Topology in the Sierpiński-Hofstadter problem
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Topology in the Sierpi´nski-Hofstadter problem
Marta Brzezi´nska, Ashley M. Cook, and Titus Neupert Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc law University of Science and Technology, 50-370 Wroc law, Poland Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland (Dated: July 3, 2018)Using the Sierpi´nski carpet and gasket, we investigate whether fractal lattices embedded in two-dimensional space can support topological phases when subjected to a homogeneous external mag-netic field. To this end, we study the localization property of eigenstates, the Chern number, andthe evolution of energy level statistics when disorder is introduced. Combining these theoreticaltools, we identify regions in the phase diagram of both the carpet and the gasket, for which thesystems exhibit properties normally associated to gapless topological phases with a mobility edge.
Introduction — The topological character of electronicstates of quantum matter is imprinted in important uni-versal characteristics, such as quantized response func-tions, localization properties of eigenstates, and pro-tected boundary modes. Whether a system can in prin-ciple support topological phases depends on its dimen-sionality and the set of symmetries with respect to whichtopology is defined. For noninteracting but potentiallydisordered systems, this information is tabulated in theten-fold way [1], while the classification of symmetry pro-tected topological (SPT) phases in general also encom-passes interactions [2].All classifications of topological states have so far beenperformed for systems of integer spatial dimension. Forinstance, the integer quantum Hall effect, which can inmany ways be viewed as the most robust and funda-mental topological phase, exists in two-dimensional sys-tems, but not in one-dimensional systems (in three di-mensions it can exist only as a weak phase whose es-sential properties are inherited from the two-dimensionalrealization [1, 3]). However, to define the topology ofquantum states, only a notion of locality and the possi-bility to take a thermodynamic limit are required, bothof which can be defined for a general graph, not only fora regular lattice. In particular, these concepts can bedefined for a fractal lattice. Thus, a notion of topolog-ical states should also exist for quantum states definedon general graphs, including fractals with (non-integer)Hausdorff dimension. It is imperative to ask whether thequantum states on such graphs can in fact be topologicaland how the classification depends on properties of thefractals like their dimensionality or ramification number.Here, we investigate these questions by means of a casestudy on what might be considered the most natural can-didate for a topological phase of a fractal lattice: theelectronic structure in presence of a homogeneous mag-netic field. Specifically, we study lattice regularizationsof the Sierpi´nski carpet (SC) and the Sierpi´nski gasket(SG), i.e., the Sierpi´nski-Hofstadter problem. By consid-ering a magnetic flux that is homogeneous with respect tothe two-dimensional plane in which the fractals are em-bedded, we study the situation most relevant to meso- and nanoscopic experiments. An important differencebetween the gasket and the carpet is that their ramifi-cation number is finite and infinite, respectively. Thismeans that an extensive part of a gasket can be sepa-rated by just cutting a finite number of bonds, while forthe carpet this operation requires cutting a number ofbonds that tends to infinity in the thermodynamic limit.Renewed interest in the physics of fractals has beenignited by progress in experimental methods which allowto create fractal structures using, for example, molecularchains [4], atomic manipulation of molecules on the sur-face [5] or focused ion beam lithography [6]. Recent theo-retical developments include quantum transport calcula-tions [7], investigations of optoelectronic properties [8, 9],random fractal lattices [10], entanglement entropy andentanglement spectra in fractals [11], and systems withfractal boundaries [12] or fractal-like structures hostingflat bands [13]. The spectra of the SC and SG in a pres-ence of a magnetic field were studied as well [14–17],but possible topological properties of the eigenstates havenot been investigated. Topological phases on fractal lat-tices have been examined only recently within Bernevig-Hughes-Zhang (BHZ) model on the SC and SG in Ref. 18(similar considerations for completely random lattices arepresented in Ref. 19).We use a combination of approaches to identify thetopological properties of the Sierpi´nski-Hofstadter prob-lem as a function of filling and magnetic flux. First, weanalyze the localization properties of individual eigen-states on the lattice. Thereby we uncover a hierarchyof states sharply localized around “holes” of the lattice.We identify the regions in the phase diagram where suchstates dominate. Second, we use a real-space formula-tion to compute the Chern number (or Hall conductiv-ity). We find it to be sharply quantized to trivial andnon-trivial values in parts of parameter space. Finally,we add disorder to the system and study the energy levelspacing statistics. This way, we can identify regions inthe phase diagram which are separated by a plateau tran-sition from an Anderson insulating limit, indicating theirnon-trivial topology. We find good agreement betweenthese regions and the ones with non-zero Chern number, C B A C B A (a) (b) FIG. 1. Sierpi´nski (a) carpet and (b) gasket at iteration n = 4and n = 6, respectively. Black squares correspond to keptsites from underlying square (in case of SC) and triangular(SG) lattices. The summation regions included in real-spaceChern number calculations are marked with A, B, C. confirming the consistency of our results. In the follow-ing, we present each of these three approaches in succes-sion. Further details and a cross-check of our methodsfor the known Hofstadter problem on two-dimensionallattices are contained in the Supplemental Material. Model — We consider a tight-binding Hamiltonian de-scribing spinless fermions in a perpendicular orbital mag-netic field H = − t X h i,j i e i A ij c † i c j + h . c ., (1)where c † i ( c i ) is a creation (annihilation) operator on lat-tice site i and h . . . i denotes nearest neighbors. The hop-ping integral t is the only energy scale, and we set it to t = 1. The magnetic field is incorporated into the modelby the phase factors A ij = R ji A · d r with A being thevector potential satisfying relation B = ∇ × A . Hamil-tonian (1) is studied on a graph that corresponds to thelattice-regulated SC and SG (see Fig. 1). These graphshave a smallest square (for the SC) and a smallest trian-gle (for the SG), respectively. The magnetic flux per thissmallest element is chosen to be Φ, a fraction α of the fluxquantum Φ (where Φ = 2 π in units where ~ = e = 1),i.e., Φ / Φ = α . The magnetic field is assumed to behomogeneous in the two-dimensional space in which thefractal is embedded. To study the effect of disorder, weadd the on-site disorder term P i V i c † i c i to the Hamilto-nian. The coefficients V i are randomly chosen values fromthe uniform distribution in the range (cid:2) − W , W (cid:3) where W is the disorder strength in units of t . Spectral and eigenstate localization properties — InFig. 2 (a, d), we show the density of states (DOS) for sys-tems with open boundary conditions as a function of α .Discrete energy spectra E λ are smoothed using a Gaus-sian function f ( E, α ) = P λ exp n − [ E − E λ ( α )] /η o with broadening η = 0 . E = 0 and the α = 1 / ∼ . α = 1 / E = 0. Regions oflow DOS (appearing in dark blue color) host states withdistinct localization properties, which we confirm below.The spectrum of the SG [Fig. 2 (d)] has only a pointinversion symmetry about α = 1 / E = 0, whilereflection symmetries are lost for this non-bipartite lat-tice. At zero flux, the spectrum is known to be a fractalwith discrete eigenvalues [21]. The magnetic field liftsthese degeneracies. The most distinct spectral featuresare a large DOS at α = 1 / E ≈ . α = 1 /
2) slightly above zeroenergy. They reveal states sharply localized at the inter-nal edges of the fractal at different levels of the hierarchy.Groups of states of this type can be found in very closespectral proximity to one another in various places of thephase diagram. To map out these regions, we calculatea localization marker defined as B λ,l = X i ∈E l | ψ λ,i | , (2)where h i | ψ λ i = ψ λ,i and the summation is taken overthe edges E l of all internal triangles or squares at level l of the hierarchy. Therefore, B λ,l measures how muchan eigenstate | ψ λ i with an energy E λ is localized on thedifferent edges of hierarchy level l . A similar hierarchyof edge-localized states was also observed for BHZ modelin Ref. 18. With every | ψ λ i we associate a set of B λ,l for l = 0 , · · · , n . To determine where in the phase dia-gram the localization properties are most rapidly varying,we calculate the variance for each entry of the set B λ,l , l = 0 , · · · , n , across three consecutive states in the spec-trum with energies E λ − , E λ and E λ +1 and sum thesevariances over l . The results are shown in Fig. 2 (c, f) anddemonstrate that sharp changes in eigenstate localizationappear predominantly in the regions with low DOS bothfor the carpet and the gasket. We are thus led to inter-pret the regions of low DOS as made of states with edgecharacter at various levels of the fractal hierarchy. Real-space Chern number — To study potential topo-logical properties of the Sierpi´nski-Hofstadter problem,we adopt a real-space method to compute the Chernnumber introduced in Ref. 22 C = 12 πi X j ∈ A X k ∈ B X l ∈ C ( P jk P kl P lj − P jl P lk P kj ) , (3)where P is a projector onto occupied states with respectto a given Fermi level E and j, k, l are site indices in three (a) (d) (b)(e) (c)(f) FIG. 2. (a, d) Density of states in the energy-flux plane, (b, e) localization of the eigenstates, and (c, f) edge-locality markerfor the SC at iteration n = 4 and the SG at iteration n = 6, respectively. Darker regions are related to smaller density. Energyspectra reveal two gaps at E = 0 for SC and numerous gaps in case of SG. Electronic densities presented in (b, e) correspondto the time-reversal symmetric point ( α = 1 /
2) indicated by a framed part of the spectrum. The color scale corresponds tothe square modulus | ψ i | of the wave function normalized by its maximum value. (c, f) shows how the B λ,l marker changesbetween consecutive eigenstates at fixed flux. Low DOS regions are associated with largely varying localization properties ofthe eigenstates. distinct neighboring regions A , B , and C of the lattice.The regions are three neighboring sectors arranged coun-terclockwise as shown in Fig. 1. If C is quantized, itbecomes independent of the detailed choice of A , B , C in the limit of where the number of sites in each sectortends to infinity. We repeated the calculation for variouschoices of A , B , C and found that the intervals of quan-tized C discussed below are robust. In Fig. 3, we show C as a function of the Fermi energy E at fixed value offlux α = 1 / n = 4 iteration of SC (c) and the n = 6 iteration of the SG (g). We obtain the follow-ing results: (i) All fully gapped regions of the spectrum,both in the case of SG and SC, carry C = 0. (ii) Theregions of low but non-zero density of states (blue) inFig. 2 (a) for the SC correspond to stable plateaus with C ∼ ± . E = − . . . . − . E = 0 . . . . . C ∼ ± .
96 ( E = − . . . . − . E = 2 . . . . . E = − . E = 1 .
2. (iii) For the SG, non-trivial regions are lessclearly identifiable, but a clear plateau from E = 1 . . . . C ∼ . α averaged over an energy interval[ ǫ − δ, ǫ + δ ] (with δ = 0 . δ = 0 .
05 for theSG) for different system sizes, and compute the averagescaling exponent ν of the number of states in that energyrange with system size. On average, ν equals the Haus-dorff dimension d H . We show in Fig. 3 (b) that for theSC regions with (nearly) quantized Chern number con-sistently show scaling with ν < d H . This indicates thatthe normalized DOS would scale to zero in the thermo-dynamic limit in regions with quantized Chern number.For the SG, the situation is less clear except in regionsof trivial Chern number where no states are found. [seeFig. 3 (f)]. Level spacings analysis — A complementary probe oftopology can be obtained by studying the effect of lo-calization by disorder. At large disorder strength (muchlarger than the band width), all states of a system becomeAnderson localized. However, if the system is in an insu-lating state with non-trivial Hall conductivity for small (d)(c) ((cid:0)(cid:1) (b) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (e) (f)
FIG. 3. (a, e) Density of states, (b, f) scaling exponent ν of the DOS with system size as a function of E , (c, g) the Chernnumber as a function of E and (d, h) variance of level spacings in the energy-disorder plane at fixed flux α = 1 / W [white arrows in (d, h)]. Stateswith C 6 = 0 are characterized by a DOS scaling exponent ν smaller than d H in (b). A deviation from that behavior is causedby singular peaks of the DOS. disorder, the transition to the Anderson insulator hap-pens via a critical delocalized state at intermediate disor-der values [23]. To probe this transition – if present – westudy the energy level spacing statistics of the Sierpi´nski-Hofstadter problem in the presence of disorder.To determine whether states are extended or localized,we perform an energy level statistics analysis. For agiven energy ǫ and disorder realization { V i } , we find twoclosest eigenvalues satisfying E λ, { V i } < ǫ < E λ +1 , { V i } ,then calculate level spacings s ǫ,m, { V i } = E λ + m +1 , { V i } − E λ + m, { V i } , where m ∈ {− k, k } , and normalize them. Weset k = 2 as suggested in Ref. 24. This allows to in-vestigate the distribution of the level spacings and thevariance Var( s ǫ ) = h s ǫ i − h s ǫ i . The average is takenwith respect m and 10 disorder realizations for fixed ǫ .If states are delocalized, then the level spacings shouldobey the Wigner-Dyson surmise in the unitary case givenby P GUE ( s ) = s π e − π s ; if localized, they are expectedto follow a Poisson distribution P ( s ) = exp( − s ). Usingthe numerically obtained distribution of the level spac-ings for different disorder amplitudes W , we calculate thedifference between Var( s ) and the variance correspondingto P GUE [see Fig. 3 (d, h)]. Since disorder calculationsrequire exact diagonalization of the Hamiltonian repeat-edly, we focus on smaller systems (iteration n = 3 forSC and n = 5 for SG). We find that regions in energyfor which the Chern number is quantized consistentlyshow a large Var( s ) for small W , i.e., they are localized[see Fig. 3 (d, h)]. At strong disorder the systems arefully localized as well. As one follows a line of increas-ing W at constant energy, two transition scenarios canbe found, corresponding to the white and blue arrows in Fig. 3 (d, h), respectively: either there is a crossoverinto the localized region at large W without Var( s ) everbecoming close to Var( P GUE ) = 0 . W is separated by a delocalized region withVar( P GUE ) = 0 .
178 from the localized states at large W . These two scenarios are in correspondence with theChern numbers computed in the absence of disorder: Theformer is found for regions with trivial quantized Chernnumber, the latter for non-trivial quantized Chern num-ber. Conclusions — We have investigated topological elec-tronic properties of two fractal lattices, the Sierpi´nskicarpet and gasket in a external magnetic field. By per-forming level spacings analysis, Chern number calcula-tions and by investigating localization properties of in-dividual eigenstates, we identified states with non-trivialtopology that show characteristics similar to the quan-tum Hall effect. They do, however, occur on graphs withnon-integer Hausdorff dimension. Our results, whichstrongly suggest the existence of quantum Hall-typestates on fractals, call for an extension of the classifi-cation of topological states to such more general graphs.The example we investigated is in particular tailoredto challenge the following sharp distinction by dimen-sionality: Long-range entangled phases (to which the in-teger quantum Hall effect belongs in the terminology ofRef. [25]) have been proven to not exist in one dimen-sion [26]. It is thus imperative to ask what dimensionalproperties a graph must have in order to support long-range entangled ground states of local Hamiltonians.Finally, we emphasize that topological states on fractallattices may provide a way to understand so-called frac-ton topological order in three-dimensional systems [27–30]. Fracton models have been studied as codes withlarge ground state manifolds in which quantum infor-mation may be stored. In some fracton models, oper-ators that create excitations have support on a fractalsubset of the three-dimensional lattice. From more con-ventional topological orders, we know that the opera-tors that create excitations can be thought of carryinga lower-dimensional SPT phase. It would be interest-ing to investigate whether the same picture holds for thefractal case. Similar considerations may apply to therelated fractal symmetry breaking states investigated inRef. [31].
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Creating the lattices and flux distribution
To construct the Sierpi´nski carpet (with d H ≃ . L ( n ) = 3 n sites alongouter edge and in every iterative construction step n we remove (1 − (8 / n ) · n sites. From Pascal’s triangle moduloprime number m embedded in triangular lattice with 2 n + 1 rows, we can create a series of fractal triangular latticeswith Hausdorff dimension d H = 1 + log m (cid:0) m +12 (cid:1) ( m = 2 gives the Sierpi´nski gasket with d H ≃ . π . / - / - / - / - / - / - / - / - / - (a) (b) FIG. 4. Phase distribution on 4 × A ij phase between site i and j is equal to the number shown above the bond in 2 π units. A phase acquired with the respect to the direction pointedby arrows has a positive sign. Level statistics
In Fig. 6 and 5 we show the level spacings distributions for three values of ǫ at W = 1 , , W , we mark points on the phase diagrams with greensquares, which correspond to the histograms below.In Figs. 7 and 8 we present the Var( s ) − Var( P GUE ) in the energy-flux plane at fixed W = 1 , , α = 1 / W = 1 W = 3 W = 5(b) (c) (d)(e) (f) (g)(h) (i) (j) (cid:8) = (cid:9) = - . (cid:10) = - . (a) FIG. 5. (a) Energy levels for n = 3 carpet in the absence of disorder and phase diagram from 3 at α = 1 /
4. (b)–(j) distributionof the level spacings. Histograms are related to the spacings around (h, i, j) ǫ = − .
5, (e, f, g) - to ǫ = − . ǫ = 0. Calculations were performed for three disorder strengths W = 1 (b, e, h), W = 3 (c, f, i) and W = 5 (d, g, j) W = 1 W = 3 W = 5(c)(f)(b) (d)(e) (g)(h) (i) (j) ϵ = ϵ = - . ϵ = - . (a) FIG. 6. (a) Energy levels for clean n = 5 gasket together with phase diagram from 3 at α = 1 /
4. (b)–(j) distribution of the levelspacings. Histograms are related to the spacings around (h, i, j) ǫ = − .