Systematic construction of square-root topological insulators and superconductors

We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of the original graph of $H_{\text{original}}$, where the subdivided graph is obtained by putting one vertex on each link in the original graph. When $H_{\text{original}}$ describes a topological system, there emerge in-gap edge states at non-zero energy in the spectrum of $H_{\text{root}}$, which are the inherence of the topological edge states at zero energy in $H_{\text{original}}$. In this case, $H_{\text{root}}$ describes a square-root topological insulator or superconductor. Typical examples are square roots of the Su-Schrieffer-Heeger (SSH) model, the Kitaev topological superconductor model and the Haldane model. Our scheme is also applicable to non-Hermitian topological systems, where we study an example of a nonreciprocal non-Hermitian SSH model.

Introduction: Topological insulators and superconductors are among the most studied fields in condensed matter physics in this decade 1,2 .They are characterized by the bulk-edge correspondence, i.e., by the emergence of topological edge states although the bulk is gapped.
Recently, a square-root topological insulator is proposed 3,4 .Its notion has also been generalized to square-root higherorder topological insulators 5 .They are characterized by the emergence of in-gap edge states at nonzero energy, which are the inherence of the topological edge states at zero energy of the original Hamiltonian 3,4 .
In this paper, we propose a general scheme to construct square-root topological insulators and superconductors from ordinary topological insulators and superconductors based on the graph theory.There is one-to-one correspondence between a tight-binding Hamiltonian and a weighted graph.A graph is composed of vertices and links.We can construct a new graph by introducing one vertex on each link, which we refer to as the subdivided graph 6,7 .We call the original graph the parent graph in contrast to the subdivided graph.Any subdivided graph is bipartite because it contains original vertices and newly added vertices.Examples are shown in Figs.1, 2 and 3, where original (new) vertices are shown in magenta (cyan).
We denote the Hamiltonian constructed on the subdivided graph as H root .We then find (H root ) 2 = H par ⊕ H res , where H par is identical to the original Hamiltonian H original up to an additive constant interpreted as a self-energy.We call H par and H res the parent and residual Hamiltonians, respectively.When H original describes a topological system, it contains zero-energy topological boundary states, producing the corresponding boundary states at nonzero energy in H root .Furthermore, zero-energy perfect-flat bulk-bands may emerge in H root as a bipartite property according to the Lieb theorem: See orange lines in Figs. 2 and 3.Because the eigenvalues are shown to be identical between H par and H res except for zeroenergy states in H res , H root is interpreted as the square-root Hamiltonian of H original .We can use the same topological index between H root and H original since the eigenvectors are identical between them.Indeed, the region of the in-gap edge states in H root is precisely the same as that of the zero-energy edge states in H original . We A square root of the model is given by 8 with E (k) = t 2 a + t 2 b + 2t a t b cos k.This is an infiniterange hopping model.
Here we recall the Dirac idea to take a square root of the Klein-Gordon equation.He obtained the Dirac equation by introducing a matrix degree of freedom.The Dirac equation has various intriguing properties such as chirality and the index theorem, which are absent in the Klein-Gordon equation.
We propose to take a square root of a Hamiltonian by increasing a matrix degree of freedom as follows: 1) We first write down a graph representation of the adjacency matrix of the original Hamiltonian H original .2) We construct a subdivided graph from the original graph.3) We construct a Hamiltonian H root on the subdivided graph.Then, we obtain (H root ) 2 = H par ⊕ H res , where H par is identical to the original Hamiltonian H original up to an additive constant, provided the hopping parameter is taken to be √ t in H root corresponding to the hopping parameter t in H original .The square-root Hamiltonian is given by H root .
We start with a Hamiltonian H original where a unit cell contains N sites connected by M hoppings.We consider a Hamiltonian on the subdivided graph, which is given by

arXiv:2005.12608v1 [cond-mat.mes-hall] 26 May 2020
It is required that , when H root is Hermitian.We have where Thus the square of the Hamiltonian, (H root ) 2 , is decomposed into a direct sum of H par and H res , which are the parent and residual Hamiltonians defined on the parent and residual graphs.
In general, H par is identical to H original up to a constant term because both of them are constructed on the same graph, where C is a positive constant obtained by calculating (H root ) 2 .This constant term can be interpreted as a self energy, as in the case of the second-order perturbation theory.The zero-energy topological edge states in H original are transformed into the in-gap boundary states at non-zero energy ± √ C in H root .The bipartite Hamiltonian has chiral symmetry, {γ, H root } = 0, with the chiral operator defined by In general, we have M > N .According to the Lieb theorem 16 , there are |M − N | zero-energy states constituting perfect-flat bulk bands.It is known [9][10][11] that the eigenvalues are identical between H par and H res except for these zero-energy states in H res and that they are non-negative.Namely, when we set we obtain {ε res The eigenvectors of (H root ) 2 are given by When the Hamiltonian H root is diagonalized by a unitary transformation U as U † H root U = H D root , the Hamiltonian (H root ) 2 is also diagonalized by the same unitary transformation as Then, the eigenvalues of H root are obtained just by taking a square root of them with the same eigenvectors, where ) Because of this property, H root is interpreted as the square-root Hamiltonian of H original .
An important observation is that the topolgocal properties are identical between H root and H original since the eigenvectors are the same.Correspondingly, the topological indices are the same.
Square-root SSH model: For the first example, we analyze the SSH model (1).The spectrum contains the zero-energy topological edge states as in Fig. 1(c).The graph of the SSH model is a simple one-dimensional graph containing two vertices in the unit cell [Fig.1(a)].The corresponding subdivided graph is a one-dimensional graph containing four vertices in the unit cell [Fig.1(b)].The square-root Hamiltonian H root is given by ( 3) with (N, M ) = (2, 2), and It is straightforward to derive (H root ) 2 = H par ⊕ H res with where Square-root Kitaev topological superconductor: The next example is a square root of the Kitaev p-wave topological superconductor model defined by [12][13][14][15] The spectrum contains the zero-energy topological edge states as in Fig. 2(c).The corresponding graph and subdivided graph  Square-root Haldane model: We next study a square root of the Haldane model.The Hamiltonian is defined on the graph in Fig. 3(a) and given by The spectrum in nanoribbon geometry contains chiral edge states as in Fig. 3(c).The subdivided graph of the honeycomb graph is shown in Fig. 3(b).The square-root Hamiltonian H root is given by the Hamiltonian (3) with (N, M ) = (2, 9) and √ 3/2, −1/2 .The parent Hamiltonian H par is found to be The chiral edge state in nanoribbon geometry emerges in H root , as shown in Fig. 3(d).Furthermore, there are 7 zero energy states in H total due to the Lieb theorem 16 with Square-root non-Hermitian SSH model: We proceed to construct a square root of a non-Hermitian SSH model by introducing the nonreciprocity γ, as illustrated in Fig. 4(a).The Hamiltonian reads [17][18][19][20][21][22] where the hopping amplitudes are different between left and right goings.The spectrum contains zero-energy edges states in the topological phase, whose real and imaginary parts are shown in Fig. 4(c) and (c').The square-root Hamiltonian H root is defined on the subdivided graph in Fig. 4(b), and given by the Hamiltonian (3) with The parent Hamiltonian H par is found to be The residual Hamiltonian is given by H res = {a ij }, where  Discussions: We have presented a systematic method to construct square-root topological insulators and superconductors based on subdivided graphs.We recall that subdivided graphs naturally arise in electric-circuits when we rewrite the Kirchhoff law into the Schrödinger equation 6,7 .Hence, it would be natural to make experimental observation of squareroot topological systems with the use of electric circuits.We start with a lattice electric circuit.In the original graph, it contains voltage at the sites, which correspond to the vertices in the graph theory.We can define currents flowing between two adjacent sites, which corresponds to links in the graph theory.Both the in-gap nonzero-energy edge states and the zeroenergy flat bands due to the Lieb theorem are to be observed by measuring impedance peaks [23][24][25] .Another possibility to realize square-root topological systems is a direct construction of lattice structures by photonic 4 or acoustic systems 26,27 .

FIG. 1 :
FIG. 1: Illustration of (a) the graph and (b) the subdivided graph for the SSH model HSSH.The green rectangles represent the unit cells.(c) Energy spectrum of HSSH and (d) Hroot in unit of ta as a function of t b /ta.The topological edge states are marked in magenta.In-gap edge states are on the curve E = ± |ta| + |t b | for Hroot.
res is the Rice-Mele model.In-gap edge states appear at E = ± |t a | + |t b | for |t b | > |t a |, as illustrated in Fig.1(d), whose origin is the topological zero-energy states in the SSH model [Fig.1(c)].

FIG. 2 :
FIG. 2: Illustration of (a) the graph and (b) the subdivided graph for the Kitaev model HKitaev.(c) Energy spectrum of HKitaev and (d) Hroot in unit of t as a function of ∆/t.The topological edge states are marked in magenta.In-gap states are on the curve E = ± 2 |t| + 2 |∆| for Hroot.Lieb perfect-flat bulk-bands are marked in orange.
where ∆ = e −iπ/4 √ ∆.We calculate (H root ) 2 = H par ⊕ H res .The parent Hamiltonian H par is found to be the Kitaev Hamiltonian 12-15 with µ = 0 and the addition of a constant term 2 |t| + 2 |∆|.Ingap edge states appear in H root at E = ± 2 |t| + 2 |∆| as in Fig.2(d), which are transformed from the zero-energy topological states in the Kitaev model[Fig.2(c)].Furthermore, there are perfect-flat bulk-bands at zero energy in H root due to the Lieb theorem 16 with |M − N | = 2.

FIG. 3 :
FIG. 3: Illustration of (a) the graph and (b) the subdivided graph for the Haldane model HHaldane.(c) Energy spectrum of HHaldane and (d) Hroot in unit of t as a function of the momentum k.The chiral edge states are marked in magenta.Lieb perfect-flat bulk-bands are marked in orange.We have set λ = 0.2t/(3 √ 3).

FIG. 4 :
FIG. 4: Illustration of (a) the graph and (b) the subdivided graph for the nonreciprocal non-Hermitian SSH model H non SSH .(c) Real and imaginary parts of the energy spectrum of H non SSH and (d) Hroot in unit of ta as a function of t b /ta.The topological edge states are marked in magenta.In-gap states are on the curve E = ± |ta| + t 2 b − γ 2 for Hroot.We have set γ = ta/4.
10d ε j ≥ 0. It follows from (6) that the eigenvectors of H par and H original are the same, H original |ψ par j = (ε j − C)|ψ par j .Furthermore, the eigenvectors |ψ res j of H res are obtained from those of H par as10 root ) 2 = H par ⊕ H res , we obtain