Electronic properties of the Octagonal tiling in a magnetic field

Authors

address (Dated: October 21, 2016)

The electronic properties of the Octagonal tiling, a quasiperiodic tiling of codimension 2, are studied numerically and theoretically. The energy spectrum as a function of the magnetic flux through the system (the Hofstadter butterfly) displays a rich gap structure, which is being analysed. A relevant parameter of the system, the effective mass, is identified and studied numerically. The magnetic susceptibility is studied.

I. INTRODUCTION

Definition of quasicrystals ; remarquable physical properties 1. Studied a lot during the 80’s^{2}, with a revival of interest today : new experimental and numerical tools, which allow to study the still ill understood magnetic and electronic properties of quasiperiodic structures. Moreover, quasiperiodic tilings of codimension 2, such as the Octagonal tiling, have not been studied much.

II. THE MODEL

A. Construction of the Octagonal tiling

The Octagonal tiling (or the Amman-Benker tiling, discovered by Amman^{?} ) is a two-dimensional quasiperiodic tiling, whose diffraction pattern has an 8-fold symmetry. It is composed of two types of square and diamond tiles, of identical edge lengths \sqrt{2} (see fig. (1.a)). It can be obtained by the cut and project method^{3} : a subset of the Z_{4} lattice (points and edges between nearest neighbors) is projected on a two dimensional plane of R_{4} directed by irrational vectors ; since we project a fourdimensional set on a two-dimensional plane, we say that the tiling is of codimension two. The Octagonal tiling is infinite, but one can build a series of so called periodic approximants (periodic tilings) which converge asymptotically to the quasiperiodic structure. To simplify, we will call periodic approximant the unit cell of such a periodic tiling. We choose here the family of approximants described in^{3} ; however other families are available in the literature (see for example^{4}). The approximants are constructed by using rational planes for the projection space. Their properties (number of tiles and size) are related to the Octonacci sequence :

\{ \begin{matrix}O_{1} = 1 \\ O_{2} = 2 \\ O_{k+1} = 2O_{k} + O_{k−1} \end{matrix}

The Octonacci sequence verifies : lim_{n→∞} \frac{O_{k+1}}{O_{k}} = 1+\sqrt{2}, where λ = 1 + \sqrt{2} is the silver number. The periodic approximant of order k contains N_{k} = O_{2k+1}+O_{2k} vertices.

Note that the Octagonal tiling is composed of tiles of relative surfaces \sqrt{2}, which is an issue for the problem we

(a)  (b) (c)  (d)

FIG. 1: Approximant of order 3 of the octagonal tiling, (a) in its original form and (b) after a deformation of the edges, with r_{s} = 2.

(c) Single phason flip, (d) tiling after \frac{N^{2}}{2} flips.

wish to study, namely the spectral properties of the quasicrystal in the presence of a uniform magnetic field perpendicular to the tiling (see details in the next section). Irrational areas of tiles mean that the energy spectrum is not periodic with the magnetic flux, and we would have to scan the tiling for all values of the flux f ∈ R_{+} in order to obtain the full description of the energy spectrum. To bypass this difficulty, we slightly modify the projection direction, and obtain a tiling with the same connectivity, constituted of three types of tiles (small squares, big squares and rhombi), s.t. the big square and rhombus tiles have the same surface, and that the ratio r_{s} of the surfaces of the big square and small square tiles is rationnal (see fig. (1.b)). This transformation does not change the connectivity between the sites nor the symetries and patterns of the lattice.

In order to study the influence of structural disorder on the spectral properties of the quasicrystal, we also construct random tilings obtained from the Octagonal tiling by flipping tiles at random (see fig. (1.c), (1.d)). In order to reach maximum disorder for a tiling of N sites, one needs to perform \frac{N^{2}}{2} flips^{5}.

B. Tight-binding model

We now wish to study the electronic properties of the quasicrystal. We consider a polarized electron evolving on the lattice, in a quasiperiodic potential. For simplicity, we choose the tight-binding model : the particle is allowed to hop between nearest neighbor sites. In the absence of magnetic field, the corresponding Hamiltonian is

H = ∑_{⟨i,j⟩} t_{i,j}|i⟩⟨j|  (1)

where ⟨i, j⟩ denotes a couple of sites, |i⟩, |j⟩, closest neighbours.

The energy spectrum of such a system has been studied in^{6}. Here, we study the effect of an external uniform magnetic field B, perpendicular to the tiling. In this case, one must add a Peierls phase term^{7} to the coupling amplitude :

t_{i,j} → te^{−iθ_{i→j}} = te^{−i2π ∫_{i}^{j} dl·A}

where A is a potential vector associated to the magnetic field B = ∇ × A. Due to the magnefic field, gaps appear in the energy spectrum ; in the case of the square lattice, the graph of the spectrum as a function of the magnetic flux is called the Hofstadter butterfly^{8}. By analogy, we shall use the term Hofstadter butterfly or butterfly for the Octagonal tiling. We aim at understanding, theoretically, the spectral properties of the (infinite size) Octagonal tiling. However, numerically, we can only modelize finite size systems ; we will thus use periodic approximants for the numerics.

In the following, we set t_{i,j} = −1, ℏ = 1, e = −2π so that the elementary flux ϕ_{0} = \frac{h}{|e|} = 1, and a = 1 the length of the smallest distance between nearest neighbors. We introduce the flux f = a^{2}B = B the magnetic flux through a small tile (small square or diamond), and f̄ = \frac{A_{k}}{N_{k}} B ≈ 1.2f the average flux through a tile.

In open boundary conditions, any value of the magnetic flux is permitted. However, the spectrum is poluted by the edge modes ; to avoid this, we choose periodic boundary conditions. In this case, we need to take into account the effect of the magnetic translation operator^{9}. In fact, since the potential vector A is not periodic, the Hamiltonian (1) does not commute with the translation operator T_{a} even if the periodic tiling is invariant under the translation by the vector a. To restore the translation invariance, one needs to add a gauge term to the potential vector : A(r) → A(r) + χ(r). A standard calculation shows that χ(r) = ∫_{0}^{r} dl · A(l) works, i.e. e^{2iπχ(r̂)}T_{a}HT_{−a}e^{−2iπχ(r̂)} = H. The energy spectrum of the Hamiltonian is now periodic in the magnetic flux f, and we can therefore restrict our study to one period : f ∈ [0, 1]. Furthermore, the spectrum is unchanged if we reverse the field’s direction (f → −f), and, since the lattice is bipartite, the spectrum is symmetric with respect to 0 (rigorously, this symmetry is broken in periodic boundary conditions due to odd-length paths encircling the torus, but these effects can be neglected for large enough approximants).

FIG. 2: Hofstadter butterfly for the order 5 approximant, periodic boundary conditions.

FIG. 3: Hofstadter butterfly for the disordered order 4 approximant, periodic boundary conditions.

III. HOFSTADTER BUTTERFLY, WANNIER DIAGRAM AND GAP LABELING

In this section, we present numerical results and discuss the gap structure of the energy spectrum.

A. Hofstadter butterfly

As discussed in the previous section, in periodic boundary conditions, we can restrict our study of the energy spectrum to f ∈ [0, 1]. The butterfly obtained (fig. 2) has the symmetries predicted from the symmetries of the system (symmetry of the spectrum w.r.t. 0, and f → 1 − f symmetry due to both the invariance of the spectrum under f → −f and the 1−periodicity). It displays a very rich gap structure, which could not be observed in open boundary conditions. Furthermore, we observe the same gaps for all the approximants numerically accessible (up to image resolution), and we predict that the butterfly of the infinite tiling has the same gap structure.

As expected from the presence of magnetic field, we observe Landau levels near f = 0, which broaden due to the lattice potential, both in the ordered (fig. 2) and disordered (fig. 3) cases (we will discuss the broading quantitatively in the next section). Note that the Landau levels are the only gaps which survive the structural disorder (fig. 3), and are not wiped out by it ; therefore, the structural disorder induced by the flips is a weak disorder.

B. Gap labeling and Wannier diagram

To characterize the gaps in the Hofstadter butterfly, we turn to the Wannier diagram (fig. 4), namely the normalized integrated density of states N (E, f̄) (NIDoS) in the gaps as a function of the average magnetic flux through one tile f̄ : if a gap opens at the energy E, it correpsonds to a point (f̄, N (E, f̄)) in the Wannier diagram.

FIG. 4: Wannier diagram for the approximant of order 4, PBC. X-axis : average flux per tile \overline{f} = \frac{A_{k}}{N_{k}} f ; Y-axis : normalized integrated density of states N (E, f̄).

It is well known that the slopes of the lines of the Wannier diagramm are connected to the Hall conductivity inside a gap, as given by the Streda formula^{10}^{11} : σ_{H} = e\frac{∂N (E,f̄)}{∂f̄} = \frac{e^{2}}{h} ν, where ν is a topological invariant integer, called the Chern number. Integrating this formula, one finds :

N (E, f̄) = νf̄+_{0}

where N_{0} is a constant which depends on the gap. As argued by^{12} and^{13}, the gap labeling theorem implies that the gaps can be labelled using two integers (p, q) in the thermodynamic limit, and that N_{0} = ^{\frac{p+q\sqrt{2}}{8}}. We argue that, for finite size approximants, one needs three integers to label a gap, but that only two remain at the thermodynamic limit. In fact, adapting the argument of^{14} and using the Octonacci instead of the Tribonacci sequence, we find that, for an approximant of order k, N_{0} is of the form :

N_{0} = \frac{pA_{k} + qB_{k} + rC_{k}}{N_{k}}

with (p, q, r) ∈ Z independent of k, and lim_{k→∞} \frac{C_{k}}{N_{k}} = 0 so that only two integers remain in the thermodynamical limit. We furthermore predict :

N_{0} = \frac{p + q\sqrt{2}}{2}

i.e. the integers required to label the gaps correspond to those of the labeling of^{12}^{13} which are multiples of four.

This labeling allows us to classify the gaps, according to ν, which is related to the response to an applied magnetic field, and q, which accounts for the quasiperiodic order. The integer p insure that the IDoS is well normalized (N (E, f̄) ∈ [0, 1]). First, gaps labelled by ν ≠ 0, q = 0, which correspond to the main IQHE gaps and are the only gaps to survive disorder (cf. previous section and fig.2, 3). Second, gaps labelled by ν ≠ 0, q ≠ 0, which form fans reminding the IQHE gaps and are seperated by Landau levels ; they appear at some specific values of f̄, related to the Octonacci sequence, similarly to what happens at rational fluxes of the Hofstadter butterflies of periodic tilings^{15}^{8}. Finally, gaps labelled by ν = 0, q ≠ 0, which also disappear with disorder and are therefore also associated to the quasiperiodic order. Although it is expected, at f = \frac{1}{2}, to find gaps with a nul Chern ν number because of the time-reversal symmetry, the remarkable point here is to find such gaps at other values of f. Such gaps seldom appear and have so far ony been observed in the Lieb tiling^{16}, the dice tiling^{17} and the Rauzy tiling^{14}. The condition of existence and the physics of these gaps are still open questions.

IV. LANDAU LEVELS AT BAND EDGES

We focus here on the Landau levels which appear at band edges. Remind that the energy levels of an electron evolving in a two-dimension plane of surface A are, in the presence of a uniform magnetic field B :

E_{n} = ℏω (n + \frac{1}{2}) , n ∈ ℕ

where ω = \frac{eB}{m} and m is the effective mass of the electron ; each level has a degeneracy \frac{|B|A}{ϕ_{0}}. In the presence of a lattice potential, the levels broaden as the flux increases^{18}^{19}; however, in the case of the Octagonal tiling, the barycentre of the levels in the butterfly can be fitted by a family of lines expressed by :

E_{n} = \frac{2πf}{m} (n + \frac{1}{2})

with \frac{1}{m} = 2.9(1), a value stable for approximants up to k = 8 (1607521 sites) ; we predict that this value remains stable in the thermodynamic limit, which defines an effective mass for the Octagonal tiling.

It is known that, in the presence of a periodic potential, the energy levels tend to broaden slowly^{18},^{19}. A similar slow broadening, e.g. \frac{ΔE_{n}}{ℏω} ∝ e^{− \frac{1}{f}} with ΔE_{n} the width of the n − th level and ℏω the energy difference between two levels) has been observed for quasicrystals as well^{14}. Here, we observe a much more important broadening : lim_{f→0} \frac{ΔE_{n}}{ℏω} = C where C is a constant, C ∈ [0.1, 0.2]. In the disordered case, it has been observed for the Rauzy tiling that the broadening diverges to infinity^{?} .

V. EFFECTIVE MASS IN ZERO FIELD

As discussed in the previous section, in the zero field limit, the effective mass of an electron submitted to the quasiperiodic potential is given by the average slope of the Landau levels.

We aim here at indentifying an effective mass in zero field. There exist several ways to define such a quantity (see complementary material in^{14}). A first approach consists in mesuring the density of state (DoS) at the band edge. At zero temperature, the DoS is : ρ_{T =0}(E) = ∑_{n} δ(E − E_{n}), with {E_{n}} the energy spectrum, which is not analytical ; at finite temperature T however, the DoS is given by : ρ_{T} (μ) = − ∑_{n} n′_{F} (E_{n}−μ) with n_{F} (E) = \frac{1}{1+e^{βE}} the Fermi distribution function at chemical potential μ = 0. The DoS at finite temperature is thus smooth at the band edge, and can be approximated by the DoS associated to a parabolic dispersion relation in 2D, a constant equal to \frac{m_{ρ}}{2π}. We find for the Octagonal tiling : \frac{1}{m_{ρ}} = 2.9(1), as for the effective mass calculated with the Landau levels : m = m_{ρ}.

Another approach uses the so called effective mass tensor^{20}. In the case of another quasiperiodic tiling^{14}, both the Landau levels slope and the effective mass tensor gave consistent values of the effective mass.

The idea is to calculate the effective mass of an electron of energy ϵ(k) close to the energy minimum. If the minimum is reached at k = k_{0}, then the linear term in k − k_{0} vanishes when we expand ϵ(k):

ϵ(k) = ϵ(k_{0}) − α(k − k_{0})^{2} + O((k − k_{0})^{3})

and we define an effective mass m_{T} s.t. : α = \frac{ℏ^{2}}{2m_{T}} . The electron thus responds to a driving fiels as if it had a mass m_{T}. To derive the effective mass, we calculate the effective mass tensor, derived from Bloch’s Hamiltonian :

H_{0}(k) = e^{ik·r̂}H_{0}e^{−ik·r̂}

where H_{0} is the Hamiltonian of the periodic boundary conditions problem, at zero magnetic field. To obtain the effective mass tensor, we develop the dispersion relation around the center of the first Brillouin, for the first energy subband :

E_{1}(k) = E_{1}(0) + \frac{1}{2}α_{i,j}k_{i}k_{j} + O(k^{2})

where the α_{i,j} define the effective mass tensor :

α = (\begin{matrix}α_{x,x} & α_{x,y} \\ α_{x,y} & α_{y,y} \end{matrix})

and the effective mass m_{T} is given by :

\frac{1}{m_{T}} = \sqrt{det α}

Up to a constant, it can be shown that m_{T} is equal to the Thouless conducance^{21}, which measures the stiffness of the ground state energy when one modifies the boundary conditions. More precisely, we wish to quantify how the energy spectrum changes when we pass from periodic boundary conditions, with a system invariant under the translation along the vector L = Lx̂ : ψ(r + L) = ψ(r), to a more general problem where the phase is shifted from a factor K · L : ψ(r + L) = e^{iK·L}ψ(r). Formally, this amounts to study the spectrum of the operator e^{iK·r}He^{−iK·r}. If \frac{1}{m_{T}} is close to zero, then the groundstate energy does not change a lot when we vary the boundary conditions ; if it is large, then the groundstate energy is less stiff. The stiffness of the groundstate energy can be used as a criteria for localization ; indeed, if the groundstate wave function is very localized, then we expect that it won’t be much affected by a twist of the bondary conditions, unlike an extended groundstate. This has been discussed quantitatively in^{21}.

For the Octagonal tiling, unlike m and m_{ρ}, m_{T} does not converge to a finite value as we increase the size of the approximant ; instead, m_{T} goes to zero with the size of approximant, as can be intuited from the numerical results in table 1.

\begin{matrix}N_{k} & 239 & 1 393 & 8 119 & 47 321 & 275 807 & 1 607 521 \\ \frac{1}{m_{T}} & 2.9465 & 2.9173 & 2.8861 & 2.8530 & 2.8225 & 2.7891 \end{matrix}

TABLE I: Inverse effective masses \frac{1}{m_{T}} obtained with the effective mass tensor, for approximants of orders k going from 3 to 8

More precisely, we find that, for the k − th approximant : ln (\frac{1}{m_{T}} ) = c − 0.00624 ln(N_{k}), with c a constant. This is consistent with the nature of the ground state of the system, which we find to be extended algebraically as shown by the inverse participation ratio I : for an approximant of N_{k} sites, I ∝ _{\frac{1}{N_{k}^{β}}} with β = 0.987.

VI. CONCLUSION

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