Zero-energy modes in super-chiral nanographene networks of phenalenyl-tessellation molecules
ZZero-energy modes in super-chiral nanographene networks of phenalenyl-tessellationmolecules
Naoki Morishita ∗ and Koichi Kusakabe † Graduate School of Engineering Science, Osaka University,Toyonaka, Osaka 560-8531, Japan (Dated: February 9, 2021)We have derived a general rule for the appearance of zero-energy modes in super-chiral defectivenanographene. This so-called “super-zero-sum rule” defines the appearance of zero modes in anew class of materials, which we call polymerized phenalenyl-tessellation molecules (poly-PTMs).Through theoretical modeling of the electronic states in these molecular forms, we provide concretesolutions for achieving the quantum-spin systems needed in quantum-information devices. The two-dimensional graph of electronic π -orbitals in the poly-PTM possesses a number of localized zeromodes equivalent to that of vacancies in PTMs. In addition to the modes confined to each PTM,another type of zero mode may appear according to the super-zero-sum rule supported by super-chirality. Since the magnetic interactions among quantum spins in the zero modes are determinedby how they appear (which is governed by the super-zero-sum rule), our rule is indispensable fordesigning quantum-information devices using electron zero modes in poly-aromatic hydrocarbonsand defective graphene with vacancies. I. INTRODUCTION
The unconventional electronic structure of graphene[1,2] has attracted a great interest from researchers becauseof the emergence of the massless Dirac fermion[3, 4]. Oneunique characters of the Dirac fermion system is the ap-pearance of zero-energy modes, which are typically foundat the edges[5–15] and the vacancies[16–20] of graphene.Because the zero modes at Fermi energy are naturallyhalf-filled (with one electron occupying each mode at thecharge neutral point), and since unpaired electrons canbecome magnetically active, the physics and chemistryarising from the zero modes have been highlighted, cre-ating a vast field in science[21, 22]. Indeed, since a single-electron spin may appear in each zero mode orbital be-cause of the electron correlation effect, electron spins cancreate a Heisenberg spin system as a whole, such thatuseful magnetic functions can appear in graphene withzero modes.Zero modes have potential uses as quantum bits, sincea single-electron spin may appear in the orbital and un-paired electrons may show a nontrivial correlation be-cause of the strong correlation effects. There are severaltypes of zero modes supporting localized electrons, whichcoexist with Dirac electrons. Therefore, it is importantto find a general rule for controlling the appearance ofzero modes. One example is Shima and Aoki’s classifi-cation of the appearance of gapless linear dispersion andflat bands[23]. The appearance of the flat band at thezero-energy indicates that a huge number of zero modesexist in their honeycomb networks. However, a rule basedon crystal lattice symmetry is not applicable to general ∗ [email protected] † [email protected] molecular structures and networks with a lower symme-try than that of the periodic honeycomb lattice.Singular enhancement in the density of state aroundthe Dirac point has been discussed as an effect of a chiraldefect (e.g. atom vacancies), which are often assumedto be randomly distributed[24]. The related density ofstates in a model with structural defects have been dis-cussed by specifying several examples[25]; however, with-out finding a control rule for certifying zero-energy eigen-states in carbon networks, this discussion is inadequatefor purposes such as designing devices for quantum com-puters, for which more precise control of the electronspins would be required.For a definite series of nanographene networks, theauthors reported a concrete example of a rule to ob-tain the zero modes, even for a system with no sublat-tice imbalance[26, 27]. Although this work was followedby consideration of networks with decorated edges[28],two types of zero modes are defined for our vacancy-centered systems, successfully realizing an embedded lo-calized zero mode as an eigenstate. It was discussed thata well-defined boundary condition of a supposed zero-energy eigenstate becomes artificial, which is given onlyby a mode expansion fomula[29]. Our systems, however,provided a means of defining the true Dirichlet bound-ary condition for the zero mode wavefunction ψ ( r ) as ψ ( r ) | boundary = 0[26]. We have also recently discovered aclass of graphene nanomolecular structures that we callphenalenyl-tessellation molecules (PTMs)[30]. When aPTM contains atomic defects, two types of special non-bonding molecular orbitals appear at the zero-energylevel; one is a vacancy-originated localized zero mode,and the other is an extended Dirac zero mode.This paper reports a general rule for the appearanceof the Dirac and localized zero modes on a graph of π -electron systems of a class of PTM-based poly-aromatichydrocarbons. Our new rule generally holds in molecu- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b lar and periodic systems, even with low spatial symmetry.However, we discovered the relevance of generalized chiralsymmetry in a super-structure (or super-chiral symme-try) in super networks of PTMs.A quantum spin in the zero mode has the potential tobe used as an easy-to-access quantum bit. Furthermore,plenty of zero modes can be created in a two-dimensional π -orbital lattice of graphene structure. If the distribu-tion of spins and the connections among them can beengineered to yield a desired spin system, these mate-rials may be useful as units for quantum computation.Each unit that possesses a unique unitary transformationin its time-evolution step may work as a computationalelement. Our method of producing zero modes is flex-ible enough to design quantum devices using graphenenanostructures; therefore, our rule is indispensable forthe design of quantum-information devices. II. METHODA. Construction of poly-PTM structures
PTMs comprise phenalenyl unit (PU) tiling. Figure1 shows a typical example (vacancy-centerd hexagonalarmchair nanographene, VANG[26]). PTMs have fringesthat form a double zig-zag corner (DZC), where two con-secutive zig-zag edges are indicated by dotted lines inFig. 1.Let us consider the effect that adjusting the atomicpositions on an imaginary honeycomb lattice has uponPTMs, as shown in Fig. 2. PTMs can be categorizedinto two groups, α -PTM and β -PTM, where the centerof each component PU of an α -PTM(a β -PTM) is al- FIG. 1. Vacancy-centered hexagonal armchair nanographene(VANG) is a typical example of a phenalenyl-tessellationmolecule (PTM). Each dodecagonal carbon skeleton unit col-ored in light or dark gray is a phenalenyl unit (PU). Threedouble zig-zag corners (DZCs) are indicated by dotted lines.(a) A vacancy-originated localized zero mode (red and blue)on the B-sites and (b) a √ ×√ ways located at an A-site (a B-site), as denoted by white(black) circles.In this paper, we consider a polymerized PTM struc-ture of α - and β -PTM. In our method for constructinga polymerized PTM, connections ( σ -bonds of carbons)between PTMs are created only at DZCs (shown by bluebonds in Fig. 2). We call a π -network system definedin this way a poly-PTM . We notice that an α -PTM anda β -PTM can be placed in close proximity to each otherto create two bonds at their DZCs in a poly-PTM sys-tem. Two α -PTMs are not directly connected, nor aretwo β -PTMs.Let us define the number of π - π connections, N C , be-tween each α - and β -PTM. In Fig. 2, N C = 4 betweenthe α - and β -PTMs, whereas N C = 2 among the other α - and β -PTM pairs. B. Tight-binding model and the zero-sum rule forpoly-PTMs
Below, we consider a single-orbital tight-binding model(TBM) with a constant non-zero nearest neighbor trans-fer t (cid:54) = 0 and an on-site energy ε = 0. The energy eigen-wavefunction of the TBM is expressed by the amplitude ψ i,l at the i -th site belonging to the l -th sublattice ( l = A or B), forming vectors Ψ A = t ( ψ , A , ..., ψ N A , A ) andΨ B = t ( ψ , B , ..., ψ N B , B ). The site index i runs from 1 to N A ( N B ) where N A ( N B ) is the number of sites in sublat-tice A(B).The Schr¨odinger equation with eigen-energy E is givenin matrix form by (cid:18) H A ← B H B ← A (cid:19) (cid:18) Ψ A Ψ B (cid:19) = E (cid:18) Ψ A Ψ B (cid:19) . (1) FIG. 2. A polymerized PTM (poly-PTM) system comprising α -PTMs (light gray) and β -PTMs (dark gray) on an imagi-nary honeycomb lattice. The two types of PTMs are con-nected by bonds (blue lines) via their respective DZCs (redlines). In the background bipartite honeycomb lattice, theA-sites are colored in white, whereas B-sites are black. Here, H A ← B (= H † B ← A ) represents a transfer matrix froma B-site to an A-site. Equation (1) gives the next linearequation when E = 0: (cid:26) H B ← A Ψ A = 0 ,H A ← B Ψ B = 0 . (2)Equation (2) indicates that, when a zero-energy modeΨ A(B) with E = 0 exists on one of A(B)-sites (i.e. the i A(B) -th site), the sum of the value of Ψ
B(A) on the B(A)-sites connected to this site must be equal to zero. We callthis the zero-sum rule .Conversely, if there exists a wavefunction Ψ
A(B) satis-fying the zero-sum rule at each i B(A) -th site in the wholesystem, every element of the resulting vector given bythe left side of Eq. (1) becomes zero vector. Then Ψ
A(B) satisfies this equation with the eigenvalue E = 0. Thus,finding a nontrivial wavefunction that satisfies the zero-sum rule in the entire system is equivalent to finding thesolution of a nontrivial wavefunction Ψ A(B) with E = 0 .Figure 1 shows examples of the zero modes that sat-isfy the zero-sum rule on the A(B)-sites. One type isa vacancy-originated localized zero mode type, which isfound on the B-sites (red and blue), and the other isan extended Dirac zero mode, which is found on the A-sites (orange and green). We can see that the former haslarger wavefunction amplitudes at the B-sites around thevacancy than at those on the periphery, whereas the lat-ter has perfectly uniform √ ×√ FIG. 3. An exemplary poly-PTM comprising α -, α -, and β -PTMs. We can see that the localized zero mode (LZM) inthe α -PTM satisfies the zero-sum rules on the two A-sitesmarked with stars at the DZC by taking the amplitudes ofthe wavefunction outside the α -PTM to be zero. III. RESULTSA. An example of a poly-PTM
Figure 3 is an exemplary poly-PTM comprising two α -PTMs and one β -PTM. Two types of wavefunctions with E = 0 can appear in a poly-PTM. One is the localizedzero mode (LZM) type. In the example, it is perfectlylocalized in the α -PTM (red and blue circles on the B-sites). Large LZM amplitudes appear mainly around thevacancy of the α -PTM. Furthermore, the distribution ofthe wavefunction amplitudes is not uniform. The othertype is the Dirac zero mode (DZM) ; in this example, itexpands over the α -, α -, and β -PTMs (orange andgreen circles on the A-sites). On the A-sites at the α -and α -PTMs, the DZM wavefunction has the specialcharacteristic of uniform √ ×√ β -PTM.Such examples obtained by numerical simulations sug-gest that LZMs appearing in a poly-PTM system can becharacterized as wavefunctions lying on the B(A)-sites ofthe α ( β )-PTMs, and that each LZM is confined withinone of the PTM subgraphs. An LZM or a non-uniformpart of a DZM can exist on the B(A)-sites of the α ( β )-PTMs. The DZM has a perfectly uniform √ ×√ β ( α )-PTMs. To prove thesestatements, we start by restating rules of the appearanceof the zero modes on a single isolated PTM and thendeclare the rules for a general poly-PTM. B. PTM Rules
PTM Rule 1.
When there are n vacancies in a singleisolated α ( β )-PTM, there are n LZMs, which correspondto the vacancy-induced zero modes in the α ( β )-PTM ap-pearing on the B(A)-sites. Thus, when there is no va-cancy, there is no LZM. PTM Rule 2.
For any single isolated PTM, there isone and only one DZM with a uniform √ ×√ -shapeddistribution of wavefunction amplitudes. The non-zeroamplitudes of the DZM are only on the A(B)-sites of theisolated α ( β )-PTM. For each PU, the amplitude of thisDZM solution is only zero at the center site; otherwise,it is non-zero.Proof of PTM Rules 1 and 2. The proof of the PTMRules can be immediately derived from the proof givenin Ref. [30] and our definition of the A(B)-sites of the α ( β )-PTM.derived.Now, let us consider a poly-PTM system for whichthere are three rules. C. Poly-PTM Rules
Poly-PTM Rule 1.
In a poly-PTM, when there are n vacancies in an α ( β )-PTM, there are n LZMs, which cor-respond to the vacancy-induced zero modes in the singleisolated α ( β )-PTMappearing on the B(A)-sites. Poly-PTM Rule 2.
In a poly-PTM, if there is a wave-function with zero-energy and if and only if non-zero am-plitudes of the function are found on the A(B)-sites in a α ( β )-PTM, the wavefunction has the √ ×√ -shaped dis-tribution within the PTM, as described by the PTM Rule2. Thus, the value of the wavefunction is represented bya single parameter ϕ as a complex-valued factor in each α ( β )-PTM. Here, we introduce a schematic representation of arenormalized graph of the poly-PTM system: a weighted super-graph . According to our definition of a poly-PTM,the graph is bipartite. Figure 4 shows an exemplarysuper-graph for the poly-PTM system in Fig. 2. EachPTM is shown as a node, at which there may be va-cancies. Using this bipartite super-graph of nodes, it ispossible to understand how the zero modes appear in apoly-PTM system. A generalized zero-sum rule is thennaturally introduced, which we call the super-zero-sumrule . The third rule of poly-PTM governs the DZM’sappearance.
Poly-PTM Rule 3.
When
M α -PTMs and
L β -PTMscomprise a poly-PTM, the connections among them de-fines a bipartite super-graph of α - and β -PTMs. At eachnode of the super-graph of α - and β -PTMs, an ampli-tude ϕ α i ( ϕ β j ) is assigned, where i = 1 , · · · , M and j = 1 , · · · , L . Considering the super-zero-sum rule of ϕ l ( l ∈ α i , or l ∈ β j ) on the super-graph, if there existsa super-zero mode that satisfies the super-zero-sum rule,then there is a corresponding DZM in the poly-PTM. Theuniform √ ×√ -shaped form of the DZM appears on theA(B)-sites of each α ( β )-PTM subgraph. Proofs of these Poly-PTM Rules are given in the Ap-pendix.In summary, the LZMs localized in a PTM appear ac-cording to the number of vacancies in each PTM sub-graph. The DZMs expanding over multiple PTMs ap-pear according to the super-zero-sum rule in the bipar-tite super-graph of a poly-PTM system made of α - and β -PTMs. IV. APPLICATIONA. Ground states
The super-zero-sum rule governing the appearance ofDZMs becomes critical when discussing the spin groundstate. In the following discussion, we follow well-known Ovchinnikov’s rule[31] for alternating hydrocar-bon systems and Lieb’s theorem[32] for repulsive on-site
FIG. 4. A bipartite super-graph of the poly-PTM systemin Fig. 2. Each α ( β )-PTM is expressed in light (dark) gray,and the links of the super-graph are black lines. Vacanciesare indicated by white circles. Coulomb interaction within the π -orbitals. Typical ex-amples are shown in Fig. 5. Figure 5 (a) presents an α -PTM with two vacancies. Two LZMs originate fromthe vacancies, and there is also one DZM from the super-zero-sum rule. According to this example, when we fo-cus on the subspace of E = 0, two parallel spins andone anti-parallel spin appear in the two LZMs and theone DZM. In the ground state, one of the parallel spinscomprises a singlet with the anti-parallel one, making thetotal ground state a doublet for which the total magneticmoment becomes S = 1 /
2. The same result is mentionedin our previous paper[30]. In Fig. 5 (b), a β -PTM isconnected to the example mentioned above. No DZMoccurs according to the super-zero-sum rule; therefore,there appear two parallel spins in the LZMs on the B-sitesand in the ground state. These make up a spin tripletwith the total magnetic moment of S = 1. These exam-ples show the importance of controlling the appearanceof both LZMs and DZMs using the super-zero-sum ruleto determining the spin ground state and the low-energyeffective spin Hamiltonian. Thus, our rule enables mag-netic molecules and magnetic nanocarbon systems to bedesigned rather freely, having advantages compared withother attempts[33, 34]. This may allow the material re-alization of quantum-computation-device structures. B. Discussion from the viewpoint of theShima-Aoki theorem
Let us consider the example of a periodic poly-PTMsystem comprising VANG[26, 27] as α - and β -PTM.When VANG forms a two-dimensional hexagonal lattice(Fig. 6(a)), we have two flat bands at E = 0 in the bandstructure (Fig. 6(b)), which correspond to the vacancy-induced LZMs. There appear two additional zero modesat the K-point in the Brillouin zone derived from the FIG. 5. (a) An α -PTM with two vacancies. Two parallelspins and one anti-parallel spin occur in the two LZMs andone DZM. The total ground state becomes a doublet with S = 1 /
2. (b) A β -PTM is connected to the α -PTM. TheDZM disappears, and the ground state becomes triplet with S = 1. √ × √ FIG. 6. (a) A super-bipartite honeycomb lattice of VANG.Each α ( β )-PTMis given by a VANG molecule. (b) The bandstructure of the VANG-VANG hexagonal lattice. Two flatLZMs degenerate at E = 0, and the linear DZMs appearaccording to the super-zero-sum rule rule at the K-point. FIG. 7. (a) A super-poly-PTM structure; (b)the super-graph; and (c) the corresponding bipartite super-super-graph.(d) The zero mode that appears across multiple super-PTMsshown in (a) is governed by a super-super-zero-sum rule. Shima–Aoki theorem, because of there being a vacancysite at each VANG, the VANG–VANG super-bipartitehoneycomb-lattice structure is classified as type A [23].Although a normal A structure should have a semicon-ducting band gap and m ( ≥
0) flat bands according to thetheorem, the system may show the gapless Dirac cone,since an accidental degeneracy is caused by the LZM ofthe VANG. Accidental degeneracy occurs within the zoneinterior. In this system, two zero-energy representationsof the E symmetry appear at the K-point. These factsby LZMs of VANG forming a zero-energy flat band arereally designed by our new Poly-PTM Rules. As a re-sult of the super-honeycomb graph of VANG, the gaplessDirac mode remains.Of course, we can consider other PTM suructures with C symmetry with a vacancy at the center besides aVANG super-atom. Therefore, there is a scheme forsystematically constructing poly-PTM honeycomb struc-tures that constitute exceptional cases of the Shima-Aokitheorem. C. Construction of a poly-PTM as real-spacerenormalization group operation
Here, let us mention that the Hamiltonian mappingdescribed in this paper is a real-space renormalization-group operation[35, 36]. Such an operation can be ap-plied recursively. As shown in Fig. 7(a), we can constructa super-poly-PTM structure in the scope of a poly-PTMwith a super-graph (Fig. 7(b)), and we can consider acorresponding super-super-graph (Fig. 7(c)). A super-super-zero-sum rule on a super-super-graph dominatesthe zero mode that appears across multiple super-PTMs(Fig. 7(d)). The conclusions of these discussions deepenour understanding of the relationship between the nanos-tructures and zero modes of graphene.
V. CONCLSIONS
We found that each PTM with n vacancies will alsohave n LZMs and that there are also DZMs derived from √ × √ ACKNOWLEDGMENTS
The authors thank S. Teranishi and Y. Wicaksonofor our illuminating discussions and valuable com-ments. This work is partly supported by KAKENHI No.K034560.
Appendix: Proofs of Poly-PTM Rules
Proof of Poly-PTM Rule 1.
By letting the amplitudes ofthe wavefunction be zero outside of the PTM, the zero-sum rules on the A(B)-sites at the DZCs, as well as thoseinside the PTM, are satisfied by the n LZMs in the PTMthat correspond with those in the isolated PTM. An am-plitude of zero naturally satisfies the zero-sum rules re-quired on the A(B)-sites outside the PTM, such that the n LZMs in the PTM satisfy all zero-sum rules and appearin the poly-PTM system.
Proof of Poly-PTM Rule 2.
Since each α ( β )-PTM isconnected to the outside only at the DZCs, the zero-sumrule on each B(A)-site in the α ( β )-PTM does not includethe amplitude of the wavefunction on the A(B)-site out-side the PTM.Suppose that a wavefunction with zero-energy is foundin the poly-PTM. When its non-zero amplitudes arefound on A(B)-sites in one of the α ( β )-PTM, the zero-sum rules on the B(A)-sites in the PTM are satisfiedby the wavefunction. The zero-sum rule equations aregiven by a partial set of those in the whole poly-PTM.Note that this partial set is identical to the full set of thezero-sum rule equations in an isolated α ( β )-PTM. Thismeans that the partial set, a necessary condition for the zero mode of the poly-PTM, becomes a sufficient condi-tion for the zero-sum rules of the single isolated PTM,but not for the wavefunction to be the zero mode of thepoly-PTM.Because of PTM Rule 2, there can be only √ ×√ α ( β )-PTM subgraph, which is the unique zero mode ap-pearing on the A(B)-sites of α ( β )-PTM. The shape of themode is also unique. Therefore, the zero mode wavefunc-tion of the poly-PTM has a √ ×√ α ( β )-PTM. From the uniformity of the √ ×√ α ( β )-PTMcan be parametrized by a local phase factor, ϕ .Before moving on to the proof of Poly-PTM Rule 3,let us consider an example. First, we define an arrowdiagram. Let us focus upon the wavefunctions on theA-sites. The zero-sum rule a j + a k + a l = 0 (A.1)on each B-site for a wavefunction on the A-site is drawnas shown on the left-hand side of Fig. 8(a). Here, a j , a k , and a l represent the values of the wavefunction onthe j -, k -, and l -th A-sites around the i -th B-site. Theright-hand side of Fig. 8(a) shows an arrow diagram:three arrows are pointing to the i -th B-site. Using thisstructure as a local diagram, we can write the whole zero-sum rule in a global diagram.A poly-PTM is shown in Fig. 8(b). In this system,we have two zero modes on the A-site of the poly-PTM, FIG. 8. (a) Left: zero-sum rule on the i -th B-site. Right:The rule is expressed by arrows forming a local arrow dia-gram. (b) Zero-sum rules for the entire β -PTM are shown bygreen, red and blue arrows. (c) Defined super-graph. Eachconnection is assigned a weight of 2 t (= N C t ). which are finally found to be DZMs satisfying the zero-sum rules at the B-sites. We focus upon the β -PTM atthe center of the network; then, the zero-sum rules of theentire β -PTM can be drawn by the arrow diagram. Thezero-sum rule at the center of each PU is colored green.Then, we see that the zero-sum rules at the periphery ofeach PU can be alternately colored red and blue. Conse-quently, the edges of the β -PTM are always colored redand blue as well. This is a common property of generalPTM.In this example, there are 11 A-sites and 12 B-sitesin the β -PTM. When we consider the zero-sum rulesat every B-sites in the β -PTM, we obtain 12 linearlyindependent equations, which are linear equations of thezero mode wavefunction. The values of these modes areindicated as a β ,l with l = 1 , · · · , N β, A = 11 for the 11A-sites. In the zero-sum rules, there appear six moremode values. In the adjacent α -, α - and α -PTMs,there are 2 × a α i ,l ≡ a il appear with i = 1 , · · · , M = 3. Relevant A-sites in each α -PTM arethose at a DZC. Thus, we renumber the A-sites in α -PTM, and l = 1 , a β ,l and a il for 17(= 11 + 6)) A-sites.We will show that, when the values of the zero modeat the exterior A-sites in the adjacent α -PTMs (i.e. α -, α -, and α -PTMs) are determined as boundary condi-tions for the interior, a β ,l can be uniquely determinedto obtain the full solution. The wavefunction of the α -PTMs can be determined using the super-zero-sum rule introduced below.Here, under Poly-PTM Rule 2, we can concisely ex-press the relative value of the √ × √ α i PTM. Considering the possible DZMon A(B)-sites, let us introcduce a number ϕ α ( β ) i to each α ( β )-PTM, where ϕ α ( β ) i is complex-valued and can bezero. This value represents a factor of the √ ×√ α ( β )-PTM. Then,we can also derive the following relationships from the √ ×√ α -, α - and α -PTMs: a = − a := ϕ α ,a = − a := ϕ α ,a = − a := ϕ α . (A.2)Since these three conditions are determined by the zero-sum rules in the α i -PTMs, the constraints are inde-pendent of the 12 above-mentioned equations. There-fore, the number of degrees of freedom (DOF) remain-ing for the wavefunction with respect to a β ,l and a il is17 − − α -, α -and α -PTMs, as expressed by ϕ α i . By adding each sideof the corresponding zero-sum rule [Eq. (A.1)] shown in a red diagram and subtracting each zero-sum rule equationshown in the blue diagram, we obtain: a + a + a − ( a + a + a ) = 0 . (A.3)This simplification is always possible because the a β ,l atthe A-sites on the periphery of all PUs in the β -PTMappears twice in zero-sum rule equations at the B-siteson the periphery: once in the equation for the red-coloreddiagram and again for the blue-colored diagram. There-fore, adding the equations with alternating positive andnegative factors along the circumference of PUs into the β -PTM cancels-out a β ,l in the final expression. Alongthe way, a i and a i remain for concluding Eq. (A.3).From Eqs. (A.2) and (A.3), we can derive the followingrelationship: 2 ϕ α + 2 ϕ α + 2 ϕ α = 0 . (A.4)Note that the coefficient for each ϕ in Eq. (A.4) corre-sponds to N C = 2.For the present example, there are two linearly inde-pendent solutions satisfying Eq. (A.4). Indeed, we knowthat the three-dimensional vectors following one con-straint making the vector be normal to one direction aregiven in a two-dimensional vector space. Consequently,there are two DZMs. Here, we need to certify the exis-tence of a unique solution for each of determined set of ϕ α i ( i = 1 , X a = b remains for the determinationof a = t ( a β , , · · · , a β , ), where a non-trivial vector b isgiven by ϕ α i . The matrix X is regular because the trans-formation from the original 12 zero-sum rule equations islinear.Equation (A.4) expresses the relationship among ϕ α , ϕ α and ϕ α with DOF of two for the values of the wave-functions on the α -, α -, and α -PTMs connected to the β -PTM. This can be regarded as a super-zero-sum rule on the super-graph defined by α -, α -, α -, and β -PTMsand the connections among them, as shown in Fig. 8(c).Two linearly independent solutions of the super-zero-sumrule expressed in Eq. (A.4) are the super-zero modes that appear in the system shown in Fig. 8(c). Using thissuper-zero-sum rule, we can make a proof of Poly-PTMRule 3. Proof of Poly-PTM Rule 3.
Let us consider a β -PTMwith n vacancies. Suppose that this PTM is connectedto M ( >
1) adjacent α -PTMs. Let N C among them be N C , N C , ..., N C M . There are N β, A A-sites in the β -PTM; in the adjacent α -PTMs, there are (cid:80) Mi =1 N C i A-sites that connect to the DZCs of the β -PTM. When wefocus upon the wavefunctions on the A-sites, there are N β, B linearly independent zero-sum rule equations on theB-sites in the β -PTM; by the Poly-PTM Rule 2, thereare (cid:80) Mi =1 ( N C i −
1) linearly independent equations fromthe √ ×√ α -PTMs. In total, the wavefunction on the A-sites has N β, A + (cid:80) Mi =1 N C i − N β, B − (cid:80) Mi =1 ( N C i −
1) = n + M − N β, A − N β, B = n − β -PTM[30]. Since n DOFs originate fromthe vacancies corresponds to the LZMs, the remaining M − √ ×√ α -PTMs becomes as follows: t M (cid:88) i =1 N C i ϕ α i = 0 . (A.5)When we focus upon the wavefunctions on the B-sitesin an α -PTM which has L adjacent β -PTMs, based thesame discussion, the relationship among the √ × √ t L (cid:88) j =1 N C j ϕ β j = 0 . (A.6)When we consider a poly-PTM comprising N α α -PTMs and N β β -PTMs, we can define a super-graphcorresponding to the poly-PTM structure by assigninga weight tN C k − l to each link between the k -th α -PTMand the l -th β -PTM, where N C k − l is the number of theconnections.By considering Eqs. (A.5) and (A.6) on all β - and α -PTMs in a poly-PTM, respectively, we obtain the fol-lowing equations: (cid:26) H β ← α Φ α = 0 ,H α ← β Φ β = 0 . (A.7)Here, Φ α = t ( ϕ α , ..., ϕ α Nα ) and Φ β = t ( ϕ β , ..., ϕ β Nβ ). H β ← α (= H † α ← β ) can be seen as an effective transfer ma-trix from α ( β )-PTM to a β ( α )-PTM, which is definedas ( H β ← α ) lk = tN C k − l . Equation (A.7) expresses rela-tionships among the values of the √ ×√ α - and β -PTMs in the poly-PTM. This is the super-zero-sum rule on the super-graph, and thesolution is the super-zero mode.According to the zero-sum rule, the determinationequations for the wavefunctions at the A(B)-sites in the β ( α )-PTM subgraph are mutually independent. For ex-ample, for the A-sites in the β -PTM subgraph, we have N β, B = N β, A − n + 1 ≤ N β, A + 1. There appears to beone super-zero-sum rule equation per β -PTM accordingto a linear transformation. The other N β, A − n equationsdetermine the wavefunction on these sites since a set ofsimultaneous linear equations can be solved if the numberof equations is not larger than the number of variables.So, if Eq. 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