Fractal defect states in the Hofstadter butterfly
FFractal defect states in the Hofstadter butterfly
Yoshiyuki Matsuki, ∗ Kazuki Ikeda, † and Mikito Koshino ‡ Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
We investigate the electronic properties in the Bloch electron on a square lattice with vacanciesin the uniform magnetic field. We show that a single vacancy site introduced to the system createsa defect energy level in every single innumerable fractal energy gap in the Hofstadter butterfly.The wavefunctions of different defect levels have all different localization lengths depending on theirfractal generations, and they can be described by a single universal function after an appropriatefractal scaling. We also show that each defect state has its own characteristic orbital magneticmoment, which is exactly correlated to the gradient of the energy level in the Hofstadter diagram.Probing the spatial nature of the defect-localized states provides a powerful way to elucidate thefractal nature of the Hofstadter butterfly.
Introduction
The Hofstadter butterfly is the energyspectrum of Bloch electrons moving in a two-dimensionallattice under a uniform magnetic field, which is character-ized by a nested fractal band structure [1–7]. It has beenactively studied in condensed matter physics [8–13], andalso from a wide variety of perspectives including math-ematics [14–20] and quantum geometry [21, 22]. Exper-imentally, the evidence of the fractal nature of the Hof-stadter spectrum was found in various systems, such asGaAs/AlGaAs heterostructures with superlattices [23–25], ultracold atoms in optical lattices [26–28], graphene-based moir´e superlattices [29–31], photons with the su-perconducting qubits [32] and one-dimensional acousticarray [33–35].Currently, however, the experimental observation ofthe butterfly is mostly limited to the measurement of thespectral structure and the transport properties. Actually,richer fractal information is encoded in the wavefunctionsof the Hofstadter butterfly, but it is generally consideredto be difficult to access in experiments. The character-istic spatial property of each wavefunction is generallyaveraged out in the physical observables due to the sum-mation over the Bloch momentum.In this paper, we theoretically propose that the spatialstructure in the Hofstadter system can be elucidated byintroducing a point defect to the system. In an electronsystem under a magnetic field, generally, a point disorderpotential gives rise to defect localized states in the energygaps between Landau levels [36–42]. The effect of latticedefects on the Hofstadter spectrum was investigated insome past works [43–50], and the in-gap defect levels werefound at a certain magnetic flux [50]. However, it has notbeen clear how the self-similar nature is manifested in thedefect localized states.In this letter, we study the Hofstadter problem withvacancy defects in a square lattice to investigate the frac-tal properties of defect states. We show that a singlevacancy site introduced to the system creates a defectenergy level in every single innumerable fractal energygap in the Hofstadter butterfly. We find that the wave-functions of different defect levels have all different lo-calization lengths depending on their fractal generations, and importantly, the localization length of any levels canbe approximately described by a single universal curveafter an appropriate fractal scaling. We also find thatthe defect states are accompanied by an orbital magneticmoment due to rotating electric current, and its magni-tude exactly coincides with the gradient of the energygap in the Hofstadter diagram. These results give a newquantitative perspective on the spatial fractal nature ofthe Hofstadter butterfly, and provide a powerful way toelucidate the fractal nature of the Hofstadter butterflyby probing the defect states. (b)(a) x
30 supercell
Formulation
We consider a square lattice with asingle-site defect as illustrated in Fig. 1. We assume thatthe system is periodic with N × N supercell and eachsupercell includes a single vacancy site. The system isunder a uniform magnetic field B perpendicular to thesystem. Let φ = Ba / ( h/e ) be the number of magneticflux quanta per a 1 × a = 1 is thespacing between the lattice points. In what follows, weconsider a single orbital tight-binding Hamiltonian, H = − t (cid:88) (cid:104) m,n (cid:105) m,n (cid:54) = d e iθ mn c † m c n , (1)where t ( >
0) is the hopping parameter, (cid:104) m, n (cid:105) is a pairof the nearest neighbor sites, c † n ( c n ) is the creation (an-nihilation) operator at site n , d is the site of defects, θ mn = − ( e/ (cid:126) ) (cid:82) mn A · d (cid:96) is the Peierls phase [51], and A = (0 , Bx,
0) is the vector potential. We take t = 1throughout this letter. In a perfect lattice without a de- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (i) (iii) (ii) Magnetic flux E ne r g y (a) R
30 superlattice with a single-site defect, which is plotted against the magnetic flux φ .Thered and black dots represent the bulks states and the defect-localized states, respectively. The labels (i), (ii) and (iii) correspondto the wavefunctions shown in Fig. 3. (b) Subcell decomposition of the Hofstadter butterfly (see the text). The states (i), (ii)and (iii) belong to the positive gradient principal gaps of the main spectrum, R and C R , respectively. fect under a rational magnetic flux φ = p/q ( p, q : coprimeintegers), the energy band splits into q subbands [5].The number of total magnetic fluxes penetrating anentire supercell is Φ = N φ . For Φ = P/Q with co-prime integers P and Q , the eigenstates of the systemcan be taken as magnetic Bloch states, which satisfy thefollowing conditions [3, 52, 53]: ψ ( r + L ) = e ik x L e − i πeBL y ψ ( r ) , (2) ψ ( r + L ) = e ik y L ψ ( r ) , (3)where L = ( QN a,
0) and L = (0 , N a ) are the primi-tive lattice vectors of the magnetic unit cell. Then theeigen-energies and eigen-wavefunctions can be obtainedby diagonalizing QN × QN Hamiltonian matrix. Wealso perform similar calculation and analyses for a honey-comb lattice with periodic vacancies, which is presentedin Supplementary Information.
Fractal defect states
Fig. 2(a) shows the energy spec-trum of 30 ×
30 superlattice with a single-site defect, plot-ted against the magnetic flux φ . The red and black dotsrepresent the bulk states and the defect-localized states,respectively. Here the defect-localized states are iden-tified by the condition that the wave amplitude withinseven-site distance from the defect point is more than60% of the total amplitude. We observe that a defectlevel exists in every single gap, indicating that the spec-trum of the defect states inherits the nested fractal struc-ture of the Hofstadter butterfly.The left panels in Fig. 3 represent the squared wave-functions of defect levels (i), (ii) and (iii) in Fig. 2, whichare taken from different minigaps of the Hofstadter but-terfly. Here the eigenstates are calculated for 40 × ≤ φ ≤ L n , R n and C n ( n ∈ Z ), respectively. The gap struc-ture of each subcell plotted against the local variable x (0 ≤ x ≤
1) is identical to that of the main spectrumplotted against the magnetic flux φ (0 ≤ φ ≤ φ and the local variable x for the subcell X n ( X = L, R, C ) are related by φ = φ X n ( x ), where φ L n ( x ) = φ R n ( x ) = ( n + x + 2) − , (4) φ C n ( x ) = [2 + 1 / ( n + x )] − . (5)In Fig. 2(b), we present an axis of the local variable x for R subcell, where x = 0 , / , φ = 1 / , / , /
3, respectively. The main spectrum canalso be regarded as a single subcell, where the local vari-able is the magnetic flux itself, i.e., φ = x . By repeatingthis addressing scheme, we can define subcells in highergenerations. For instance, X m Y n refers to subcell Y n insubcell X m in the main spectrum. The relation betweenthe local variable x for subcell X m Y n and the global vari-able φ is given by φ = φ X m ( φ Y n ( x ))( ≡ φ X m Y n ( x )).For each subcell, we define the positive (negative) prin-cipal gap as the diagonal gap running from the lower (up-per) left corner to the upper (lower) right corner of the (c) (b) (a) Wave amplitude = 1 /
5) = 5 /
5) = 5 /
5. The global variables φ for (i), (ii)and (iii) are 1 / φ R (1 /
5) = 5 /
11 and φ C ( φ R (1 / /
21, respectively. For each state, we define the localiza-tion length ξ by ξ = (cid:80) i | r i − r | | ψ ( r i ) | , where ψ ( r i )is the wave amplitude at site r i , and r is the vacancyposition. Here we show that the ratio of ξ ’s of differentsubcells is approximately equal to the ratio of the de-nominators of φ of those states. For the states (i), (ii)and (iii) in Fig. 3, for instance, this claims that the ra-tio of the ξ ’s of the three states is 5 : 11 : 21. Indeed, R eno r m a li z ed l o c a li z a t i on l eng t h Local variable x Lo c a li z a t i on l eng t h Local variable x (a)(b) R
5) = 5 /
11 have similar structures with lengthscales of 5 : 11. Naturally, the defect states take over thesame scaling feature.Let ξ ( φ ) be the localization length of the defect statein the principal gap of the main spectrum at the flux φ , and ξ X n ( x ) be that of subcell X n at local variable x .According to the argument above, we have the relation ξ ( p/q ) /q = ξ X n ( p/q ) /D [ φ X n ( p/q )], where D [ φ ] is the de-nominator of φ . Using Eqs. (4) and (5), this immediatelyleads to the relation between ξ and ξ X n , ξ ( x ) = φ R n ( x ) ξ R n ( x ) = φ L n ( x ) ξ L n ( x ) , (6) ξ ( x ) = φ C n ( x ) n + x ξ C n ( x ) , (7)which is a key finding of this work. Note that, al-though the denominator D [ x ] is not a continuous func-tion of x , the scaling ratio ξ X n ( x ) /ξ ( x ) is a continu-ous function of x . Similarly, the relation for higherfractal generations can be obtained from ξ ( p/q ) /q = ξ X m Y n ··· ( p/q ) /D [ φ X m Y n ··· ( p/q )]. For C R subcell, forinstance, it gives ξ ( x ) = φ C R ( x ) ξ C R ( x ).In Fig. 4 , we plot (a) the localization lengths( ξ, ξ R , ξ R , ξ C R ) and (b) the renormalized values( ξ, φ R ξ R , φ R ξ R , φ C R ξ C R ) as functions of the lo-cal variable x . Here the solid and dashed curves rep-resent the positive and negative principal gaps, respec-tively. The two curves are identical for the main spec-trum due to the electron-hole symmetry. We see thatthe renormalized localization lengths [Fig. 4(b)] quanti-tatively match in a wide range of x . The ξ ( x ) diverges at x = 0 in proportion to 1 / √ x , corresponding to the factthat the length scale in a weak magnetic field is given bythe magnetic length (cid:112) (cid:126) / ( eB ). We have the same featurein x = 1 symmetrically. At x = 1 /
2, we notice that the ξ ( x ) diverges only in the negative gaps of the subcells,while it remains finite in the positive gaps. This links tothe fact that the negative gaps close while the positivegaps are open at x = 1 /
2. For the main spectrum andany subcells centered at E = 0, the positive and negativegaps both closes at x = 1 / R and C R seem to follow a different curve from therest in the limit of x →
0, and this is related to the factthat the gap is approaching the Dirac point of the mag-netic Bloch band at E = 0. The origin of the differentscaling nature is argued in Supplemental Information. Magnetic moment
Generally, the in-gap defect statesin a time-reversal-symmetry broken system are accompa-nied by an orbital current circulation [50, 55]. Here wefind that the magnetic moment created by the orbital cur-rent of the defect states in our system is precisely relatedto the gradient of fractal defect states on the Hofstadterdiagram. The local electric current J nm from site m tosite n is calculated by J nm = i ( − e ) t (cid:126) ( e iθ mn ψ ∗ n ψ m − c . c . ) . (8)The right column in Fig. 3 shows that the local currentcalculated for the defect states (i), (ii) and (iii). Thethickness of and the size of arrows is proportional to theabsolute value of current. Here the current rotates in theclockwise direction, i.e., it has a negative magnetic mo-ment. Actually, the direction of the current synchronizes (a)(b) Magnetic flux
30 superlattice with (a) consec-utive two-point defect and (b) consecutive three-point defect. with the gradient of the defect energy level in the Hof-stadter diagram, because the orbital magnetic momentis given by m = − dE/dB (see Supplementary Informa-tion for the proof). In other words, a defect state in theHofstadter butterfly precisely tunes its own current cir-culation, in such a way that the energy level stays insidethe fractal gap in changing magnetic field. Multi-point vacancies
We also consider various multi-point vacancies in Fig. 5; (a) consecutive two-point defectand (b) three-point defect. The corresponding spectraof 30 ×
30 superlattice are shown in Fig. 5, where weobserve that the number of defect states in every singlefractal gap matches that of the missing sites in the defect.When the number of missing atoms becomes larger, weexpect that more and more defect levels fill in the energygaps, and they eventually form quantum Hall edge statescirculating around the hole. The fractal defect levels canbe regarded as the quantized version of the quantum Halledge state in the atomic limit. Therefore the emergenceof the defect levels in the fractal gaps may be viewed asa sort of the bulk-edge correspondence [56, 57] requiringthe existence of the edge states in a bulk gap with a non-zero Chern number. However, it should also be notedthat the number of the defect states in each gap is notat all related to the Chern number, but it just coincideswith the number of missing sites for any gaps.
Conclusion and Discussions
We have reported thatthe states localized around the defects fractally appearin every single band gap of the Hofstadter butterfly. Thedefect states in different energy gaps have all differentlength scale in the spatial decay, while they follow a uni-versal curve after the appropriate fractal scaling. Eachdefect state has its own characteristic magnetic moment,which is exactly linked to the gradient of the correspond-ing bulk energy gap in the Hofstadter diagram.While the previous observations of Hofstadter butterflyhave been mainly conducted by spectroscopic/transportmeasurements of the energy gap structure, our work pro-vides a powerful method to observe the fractal nature inthe wavefunction by measuring the spatial decay of thedefect states using scanning tunneling spectroscopy.
Acknowledgement
We thank Kin-ya Oda for fruit-ful discussions. This work was supported in partby Grant-in-Aid for JSPS Research Fellow, No.JP19J20559, JP19J11073, JSPS KAKENHI Grant Num-ber JP20H01840 and JP20H00127 and by JST CRESTGrant Number JPMJCR20T3, Japan. ∗ [email protected] † [email protected] ‡ [email protected][1] P. G. Harper, Proceedings of the Physical Society. SectionA , 879 (1955).[2] P. G. Harper, Proceedings of the Physical Society. SectionA , 874 (1955).[3] J. Zak, Phys. Rev. , A1602 (1964).[4] M. Y. Azbel, Zh. Eksp. Teor. Fiz. , 929 (1964).[5] D. R. Hofstadter, Phys. Rev. B , 2239 (1976).[6] G. H. Wannier, physica status solidi (b) , 757 (1978).[7] G. H. Wannier, G. M. Obermair, and R. Ray, physicastatus solidi (b) , 337 (1979).[8] F. H. Claro and G. H. Wannier, Phys. Rev. B , 6068(1979).[9] R. Rammal, J. Phys. France , 1345 (1985).[10] D. Pfannkuche and R. R. Gerhardts, Phys. Rev. B ,12606 (1992).[11] G. Gumbs, D. Miessein, and D. Huang, Phys. Rev. B , 14755 (1995).[12] M. 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Yoshiyuki Matsuki, Kazuki Ikeda, and Mikito Koshino Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
THE ORIGIN OF UNUSUAL SCALING CURVESOF LOCALIZATION LENGTH
In the main article, we demonstrated that the local-ization length of defect state in each minigap of the Hof-stadter butterfly is approximately described by a singleuniversal function when it is appropriately scaled. How-ever, there are some exceptional gaps in the spectrum,where the localization length follows a different curve.An exceptional scaling behavior occurs in the princi-pal gaps of any subcells ending with C , i.e., XY · · · C .Figure 6(a) shows the renormalized localization length[Eqs. (6) and (7) of the main text] of the positive princi-pal gap of the main cell, C , C C and C C C subcells[ ξ , ( φ C /x ) ξ C , ( φ C C /x ) ξ C C , ( φ C C C /x ) ξ C C C ] asfunctions of the local variable x . We observe that thecurves slightly shift in relative to each other in the limitof x →
0, whereas they precisely match in x (cid:38) / C , C C and C C C subcell correspond tothe second, third and fourth lowest gaps of the Landau-level spectrum, respectively [Fig. 6(c)]. This is in con-trast to any subcells NOT ending with C , where theprincipal gap definitely connects to the lowest gap (thegap just above the lowest Landau level) in the mag-netic Bloch subband at x = 0. The characteristic lengthscale of the n -th Landau level wavefunction ϕ n is givenby (cid:112) n + 1 / l B , where l B = (cid:112) (cid:126) / ( eB ) is the magneticlength. Accordingly, the defect-localized state which ex-ists between the Landau levels n − n should havethe length scale of the order of ∼ √ n l B , and the depen-dence on n results in the different scaling curves. Indeed,the localization lengths of the main cell, C , C C ,and C C C in the region of x < .
01 in Fig. 6(a) are verywell (within 1%) fitted by ξ ≈ . √ n − . l B , (9)with n = 1 , , l B ≈ / √ πx (in units of the lattice constant) in thisregion.Similarly, the same scaling rule is applicable to XY · · · Z ( C ) n subcells. In Fig. 6(b), we plot the renor-malized localization length of the principal gaps of R , R C and R C C subcells by dashed curves. In the limit x →
0, the three curves perfectly match with those of themain cell, C and C C , respectively (solid curves).Another exceptional case occurs in a gap connected tothe Dirac point of the magnetic Bloch band, such as thenegative principal gaps of R and C R mentioned in the main text. If we take R subcell, for instance, the gapleads to the point of φ = 1 / E = 0 of the maindiagram in the limit of x →
0, where a pair of magneticBloch bands are touching just like in graphene due to theelectron-hole symmetry of the model. Now the wavefunc-tion of the Landau level n in graphene is composed of the ϕ n − and ϕ n at different sublattices [A1,A2], so that itslength scale is just in the middle of those of ( n − n -th Landau levels in the conventional massive electron.Accordingly, the defect-localized states should have theintermediate localization length compared to the massivesystem. Indeed, the localization length of defect statesin R cell for small x is given by ξ ≈ . l B , which isbetween the values of n = 1 and 2 in Eq. (9). In Fig. 4(b)in the main text, we see that the C R subcell also fol-lows the same curves as R . This is because its negativeprincipal gap leads to the point of φ = 1 / E = 0,which is also the Dirac point. THE GRADIENT OF DEFECT STATES ANDTHE MAGNETIC MOMENT
We prove that the gradient of a defect energy levelin the Hofstadter diagram coincides with the magneticmoment created by the local electric current. For thispurpose we show that the magnetic moment m obeysEq. (18) in the presence of a generic potential V ( r ) inquantum mechanics. To show this formula we considerthe system described by the Hamiltonian, H = 12 m ( − i ∇ + e A ) + V ( r ) ( (cid:126) = 1) . (10)Considering a small change in the magnetic field, we ob-tain the perturbed Hamiltonian H = 12 m ( − i ∇ + e ( A + δ A )) + V ( r ) (11)= H −
12 ( J · δ A + δ A · J ) ≡ H + δH, (12)where J is the current operator J = ( − e ) i [ r , H ] = ( − e ) − i ∇ + e A m . (13)The variation of the energy δE within the first orderperturbation is δE = (cid:104) ψ | δH | ψ (cid:105) = − (cid:104) ψ | ( J · δ A + δ A · J ) | ψ (cid:105) , (14)where ψ is the eigenfunction of H . The first term inthe most right hand side in (14) is evaluated as follows: (cid:104) ψ | ( J · δ A ) | ψ (cid:105) = (cid:90) d r d r (cid:48) ψ ∗ ( r ) J ( r ) δ ( r − r (cid:48) ) ψ ( r ) · δ A ( r (cid:48) ) . (15)We can evaluate the second term in a similar way, andfinally obtain δE = (cid:90) d r (cid:48) j ( r (cid:48) ) · δ A ( r (cid:48) ) . (16)Here we used the local current operator j , j ( r (cid:48) ) ≡ − (cid:90) d r ψ ∗ ( r )( J ( r ) δ ( r − r (cid:48) )+ δ ( r − r (cid:48) ) J ( r )) ψ ( r ) . (17)By using the expression of the magnetizing current j = ∇ × m , one can find that the magnetic moment m obeys m = − dEd B . (18) THE RELATIONSHIP BETWEEN THEDISTANCE OF DEFECT SITES AND THEFRACTAL ENERGY SPECTRUM
Here we discuss the relationship between the distancesof vacancy sites and the fractal energy levels in the Hof-stadter diagram. For this purpose, we consider the two-point vacancies illustrated in Fig.7: (a) consecutive two-point defect, (b) two split defects. The correspondingspectra of 30 ×
30 superlattice are shown in Fig.7(a) and(b). By comparing Figs.7(a) and (b), we notice that thetwo defect energy levels get closer as the distance be-tween the two defect sites gets further away. This is aconsequence of the hybridization of the defect states oftwo single vacancy sites, where the coupling strength ex-ponentially decreases as the distance increases. A similareffect is also found in defect states in graphene [A3]. Wesee the same tendency consistently in all the fractal gaps.
Magnetic flux E ne r g y
Main (b) R C
0, respectively.Similarly, the positive principal gap of R C and R C C sub-cells correspond to the second and third Landau-level gaps,respectively, of the magnetic Bloch band in φ → / (a)(b) E ne r g y E ne r g y Magnetic flux
30 superlatticewith (a) consecutive two-point defect and (b) two split defects.
Fractal Structure and Wavefunction on a honeycomblattice with a defect (a) (b) x
20 supercel
Here we consider defect-localized states on a honey-comb superlattice with a single-site defect (Fig. 8). Weassume that the system is periodic with N × N supercelland every single supercell includes a single vacancy site.Now we take the super period N = 20, which is taken tobe sufficiently large to avoid interference between defect-localized states in neighboring cells. The Bloch electron wavefunctions obey the magnetic Bloch condition: ψ ( r + L ) = e ik x L e − i πeBL y ψ ( r ) , (19) ψ ( r + L ) = e ik y L ψ ( r ) , (20)where L = ( √ N a/ ,
0) and L = (0 , N a ) are the prim-itive lattice vectors of the magnetic unit cell. (i) (iii) (ii) Magnetic flux E ne r g y (a)
20 superlattice with asingle-site defect, which is plotted against the magnetic flux φ . The red and black dots represent the bulks states and thedefect-localized states, respectively. The labels (i), (ii) and(iii) correspond to the wavefunctions shown in Fig. 10. (b)Subcell decomposition of the Hofstadter butterfly. The states(i), (ii) and (iii) belong to the positive gradient principal gapsof the main spectrum, H and H , respectively. FIG.9(a) shows the energy spectrum of 20 ×
20 super-lattice in a honeycomb lattice with a single defect, plot-ted against the magnetic flux φ . The red and black dotsrepresent the bulk states and the defect-localized states,respectively. Here the defect-localized states are iden-tified by the condition that the wave amplitude within2 √ a from the defect point is more than 50% of thetotal amplitude. We observe that a defect level exists0in every single gap, indicating that the spectrum of thedefect states inherits the nested fractal structure of theHofstadter butterfly, same as the square lattice case. (c) (b) (a) Wave amplitude = 1 /
5) = 5 /
5) = 5 /
To compare uniformly the localization lengths in dif-ferent fractal generation gaps, we again define the subcell decomposition and the local variable y in a honeycomblattice. As shown in Fig.9(b), the main energy spectrumin 0 ≤ φ ≤ H n ( n = 0 , ± , · · · ). In fact, the gap structure of eachsubcell plotted against the local variable y (0 ≤ y ≤
1) isidentical to that of the main spectrum plotted against themagnetic flux φ (0 ≤ φ ≤ φ and local variable y is φ H n ( y ) = ( n + y + 2) − . (21)In the main spectrum, the global variable φ correspondsto the local variable y since the main spectrum can beviewed as a single subcell. In the following, we show theamplitude and the electric current of the correspondingdefects states of different subcells at the same local vari-able y .The left panels in Fig.10 show the distribution of thewave amplitude of three defect levels (i), (ii) and (iii)indicated in Fig. 9, which are taken from the positiveprincipal gaps of the main spectrum, H and H , re-spectively, with the same local variable y = 1 /
5. Fromthe equation (21), the global variables φ for (i), (ii) and(iii) are 1 / φ H (1 /
5) = 5 /
11 and φ H (1 /
5) = 5 /
16, re-spectively. We can observe that the defect wavefunctionslocalize around the defect with different length scales.Moreover, the right panels in Fig.10 represent that thelocal current calculated for the defect states (i), (ii) and(iii). As discussed above, the magnetic moment createdby the local current coincides with the gradient on theHofstadter diagram (Fig.9). Actually, the current rotatesin the clockwise direction, and the direction of the currentsynchronizes with the gradient of the defect energy levelin the Hofstadter diagram. The same properties as thesquare lattice case (the localization around the defect, thefractality of localization length, and the correspondencebetween magnetic moment and gradient of defect states)are observed in a honeycomb lattice case.[A1]
N. H. Shon and T. Ando, Journal of the Physical Soci-ety of Japan, , 2421-2429 (1998).[A2] Y. Zheng and T. Ando, Phys. Rev. B , 245420(2002).[A3] A. L. C. Pereira and P. A. Schulz, Phys. Rev. B78