Effect of coordination on topological phases on self-similar structures
EEffect of coordination on topological phases on self-similar structures
Saswat Sarangi and Anne E. B. Nielsen ∗ Max-Planck-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany
Topologically non-trivial phases have recently been reported on self-similar structures. Here, weinvestigate the effect of local structure, specifically the role of the coordination number, on thetopological phases on self-similar structures embedded in two dimensions. We study a geometrydependent model on two self-similar structures having different coordination numbers, constructedfrom the Sierpinski Gasket. For different non-spatial symmetries present in the system, we nu-merically study and compare the phases on both the structures. We characterize these phases bythe localization properties of the single-particle states, their robustness to disorder, and by using areal-space topological index. We find that both the structures host topologically non-trivial phasesand the phase diagrams are different on the two structures. This suggests that, in order to extendthe present classification scheme of topological phases to non-periodic structures, one should use aframework which explicitly takes the coordination of sites into account.
I. INTRODUCTION
After the discovery of the quantum Hall effect, thestudy of topological phases has been one of the leadingresearch areas in condensed matter physics. In non-interacting electronic systems, topologically non-trivialphases are usually identified by the presence of gaplessboundary modes highly robust to weak disorders, andare characterized by relevant topological invariants [1, 2].These phases are well understood for translationallyinvariant systems, as the presence of a well-definedmomentum eigenbasis gives a natural setting to de-scribe the topology of bulk wavefunctions. Systematicclassification of topological phases on non-interactingtranslationally invariant systems has been done in termsof both non-spatial and spatial symmetries [3–8].Although translational invariance is a necessarycondition for the presence of a well-defined momentumeigenbasis, it turns out that this is not a necessarycondition for the existence of topological phases. Topo-logical phases have been reported in quasiperiodic,quasicrystalline, and amorphous systems [9–11] whichonly preserve the notion of a well-defined “bulk” and“boundary”, as defined in regular lattice systems withopen “boundary”. Also recently, properties associatedwith topological phases have been reported on finitetruncations of fractals like the Sierpinski Gasket andSierpinski Carpet [12–16] which even lack this notionof “bulk” and “boundary”. Although there have beensome speculations [12, 13, 15], the factors affecting thetopological properties of systems without a precise bulk-boundary distinction, are yet to be clearly identified. Inan attempt to identify one such factor, here we studythe effect of coordination on the topological properties ofnon-interacting Hamiltonians on self-similar structures. ∗ On leave from Department of Physics and Astronomy, AarhusUniversity, DK-8000 Aarhus C, Denmark
The way the sites are coordinated locally on a latticeplays an important role in determining which topologicalphases the lattice can host. To see this, consider a generaltwo-orbital nearest-neighbor tight-binding model on a 2DBravais lattice, similar to what is considered in [5], givenby H tb = (cid:88) R ,< r >,α,β t ( r )( ψ † α ( R ) f ( θ r ) ψ β ( R + r )) , (1)where R specifies the position vectors for the sites, r specifies the relative vectors between two sites, { α , β } label the two orbitals, and (cos( θ r ) , sin( θ r )) = r / | r | .The function f ( θ r ) is any function such that H tb isHermitian. The matrix elements of the correspondingBloch Hamiltonian H tb ( k ), which essentially determinethe band topology, encode the information about thelocal structure of the lattice as they involve a sum overall nearest neighbors. This is how local properties likecoordination comes into the picture. As the form of H tb is entirely determined by the crystal symmetry of theunderlying lattice [5], crystal symmetries are used fortopological classification of such systems. On some twodimensional lattices, the graph of the model, formed byidentifying the sites as the vertices and the non-zerohoppings as the edges, forms a regular tiling of the twodimensional space. For such cases, the coordinationnumber is uniquely determined by the crystal symmetry.Examples of such cases are nearest neighbor modelson triangular, square and hexagonal lattices. But onself-similar structures, to the best of our knowledge,no such correspondence has been established betweencoordination and spatial symmetries.The idea of coordination is also crucial for the distinc-tion between “bulk” and “boundary” on regular lattices.But, self-similar structures lack a clear distinctionbetween bulk and boundary. However coordinationnumber, and hence the notion of coordination, is welldefined for self-similar structures, as those are specialgraphs like regular lattices. For this study, we firstconstruct two different self-similar structures from the a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Sierpinski Gasket (SG), with different coordinationnumbers, which have the same Hausdorff dimension.We then numerically study a geometry dependentnon-interacting nearest neighbor Hamiltonian on bothstructures by looking at certain observables of interest.The rest of the paper is organized as follows. In sec-tion II, we describe the construction of the two differentfractal structures mentioned in the previous paragraph.We define the model Hamiltonian and the observableswe are looking at in section III. In section IV, we presentand compare various properties of the Hamiltonian onboth the structures. Finally, in section V, we concludewith a summary of our results and discuss some of theremaining open questions on the subject.
II. CONSTRUCTION OF FRACTALSTRUCTURES
We construct self-similar structures with two differentcoordination numbers from the SG. First we constructthe SG by a recursive procedure starting from anequilateral triangle. We divide it into four equilateraltriangles of equal area, remove the central triangle, andrepeat the procedure infinitely for each of the remainingtriangles. We call the structure generated after g iterations for ‘SG with generation g ’ and the trianglesremoved in a particular iteration for ‘triangles belongingto generation g ’.For the first structure (shown in Fig. 1), we identifythe vertices of the triangles in each generation of theSG with the sites, and the edges with the bonds. Thisgives a self-similar structure in which, all sites exceptthe three corner sites (marked in yellow in Fig. 1), havecoordination number 4. We denote this structure by‘SG-4’. Tight binding models on this type of structurehave been extensively studied using real space renormal-ization methods [17–19].For the second structure (also shown in Fig. 1),we identify the centroids of the smallest triangles ineach generation of the SG with the sites, and connectthe nearest neighbors. This also gives a self-simliarstructure. But in this case, in each generation, all sitesexcept the three corner sites, have coordination number3. We denote this structure by ‘SG-3’. Notice that thefirst generation of the SG-4 is obtained from the zerothgeneration of the SG, whereas the first generation of theSG-3 is obtained from the first generation of the SG.Due to the self-similar nature of the SG-3 and the SG-4, for each structure, we can remove certain specific sitesfrom a given generation ‘ g ’ so that the structure with theremaining sites resembles that of generation ‘ g − g ’. This is illustrated in Fig. 2. In both the FIG. 1.
Schematics for the construction of the self-similar struc-tures. The shaded regions are the finite truncations of the SG fordifferent generations. The blue and red dots indicate the positionsof the sites for the structures SG-4 and SG-3, respectively. Theblack solid lines represent the bonds between the sites. The threecorner sites are marked with an additional yellow dot in both thestructures.
FIG. 2.
Schematics showing the self-similarity of (a) SG-4 and(b) SG-3. The sites of the 3rd generation are marked in red forboth the structures. When these sites are removed, the remainingstructure resembles that of the 2nd generation. structures, in each generation, only the three corner sitesof the SG-3 and the SG-4 are two coordinated, but weexpect this to not affect the physics when we are far fromthe corner sites. Notice that both the structures havethe same Hausdorff dimension as the SG. For numericalcalculations, we carry out the constructions mentionedabove, but with a finite number of iterations for the SG,which gives us structures with finite number of sites forthe SG-3 and the SG-4.
III. MODEL AND APPROACH
We study the fermionic, generalised Bernevig-Hughes-Zhang (
BHZ ) model, on the self-similar structures men-tioned in section II. We choose to study this model be-cause the
BHZ model is known to host topologically non-trivial quantum spin Hall insulating phases on transla-tionally invariant lattice systems [20]. Also, this modelcan be easily generalized to make it depend on the geom-etry of the underlying motif [10, 12]. We define the modelin the following way. Each site has two orbital degrees offreedom, denoted by α = { c, d } , and two spin degrees offreedom, denoted by σ = ±
1. We consider only nearestneighbor hopping. The Hamiltonian is given byˆ H BHZ = M (cid:88) jσ ˆ ψ † jσ τ z ˆ ψ jσ − t (cid:88)
2, this modelhas been studied on a fractal structure which is closelyrelated to SG-4, but with different boundary conditions[12].We numerically study the systems by primarily lookingat the localization, dynamics and the topological natureof the single-particle states at half-filling. For the numer-ical computations, we use KWANT code [22]. A singleparticle state denoted by label n can be written as | ψ n (cid:105) = (cid:88) jα ϕ n,jα | jα (cid:105) (4)where {| j (cid:105)} denotes the basis vectors in the site basis. Westudy the localization of single particle states by looking at the density at any site j , given by ρ n ( j ) = (cid:88) α | ϕ n,jα | . (5)Given that it is unclear how to have a sharp distinctionbetween bulk and edge states in the case of fractalsystems, we define ‘bulk-like’ and ‘edge-like’ states asfollows. An eigenstate is a bulk-like state if it has finiteprobability density on sites which enclose the trianglesbelonging to more or less every generation of the SG. Onthe other hand, an eigenstate is an edge-like state, if it islocalized on sites which enclose the triangles belongingentirely to a particular generation of the SG.We use Kitaev’s topological index to study the topo-logical properties of the systems, which relies solely onthe real space description of the system [23]. We firstchoose a subsection X of the fractal and divide it intothree parts, A , B and C , as shown in Fig. 3. We use thefollowing expression for the real space Chern number ν ( P ) = 12 πi (Tr( AP BP CP ) − Tr(
AP CP BP )) (6)where P = (cid:80) k | ψ k (cid:105) (cid:104) ψ k | is the projector onto the desiredeigen states. A, B, C are diagonal matrices with A = ˜ A ⊗ N orb B = ˜ B ⊗ N orb C = ˜ C ⊗ N orb (7)where ˜ A, ˜ B, ˜ C denote the projectors into the sectors A , B , C (as shown in Fig. 3) respectively, and N orb is thenumber of orbitals per site which is 2 in this case.We also check the dynamics of the states close to theFermi energy. To do this, we project a single particlestate, initially localized in the c orbital of one of the sitesof the fractal, onto a part of the eigenbasis defined by E min < E < E F , and then time evolve under ˆ H . Here, E denotes the eigen-energies of the Hamiltonian and E F denotes the Fermi energy. E min is chosen such that theenergy range, ( E min , E F ), is small enough to look at thestates near the Fermi energy but also large enough toencompass all the edge-like states below the Fermi energy.All the computations have been done with E min = − . W = (cid:88) j ˆ ψ † j ˜ W j ˆ ψ j (8)where ˜ W j = diag( (cid:15) cj , (cid:15) dj ) and (cid:15) cj , (cid:15) dj are random numbersdrawn from a uniform random distribution with mean µ = 0 and variance W . The total Hamiltonian underwhich the system is time evolved then becomes ˆ H dis =ˆ H + ˆ W . (a) (b) FIG. 3.
Partitions of the 6th generation of (a)
SG-4 and (b)
SG-3,for the real space Chern number calculation. The regions A , B and C are marked in red, green, and blue respectively. The subsectionis X = A ∪ B ∪ C IV. RESULTS
The Hamiltonian in Eq. (3), can be rewritten in thefollowing block form in orbital ⊗ site notationˆ H = ˆΨ † H ˆΨ , (9)ˆΨ = (cid:18) ˆ C ˆ D (cid:19) , H = (cid:18) M − tH λ ∆ λ ∆ † − ( M − tH ) (cid:19) . Here, ˆ C =(ˆ c , ˆ c , ..., ˆ c N s ) T and ˆ D =( ˆ d , ˆ d , ..., ˆ d N s ) T ,where N s is the total number of sites. ∆ jk = − ie − iθ jk and H jk = 1, if j, k are nearest neighbors connected bya bond as shown in Fig. 1, and otherwise zero. FromEq. (9), it is easy to see that this model has a charge-conjugation symmetry for all values of M, t, and λ , givenby P − H P = − H . (10)Here P = τ x K , where K is the complex conjugation oper-ator. A consequence of this symmetry is the spectra be-ing symmetric around zero energy. Apart from this, theHamiltonian has other non-spatial symmetries for cer-tain specific parameter values. So we break our resultsinto three parts, specifically focusing on three particu-lar parameter regimes, each having different symmetryproperties. A. t (cid:54) = λ = 0 For λ = 0, the Hamiltonian in Eq. (9) becomes H = τ z ⊗ ( M − tH ) (11)which is block diagonal and decouples into two single or-bital tight-binding models. This is well studied in theliterature on the SG-4 [17–19]. The spectrum of themodel (shown in Fig. 4) is symmetric about E = 0, asexpected, due to the charge-conjugation symmetry (10)of the model. It is already known for SG-4 that thespectrum is self similar and has infinitely many gaps inthe infinite g limit. We find that the spectrum of SG-3 E n e r g y −4−3−2−101234 Index (n/N)0 0.2 0.4 0.6 0.8 1 SG-3: N=1458 E n e r g y −4−3−2−101234 Index (n/N)0 0.2 0.4 0.6 0.8 1 SG-4: N=732 (a) (b)
FIG. 4.
Spectrum of H for λ = 0, M = 0, and t = 1 on (a) SG-4with g = 6 and (b) SG-3 with g = 6. N = 2 N s denotes the totalnumber of eigenstates where N s is the total number of sites. is also self-similar with infinitely many gaps in the infi-nite g limit. We confirm this numerically by computingthe spectrum for different g values, and analytically byfollowing the renormalisation procedure done in [17, 18].For M = 0, as seen in Fig. 4(b), we see a very high degen-eracy at zero energy in case of SG-3, which is not seen incase of SG-4. The model has the symmetry that τ z com-mutes with the Hamiltonian (11), but this only gives riseto a twofold degeneracy. The large degeneracy is hencea consequence of the spatial arrangement of the sites inthe underlying structure and not due to any non-spatialsymmetry of the Hamiltonian. In this particular regime,however, the Hamiltonian does not host any topologicalphases on either of the structures as H does not hostany topological phase. A nonzero mass term M , simplyopens up a trivial gap in the spectra. B. λ (cid:54) = t = 0 Now we consider the case when we only have the on-site term, c → d hoppings, and d → c hoppings. Thenthe Hamiltonian matrix H in Eq. (9) reduces to H = M τ z + λ (cid:18) † (cid:19) def = M τ z + λH xy . (12)We start by studying H for M = 0. We see that everyenergy level is at least doubly degenerate on both thestructures. This is because H xy has an additionalorbital symmetry given by τ z H xy τ z = − H xy alongwith the charge-conjugation symmetry (10). Hence,the system possesses time-reversal symmetry givenby T − H xy T = H xy , where T = iτ y K , which resultsin the Kramers degeneracy. If | ψ (cid:105) = ( C, D ) T , where C = ( c , c , .., c N s ) T and D = ( d , d , .., d N s ) T , is aneigenstate of H xy , then T | ψ (cid:105) = ( − D ∗ , C ∗ ) T is also aneigenstate of H xy . Also, ψ and T ψ are orthogonal toeach other as (cid:104) ψ | T ψ (cid:105) = 0.We find that the spectrum of H xy on SG-4 hostshighly degenerate levels at the Fermi energy (Fig. 5(a)),which is not present in the case of SG-3. The Chern E n e r g y −3−2−10123 Index (n/N)0 0.2 0.4 0.6 0.8 1 SG-4: N=732(a) E n e r g y −3−2−10123 Index (n/N)0 0.2 0.4 0.6 0.8 1 SG-3: N=1458(b) −0.00100.001 −0.200.2
FIG. 5.
Spectrum of H xy on (a) SG-4 with g = 6 and (b) SG-3with g = 6. N = 2 N s denotes the total number of eigenstates where N s is the total number of sites. For (a), the inset shows the highlydegenerate levels (flat band) at E = 0. For (b), the inset shows thewhole range of edge-like states near zero energy. number for the collection of degenerate levels at E F turns out to be zero, when computed using Eq. (6). OnSG-3, H xy hosts doubly degenerate zero energy states.Interestingly, these zero energy states are edge-likestates, completely localized on the sites present on thetriangle of the 1st generation. In fact, we observe thatall states close to zero energy, shown in the inset ofFig. 5(b), are edge-like states. A few examples of suchstates are shown in Fig. 6. In this case also, we find theChern number to be zero, when computed by projectingonto the filled states (half-filling). However, looking atthe dynamics of the edge-like states close to the Fermienergy, we find two modes of opposite chirality beingpresent in the system. We also find this characteristicin the dynamics being robust to weak disorders. As thepresence of robust helical edge-states is a signature of Z topological order, we expect that this might be thecase here. H xy is topologically non-trivial on SG-3 andthe Chern number being zero is a consequence of thetime-reversal symmetry in the system.The M τ z term creates a gap in the spectra of H (shownin Fig. 9) on both the structures. For SG-4, the flat-band at zero energy splits into two flatbands with ener-gies M and − M . Addition of a M τ z term breaks thetime-reversal symmetry of H , since T − ( M τ z + H xy ) T =( − M τ z + H xy ). However, we still find the spectra of H on SG-3 to consist of doubly degenerate states as in thecase of H xy . This double-degeneracy is independent ofthe fractal structure and is due to non-spatial symmetriesof H xy as shown in Appendix A. C. t = λ (cid:54) = 0 Switching on both [ c → c , d → d ] and [ c → d , d → c ] hoppings brings a lot of interesting physicsinto the picture. From Fig. 10, we find that H hoststopological phases on both the structures. In the regime0 < ∼ M < ∼ .
5, both SG-3 and SG-4 host topologicalphases with the same Chern number and support edge-
FIG. 6.
Few examples of edge-like eigenstates of H xy on SG-3,close to zero energy. The color bar represents the relative densityper site of an eigenstate, | ψ n (cid:105) , defined by ρ n ( j ) / max( ρ n ( j )). FIG. 7.
Time evolution of a state, initially localised in the c or-bital of one of the corner sites, evolved under ˆ H . The initial stateis projected onto a sector defined by − . < E <
0. The colorbar represents the relative density per site of an eigenstate, | ψ n (cid:105) ,defined by ρ n ( j ) / max( ρ n ( j )). like states. For SG-4 with different boundary conditions,similar edge-like states were reported [12], which wererobust against random onsite disorder, and possesseda chiral nature. In our case also, we find the same forboth SG-3 and SG-4.In the regime, − < ∼ M < ∼ − .
2, SG-3 and SG-4 hostdifferent topological phases, characterized by differentChern numbers. For SG-4, there are many level crossings
FIG. 8.
Time evolution of a state, initially localised in the c orbitalof one of the corner sites, evolved under ˆ H dis with W = 0 .
1. Theinitial state is projected onto a sector defined by − . < E < | ψ n (cid:105) , defined by ρ n ( j ) / max( ρ n ( j )). E n e r g y E n e r g y FIG. 9.
Part of the spectra of Mτ z + H xy as a function of M on(a) SG-4 and (b) SG-3. The Mτ z term breaks the time-reversalsymmetry and opens a gap proportional to M in the spectrum of H xy . in the regime − < ∼ M < ∼ − . M ≈ − .
24, which seems tobe one of the transition points from a topological phaseto a trivial phase.
V. CONCLUSION
We have explored the properties of a geometrydependent Hamiltonian on two different finite fractalstructures (SG-3 and SG-4) which only differ in the waythe sites are coordinated. The Hamiltonian has differentnon-spatial symmetries for different parameter regimes. C he r n N u m be r ( ν ) −2−101 Mass Term (M/λ)−4 −3 −2 −1 0 1 2 3 4 g=5 g=6 g=7 Triangular Lattice (a) −0.02−0.0100.010.02−2 −1.8 −1.6 −1.4 −1.2 −1 C he r n N u m be r ( ν ) −2−101 Mass term (M/λ)−4 −3 −2 −1 0 1 2 3 4 g=4 g=5 g=6 Triangular Lattice (b) −0.00500.005−1.4 −1.35 −1.3 −1.25 −1.2 −1.15 −1.1 FIG. 10.
Real space Chern number for H in the regime λ = t on (a)SG-4 and (b) SG-3. The computation is done using Eq. (6), whichstrongly depends on the system size. We do a system size scalingby looking at Chern numbers for different generations g . The insetin each plot shows the first two energy levels closest to the Fermienergy, for different generations. The legend for the insets are thesame as that for the main plots. The inset of (a) shows numerouslevel crossings for SG-4, which increase with g . The inset of (b)shows a single level crossing for SG-3. E n e r g y � � � � � � �������������� ������������������������������������� � � � � � � FIG. 11.
Part of the spectra for H in the regime λ = t on (a) SG-4with g = 7, and (b) SG-3 with g = 6. The regions of the spectrawhich host edge-like states are pointed out for both the structures.These regions correspond to the topological regions in Fig 10. We study the systems in each of these parameter regimesseparately. We find that the topological properties ofthis Hamiltonian are significantly different on the twostructures.In the regime t = λ (cid:54) = 0, where only charge-conjugation symmetry is present, the half-BHZ modelcan host both topologically trivial and non-trivial phasescharacterized by a non-zero real-space Chern number,on both the structures. For both SG-3 and SG-4, we findchiral edge-like eigenstates close to the Fermi energy forthe parameter regimes corresponding to the topologicallynon-trivial phases. However, the phases obtained foreach of the structures are different which is evidentfrom their phase diagram (Fig. 10). In the regime λ (cid:54) = t = 0, where all the three symmetries (time-reversal,charge-conjugation, and orbital symmetry) are present,we find the existence of non-trivial doubly degenerateedge-like eigenstates with opposite chiralities near theFermi energy on SG-3. No such chiral edge-like statesare present in case of SG-4 for this particular parameterregime. Instead, a highly degenerate zero energy bandis present in SG-4 which we expect to be topologicallytrivial.As the distinguishing factor between the two structuresis their coordination number, we arrive at the conclu-sion that the topological properties on self-similar lat-tice systems depend significantly on the way the sitesare coordinated. The description of topological phasesin translationally invariant non-interacting systems doesnot explicitly take the coordination into account. There,coordination is taken implicitly into account in the ma-trix elements of the corresponding Bloch Hamiltonians.But that is not possible for systems which lack transla-tional symmetry. The results of this work suggest that,in order to extend the present classification scheme, it isimportant to use a framework which explicitly takes thecoordination of sites into account. Perhaps one way tolook at such systems is to use a framework of graphs. ACKNOWLEDGMENTS
We thank Aniket Patra, Adhip Agarwala and BlazejJaworowski for useful discussions.
Appendix A: Two-fold Degeneracy in Mτ z + H xy The two-fold degeneracy in H xy is a consequence ofthe fact that T − H xy T = H xy . Adding a mass term, M τ z , breaks this symmetry. However, eigenstates of M τ z + H xy still form degenerate pairs. Consider aneigenstate | ψ (cid:105) of H xy with eigenvalue (cid:15) . Due to thesymmetry τ z H xy τ z = − H xy , we have that τ z | ψ (cid:105) is alsoan eigenstate of H xy but with eigenvalue − (cid:15) . Notice thataddition of the M τ z term also breaks this symmetry.Here, we analytically show that the effect of the M τ z term is to hybridize | ψ (cid:105) and τ z | ψ (cid:105) .We assume an ansatz eigenstate of M τ z + H xy of theform α | ψ (cid:105) + βτ z | ψ (cid:105) , with eigenvalue E . We have thefollowing equation (cid:16) M τ z + H xy (cid:17) ( α | ψ (cid:105) + βτ z | ψ (cid:105) ) = (cid:16) βM + α(cid:15) (cid:17) | ψ (cid:105) + (cid:16) αM − β(cid:15) (cid:17) τ z | ψ (cid:105) = E ( α | ψ (cid:105) + βτ z | ψ (cid:105) ) . (A1)For (cid:15) (cid:54) = 0, we have that | ψ (cid:105) and τ z | ψ (cid:105) are orthogonalbecause they are eigenstates of H xy with different eigen-values. Defining (cid:15) (cid:48) = (cid:15)/M and E (cid:48) = E/M , and equatingthe coefficients of | ψ (cid:105) and τ z | ψ (cid:105) in Eq. (A1), we get β + α(cid:15) (cid:48) = αE (cid:48) ; α − β(cid:15) (cid:48) = βE (cid:48) . (A2)Solving the pair of equations in (A2) for α , β and E , weget α ± β ± = (cid:15) (cid:48) ± (cid:112) (cid:15) (cid:48) ; E ± = ± (cid:112) M + (cid:15) . (A3)So we have shown that α ± | ψ (cid:105) + β ± τ z | ψ (cid:105) are eigenstatesof M τ z + H xy with α ± , β ± satisfying Eq. (A3).Now, as P − ( M τ z + H xy ) P = − ( M τ z + H xy ), with P = τ x K , we have that P | Φ (cid:105) is an eigenstate of the Hamilto-nian, M τ z + H xy , with eigenvalue − ξ , if | Φ (cid:105) is an eigen-state with eigenvalue ξ . Hence, | Ψ + (cid:105) = α + | ψ (cid:105) + β + τ z | ψ (cid:105) and P | Ψ − (cid:105) = P (cid:16) α − | ψ (cid:105) + β − τ z | ψ (cid:105) (cid:17) are both eigen-states of the Hamiltonian, M τ z + H xy , with the sameeigenvalue E + . Notice that the states, | Ψ + (cid:105) and P | Ψ − (cid:105) ,are orthogonal to each other, as | ψ (cid:105) , P | ψ (cid:105) , τ z | ψ (cid:105) and P τ z | ψ (cid:105) = − T | ψ (cid:105) , are mutually orthogonal. The states | ψ (cid:105) and P τ z | ψ (cid:105) are orthogonal to the states P | ψ (cid:105) and τ z | ψ (cid:105) as they are eigenstates of H xy , a Hermitian oper-ator, with different eigenvalues. 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