v}}_1+n{\mathbf{v}}_2|(m,n)\in Z^2,\hspace{0.17em}n\ge n_{\text{min}}^A\}\\ \hspace{1em}\cup \{{\mathbf{v}}_B+m{\mathbf{v}}_1+n{\mathbf{v}}_2|(m,n)\in Z^2,\hspace{0.17em}n\ge n_{\text{min}}^B\}.\end{array}$$ [4]

Our results depend only on $n_{\text{min}}^A-n_{\text{min}}^B$, because they are independent of the shift of $n_{\text{min}}^A$ and $n_{\text{min}}^B$ together by an arbitrary integer; this amounts to an integer translate of ${\mathbb{H}}_{♯}$ in the direction ${\mathbf{v}}_2$. One can show that $n_{\text{min}}^A-n_{\text{min}}^B\in \{0,s_2\}$. If $n_{\text{min}}^A=n_{\text{min}}^B$, we say that ${\mathbb{H}}_{♯}$ is a balanced terminated honeycomb (in this case, for each parallelogram, mentioned before, which overlaps with ${\mathbb{H}}_{♯}$, its A site and B site lie inside ${\mathbb{H}}_{♯}$); otherwise, we say that ${\mathbb{H}}_{♯}$ is an unbalanced terminated honeycomb (in this case, there are parallelograms whose A site lies in ${\mathbb{H}}_{♯}$ but whose B site does not lie in ${\mathbb{H}}_{♯}$, or else whose B site lies in ${\mathbb{H}}_{♯}$ but whose A site does not lie in ${\mathbb{H}}_{♯}$).

ZZ-Type and AC-Type Edges

We present geometric and arithmetic characterizations of the types of edges that arise via our honeycomb termination procedure.

First, by translation invariance with respect to the vector ${\mathbf{v}}_1$, there is a row of A sites in ${\mathbb{H}}_{♯}$ with minimal distance $D_A\ge 0$ to the line l and, similarly, a row of B sites in ${\mathbb{H}}_{♯}$ with minimal distance $D_B\ge 0$ to the line l.

We say that ${\mathbb{H}}_{♯}$, the terminated structure arising from l, or simply its edge, is of ZZ type if $D_A\ne D_B$, and we say that it is of AC type if D_A = D_B. It can be proven that ZZ-type edges are those for which the integers defined in Eq. 1 satisfy $a_{11}-a_{12}=\pm 1\hspace{0.17em}\text{mod}\hspace{0.17em}3$ (i.e., $s_2=\pm 1$), and AC-type edges are such that $a_{11}-a_{12}=0\hspace{0.17em}\text{mod}\hspace{0.17em}3$ (i.e., $s_2=0$). Note that ZZ-type edges could be balanced or unbalanced, whereas the AC-type edges could only be balanced.

The best known edges correspond to the choices 1) ${\mathbf{v}}_1=\stackrel{°}{{\mathbf{v}}_1}$ (or, equivalently, ${\mathbf{v}}_1=\stackrel{°}{{\mathbf{v}}_1}-\stackrel{°}{{\mathbf{v}}_2}$), the classical ZZ edges (balanced/ordinary or unbalanced/bearded); and 2) ${\mathbf{v}}_1=\stackrel{°}{{\mathbf{v}}_1}+\stackrel{°}{{\mathbf{v}}_2}$, the classical AC edge; see Fig. 2.

Fig. 2.

${\mathbb{H}}_{♯}$ for the classical ZZ and AC edges, balanced and unbalanced. A sites are in blue, and B sites are in red. The edge, l, is indicated with a dashed line.

The Tight-Binding Hamiltonian, $H_{♯}$, Associated with ${\mathbb{H}}_{♯}$

Our Hamiltonian acts on “wave functions” $\psi ={({\psi }_{\omega })}_{\omega \in {\mathbb{H}}_{♯}}$ with each ${\psi }_{\omega }\in C$; ${\psi }_{\omega }$ is the quantum amplitude of ψ at site ω. We define the Hamiltonian $H_{♯}$ by the formula

$${\left(H_{♯}\psi \right)}_{\omega }={\displaystyle \sum _{\omega \prime \in \text{NN}(\omega )\cap {\mathbb{H}}_{♯}}{\psi }_{\omega \prime }}.$$ [5]

We may Fourier analyze $H_{♯}$ by virtue of the invariance of ${\mathbb{H}}_{♯}$ under translation by the vector ${\mathbf{v}}_1$. Given $k\in [0,2\pi )$, a k-pseudoperiodic wave function is a vector $\psi ={({\psi }_{\omega })}_{\omega \in {\mathbb{H}}_{♯}}$ such that ${\psi }_{\omega +{\mathbf{v}}_1}=e^{ik}{\psi }_{\omega }$ for all $\omega \in {\mathbb{H}}_{♯}$. There is a natural l² norm on k-pseudoperiodic wave functions, given by summing $|{\psi }_{\omega }{|}^2$ over equivalence classes modulo translations by multiples of ${\mathbf{v}}_1$. We write $l_k^2({\mathbb{H}}_{♯})$ to denote the space of k-pseudoperiodic wave functions with finite l² norm. The Hamiltonian $H_{♯}$ maps $l_k^2({\mathbb{H}}_{♯})$ to itself as a bounded self-adjoint operator. We write $H_{♯,k}$ to denote the restriction of $H_{♯}$ to $l_k^2({\mathbb{H}}_{♯})$.

We are interested in the spectral theory of the operator $H_{♯}$ acting in the space $l^2({\mathbb{H}}_{♯})$. By translation invariance with respect to ${\mathbf{v}}_1$, this can be reduced to the spectral theory of $H_{♯,k}$ for $k\in [0,2\pi )$. In particular, we want to understand eigenvectors of $H_{♯,k}$, which are called edge states, that is, pairs $(E,\psi )$ such that $H_{♯,k}\psi =E\psi$ and $0\ne \psi \in l_k^2({\mathbb{H}}_{♯})$. By definition, an edge state is plane-wave-like under translation by ${\mathbf{v}}_1$ (parallel to the edge), but the amplitudes ${\psi }_{\omega }$ decay to zero as the distance from ω to the edge tends to infinity. If E = 0 is an edge state eigenvalue for k ranging over a subinterval of $[0,2\pi )$, then we refer to the corresponding family of edge states as a zero energy/flat band of edge states of $H_{♯}$.

We next state one of the main results of this article.

Theorem 1:

Complete classification of zero-energy/flat-band edge states of $H_{♯}$ for rational edges. Assume $k\in [0,2\pi ]$.

• •For any AC edge (D_A = D_B), there are no zero-energy edge states.
• •All ZZ edges ($D_A\ne D_B$) give rise to a flat band of zero-energy edge states for k varying in a proper quasi-momentum subset of $[0,2\pi ]$. These states are supported either exclusively on the A sites of ${\mathbb{H}}_{♯}$ (A site ES) or exclusively on the B sites of ${\mathbb{H}}_{♯}$ (B site ES). Whenever $H_{♯,k}$ has a zero-energy edge state, the space of such states is one-dimensional.

A complete classification is given in Table 1 $(I=(\frac{2\pi }{3},\frac{4\pi }{3}))$.

Table 1.

Description of all zero-energy edge states

	DA < DB	DB < DA
Balanced	A-site ES for $k\in I$	B-site ES for $k\in I$
Unbalanced	A-site ES for	B-site ES for
	$k\in [0,2\pi ]\text{}\setminus \overline{I}$	$k\in [0,2\pi ]\text{}\setminus \overline{I}$

Representation Formulae for Zero-Energy/Flat-Band Edge States

Each edge state is expressible as a linear combination of exponential solutions of the bulk difference equations, which 1) decay into the bulk and 2) satisfy the boundary conditions that ${\psi }_{\omega }$ vanishes at all vertices of $\mathbb{H}$ which are outside ${\mathbb{H}}_{♯}$. These decaying exponential solutions are associated with the roots of an appropriate polynomial, which depends on the edge parameters $(a_{ij})$: $p_{+}(\zeta ;k)$ for B site edge states and $p_{-}(\zeta ;k)$ for A site edge states. In particular, $p_{+}(\zeta ;k)$ and $p_{-}(\zeta ;k)$ are polynomials of degree $n_3-n_1$, where $n_{\nu }$ ($\nu =1,2,3$) are introduced in the detailed mathematical setup below.

In Theorem 1, edge states occur only if we are in the ZZ case; $a_{11}-a_{12}\ne 0\hspace{0.17em}\text{mod}\hspace{0.17em}3$. Let $r=r(k)$ denote the number of zeros of the relevant polynomial inside the open unit disk and denote these roots by

$${\zeta }_j={\zeta }_j(k),\hspace{1em}1\le j\le r(k).$$ [6]

For $k\notin \{2\pi /3,4\pi /3\}$, we have proved in ref. 3 that

$$r(k)=\{\begin{array}{l}-n_1-s_2{\mathbb{1}}_{(2\pi /3,4\pi /3)}(k),\hspace{1em}\text{for} p_{+}\hfill \\ n_3+s_2{\mathbb{1}}_{(2\pi /3,4\pi /3)}(k),\hspace{1em}\text{for} p_{-},\hfill \end{array}$$ [7]

where $1_S(k)$ denotes the indicator function of the set S. The cases $k=2\pi /3,4\pi /3$ are dealt with separately in ref. 3.

Furthermore assume that k lies in an appropriate subinterval of $[0,2\pi ]$ for which there are edge states; see Table 1 in Theorem 1. In the following result, we let $n_{\text{min}}^A,\hspace{0.17em}{n}_{\text{min}}^B$ be as in Eq. 4 and let $n_{\nu }$, for $\nu =1,2,3$, be as defined in the detailed mathematical setup below.

Theorem 2

(Representation formulae).Denote the nonzero components of a zero-energy edge state ψ by

$${\psi }_{{\mathbf{v}}_I+m{\mathbf{v}}_1+n{\mathbf{v}}_2}=e^{\mathit{imk}}\mathrm{\Psi }(n;k)\hspace{1em}\text{for}\hspace{0.17em}m\in \mathbb{Z},\hspace{0.17em}n\ge n_{\text{min}}^I,$$

where I = A or B, according to whether the edge state is supported on A sites or B sites. Let $n_{\text{base}}$ be defined by

$$n_{\text{base}}=\{\begin{array}{l}{n}_{\text{min}}^B-n_3,\hspace{1em}\text{if}\hspace{1em}I=A\hfill \\ n_{\text{min}}^A+n_1,\hspace{1em}\text{if}\hspace{1em}I=B.\hfill \end{array}$$ [8]

Then,

$$\mathrm{\Psi }(n;k)=c\tilde{\mathrm{\Psi }}(n),$$

where c is an arbitrary complex constant, and $\tilde{\mathrm{\Psi }}(n)$ is given by the three equivalent formulas for $n\ge n_{\text{base}}$,

$$\tilde{\mathrm{\Psi }}(n)={\displaystyle \sum _{\stackrel{{\mathcal{l}}_1+\dots +{\mathcal{l}}_r=n-n_{\text{base}}+1}{{\mathcal{l}}_1,\dots ,{\mathcal{l}}_r\ge 1}}{\zeta }_1^{{\mathcal{l}}_1-1}}\cdots {\zeta }_r^{{\mathcal{l}}_r-1},$$ [9]

$$\tilde{\mathrm{\Psi }}(n)=\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }{e}^{i(n-n_{\text{base}}+1)\omega }}{\displaystyle \prod _{j=1}^r{\left(e^{i\omega }-{\zeta }_j\right)}^{-1}}d\omega ,$$ [10]

$$\tilde{\mathrm{\Psi }}(n)={\displaystyle \sum _{j=1}^r\frac{{\zeta }_j^{n-n_{\text{base}}}}{{\displaystyle \prod _{\mathcal{l}\in \{1,\dots ,r\}\setminus \left\{j\right\}}\left({\zeta }_{\mathcal{l}}-{\zeta }_j\right)}}}\text{}.$$ [11]

For $n<n_{\text{base}},\hspace{0.17em}\tilde{\mathrm{\Psi }}(n)=0$.

For the normalization, we have

$$\left|\right|\tilde{\mathrm{\Psi }}|{|}_{l^2(\mathbb{Z})}^2={\displaystyle \sum _{j=1}^r\frac{{\zeta }_j^{r-1}}{1-|{\zeta }_j{|}^2}}\hspace{0.17em}{\displaystyle \prod _{\mathcal{l}\in \{1,\dots ,r\}\setminus \left\{j\right\}}\frac{1}{({\zeta }_{\mathcal{l}}-{\zeta }_j)(1-\overline{{\zeta }_j}{\zeta }_{\mathcal{l}})}}.$$ [12]

Note that this representation formula reduces to the standard formula for the zero-energy edge states in the case of the classical ZZ edges (for which r = 1 and ${\zeta }_1={(1+e^{ık})}^{\pm 1})$.

Nonzero-Energy (Dispersive) Edge States

We have designed and implemented a numerical method to compute edge states and their associated dispersion curves. All computations were performed in MATLAB using the computer code (6). A brief review of numerical investigations, with focus on $E\ne 0$ edge states, appears below; see also ref. 3. For ${\mathbb{H}}_{♯}$ terminated by a rational edge other than the standard ZZ and AC, we find nonzero-energy, nonflat-band (hence dispersive) edge states as shown in Fig. 3; in each plot, the intersection of the dark region with the vertical line corresponding to a fixed k is the spectrum of $H_{♯,k}$. In particular, the curved lines bifurcating out of the (Dirac) points, $(E,k)=(0,2\pi /3)$ and $(0,4\pi /3)$ (Fig. 3, Middle and Right), and $(E,k)=(0,0)$ and $(E,k)=(0,2\pi )$ (Fig. 3, Left), parametrize non-flat-band edge states.

Fig. 3.

The $l_k^2$ spectrum of ${\mathbb{H}}_{♯,k}$ versus k for several choices of edges: (Left) $(a_{11},a_{12})=(4,1)$, an AC-type edge; (Middle) $(a_{11},a_{12})=(6,1)$, a ZZ-type balanced edge; and (Right) $(a_{11},a_{12})=(8,1)$, a ZZ-type unbalanced edge. Above each plot is a diagram of the associated edge with the bonds that connect the sites in the structure closest to the boundary (also called frontier sites and represented with filled circles) to the closest sites outside the structure (represented with empty circles).

Relation to Previous Work

The tight binding edge Hamiltonian has most commonly been studied for the classical ZZ and AC edges; see Fig. 2. In ref. 7, it is proved that the classical AC edge supports no edge modes (zero or nonzero energy).

There are studies, in the physics literature, of rational edges (4, 5, 8). The definitions of edges used in these works differ from ours. Let us now describe these classes of edges, and contrast them with the class of edges studied in this article. Recall that our edges are boundaries of structures ${\mathbb{H}}_{♯}\subset \mathbb{H}$, comprising all honeycomb vertices in a closed half-space determined by a line parallel to ${\mathbf{v}}_1$, where ${\mathbf{v}}_1$ is any vector in the triangular lattice Λ. Here, we shall refer to such edges as half-space termination edges.

The notion of minimal edge was introduced in ref. 4. Minimal edges have the following properties:

• •the structure is periodic with period vector ${\mathbf{v}}_1=a_{11}\stackrel{°}{{\mathbf{v}}_1}+a_{12}\stackrel{°}{{\mathbf{v}}_2}$, where a₁₁, a₁₂ are positive integers;
• •no site of ${\mathbb{H}}_{♯}$ has two nearest neighbors in $\mathbb{H}\setminus {\mathbb{H}}_{♯}$;
• •no site of $\mathbb{H}\setminus {\mathbb{H}}_{♯}$ has two nearest neighbors in ${\mathbb{H}}_{♯}$; and
• •within a period, there are precisely $a_{11}+a_{12}$ frontier sites, that is, sites of ${\mathbb{H}}_{♯}$ with neighbors in $\mathbb{H}\setminus {\mathbb{H}}_{♯}$.

See Fig. 4, Left for an example of a minimal edge. It is suggested in ref. 4 that such minimal edge structures are energetically preferred. In general, a minimal edge need not be of the half-space termination type studied here.

Fig. 4.

The $l_k^2$ spectrum of ${\mathbb{H}}_{♯,k}$ versus k for the ZZ-type edge $(a_{11},a_{12})=(3,2)$, (Left) balanced and (Right) unbalanced. Note that the balanced edge is minimal, whereas the unbalanced one is not.

The class of modified edges, arising from the periodic attachment of atoms and bonds to minimal edge atoms at frontier sites of ${\mathbb{H}}_{♯}$, is studied in ref. 8. The edges studied here may be either minimal or modified (Fig. 4).

In ref. 5, edges which arise from a periodic pattern of displacements of a selected dimer (pair of nearest neighbor sites) are studied, with period vector ${\mathbf{v}}_1=a_{11}\stackrel{°}{{\mathbf{v}}_1}+a_{12}\stackrel{°}{{\mathbf{v}}_2}$ with $a_{11},a_{12}\in \mathbb{Z}$. In the case where $a_{11},a_{12}>0$, this class of edges is asserted to be precisely the class of minimal edges, as defined in ref. 4. There is overlap between our class of half-space termination edges and those discussed in ref. 5, but neither class includes the other.

We now compare our results with those of refs. 4, 5, and 8. The main goal of ref. 4 is to derive continuum boundary conditions for an effective Dirac operator, associated with a minimal rational edge. Toward this goal, they consider the tight binding model for parallel quasi-momentum $k\approx 0$. The article (5) postulates a bulk-edge correspondence: For a fixed edge, the dimension of the subspace of zero-energy edge states is equal to the winding number of the Zak phase along a one-dimensional Brillouin zone determined by the edge orientation. The authors of ref. 5 apply this approach to obtain an expression, derived previously in ref. 4, for the density of edge states. The reader should note that the results of ref. 5 are displayed in terms of a scaled (edge dependent) parallel quasi-momentum range, while the range of parallel quasi-momenta in the present article is fixed to be $[0,2\pi ]$. There appears to be agreement between our rigorous results and the results in ref. 5 for those edges in the overlap of our studies. To our knowledge, no previous articles rigorously address, for a general class of rational edges, the questions of which parallel quasi-momentum ranges support zero-energy edge states; when they exist, the dimensionality of the eigenspaces; whether the edge states are supported on A or B sublattice sites; or explicit formulae for zero-energy edge states.

Numerical studies in ref. 8 indicate that a flat band, for a minimal edge, can give rise to nonzero-energy edge state curves when additional sites and bonds are attached to form a modified edge. Our numerical investigations give strong evidence that nonzero-energy edge state curves arise in minimal structures themselves.

Detailed Mathematical Setup and Ideas behind Our Results

We first give an explicit formulation of the edge state eigenvalue problem for the Hamiltonian ${\mathbb{H}}_{♯}$ defined in Eq. 5. By Eq. 3b, there are integers k₁ and k₂ such that $a_{11}-a_{12}=3k_2+s_2$ and $a_{22}-a_{21}=3k_1+s_1$, where $s_1,s_2\in \{-1,0,1\}$. Now set ${\tilde{m}}_1=k_1,\hspace{0.17em}{\tilde{n}}_1=k_2,\hspace{0.17em}{\tilde{m}}_2=k_1+a_{21},\hspace{0.17em}{\tilde{n}}_2=k_2-a_{11},\hspace{0.17em}{\tilde{m}}_3=k_1-a_{22}$, and ${\tilde{n}}_3=k_2+a_{12}$. Except for the classical ZZ case (${\mathbf{v}}_1=\stackrel{°}{{\mathbf{v}}_1})$, which we analyze separately, the integers ${\tilde{n}}_1,{\tilde{n}}_2$, and ${\tilde{n}}_3$ are all distinct. We define $m_{\nu }={\tilde{m}}_{\sigma (\nu )},\hspace{0.17em}{n}_{\nu }={\tilde{n}}_{\sigma (\nu )}),\hspace{0.17em}\nu =1,2,3$, where σ is the permutation of {1, 2, 3}, such that

$${\tilde{n}}_{\sigma (1)}<{\tilde{n}}_{\sigma (2)}<{\tilde{n}}_{\sigma (3)}.$$

The three nearest neighbors in $\mathbb{H}$ to the A site ${\mathbf{v}}_A+m{\mathbf{v}}_1+n{\mathbf{v}}_2$ are the three B sites,

$${\mathbf{v}}_B+(m+m_{\nu }){\mathbf{v}}_1+(n+n_{\nu }){\mathbf{v}}_2,\hspace{1em}\nu =1,2,3,$$

and the three nearest neighbors in $\mathbb{H}$ to the B site ${\mathbf{v}}_B+m{\mathbf{v}}_1+n{\mathbf{v}}_2$ are the three A sites,

$${\mathbf{v}}_A+(m-m_{\nu }){\mathbf{v}}_1+(n-n_{\nu }){\mathbf{v}}_2,\hspace{1em}\nu =1,2,3.$$

A wave function in $l_k^2$ may be written in the form

$$\psi ({\mathbf{v}}_A+m{\mathbf{v}}_1+n{\mathbf{v}}_2)=e^{\mathit{imk}}{\psi }^A(n),\hspace{1em}n\ge n_{\text{min}}^A$$ [13a]

$$\psi ({\mathbf{v}}_B+m{\mathbf{v}}_1+n{\mathbf{v}}_2)=e^{\mathit{imk}}{\psi }^B(n),\hspace{1em}n\ge n_{\text{min}}^B,$$ [13b]

where $m\in \mathbb{Z}$.

The Edge State Eigenvalue Problem

The wave function in Eq. 13 is an edge state for energy E if and only if ψ^A, ψ^B satisfy the difference equations and boundary conditions,

$${\displaystyle \sum _{\nu =1,2,3}{e}^{i{m}_{\nu }k}}{\psi }^B(n+n_{\nu })=E{\psi }^A(n),\hspace{1em}n\ge n_{\text{min}}^A$$ [14a]

$${\psi }^B(n)=0,\hspace{1em}n<n_{\text{min}}^B,$$ [14b]

and

$${\displaystyle \sum _{\nu =1,2,3}{e}^{-i{m}_{\nu }k}}{\psi }^A(n-n_{\nu })=E{\psi }^B(n),\hspace{1em}n\ge n_{\text{min}}^B$$ [15a]

$${\psi }^A(n)=0,\hspace{1em}n<n_{\text{min}}^A.$$ [15b]

On the Proof of Theorem 1 on Zero-Energy/Flat-Band Edge States

When E = 0, Eqs. 14 and 15 decouple. We shall discuss Eq. 14 here, the equation governing edge states which are supported on B sites; Eq. 15, governing edge states which are supported on A sites, is addressed analogously.

Solutions of Eq. 14 are related to roots, $\zeta =\zeta (k)$, located within the open unit disk, of the indicial equation

$${\displaystyle \sum _{\nu =1,2,3}{e}^{i{m}_{\nu }k}}{\zeta }^{n_{\nu }}=0,$$ [16]

or, equivalently, of the $n_3-n_1$ degree polynomial equation: $p_{+}(\zeta ,k)\equiv 1+e^{i(m_2-m_1)k}{\zeta }^{n_2-n_1}+e^{i(m_3-m_1)k}{\zeta }^{n_3-n_1}=0$.

As in Eq. 6, we denote these roots by ${\zeta }_j={\zeta }_j(k),\hspace{0.17em}j=1,\dots ,r(k)$, where we recall that $r=r(k)$ is the number of roots of the polynomial equation [16] inside the open unit disk. Since all the roots can be proven to be simple, any solution of Eq. 14 is given by

$${\psi }^B(n)={\displaystyle \sum _{j=1}^r{A}_j}{\zeta }_j^n,\hspace{1em}\text{for}\hspace{1em}n\ge n_{\text{min}}^A+n_1,$$ [17a]

$${\psi }^B(n)=0,\hspace{1em}n<n_{\text{min}}^A+n_1,$$ [17b]

where $(A_1,\dots ,A_r)$ is an arbitrary solution of the linear algebraic system

$${\displaystyle \sum _{j=1}^r{\zeta }_j^n}{A}_j=0,\hspace{1em}{n}_{\text{min}}^A+n_1\le n<n_{\text{min}}^B.$$ [18]

Note that $n_{\text{min}}^A+n_1\le n_{\text{min}}^B$. Indeed, we have already observed that $n_{\text{min}}^B-n_{\text{min}}^A\in \{0,-s_2\}$, and we prove, in appendix A.4 of ref. 3, that $n_1\le -1$. Thus, nontrivial B-site edge states exist for those $k\in [0,2\pi ]$ for which Eq. 18 has a nontrivial solution $(A_1,\dots ,A_r)$. Since the number of independent equations in Eq. 18 is $n_{\text{min}}^B-n_{\text{min}}^A-n_1$ and there are $r(k)$ unknowns, a nontrivial B-site edge state for energy E = 0 exists if and only if

$$r(k)>n_{\text{min}}^B-n_{\text{min}}^A-n_1.$$ [19]

Furthermore, if Eq. 19 holds, then the dimension of the zero energy eigenspace is equal to $r(k)-[n_{\text{min}}^B-n_{\text{min}}^A-n_1]$. The expression for $r(k)$ is displayed in Eq. 7. To deduce Theorem 1 from Eq. 19, we use the expression for $r(k)$ and the following formula, proved in (3), for $D_B-D_A$ in terms of $s_2,n_{\text{min}}^B$ and $n_{\text{min}}^A$:

$$D_B-D_A=\frac{\sqrt{3}}{2}|{\mathbf{v}}_1{|}^{-1}\left(\frac{1}{3}{s}_2+n_{\text{min}}^B-n_{\text{min}}^A\right)\text{}.$$

An analogous argument yields the result for A-site edge states.

On the Proof of Theorem 2 on Representation Formulae for Zero-Energy/Flat-Band Edges States

Here, we explain our derivation of our explicit formulae for zero-energy edge states. Eq. 9 can be reduced to the following result. Let ${\zeta }_1,\dots ,{\zeta }_r$ denote distinct complex numbers. Then, there exist $A_1,\dots ,A_r\in C$ such that,

$$\text{for} \text{all}\hspace{0.17em}n\ge 1,\hspace{1em}{\displaystyle \sum _{\stackrel{k_1+\dots +k_r=n}{k_1,\dots ,k_r\ge 1}}{\zeta }_1^{k_1}}\cdots {\zeta }_r^{k_r}={\displaystyle \sum _{j=1}^r{A}_j}{\zeta }_j^n.$$ [20]

Note, in particular, that this expression vanishes for $1\le n<r$. Eq. 20 is proven by induction on the number r. Eq. 10 follows from Eq. 9 by a discrete Fourier transform, and, finally, Eq. 11 is obtained from Eq. 10 by residue calculation.

Remarks on the Numerical Investigation of $E\ne 0$ Edge States

When $E\ne 0$, Eqs. 14 and 15 are no longer uncoupled. As in the E = 0 case, solutions are represented as a linear combination of exponential solutions, ${\zeta }^n\xi$, where $\xi =({\xi }^A,{\xi }^B)\in C^2\setminus \left\{0\right\}$ and $\zeta \in C$ such that the following equations hold.

$$P_{+}(\zeta ,k)·P_{-}(\zeta ,k)=E^2,\hspace{1em}\text{where}$$ [21a]

$$P_{+}(\zeta ,k)=\text{}{\displaystyle \sum _{\nu =1,2,3}}{e}^{i{m}_{\nu }k}{\zeta }^{n_{\nu }},\hspace{0.17em}{P}_{-}(\zeta ,k)=\text{}{\displaystyle \sum _{\nu =1,2,3}}{e}^{-i{m}_{\nu }k}{\zeta }^{-n_{\nu }},$$ [21b]

with $m_{\nu },n_{\nu }$ as defined earlier; when ζ solves Eq. 21, there exists a nonzero vector $({\xi }^A,{\xi }^B)\in C^2$ that satisfies

$$\left(\begin{array}{cc}-E& P_{+}(\zeta ,k)\\ P_{-}(\zeta ,k)& -E\end{array}\right)\left(\begin{array}{c}{\xi }^A\\ {\xi }^B\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)\text{}.$$ [22]

If E is not in the essential spectrum (9) of ${\mathbb{H}}_{♯}(k)$, we can prove that Eq. 21 has $n_3-n_1$ roots inside the open unit disk (and no roots on the unit circle); we denote them by ${\zeta }_1,\dots ,{\zeta }_{n_3-n_1}$ (not to be confused with the roots listed in Eq. 6) and denote the corresponding ξ as $({\xi }_j^A,{\xi }_j^B)$. The ζ_j, ${\xi }_j^A,{\xi }_j^B$ depend on E and k. The vector $({\xi }_j^A,{\xi }_j^B)$ is defined up to a multiplication by a nonzero complex scalar. We assume that the ζ_j are all distinct. The vector $({\xi }_j^A,{\xi }_j^B)$ can then be taken to depend analytically on E as E varies in a small complex disk.

The analog of Eq. 18, governing edge states whose energies, E, are not constrained to be zero, is the $(n_3-n_1)\times (n_3-n_1)$ system of homogeneous linear equations

$${\displaystyle \sum _{j=1}^{n_3-n_1}{\zeta }_j^n}{\xi }_j^A·A_j=0,\hspace{1em}\text{for}\hspace{1em}{n}_{\text{min}}^B-n_3\le n<n_{\text{min}}^A$$ [23a]

$${\displaystyle \sum _{j=1}^{n_3-n_1}{\zeta }_j^n}{\xi }_j^B·A_j=0,\hspace{1em}\text{for}\hspace{1em}{n}_{\text{min}}^A+n_1\le n<n_{\text{min}}^B$$ [23b]

for unknowns $A_1,\dots ,A_{n_3-n_1}$. An edge state with quasi-momentum $k\in [0,2\pi ]$ and energy E occurs if and only if the determinant of the coefficient matrix of the linear system Eq. 23, $\mathrm{\Delta }(E,k)$, is equal to zero.

We have numerically investigated the edge state eigenvalue problem, for different choices of rational edge termination, by studying the function $(E,k)\mapsto \mathrm{\Delta }(E,k)$ over a discrete grid with respect to E and k of different resolutions. The discrete values of k vary in the interval $I_k=[0,2\pi ]$ in Figs. 3 and 4 and in the interval $I_k=[0,\pi ]$ in Figs. 5 and 6; those of E vary between rigorous bounds on the spectrum of $H_{♯}$. More precisely, given $(k,E)$, we may carry out the following algorithm:

Fig. 5.

Plots of $(k,E)\mapsto \log |\mathrm{\Delta }(E,k)|$ for $k\in (0,\pi )$ ($N_k=1,000$ points) and $E\in (-E_{\text{lim}},E_{\text{lim}})$ ($N_E=1,000$ points) for various AC-like edges with $a_{12}=1$.

Fig. 6.

Plots of $(k,E)\mapsto \log |\mathrm{\Delta }(E,k)|$ for $k\in (0,\pi )$ (N_k = 600 points) and $E\in (-E_{\text{lim}},E_{\text{lim}})$ (N_E = 600 points) for various ordinary ZZ-like edges with $a_{12}=1$.

• •Compute the $2(n_3-n_1)$ roots of Eq. 21. The roots are calculated by computing the eigenvalues of the associated companion matrix. We verify the assumption that the roots, ζ, are distinct.
• •Deduce whether E is in the essential spectrum of $H_{♯}(k)$ (if one of the roots lies on the unit circle) or not.
• •Compute, for each root $\zeta (k,E)$ inside the unit circle, a vector satisfying Eq. 22.
• •Construct the matrix $\mathbb{M}(k,E)$ appearing in Eq. 23, and compute its determinant $\mathrm{\Delta }(k,E)$.

We make a heat map of the function $(k,E)\mapsto \log |\mathrm{\Delta }(k,E)|$ over the regular $(N_k+1)\times (N_E+1)$ grid of points of $I_k\times (-E_{\text{lim}},E_{\text{lim}})$ where $N_k,N_E=1,000$. In Fig. 3, the dark areas correspond to the essential spectrum. Outside of the dark areas, we seek edge state curves by studying where $(k,E)\mapsto \log |\mathrm{\Delta }(k,E)|$ takes on very large negative values. The function $E\mapsto |\mathrm{\Delta }(k,E)|$ appears to vanish at E = 0 for $k\in (2\pi /3,4\pi /3)$ for the ZZ-type edges (Fig. 3, Middle and Right) but not to vanish at E = 0 for all $k\in (0,2\pi )$ for the AC-type edge (Fig. 3, Left). This illustrates Theorem 1.

To confirm the existence of an edge state near a particular point $(k,E)=(k_0,E_0)$, we compute the winding number of the mapping $E\mapsto \mathrm{\Delta }(k_0,E)$ along a sufficiently small circle about E₀, which makes sense because the mapping is analytic. In all the investigated cases, the winding number is equal to one, implying that there exists a simple root of $E\mapsto \mathrm{\Delta }(k_0,E)$ near E₀.

Finally, we study the behavior of the spectrum as the length of the period vector increases. We consider, first, a sequence of AC-type edges defined by $a_{12}=1$ as a₁₁ increases. We observe the presence of multiple dispersive (nonflat) edge state curves bifurcating from $(k,E)=(0,0)$ (and from $(k,E)=(2\pi ,0)$), and we find that the number of curves increases when a₁₁ increases. Fig. 5, showing the cases $a_{11}=10$ and $a_{11}=22$, illustrates the increasing complexity of the dispersion curves. Similar observations hold for balanced ZZ-like edges; see Fig. 6. Note that, as a₁₁ tends to infinity, the sequence of edges studied (a₁₁ increasing and $a_{12}=1$) tends to the balanced classical ZZ edge. Although the classical ZZ edge has a single dispersion curve, which is flat only over a limited range of k, the nearly flat dispersion curves in Figs. 5 and 6 extend over all $k\in (0,2\pi )$. Note, however, that the definition of quasi-momentum depends on the edge. On the other hand, we have studied a sequence of edges for which a₁₁ and a₁₂ are two consecutive Fibonacci numbers, and we find no evidence for such increasing complexity.

Summary and Open Questions

We have completely analyzed the question of which rational edges give rise to zero-energy/flat-band edge states. We have given formulae for these edge states when they exist. Finally, we have given a general criterion for the existence of edge states (dispersive or nondispersive) that we have implemented numerically.

Many natural open questions arise from this study (for an extended list, see ref. 3), and we mention two of them. Note that our results do not determine all rational edges that give rise only to zero-energy edge states. Indeed, while the classical armchair edge has no edge states (flat band or dispersive), we have shown that, more generally, an edge of “AC type” produces no zero-energy edge states, but it can produce dispersive edge state curves of nonzero energy. A mathematically rigorous understanding of precisely which edges give rise to dispersive edge states, in particular, an understanding of the bifurcation of the dispersive curves in terms of the values $(a_{11},a_{12})$, would be of interest. Finally, the main motivation of the present study was to understand transport along irrational edge terminations, and we hope to use the results presented here as a first step.

Data, Materials, and Software Availability

Computer code has been deposited in Open Science Framework (https://osf.io/6z874/) (6).