# Topological Kirchhoff Law

and Bulk-Edge Correspondence for Valley-Chern and Spin-Valley-Chern
Numbers

###### Abstract

The valley-Chern and spin-valley-Chern numbers are the key concepts in valleytronics. They are topological numbers in the Dirac theory but not in the tight-binding model. We analyze the bulk-edge correspondence between the two phases which have the same Chern and spin-Chern numbers but different valley-Chern and spin-valley-Chern numbers. The edge state between them is topologically trivial in the tight-binding model but is shown to be as robust as the topological edge. We construct Y-junctions made of topological edges. They satisfy the topological Kirchhoff law, where the topological charges are conserved at the junction. We may interpret a Y-junction as a scattering process of particles which have four topological numbers. It would be a milestone of future topological electronics.

Topological insulator is one of the most fascinating concepts found in this decadeHasan (); Qi (). It is characterized by topological numbers such as the Chern () number and the index. When the spin s is a good quantum number, the spin-Chern () number replaces the role of the indexProdan (); Sheng (); Yang (). We consider honeycomb lattice systems. Electrons resides either in the or valley in the low-energy Dirac theory. Accordingly we can define the valley-Chern () numberFang11 (); Fang13 (); Li () and the spin-valley-Chern () numberFang11 () in the Dirac theory. This valley degree of freedom leads to valleytronicsRycerz (); Qiao10 (); Qiao11 (); Qiao12 (); Qiao13 (); Xiao07 (); Xiao12 (); Cao (); Jung (); Yao (); Tse (); Ding (). However, the and numbers are ill-defined in the tight-binding model because the topological numbers are defined by the summation of Berry curvatures over the entire Brillouin zone. Namely, a state is indexed by the two topological numbers in the tight-binding model, while it is indexed by the four topological numbers in the Dirac theory.

The are four independent spin-valley dependent Chern numbers in the Dirac theory of honeycomb systems. Each Chern number can be controlled independently by changing the sign of spin-valley dependent Dirac masses. There are 16 types of topological insulators, as shown in the table 1. They are quantum anomalous Hall (QAH) insulator, four types of spin-polarized QAH (SQAH) insulators, quantum spin Hall (QSH) insulator and the band insulator with charge-density-wave (CDW) or antiferromagnetic (AF) order. The CDW and AF insulators are regarded trivial in the tight-binding model.

QAH | ||||||||

SQAH | ||||||||

SQAH | ||||||||

SQAH | ||||||||

SQAH | ||||||||

QSH | ||||||||

CDW | ||||||||

AF |

In this paper, we study the bulk-edge correspondence with respect to the and numbers by examining the boundary of two insulators which have the same and numbers but different and numbers. First we show that gapless edge states appear though they are trivial in the tight-binding model. Furthermore, we show that they are as robust as the topologically protected edges.

We propose a topological electronics based on the edge states in the Dirac theory. We are able to assign four topological numbers to each edge states. By joining three different topological insulators at one point, we can construct a Y-junction made of topological edge states. The edge states at the junction satisfies the conservation of four topological numbers, which we call the topological Kirchhoff law. We can change the connectivity of edge states by changing the topological property of bulk insulators, for instance, by applying electric field. The process may be interpreted as a pair annihilation of two Y-junctions.

Hamiltonian: The honeycomb lattice consists of two sublattices made of and sites. We consider a buckled system with the layer separation between these two sublattices. The states near the Fermi energy are orbitals residing near the and points at opposite corners of the hexagonal Brillouin zone. The low-energy dynamics in the and valleys is described by the Dirac theory. In what follows we use notations , , in indices while for , for ,, and for in equations. We also use the Pauli matrices and for the spin and the sublattice pseudospin, respectively.

We have previously proposed a generic Hamiltonian for honeycomb systemsEzawa2Ferro (), which contains eight interaction terms mutually commutative in the Dirac limit. Among them four contribute to the Dirac mass. The other four contribute to the shift of the energy spectrum. We are able to make a full control of the Dirac mass and the energy shift independently at each spin and valley by varying these parameters, and materialize various topological phasesEzawaQAHE (); EzawaPhoto ().

By taking those affecting the Dirac mass, the tight-binding model is given byKaneMele (); LiuPRB (); Ezawa2Ferro (),

(1) | |||||

where creates an electron with spin polarization at site , and run over all the nearest/next-nearest neighbor hopping sites. We explain each term. The first term represents the nearest-neighbor hopping with the transfer energy . The second term represents the SO couplingKaneMele () with . The third term is the staggered sublattice potential termEzawaNJP () with in electric field . The forth term is the Haldane termHaldane () with . The fifth term represents the antiferromagnetic exchange magnetizationEzawa2Ferro (); Feng () with .

We give typical sample parameters though we treat them as free parameters. Silicene is a good candidate, where eV, meV and Å. The Haldane term might be induced by the photo-irradiation, where with the frequency and the dimensionless intensityEzawaPhoto (); Kitagawa01B (); Oka (). The antiferromagnetic exchange magnetization might be induced by certain proximity effects. The second candidate is provskite transition-metal-oxide grown on [111]-direction, which has antiferromagnetic order intrinsicallyHu (). This material has also the buckled structure as in the case of silicene. Parameters are eV, meV, , meV for LaCrAgO.

The low-energy Hamiltonian is described byEzawa2Ferro ()

(2) | |||||

where is the Fermi velocity. The coefficient of is the mass of Dirac fermions in the Hamiltonian,

(3) |

The band gap is given by .

Topological numbers: We consider the systems where the spin is a good quantum number. The summation of the Berry curvature over all occupied states of electrons with spin in the Dirac valley yieldsHasan (); Qi (); Diamag ()

(4) |

There are four independent spin-valley dependent Dirac masses determined by the four parameters , , and . Accordingly, we can define

(5) | |||||

(6) |

and

(7) | |||||

(8) |

It is to be emphasized that and are not defined in the tight-binding model.

The possible sets of topological numbers are , , , , up to the overall sign in the tight-binding model. They are the trivial, QAH, QSH and two types of SQAH insulators, respectively. They are further classified into subsets according to the valley degree of freedom in the Dirac theory. Trivial insulators are divided into two; one with CDW order and the other with AF ordersEzawa2Ferro (). Each type of SQAH insulators are divided into two: There are four types in all, which we denote by SQAH with . All of them are summarized in Table 1.

Bulk-edge correspondence: The most convenient way to determine the topological charges is to employ the bulk-edge correspondence. When there are two topological distinct phases, a topological phase transition may occur between them. It is generally accepted that the band gap must close at the topological phase transition point since the topological number cannot change its quantized value without gap closing. Note that the topological number is only defined in the gapped system and remains unchanged for any adiabatic process. Alternatively, we may consider a junction separating two different topological phases in a single honeycomb systemEzawaNJP (). Gapless edge modes must appear along the boundary. We may as well analyze the energy spectrum of a nanoribbon in a topological phase, because the boundary of the nanoribbon separates a topological state and the vacuum whose topological numbers are zero: See Fig.1(a).

CDW-AF junction: We first investigate the trivial insulator in the tight-binding model, which consists of two subsets (CDW and AF) in the Dirac theory. It is well known that a nanoribbon made of either the CDW insulator or the AF insulator has no gapless edge modes, as is regarded to be a demonstration of their triviality: See Fig.1(b) and (c). One may wonder how they can be topological in the Dirac theory without gapless edge modes in view of the bulk-edge correspondence. The answer to this problem is that the and numbers are not defined in the vacuum. Indeed, only the charge and the spin are well defined to be zero in the vacuum. Note that the gap needs not close at such a boundary because and are defined only inside of the nanoribbon. This explains the absence of gapless edge modes in Fig.1(b) and (c).

We investigate the junction made of the CDW and AF insulators, whose topological numbers are and , respectively. On one hand, we expect no gapless edge modes in the tight-binding model. On the other hands, there should be gapless edge modes in the Dirac theory. We ask how these two properties are compatible.

To answer this problem we study a hybrid nanoribbon by separating a nanoribbon into two parts, one in the CDW phase and the other in the AF phase: See Fig.1(d). Only the and numbers change across the boundary separating these two regions. We have calculated the band structure of such a hybrid nanoribbon, whose result we display in Fig.1(e). We find one edge state crossing the Fermi energy twice. It is a manifestation of the fact that the and numbers do not change. On the other hand, when we concentrate on the vicinity of the and points, there are well defined edge states. It is a manifestation of the fact that the and numbers change at the junction.

We proceed to argue how strongly the and numbers are protected. The word "topologically protected" means that the edge states are robust against perturbations whose magnitude is less the bulk gap . Indeed, the bulk gap may close by perturbations stronger than it, invalidating the topological arguments at all. How robust is the edge mode at the CDW-AF junction? We have checked that the edge mode takes the extremal energy at the point with

(9) |

It costs the energy to remove the edge states. Now, the topological edge states are protected by the bulk energy gap

(10) |

We find since is the order of eV, while is the order of meV. It is very robust against perturbations.

AF-AF junction: The second example is given by the junction made of AF with and AF with . The phase boundary is an antiferromagnetic domain wall with the magnetization reversed at a line defect. Let us take the line along the axis. The junction is created by introducing the order parameter such that for and for . To investigate the edge state located at , we calculate the band structure of a hybrid nanoribbon composed of the AF phase and the AF phase. We present the result in Fig.1(f). We see clearly gapless edge modes highly enhanced at the point. The extremal energy of the gapless edge mode is given by (9), while the bulk gap is given by (10) also in this case. The edge is very robust.

SQAH-SQAH junction: We next investigate the junction made of two different SQAHs. As an explicit example we take SQAH with and SQAH with . It is not appropriate to use a hybrid nanoribbon in the present case since gapless edge modes appear even for a simple nanoribbon in the SQAH phase. We calculate the band structure of a nanotube geometry since no gapless edge modes appear even for a simple nanotube in the SQAH phase owing to the lack of the edge itself. We take a hybrid nanotube where one half of the nanotube is SQAH and the other half is SQAH, as illustrated in Fig.2(a). We show the result in Fig.2(b), where we see clearly a gapless edge mode highly enhanced at the point. The extremal energy of the gapless edge mode is given by (9), while the bulk gap is given by (10) also in this case. The edge is very robust as well.

Gapless edge mode in Dirac theory: We proceed to construct the Dirac theory of the gapless edge statesEzawaNJP (). They emerge along a curve where the Dirac mass vanishes, . Let us take the edge along the axis. The zero modes emerge along the line determined by , when changes the sign. We may set constant due to the translational invariance along the axis. We seek the zero-energy solution by setting with . Here, is a two-component amplitude with the up spin and down spin, . Setting , we obtain , together with a linear dispersion relation . We can explicitly solve this as

(11) |

where is the normalization constant. The sign is determined so as to make the wave function finite in the limit . This is a reminiscence of the Jackiw-Rebbi modeJakiw () presented for the chiral mode. The difference is the presence of the spin and valley indices in the wave function.

Topological Kirchhoff law: We consider a configuration where three different topological insulators meet at one point: See Fig.3. In this configuration there are three edges forming a Y-junction. It is convenient to assign the topological numbers to each edge which are the difference between those of the two adjacent topological insulators. Namely, when the topological insulator with is on the left-hand side of the one with , we assign the numbers to the boundary, as illustrated in Fig.3. The condition which edges can make a Y-junction is the conservation of these topological numbers at the junction. This law is a reminiscence of the Kirchhoff law, which dictates the conservation of currents at the junction of electronic circuits. We call it the topological Kirchhoff law.

The number of Y-junctions is given by the combination of selecting from topological insulators, i.e., . The number of topological edge states is determined by the combination of selecting from topological insulators. We have types of topological edge states. We show typical examples of Y-junctions in Fig.3.

We present an interesting interpretation of the topological Kirchhoff law. We may regard each topological edge state as a world line of a particle carrying the four topological charges. The Y-junction may be interpreted as a scattering process of these particles. In this scattering process, the topological charges conserve.

Topological electronic circuits: We can construct electronic circuits made of edge states by joining Y-junctions. Each topological edge state carries conductanceEzawaConduc (), whose magnitude is given by the Chern number in unit of . In general, the edge states carry charge , spin , valley-charge and spin-valley-charge . The present-day electronic circuits only use the charge degree of freedom. In our circuits of topological edges we can make a full use of four types of charges. This would greatly enhance the ability of information processing.

We can control the position of edge state by controlling the parameters of the bulk states. The easiest way is to apply electric field locally. Let us review the topological phase transition taking place as changesEzawaNJP () by taking , where the Dirac mass is given by . The condition implies with . It follows that for and for . For instance, the two CDW domains are made in this way in Fig.4(a). Applying only to a part in the QSH domain near the SQAH domain, we can turn this part into the CDW domain as in Fig.4(b). We have thus changed the form of circuit by a pair annihilation of two Y-junction by applying . This will open a way to topological electronics.

I am very much grateful to N. Nagaosa for many fruitful discussions on the subject. This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196.

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