What Do Bloch Electrons in a Magnetic Field Have to Do with Apollonian Packing of Circles?

Journal of Physics A: Mathematical and Theoretical

PAPER What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles?

To cite this article: Indubala I Satija 2021 J. Phys. A: Math. Theor. 54 025701

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This content was downloaded from IP address 129.6.222.101 on 16/12/2020 at 18:00 Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 54 (2021) 025701 (32pp) https://doi.org/10.1088/1751-8121/abc65c

What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles?

Indubala I Satija∗

Department of Physics, George Mason University, Fairfax, Virginia, United States of America

E-mail: [email protected]

Received 16 June 2020, revised 26 October 2020 Accepted for publication 30 October 2020 Published 16 December 2020

Abstract Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the ‘Hofstadter butterfly’. In this Apol- lonian–butterfly-connection, the integer curvatures of the circles contain ina convoluted form, the topological quantum numbers of the butterfly graph—the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. In this mapping, Farey tree hierarchy plays the central role, revealing how the geometry and the number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.

Keywords: Hofstadter butterfly, Apollonian gasket, quantum Hall effect

(Some figures may appear in colour only in the online journal)

1. Introduction

The ‘Hofstadter butterfly’ [1, 2] is a quantum fractal representing the energy spectrum of two- dimensional spinless electron gas in a square lattice, subjected to a transverse magnetic field. It is a physical model for topological states of matter known as the integer quantum Hall states [3]. The ‘butterfly graph’ is composed of a nestedet s of images where each sub-image, although somewhat distorted, is a replica of the original spectrum resembling a butterfly. Here we show that the number-theoretical aspect of this fractal is intimately related to an abstract fractal known as the Apollonian gasket [4, 5]. Named in honor of Apollonius of Perga who studied

∗Author to whom any correspondence should be addressed.

Figure 1. Hofstadter butterfly (left) and IAG (right). The gaps—the forbidden regions of the spectrum are labeled with integers that represent the quantum numbers of Hall conductivity. In the Apollonian packing, each circle is labeled by its curvature, an integer, which is the inverse of the radius.

geometrical problem of mutually tangent circles before 300BC, Apollonian gasket describes a nested set of configurations of four mutually tangent circles. The relationship between the butterfly graph and the IAG which we refer as the Apollonian–butterfly-connection or ABC, associates every butterfly in the butterflyraph g with an integral Apollonian gasket or IAG—an Apollonian packing where all circles have integer curvatures. Figure 1 shows the Hofstadter butterfly and the IAG—the two fractals that are labeled by integers. In the butterfly graph, these integers are topological quantum numbers of Hall conductivity [3]. In IAG, the integers represent the integer curvatures of the circles. ABC is a statement about a convoluted relationship between these two sets of integers. Since its discovery, the hierarchical nature of the butterfly spectrum has been subject of numerous studies [6–10]. Recently, it was suggested that the butterfly fractal is related to IAG [11–13]. This stimulated further investigation of butterfly spectrum [14]. The highlight of this study was the proof of the ‘Farey relation’—the heart of ABC, that was known only empirically before. The Farey relation, (6) relates the magnetic flux values at the center and the boundaries of the butterfly and provides perhaps the first inkling that number theory is intertwined with the quantum mechanics of Bloch electrons in a magnetic field. In this paper, the relation between the number-theoretical properties of the butterfly spectrum and the IAG is put on a solid foundation using the Apollonian group and the conformal transformation. The self-similar scaling properties of the fractals are described in terms of the eigenvalues of the four generators of the Apollonian group and the fixed points of the conformal transformations. The transformations describing these two fractals are shown to be related as they have same scaling properties that characterize their self-similar hierarchical properties. In section 2, we will begin with a brief review of the relevant features of the butterfly spectrum. Section 3 describes the Farey relation that relates the left-edge, the right-edge and the center of every sub-butterfly in the graph. Section 4 introduces the ‘butterfly quadruplets’—the four integers that uniquely specify the number theoretical address of every sub-butterfly in the

2 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija butterfly graph. Here we also discuss the division of the butterfly graph into the ‘C-cell’ (central part) and the ‘E-cell’ (edge part). Section 5 reviews the butterfly recursions as described in reference ([14]). In section 6, we revisit the butterfly recursions, expressing them as conformal transformations that map circles to circles. Section 7 discusses butterflies that share the same flux interval but are labeled with different topological integers, referred as the butterfly siblings. Section 8 describes IAG and introduces various equivalent forms of ‘Apollonian quadruplets’. In section 9, the Apollonian group and its four generators that describe the packing of circles as well as the corresponding conformal transformations are described. In section 10, we discuss the ABC. Section 11 illustrates ABC with various examples. In appendix A, we list identities that relate various integers characterizing the butterflies. Appendix B gives a brief review of the circle inversion that is the jewel of the Apollonian packing. Appendix C describes the super-Apollonian group, that is essential for complete correspondence between the butterfly and the Apollonian. In appendix D, we review the tree of the Pythagorean triplets that encodes the hierarchical aspects of the C-cell butterflies as described in our earlier study [13].

2. The Hofstadter butterfly

The model underlying the butterfly graph is thebutterfly ‘ Hamiltonian’ that describes a simple model of spineless electrons in a two-dimensional square lattice where electrons can hop only to their neighboring sites. When subjected to a magnetic field, the Hamiltonian is given by, e e H = cos a k − A + cos a k − A , x c x y c y where B = ∇×A is the magnetic field. Here a is the lattice constant and H is defined in units of the strength of the nearest-neighbor hopping parameter. With the choice of Landau gauge, ikym (Ax = 0, Ay = Bx), and the wave function Ψn,m = e ψn(ky), the problem effectively reduces to a one-dimensional system of equations. Known as the Harper’s equation, [15],

ψn+1 + ψn−1 + 2cos(2πnφ + ky)ψn = Eψn. (1)

Ba2 The parameter φ = /e is the magnetic flux per unit cell of the lattice, measured in the unit of flux quanta /e. The Hamiltonian can also be written as [8],

H = cos x + cos p,[x, p] = iφ. (2)

This implies that the magnetic flux φ emerges as an effective Planck constant and the butterfly graph lives in a space of energy E and the Planck’s constant. The corresponding continuum Hamiltonian H = p2 + x2 describes the spectrum given by the Landau levels [16]. For rational values of the flux φ = p/q, the allowed energies of the Hamiltonian consist of q bands, with the central pair touching when q is even. The butterfly graph describes all possible integer quantum Hall states of the non-interacting fermions in two dimension. Characterizing every gap of the butterfly graph is an integerhat t represents the quantum Hall conductance corresponding to the Fermi level in that gap. In 1983 David Thouless [3] along with his collaborators showed that the Hall conductivity CH can be written as, n i f e2 e2 C = {∂ ψ∗∂ ψ − ∂ ψ ∂ ψ∗} dk dk ≡ σ . (3) H 2π kx n ky n kx n ky n x y h h n=1 T

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Figure 2. Figure illustrates the central and the edge partitioning of the butterfly, labeled respectively with the red and the green dots at the centers of the sub-butterflies. This separation is due to the two major gaps that are highlighted with dotted lines.

Here nf represents the number of filled bands as the Fermi energy lies in the gaps ofthe spectrum. The quantity in the square bracket can assume only integral values, denoted as σ. This is the Chern number—a topological quantum number of Hall conductivity. The integers σ can also be obtained as solutions of Diophantine equations [17], namely

r = pσ + qτ, ρ = σφ + τ,(4)

r where r labels the rth gap of the spectrum and ρ = q represents the electron density. The physical significance of τ remains unclear. In addition to topological integers (σ, τ) that label every gap, we can also define the quantum numbersM ( , N) associated with a band at flux p φ = q , given by the following Diophantine equation, Mq + Np = 1. (5)

Here M = τ ↑ − τ ↓ and N = σ↑ − σ↓ where σ↑(σ↓)andτ ↑(τ ↓) are the topological quantum p numbers of the upper (lower) gaps that sandwich the band at φ = q .

2.1. C-cell and E-cell butterflies Asshowninfigure2, the major gaps of the butterfly divide the graph into ‘Central’ and ‘Edge’ parts which we will refer as the C-cell and the E-cell of the butterfly as shown in figure 2.The E-cells consist of the upper and the lower parts. Throughout this paper, the C and the E-cells will be marked in red and green respectively. This color coding may be a ‘dot’ at the centers of the butterflies, or a box, or the entire butterfly. We note that the C and the E-cells of the main butterfly are ‘special’ as they have additional symmetries. For example, every sub-butterfly in the C-cell of the main butterfly has its center at E = 0 and exhibit mirror symmetry and the entire chain of these butterflies reside symmet- rically (left and right chains) about the center of the butterfly graph. Although edge-butterflies do not exhibit mirror symmetries, the upper and the lower E-cells are mirror image of each

4 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija other. In our discussion below, the term central and edge butterflies will be referred to only the C and the E-cell butterflies of the main butterfly. Below we summarize two important characteristics of the C-cell and E-cell butterflies which have been proven for the special case of the central and the edge hierarchies. For the general case of C-cell and the E-cell butterflies, theseesults r are empirical, based on observations [13]: (1) The C-cell butterfly hierarchies with center at flux value φ = pc conserve parity which qc we define as even (odd) when qc is even (odd). That is, as we zoom into the equivalent butterflies, qc retains its even or odd-ness. We note that butterflies with center at E = 0 are special case of the C-cell butterflies that exhibit even parity and horizontal mirror symmetry atall scales. The E-cell butterfly hierarchies do not conserve parity. ≡ − (2) For C-cell butterfly hierarchies, Δσ (σ+ σ−) remains constant, equal to 0 for qc- even or 1 for qc-odd. In contrast, for the E-cell butterflies, Δσ grows exponentially. This leads to an asymptotic symmetry in the C cell sub-images, in sharp contrast to the E cell butterflies that remain asymmetrical. We note that, in addition to the C and the E-cell butterflies, there are also inter-cell butterflies, butterflies whose one wing is the C-cell and another in the E-cell. However, we will not discuss these images as they are already part of the C and E-cell structures.

3. Farey tree and the butterfly fractal

As we stare at the butterfly graph, we see various sub-images. These ‘sub-butterflies’ are distorted replicas of the original graph. Each butterfly with left and right flux boundaries at pL pR pc φL = , φR = and center at φc = has been shown [8, 14] to be the renormalization of qL qR qc 1 the original butterfly graph with the corresponding boundaries at 0 and 1 and center at 2 . Intriguingly, the magnetic flux values at the butterfly boundaries and center are related as, p p + p c = L R . (6) qc qL + qR

This relation (see figure 3)whereφc is the ‘Farey sum’ of φL and φR, was originally found empirically [11, 12]. It has been now proven [14] using the RG equations describing butterfly recursions. Consequently, the ‘butterfly triplets’ pL , pc , pR which are the flux values at the qL qc qR edges and at the centers of the sub-images are ‘neighbors’ in the Farey tree, known as the friendly numbers, satisfying the equations,

|qL pR − qR pL| = 1, |qL pc − qc pL| = 1, |qR pc − qc pR| = 1. (7)

Therefore, the butterfly flux interval |Δφ|—the horizontal size of the butterfly is given by, pR pL 1 Δφ = − = . (8) qR qL qLqR An interesting perspective on butterfly sizes follows from its relation to topology. For a band at φ = pL with topological numbers (M, N), the relation [14] qL

pR = M − pL, qR = qL − N,(9) implies the existence of a butterfly, with left and right boundaries at pL and pR respectively. qL qR In other words, a band at pL , with quantum numbers (M, N) forces the existence of another qL band at pR , and these two bands together form a butterfly. It shows how the butterfly structure qR

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Figure 3. Upper panel shows the Farey tree: the tree is built up row by row. Each successive row of the tree inherits all the Farey fractions from the level above it, and some new fractions (all of which lie between 0 and 1) are added by combining certain neighbors in the preceding row using an operation called ‘Farey addition. Some examples of friendly triplets are marked in rectangular boxes. The bottom part shows the corresponding butterflies, all color coded where each Farey triplet forms the flux boundaries and center of a butterfly.

of the energy spectrum of Bloch electrons is rooted in topology. This also provides a novel way to determine topological quantum numbers of the various bands in the butterfly graph, as for every butterfly, equation9 ( ) determines the quantum numbers of the bands that form the boundaries of the butterfly.

4. Butterfly quadruplets: labeling a butterfly

We now show that every butterfly in the butterfly graph is uniquely characterized byfour integers, christened as the ‘butterfly quadruplets’ which are denoted as QB. To unveil the number-theoretical designation of butterflies, we represent the two major gaps of a butterfly by two intersecting lines (see figure 2): ρ = σ+φ − τ + and ρ = −σ−φ + τ − in (ρ − φ)plane.Here(σ+, σ−) are the magnitude of the two Chern numbers that label the two main (diagonal) gaps of a butterfly andτ ( +, τ −) are the other quantum numbers that appear in the Diophantine equation (4). The first important point to note is that the φ-coordinate of the point of intersection of these lines is τ++τ− = pc [11]. In other words, center of a butterfly corresponds to the point of σ++σ− qc intersection of the two lines with slopes-intercept given by (σ+, τ +)and(σ−, τ −) respectively.

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Figure 4. (A) Butterfly skeleton, a trapezoidal region where the energy gaps of the energy spectrum are simplified down to linear trajectories parametrized as ρ = σφ + τ. In the actual butterfly diagram, the linear trajectories become discontinuous. Panel (B) illustrates the relationship between a butterfly and its skeleton with trapezoids enclosing an edge butterfly (green) and a central butterfly (red).

Secondly, once we know the φ value at the center of the butterfly, Farey relation (7) determines the flux values at the left and the right boundaries as they are the Farey neighbors of pc . qc pL Two intersecting lines ρ = σ+φ − τ + and ρ = −σ−φ + τ − and the two parallel lines φ = qL and φ = pR in (ρ–φ) plane, determine a trapezoidal region. This trapezoid is referred as the qR butterfly skeleton. It is shown in figure 4, where the four integers (σ+, σ−, τ +, τ −) uniquely specify a butterfly. The two-dimensional (ρ–φ) plane, where every butterfly is represented by the corresponding butterfly skeleton, is the well known Wannier diagram18 [ ]. We now define the butterfly quadruplet as:

QB = {σ+, σ−, τ+ + 1, τ− − 1} (10)

The topological quantum numbers (σ±, τ ±, M, N) are related to the integers (pL, qL, pR, qR) as,

p p + p τ + τ− p− p − p M + ≡ L R = + , ≡ L R = − . (11) q+ qL + qR σ+ + σ− q− qL − qR N

Therefore, unlike the quantum numbers (M, N) of the bands forming the boundaries of a butterfly, the quantum numbersσ ( ±, τ ±) that label the gaps of the butterfly are not determined by (p , q , p , q ). We emphasize that in general, a butterfly triplet pL , pc , pR does not label L L R R qL qc qR a butterfly uniquely as the butterfly graph consists of numerous examples where multiple butterflies share the same flux interval but differ in(σ, τ) quantum numbers. This is shown in figure 8, to be discussed later. In view of the butterfly identities listed in appendix A,forthecentral and the edge butterflies, butterfly quadruplets can be written down explicitly.

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For central-butterflies:

qc pc − 1 pc + 1 σ = σ− = , τ = , τ− = , + 2 + 2 2 q q p + 1 p − 1 Q = c , c , c , c . (12) B 2 2 2 2 For edge-butterflies:

σ+ = qR, σ− = qL, τ+ = pR − 1, τ− = pL + 1

QB = {qR, qL, pR, pL}. (13)

5. Chains and nests of butterflies

The butterfly graph can be viewed as a ‘packing’ of butterflies arranged in chains, nested ad infinitum as described in our earlier studies [14]. A chain of sub-images are given by,

p0 ± jM0 φ j = , (14) q0 ∓ jN0 where the pair of quantum numbers (M0, N0) characterize the entire chain. Sub-images in the chain are nested and these nesting relationships can be repeated recursively. This is used to characterize the extent to which the pattern is self-similar as illustrated in figures 5–7). The recursive structure of the butterfly fractal, dissected into parts where each sub-image is a microcosm of the entire butterfly graph has been described as renormalization group trajectories of the butterfly integers (pL, qL, pR, qR), connecting two successive levels of the hierarchy [14]. As described in [14], we associate two matrices Bx(l)(x = L, R), with level l of a recursion, q (l) p (l) B (l) = x x . (15) x −N(l) M(l)

The recursion rules for a butterfly hierarchies can be written as:

Bx(l + 1) = RxBx(l), (16)

where the transformation matrices RL, RR are given by, ∗ ∗ ∗ ∗ − ∗ qL pL qR pR qR R = ∗ ∗ , R = ∗ ∗ ∗ . (17) L −N M R −N M + N ,

∗ ∗ ∗ ∗ The starred integers (M , N , pL, qL) are fixed by the sub-butterfly whose nesting structure is being studied as illustrated in the figures 5–7. We refer readers to the original paper [14]for various details of the recursions. In the special case, where the starting or the root configuration is the main butterfly, the renormalization equations can be expressed in terms of the renormalization of a single variable, p the magnetic flux φ = q ,givenby

∗ ∗ M φ(l) + pL ≡ φ(l + 1) = − ∗ ∗ F(φ(l)). (18) N φ(l) + qL

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Figure 5. Four levels, L = 1, 2, 3, 44 of central-butterfly recursions that results in self- similar images. Here the root is the main butterfly: start with the main butterfly and select a sub-butterfly (black box) and zoom into the sub-butterfly maintaining the relation between two successive levels throughout the nesting. Major gaps are labeled with Chern numbers. The conformal map f(z) (to be discussed later) that describes the recursive behavior of this hierarchy is also shown.

Figures 5 and 6 show butterfly nestings, described by equation (18), which we will refer as simplified butterfly recursions. An example of the general case, described by equation (16), referred as the general butterfly recursions, is shown in figure 7.

6. Butterfly recursions as conformal transformations

We now show that the renormalization trajectories, as described in section 5 can be expressed as conformal transformations [19]. This aspect of the recursions will be important in establishing the ABC. az+b A rational functions of the form w = f (z) = cz+d represent Möbius transformations, named in honor of 18th century German mathematician August Ferdinand Möbius . Also known as linear fractional transformations, they are conformal transformations z → f(z) that map circles to circles. Here the constants (a, b, c, d) can be determined in terms of two sets of triplets:

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Figure 6. Same as figure 5 for an edge-butterfly.

(z1, z2, z3)and(w1, w2, w3): ⎛ ⎞ ⎛ ⎞ z1w1 w1 1 z1w1 z1 w1 ⎝ ⎠ ⎝ ⎠ a = det z2w2 w2 1 , b = det z2w2 z2 w2 z3w3 w3 1 z3w3 z3 w3 ⎛ ⎞ ⎛ ⎞ (19) z1 w1 1 z1w1 z1 1 ⎝ ⎠ ⎝ ⎠ c = det z2 w2 1 , d = det z2w2 z2 1 . z3 w3 1 z3w3 z3 1

We first observe that the equation (18) is a conformal map with z = φ(l)andw = φl+1 ∗ ∗ − ∗ ∗ with the constants (a, b, c, d) = (M , pL, N , qL). The general case for butterfly recursions expressed in terms of two 2 × 2 matrices Bx described by equation (16) can also be rewritten as a conformal transformation by introducing complex variables hx(l)definedas,

px(l) + iqx(l) hx(l) = , (20) Ml − iNl

Using equations (16)and(17), it is easy to see that the hL(l) recursion is a conformal mapping; namely

∗ ∗ qLhL(l) + pL hL(l + 1) = ∗ − ∗ ∗ − ∗ ; (21) (qR qL)hL(l) + (pR pL)

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Figure 7. Upper panel shows three level of nesting that results in self-similar images in a C-cell of a butterfly that lacks any mirror symmetry. This is unlike figures 5 and 6 where root is the main butterfly. Lower panel shows the main butterfly and a sub-image that has same relationship with each other the nested sequences of butterflies in the upper panel. It is the lower panel that determines the constants of the transformation.

and hR and hc satisfy the following equations,

hR(l) = hL(l) + 1, hR(l) + hL(l) = hc(l). (22) − − Alternatively, by substituting M = pR pL and N = qL qR (see appendix A)in equation (16), we obtain a different form of recursions where the integers p and q decouple: ∗ − ∗ ∗ pL(l + 1) qL pL pL pL(l) = ∗ − ∗ ∗ , (23) pR(l + 1) qR pR pR pR(l) ∗ − ∗ ∗ qL(l + 1) qL pL pL qL(l) = ∗ − ∗ ∗ . (24) qR(+1) qR pR pR qR(l) These equations can be written as conformal transformations in terms of two new variables e and f:

pL(l) qL(l) el = , fl = , (25) pR(l) qR(l)

where el and fl satisfy the following recursions: ∗ − ∗ ∗ ∗ − ∗ ∗ (qL pL) el + pL (qL pL) fl + pL el+1 = ∗ − ∗ ∗ , fl+1 = ∗ − ∗ ∗ . (26) (qR pR) el + pR (qR pR)el + pR

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Finally, we observe that one can express the above two recursions more elegantly in terms of a complex variable u: ∗ − ∗ ∗ pL(l) + iqL(l) (qL pL)ul + pL u(l) = , ul+1 = ∗ − ∗ ∗ . (27) pR(l) + iqR(l) (qR pR)ul + pR We note that various forms of butterfly recursions as described above form a subset of az+b − general class of mappings f (z) = cz+d where (a, b, c, z) are integers and (ad bc) = 1. Such transformations form a group known as the ‘modular group SL(2, Z)[20].

6.1. Geometric origin of butterfly recursions Butterfly recursions expressed as conformal maps suggest that these recursions may have purely geometric origin. We now show that equation (18) can be derived by mapping two sets of butterfly triplets: namely [φL(l), φc(l), φR(l)] and [φL(l + 1), φc(l + 1), φR(l + 1)], connecting two successive levels of butterfly nesting, once the Farey relation equation (7)is taken into account. For self-similar hierarchies that start with the main butterfly, one chooses 1 (z1, z2, z3) = 0, 2 ,1 and writes (w1, w2, w3) = (φL, φC, φR). Using equation (19) along with − the Farey relation, equation (7), it is found that the constants (a, b, c, d)as:a = pR pL, b = − ∗ − − ∗ − ∗ ∗ pL, c = qR qL, d = qL.ThisgivesM = pR pL, N = qR qL and pL = pL and qL = qL. In agreement with (18). If the starting or the root is not the main butterfly, but related to the main butterfly by a conformal map, say g, the butterfly recursion (18) is modified as gFg−1 [21]. It is interesting that butterfly recursions derived by applying semi-classical methods to Harper’s equation has such a simple geometric interpretation. It shows that number theory is intricately embedded in the quantum mechanics of the problem.

6.2. Self-similarity Scaling factors characterizing self-similar properties of the butterfly graph, determined by the eigenvalues of the transformation matrices RL or RR (equation (17)), have been discussed in our earlier studies [14]. Here we note that they are also the eigenvalues of the 2 × 2matrixthat can be associated with the conformal map such as equation (16) or equations (20)–(27). The −1 eigenvalues denoted as (ζ, ζ ) describe the asymptotic scaling of integers (px, qx, M, N)and the butterfly flux interval Δφ:

J(l + 1) Δφl+1 1 lim = ζ, lim = 2 . (28) l→∞ J(l) l→∞ Δφl ζ

Here J represents the integers (px, qx, M, N). The scaling factor ζ are set of quadratic irrationals [14]givenby, (q∗ + M∗) q∗ + M∗ 2 ζ = L ± L − 1. (29) 2 2

Expressed as a continued fraction expansion, these quadratic irrationals are given by, ∗ ∗ ∗ ∗ ∗ − ζ = [n + 1; 1, n ], n = qL + M 2. (30) Therefore, every self-similar butterfly hierarchy in the butterfly graph is characterized by a single integer n∗. It is rather remarkable that nature chooses this special class of quadratic irrationals to describe self-similar characteristics of the butterfly graph. For example,

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Figure 8. Left panel shows four families of sibling-butterflies (I–IV), stacked vertically, each family shown within two dotted vertical lines. The central (red box) and non- central (green box) butterflies in each family share the same flux interval. The two integer numbers near sub-images show the two Chern numbers (σ+, −σ−) of these butterflies, distinguishing siblings from each other. Right panel shows two levels of blowups for edge (green)—center(red) butterfly siblings, corresponding to Chern numbers (7, −5) and (6, −6) respectively, shown in II in the left panel.

quadratic irrationals such as silver mean, where all entries in the continued fraction are 2 are excluded. It turns out that the C-cell butterflies are characterized by even n∗ while for the non-central hierarchies, n∗ can be even or odd. Thus, the integer n∗ that uniquely determines the scaling of the butterflies, also encodes the distinction between the C and the E-cell butterflies.

7. Butterfly-siblings

Figure 8 shows various examples where a given flux interval hosts multiple butterflies, stacked vertically in the butterfly graph. We will refer such a group of butterflies as the butterfly siblings. Characterized by different QB, members of the siblings differ in (σ±, τ ±) quantum numbers that label the two diagonal gaps of the butterflies, keeping the sum σ+ + σ− and τ + + τ − constant. The close inspection shows a very orderly arrangement of the Cherns among the siblings. Right panel in the figure shows an example of two sibling butterfly hierarchies, one residing at the center and the other at the edge of the spectrum. Although characterized by same scaling factor, they can be distinguished by the quantum numbers (σ±, τ ±). Empirical results show that Δσ = |σ+ − σ−| remains constant for C-cell butterfly hierarchies and grows exponentially with scaling factor ζ for the E-cell butterfly. Butterfly recursions described above in sections 5 and 6 involve only the integers (pL, qL, pR, qR). In the absence of σ, τ recursions, these renormalization equations do not dif- ferentiate different members of the siblings. For the general case, renormalization equations for (σ, τ)andΔσ remains an open problem.

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8. Integral Apollonian gaskets (IAG)

The story of the IAG, named after 200 BC old studies by Apollonian of Perga, begins with four mutually tangent circles. The four curvatures of these circles (κ1, κ2, κ3, κ4) satisfy the following equation, known as the Descartes theorem, found by Ren´e Descartes in 1643 [21, 22], 2 2 2 2 2 2 κ1 + κ2 + κ3 + κ4 = (κ1 + κ2 + κ3 + κ4) . (31) The quadratic relation relating four curvatures implies that given any three mutually tangent circles of curvatures (κ1, κ2, κ3), there are exactly two possible circles (see figure 9)thatare tangent to these three circles: √ κ± = (κ1 + κ2 + κ3) ± 2 κ1κ2 + κ2κ3 + κ3κ2. (32)

Denoting these two solutions as κ4, κ4,wehave

κ4 + κ4 = 2(κ1 + κ2 + κ3). (33)

Therefore, starting with three mutually tangent circles, we can construct two distinct quadruplets (κ1, κ2, κ3, κ4)and(κ1, κ2, κ3, κ4). We note that the curvature of the outer circle has to be taken negative to satisfy the equations (31)–(33). A remarkable feature of the linear equation (33) is that if the original four circles have integer curvature, all of the circles in the packing will have integral curvatures.

8.1. Apollonian quadruplets

In an Apollonian packing, it is natural to consider the quadruplets (κ1, κ2, κ3, κ4) rather than the curvatures of individual circles since every circle in the packing is a member of the quadruplet. In other words, the Apollonian packing is packing of Descartes configurations, each characterized by four integers that satisfy equation (31) and will be referred as the Descartes quadruplets, QD: QD = κ1, κ2, κ3, κ4 . (34)

The Descartes quadruplets QD are related to another the well known quadruplets, the Lorentz quadruplets QL: QL = Nx, Ny, Nz, Nt , (35)

where,

2 2 2 2 Nx + Ny + Nz = Nt . (36)

Simple algebraic manipulation of the Descartes and the Lorentz quadratic forms relates QD and QL. This relationship can be expressed as a matrix equation, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Nx 1 −1 −1 −1 κ1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜Ny⎟ ⎜0002⎟ ⎜κ2⎟ ⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ . (37) Nz 01−10 κ3 Nt 1121 κ4

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Figure 9. Given three mutually tangent circles of curvatures (κ1, κ2, κ3)(shownin black), there are two possible circles that can form a Descartes’s configuration of four mutually tangent circles: the outer circle of curvature κ4 (red) and the inner circle of cur-

vature κ4 (dotted red). The inner and the outer circles are mirror images of each other through a circle (blue) that passes through the tangency points of the three circles of curvatures (κ1, κ2, κ3). Figure on the right shows the special case where one of the circle is a straight line (κ1 = 0).

The Lorentz quadruplets are generalization of Pythagorean triplets (see appendix D)and can be constructed from two Gaussian integers (z1, z2),

z1 = m1 + in1, z2 = m2 + in2. (38)

We will refer to (m1, m2, n1, n2)asthegeneralized Euclid parameters, a generalization of the Euclid parameters (m1, m2) that determine the Pythagorean triplets as described in appendix D. The equations relating the Lorentz quadruplets to (z1, z2)are,

Therefore, with every Apollonian we can also associate a quadruplet QA, christened as the ‘Apollonian quadruplet’,

QA = {m1, m2, n1, n2}. (43)

As described later, in its simplest form, Apollonian–butterfly-connection or ABC is essen- tially a relation between the butterfly quadruplets QB and the Apollonian quadruplets QA

9. Apollonian packing

Starting with a root configuration such as the one shown in figures 1 or 10, the Apollonian packing is systematically filling in all the empty curvilinear triangular interstices to construct an Apollonian gasket [23]. As described below, there are three equivalent ways to obtain this packing: the circle inversion, the Apollonian group and the conformal mapping.

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Figure 10. Given a Descartes’s configuration of four mutually tangent circles such as (A) and (C), we can form four new configurations by reflecting through a circular mirror (dotted blue line) passing through the tangency points of three circles that are shared by

the two configurations as shown in (B) and (D). Here κ4 and κ (red circles) are mirror 4 images reflected through the blue circles. In (D), κ4 and κ4 are respectively the outer and the inner bounding circles for a configuration of three mutually tangent circles of curvatures (κ1, κ2, κ3).

9.1. Circle inversion Figure 10 illustrates a geometrical method to pack the Descartes configurations. This simple framework is based on the ‘circle inversion’ and is briefly described in appendix B.Thekeyto this geometrical construction is equation (32), which implies that given a Descartes quadruplet

QD = (κ1, κ2, κ3, κ4), one can obtain another quadruplet QD = (κ1, κ2, κ3, κ4). The two curvatures κ4 and κ4 are mirror images about a circular mirror that passes through the tangency points of (κ1, κ2, κ3). Repeating this process where a new Descartes configuration is obtained by circle inversion through any one of the circles provides a simple recipe for the Apollonian packing.

9.2. The Apollonian group From the elegant simplicity of plane geometry, we now describe a group theoretical framework to obtain the Apollonian packing of circles. The recursive filling of the space with Descartes configurations can be studied using Apollonian group where adding additional circles is accomplished by applying four matrices Si,(i = 1–4) to an a root configuration of a 2 Descartes quadruplet. These four matrices are four generators of the group where Si = 1. This relationship between the circle packing and the Apollonian group follows from equation (33) where the generators of the group are, ⎛ ⎞ ⎛ ⎞ −1222 1000 ⎜ 0100⎟ ⎜2 −122⎟ S = ⎜ ⎟ , S = ⎜ ⎟ , 1 ⎝ 0010⎠ 2 ⎝0010⎠ 0001 0001

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⎛ ⎞ ⎛ ⎞ 1000 100 0 ⎜0100⎟ ⎜010 0⎟ S = ⎜ ⎟ , S = ⎜ ⎟ . 3 ⎝22−12⎠ 4 ⎝001 0⎠ 0001 222−1

That is, we view Descartes quadruplets as column vectors v, and the Apollonian group acts by matrix multiplication, sending v to Sv. Note that only one member of the quadruplet is replaced by this act. The circle that replaces one of the circles is the mirror image of the replaced circle—mirror that passes through the tangency points of three circles that are shared by two Descartes configurations. This entire set of QD can be constructed by starting with a root quadruplet and applying four matrices Si ad infinitum. To have an ordered set of numbers in the quadruplets where the four curvatures appear in monotonically decreasing order, the four matrices Si are replaced by corresponding matrices (which we will represent as Di) so that an ordered set of quadruplets in v transforming to Dv produces an ordered set of quadruplets. We represent this relationship between Si and Di as:

Di = OˆSi. (44)

Figure 11 shows three examples of nested sets of Apollonian packings. Self-similar configurations are characterized by periodic string of the generators of the Apollonian group, denoted as P in figure. Eigenvalues of P determine the scaling of the curvatures. It should be noted that the eigenvalues of Di or various products of Di are in general complex and their determinant is either 1 or −1. However, the string of operators that describe self-similar configurations always m ± m 2 − have real eigenvalues (1, 1, λ+, λ−), m = 1, 2, 3 ...where λ± = 2 ( 2 ) 1 determines the asymptotic scaling of the curvatures. We note that any two distinct Descartes configurations can also be related by a conformal map, determined by the three tangency points of the Apollonian. These conformal maps are intimately related to the conformal maps that describe butterfly recursions as described later.

9.3. Ford circles–Ford Apollonian In an Apollonian gasket such as the one shown in figure 1, there are circles that are all tangent to the x-axis. These circles, known as Ford circles provide pictorial representation of fractions, as discovered by mathematician Lester Ford [25]. In 1938, Ford showed that every primitive p p q x y fraction q (where and are relatively prime) can be represented by a circle in the – plane p 1 with center at q , 2q2 . The key characteristic of the Ford circles is the fact that two Ford circles representing two distinct fractions never intersect and are tangent only if the two fractions are Farey neighbors. (See figure 12) Consequently, with every friendly triplet [ pL , pc , pR ], one can qL qc qR associate three mutually tangent Ford circles, that will be nicknamed as the Ford Apollonian.In this paper, we will scale all curvatures by a factor of two and hence the Ford circles represent- p 2 ing a fraction q will be represented by a circle of curvature q . Therefore, Ford Apollonian is 2 2 2 characterized by Descartes quadruplet QD = (qc, qR, qL,0)andQA = (qR, qL,0,0).FordApol- lonians are subset of Descartes configurations of figure 1 where all circles are tangent to the x-axis. They describe a special case of Descartes theorem where one of the curvatures, say κ1 is zero. From equation (32), we obtain, √ √ √ √ √ | − | κ4 = κ2 + κ3, κ4 = κ2 κ3 . (45)

17 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija

Figure 11. Examples of three chains of packing (circles are not drawn to scale), each characterized by its own string of generators and conformal transformation shown explicitly. The blue box shows Apollonians where scaling factor is unity. At the√ bottom is shown (red circles) a hierarchy characterized by the scaling ratio (2 + 3)2 = [3; 1, 2] and at the top (green circles) are nested set of Apollonians with scaling ratio √ (( 3+ 5 )2 = [2; 1]). The sequence of dotted boxes correspond to the configurations that 2 √ − 3± 5 lead to self-similar hierarchies. We note that eigenvalues of D3 are (1, 1, ( 2 )). In − 2 view of its determinant is equal to 1, it is D3 and not D3 that constitutes a self-similar hierarchy.

Figure 12. Left panel shows two sets of three mutually tangent Ford circles, represent- − PL PL+PR PR PL PL PR PR ing two sets of fractions , , and , − , . Right panel shows QL QL+QR QR QL QL QR QR p Apollonian gasket made up of Ford circles. Here each circle represents a fraction q of curvature q2 (scaled by a factor of 2). This packing of Ford circles is will be referred as Ford–Apollonian packing.

It turns out that for this case, the Descartes’s theorem is a property of homogeneous func- ± 2 2 2 2 tions of three variables. That is, given q2 q3 = q4, it can be shown that (q2 + q3 + q4) = 4 4 4 ABC 2(q2 + q3 + q4)[23]. As described below, these configurations play an important role in .

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Figure 13. (I) shows the Pappus chain (the blue circles) that divides the Apollonian packing into the central (red circles) and the edge packing (green circles). Pappus chain 1 ∞ of circles are Ford circles representing fractions n , n = 3, 4, 5 .... . Panel (II) Illus- trates the relationship between the central and the edge packing as the mirror images of each other related by inversion through the circle (dotted blue shows only section of this circle) that passes through the tangency points of Pappus chain of circles. The green (edge) and the red (center) boxes show two Descartes configuration that are mirror images: (25, 4, 9, 0) = M(28, 9, 4, 1). (III) shows any Apollonian can be divided by a Pappus chain into the central and the edge parts.

In figure 12 the two red circles correspond to (κ4, κ4)ofequation(45). We note that these circles seem to be related to topological features of the butterfly graph as, PL+PR = τ++τ− QL+QR σ++σ− − and PL PR = − M (equation (11)). QL−QR N

9.4. Butterfly recursions describe recursive structure of the Ford Apollonians

As described earlier, magnetic flux values at the left-right boundaries and the center of every butterfly form a friendly triplet, that is, the three fractions pL , pc , pR are neighbors in the qL qc qR Farey tree. Ford circle representing of these fractions are mutually tangent, forming an Apollo- nian—the Ford Apollonian. Therefore, the recursions of a geometrical fractal—an Apollonian gasket made up of Ford circles are encoded in the renormalization trajectories describing recursions of the butterfly fractal described by equations16 ( )–(17) or equivalent forms such as equations (21)–(27).

9.5. Center and edge of the Apollonian packing Figure 13 shows the central and the edge packing of circles (shown with red and green circles respectively) in an Apollonian gasket, in close analogy with the central and the edge packing of the butterflies discussed earlier. In addition, there are circles that are tangent to both the x-axis

19 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija and the upper boundary circles, shown in blue in figure. This ‘hybrid’ chain of circles, sand- wiched between the central and the edge packing is an example of ‘Pappus chain’ investigated by Pappus of Alexandria in the 3rd century AD [24]. We note that two consecutive members of the Pappus chain are tangent to each other and these tangency points lie on a circle as shown figure 13. The key feature of Pappus chain that is relevant for our studies is the fact that this chain creates two disconnected chains of inscribed circles. The upper and the lower chains will be referred as the ‘Edge-chain’ and the ‘Central-chain’ respectively. All the circles in the central-chain are tangent to the x-axis and all | − i | circles in the edge-chain are tangent to the two upper boundary circle z 2 = 1ofcurvature κ = 1. We now show that the central and edge packing of circles are related by a conformal transformation. | − i | The transformation that maps x-axis to the boundary circle z 2 = 1isgivenby,

z z → f (z) = . (46) −iz + 1

This conformal transformation, which we denote as Tce also maps the entire chain of cir- | − i | cles that are tangent to x-axis to chain of circles that are tangent to the circle z 2 = 1. Representing this transformation, by 2 × 2 matrix, we have 10 T = , (47) ce −i1 that connects the edge and the central chains. Consequently, knowing the recursions underlying packing of Ford circles, from butterfly recursions, determine the recursive structure of packing of circles that are tangent to the upper boundary circle with κ = 1. That is, given the transfor- ab mation (equation (16)) Tˆ = , describing the recursive hierarchy of Ford circles, the C cd corresponding edge packing TˆE is given by, a + ibb − Tˆ = ≡ Tˆ Tˆ T 1. (48) E c + b − i(a − d) d − ib ce C ce

The two transformations TˆC and TˆE have same trace and determinant (equal to unity) and hence same eigenvalues, but differ in eigenvectors (see below). We also note that the central and the edge-packing are mirror images of each other about a circular mirror that passes through tangency points of two consecutive members of circles in Pappus chain as shown in figure 13. It should be noted that Pappus chain remains invariant under the transformation (47) and this invariance also follows from the transformation of the p fraction q and the corresponding curvature of the Ford circles transform as,

2 p pq p 2 2 2 → + i , κ p = q → κ¯ p = (q + p − 1). (49) q p2 + q2 p2 + q2 q q

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p 1 | − i | 1 For the magnetic flux values q > 2 , the corresponding boundary circle is z (1 + 4 ) = 4 and their curvature is obtained from κ¯ p by p → (q − p). For our discussion below, we will q 1 mostly use the flux values φ 2 .

10. Apollonian–Butterfly-connection (ABC)

Butterfly fractal is a packing of butterflies where each sub-image is a microcosm of the entire butterfly graph. In this two-dimensionalE ( − φ) landscape, sub-butterflies are uniquely labeled IAG by the quadruplets QB, given by equation (10). The is a packing of four mutually tangent circles—the Descartes configuration. Eachonfiguration c is uniquely characterized by a set of four integers QA (equation (43)). We now outline a general framework to show that the butterfly fractal is intimatelyated rel to the geometrical fractal IAG. This involves pairing every butterfly to an Apollonian and this mapping is one to one. The quantum aspect of the butterfly graph is reflected in the fact that not all Descartes configurations describe a sub-butterfly in the energy spectrum. However, given a sub-butterfly, one can always find the corresponding Descartes configuration. The two sets of conformal maps that describe these two fractals are found to be related and the nesting properties of both fractals are described by the same scaling exponents. The story of the ABC begins with geometrical representation of fractions in terms of Ford circles [11]. In view of the fact that the triplet pL , pc , pR , in general describes a family of qL qc qR butterflies, each Ford Apollonian represents a family of butterfly siblings. One to one mapping between a butterfly and an Apollonian is based on the following two ansatz: (I) Ford Apollonians represent only the central butterflies. (II) For off-central butterflies, the corresponding Apollonian are obtained by appropriate conformal transformation on the Ford Apollonians. In general, conformal maps that lift Ford Apollonian above the x-axis can be constructed and the transformed Apollonian can be related to the off-centered butterflies. For example, the edge-butterflies will be represented by chain of circles that are tangent to the upper boundary circles, related to the Ford Apollonian by the equation (48). We note the in ABC, different members of butterfly siblings are related by conformal maps, thus distinguishing siblings from each other. Figures 14–18 illustrate ABC for various cases. For the special case of central and edge butterflies, the explicit formulae for the butterfly and the corresponding Apollonian quadruplets are known: For central-butterflies: q q q + 1 q − 1 Q = c , c , L , L , Q = {q , q ,0,0}. (50) B 2 2 2 2 A R L

For edge-butterflies:

QB = QA = {qR, qL, pR, pL}. (51)

We note that (see figure 14), not every Ford Apollonian represents a butterfly as butterfly fractal is a quantum fractal that imposes certain symmetry constraints on the butterflies. For →− example, E E symmetry of the butterfly graph requires qc to be an even integer for butterflies whose centers lie on the x-axis. This is reminiscent of phenomena of missing reflections of scattering of de Broglie waves in crystalline lattices.

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Figure 14. ABC illustrated—figure shows nesting for a central Descartes configurations (upper panel) where (A1 → A2 → A3) lead to asymptotically self-similar hierarchy. Lower panel shows the corresponding butterflies: (B1 → B2 → B3). Both are described√ z+1 by the conformal transformation f (z) = 2z+3 and scaling exponent ζ = 2 + 3 = [3 : 1, 2]. We note that only those Ford Apollonians where the middle circle has even curvature represent a butterfly.

Figure 15. ABC illustrated for the edge butterflies, analog of figure 14 that describes central butterflies. The butterfly and the√ corresponding Apollonian hierarchies are char- 3+ 5 acterized by the scaling exponent ζ = 2 = [2; 1]. We note that unlike the self-similar butterfly hierarchy that begins with the main butterfly, the self-similar packing of circles P 2 with = D3 begins with A1.

11. Illustrating ABC:examples

11.1. Central and edge hierarchies In the following we will first illustrate ABC using various examples from central and edge hierarchies. The quadruplets QB and QA are given by equations (50)and(51). (1) Representing main butterfly with four mutually tangent circles as shown in figure 14 is the simplest example of ABC.

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Figure 16. Left panel shows examples of butterflies (red boxes) that reside in the left and the right C-cell of an edge butterfly. The flux coordinates and the quadruplets QB (numbers inside the red parenthesis) of the butterflies are marked. The right panel shows the corresponding Apollonian and the quadruplets QD.

(2) A simple example of ABC are the central butterflies characterized by the friendly 0 1 1 triplets 1 , 2n , 2n−1 .Heren = 1, 2, 3 .... The corresponding Apollonian are formed by Pappus z chain. The butterfly and the corresponding Apollonian recursions, given by f (z) = −(2n−2)z+1 determines the asymptotic scaling factor ζ = 1. n 2n+1 n+1 (3) We now consider a chain of central-butterflies : 2n+1 , 4n+4 , 2n+3 . Self-similar hierarchy for each member of the chain and the corresponding Ford Apollonian are given by, z + n f (z) = . (52) 2z + (2n + 1) For n = 1, figure 14 illustrates ABC for this case. (4) We next consider a chain of edgebutterflies and the corresponding Apollonian, described 1 2 1 by the friendly triplet 2n+1 , 2n+3 , n+1 , n = 1, 2, 3 ..., The conformal map that describes the edge-nesting is obtained using equations (16)and(48)isgivenby,

iz + 1 f (z) = . (53) {−n + 1 + (2n + 1)i}z + (2n + 1 − i)

Figure 15 illustrates the n = 1 case. Here the recursions can also be explicitly constructed 1 2 1 with three tangency points (z1,z2, z3) = 3 , 5 , 2 of Ford Apollonian and the corresponding 3+i 10+4i 2+i (w1, w2, w3) = 10 , 29 , 5 obtained from equation (49), equation (19) determines the iz+1 conformal map as f (z) = 3iz+(3−i) . (5) As a final case, we consider the case of central-edge sibling butterflies shown in figure 8. For central butterfly, it corresponds to n = 2 of the example (3). The recursive rule for this hierarchy is given by equation (52) with n = 2. To obtain recursions for the edge, we can use the equation (48) that links center and edge packing by a conformal map. Denoting the central

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Figure 17. Upper and lower edge butterflies (green boxes), with its Chern numbers (σ+, −σ−) and the corresponding Apollonians where curvatures of the circles are marked. In the top panel, upper and lower E-cell butterflies are mirror image of each other. Bottom panel shows a generic case without mirror symmetry. In both cases, the Apollonian representing the lower edge butterfly is the mirror image—an inversion, about a circle that is the base or the lower boundary curve for the C-cell butterflies.

and edge recursion maps as fC and fE respectively, we have, z + 2 (1 + 2i)z + 2 f (z) = , f (z) = . (54) C 2z + 5 E (4 + 4i)z + (5 − 2i)

We note that the two conformal maps have same eigenvalues but differ in their eigenvector, which we denote as eC and eE respectively, ⎛ π ⎞ ⎛ ⎞ i π π cos e 8 cos ⎝ 8 ⎠ ⎝ 8 ⎠ eC = π , eE = π . (55) sin sin 8 8 Can this difference encode and shed light on the different Chern numbers of the central and edge butterfly siblings remain an open question.

11.2. C-cell and E-cell butterflies without mirror symmetries Figure 16 shows a C-cell butterflies and the corresponding Apollonian in the general case without any symmetry.

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Figure 18. On the left, upper panel marks two sub-butterflies E1 and E2 that share the same flux interval. The lower left label shows their blowup along with the Chern numbers and hence are two distinct butterflies represented by two different butterfly quadruplets. Lower right panel shows the corresponding Descartes configuration.

Although, explicit formulae to describe ABC for the general case remains unknown, the task of relating any butterfly to an Apollonian is quite straightforward. In other words, given any sub-butterfly, the corresponding Apollonian can be obtained. Below we give examples that describe this correspondence The Pappus chain divides any Apollonian into C and E cells. Using elementary but long calculations, we can determine QB, QD, QL and QA for these cases. For example, for the butterfly in the left C-cell, QB = (10, 11, 4, 4) and the corresponding QA = (15, 9, 1, 0), QD = (568, 217, 72, 9) and QL = (270, 18, 145, 307).

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Finally, we note that the hierarchical nesting properties of C-cell butterflies are also encoded in the Pythagorean tree and has been discussed in our earlier studies [13]. This includes both the even parity cases where the butterflies have their center at E = 0 and the odd parity butterflies whose centers do not reside on the E = 0 line. Appendix D provides a brief summary of this correspondence.

11.3. ABC and super-Apollonian group We note that butterfly graph consists of the upper and the lower E-cell butterflies that arein general not related by horizontal mirror symmetry as shown in figure 17. To obtain the corresponding Apollonians, we consider an extended root configuration consisting of a curvilinear triangle and its inversion through the base or the lower boundary circle representing the C- cell butterflies. This process of circle inversion is mathematically described by an operator I † † = S1S1S1. We note that it uses Si operators which are generators of the dual Apollonian group as discussed in appendix. In other words, the entire mathematical framework for ABC ‘uses’ both the Apollonian and its dual, namely the super-Apollonian group.

11.4. ABC and edge-sibling One to one mapping between a butterfly and the corresponding Apollonian, as described above exhaust almost all possible scenario with the exception of E-cell butterfly siblings. This case is illustrated in figure 18. Here two distinct E-cell butterflies share the same Pappus chain and hence the Apollonian mapping of two such butterflies lead to the same Apollonian if we follow the framework described above. To assign two different Descartes configuration to two such siblings, we use the uppermost triangular space, which has not been used earlier by the E or the C-cell butterflies. The process is fully explained and illustrated in the figure 18.

12. Summary and conclusions

Rooted in two competing length scales—the crystalline lattice and the cyclotron radius, the butterfly spectrum is a marvelous example of a physical incarnation of apparently abstract mathematics. Since its discovery, the subject has attracted a broad spectrum of physics and mathematics community [11, 29, 30]. A fascinating aspect of the butterfly tale as narrated here, comes from recognizing familiar mathematical features lurking the graph such as the Farey relation, the Pythagorean tree and the Apollonian gasket. Interrelation between the butterfly and the Apollonian fractal illustrate how these two fractals aid each other in understanding and characterizing their self-similar properties. Geometry and number theory dictates the formation of these two self-similar landscapes characterized by quadratic irrationals with a period-2 continued fraction consisting of one and another integer. Emergence of this special class of quadratic irrationals is rather remarkable. Why does the butterfly spectrum and the Apollonian gasket choose this special, measure zero subset of irrationals adds mysticism to the motion of electrons in a crystal subjected to magnetic field as well as the Apollonian packing of circles. Furthermore, Apollonian–butterfly correspondence seem to link the two solutions of Descartes configurations (equation45 ( )) to two sets of topological numbers, namely (σ, τ)and(M, N) of the butterfly. This follows from equation (11) as the locations of the inner and the outer Ford circles appear to be related to the topological quantum numbers of the (diagonal) gaps and (boundary) bands of the butterfly respectively. Possible significance of topological trait for Apollonian gaskets is intriguing and remains

26 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija open. Further analysis of the conformal maps that describe the butterfly and Apollonian recursive structure using modular group SL(2, Z) may provide further light towards some of these issues [21]. The mathematical description of the ABC as presented here leaves further room for an elegant formulation of the mapping between the two fractals. The simplicity and richness of the conformal maps such as given by equations (18)and(27) provide solid arguments for mapping part of the butterfly graph, namely the central cells of the fractal and the IAG.However,that tells only part of the ABC story as some aspects of this mapping are only empirically under- stood. With heuristic arguments and numerous examples that exhaust various scenarios from different parts of the spectrum, our discussion makes a convincing case that these two fractals are indeed related. Proof of zero measure Cantor set structure of the butterfly spectrum28 [ ], known as ‘Ten Martini problem’ had kept butterfly spectrum on mathematicians radar for several decade. More recently, in a related system ‘Question by Erdös and Szekeres’ piqued mathematicians again and perhaps many more treasures lay hidden in this system. With a new perspective centered on the role of number theory and geometry, we hope that the importance of a rigorous treatment of ABC will again attract mathematician’s attention to this iconic quantum system.

Acknowledgments

It is a great pleasure to thank Richard Friedberg, Jerzy Kocik and Michael Wilkinson for many stimulating discussions and new insights during the course of this work. Many thanks to Miguel Manacle for his help at various stages of investigating ABC. Finally, special thanks to Sam Werner for his comments and proof reading the manuscript.

Appendix A. Butterfly identities

Below we list various relationship among the integers (pL, qL, pR, qR, pc, qc, M, N, σ±, τ ±)that are associated with a butterfly. • − − (1) N = qL qR and M = pR pL [14]. • (2) σ+ + σ− = qR + qL = qc and τ + + τ − = pR + pL = pc [11]. − • qc pc 1 pc+1 (3) For the central butterflies, σ+ = σ− = 2 , pc = qL, τ+ = 2 and τ− = 2 [11]. • − (4) For the upper edge butterflies, σ+ = qR, σ− = qL, τ + = pR 1andτ − = pL + 1. For − the lower edge, σ+ = qL, σ− = qR, τ + = pL + 1andτ − = pR 1. Butterfly identities (1–3) have been proven earlier [11, 14]. To prove (4), we note that for the edge butterflies, the butterfly center consists of two bands separated by a gap, splitting the quantum number N of the butterfly into N1 and N2 where N = N1 + N2.HereN1 is the outer − − − band and N2 is the inner band. But N1 = qL (qR + qL) = qR.Now,σ+ = 0 N1 = qR. Therefore σ− = qL. − Similarly, it can be shown that τ + = pR 1andτ − = pL + 1.

Appendix B. Circle inversion

Inversion in a circle is a transformation which preserves angles. In the plane, the image of a point P with respect to a reference circle C with center O and radius r is a point P , lying on

27 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija the ray from O through P such that

OP × OP = r2. (A.1)

A key property of inversion is that the images of circles are generalized circles which means either a circle or a line (loosely speaking, a circle with infinite radius). In particular,

R2 r = r ,(A.2) d2 − r2

where r , r and R respectively are the radii of the image, the object and d is the distance between p 1 2 − the center of the object (center at q , 2q2 ) and the circle C (center at (0, 1)). This gives d 2 p2+q2−1 2 2 − r = q2 and the curvature κ = 2(p + q 1). In the special case where the object and the mirror circle touch, d = r + R and we obtain, 2 κ = κ + . (A.3) R

Appendix C. Dual and adjoint—super Apollonian group [4]

Given a Descartes quadruplets QD = (κ1, κ2, κ3, κ4), there exists another Descartes configura- ¯ ¯ tion QD = (κ¯1, κ¯2, κ¯3, κ¯4). Geometrically the Qd and its dual Qd are related as follows: given a Descartes configuration, we can obtain a related dual configuration by drawing circles through three of the tangency points of the four circles. Since there are four such trios in any quadruple, this construction gives us four new mutually tangent circles and we will denote this dual config- ¯ ¯ uration as Qd.InotherwordsQd is obtained from Qd consisting of the four circles each of which passes through the three tangency points avoiding one circle. These two dual configurations are related by an operator U: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ κ¯1 −1111 κ1 κ1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜κ¯2⎟ 1 ⎜ 1 −11 1⎟ ⎜κ2⎟ ⎜κ2⎟ ⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ ≡ U ⎝ ⎠ . (A.4) κ¯3 2 11−11 κ3 κ3 κ¯4 111−1 κ4 κ4 ¯ Figure 19 shows the relationship between QD and its dual QD along with the Apollo- nian packing and the corresponding dual packing. The Apollonian group is generated by four inversions w.r.t. those dual circles. † With the four operators Si, we next define a second set of four operations Si that correspond to inversion in one of the four circles in a Descartes configuration. Figure 20 compares and † † † contrasts Si and Si using S3 and S3. S3 changes only κ3 while with S3, κ3 remains fixed while curvatures of the other three circles change. Given an Apollonian group A with four generators (S1, S2, S3, S4), we can define a dual A¯ † † † † † Apollonian group with generators (S1, S2, S3, S4), where Si is the transpose of Si.Thetwo groups A and A¯ are related as their generators satisfy the following equation.

† Si = USiU. (A.5)

By combining A and A¯, one can form a new group, known as the super Apollonian group AS † † † † with eight generators, (S1, S2, S3, S4, S1, S2, S3, S4). The curvature of the dual circle is κ¯ =

28 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija

Figure 19. (A): Descartes configuration (dark circles) and its dual (lighter circles). (B): shows both the Apollonian packing (dark circles) and its dual (lighter circles).

† Figure 20. Illustrating the actions of S3 and S3 on a Descartes configuration

κ1κ2 + κ2κ3 + κ3κ1,whereκi (i = 1, 2, 3) are the curvatures of the 3 circles whose tangency points define the dual circle. To have an ordered set of numbers in the quadruplets where the four curvatures appear in monotonically decreasing order, the four matrices Si are replaced by corresponding matrices which we will represent as Di so that an ordered v transforming to Dv produces an ordered set of quadruplets: Di = OˆSi.

Appendix D. Pythagorean tree

A Pythagorean triple is a set of three positive integers (a, b, c) having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation,

a2 + b2 = c2. (A.6)

The triplet is said to be primitive if and only if a, b,andc share no common divisor.

29 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija

Figure 21. Pythagorean tree encodes nesting for all C-cell butterflies.

In 1934, Berggren discovered [13] that the set of all primitive Pythagorean triples has the structure of a rooted tree. In 1963, this was rediscovered by the Dutch mathematician FJM. Barning and seven years later by Hall [27] independently. Using algebraic means, it was shown that all primitive Pythagorean triplets can be generated by three matrices which we label as H1, H2 and H3 as shown in figure 21. The three matrices are given by, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −22 122 −122 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ H1 = 2 −12, H2 = 212, H3 = −212. (A.7) 2 −23 223 −223

For butterflies with their centers at E = 0, qc is always even. For such even-parity butterflies, the corresponding Pythagorean triplets are: 1 1 (n , n , n ) = q q , (q2 − q2 ), (q2 + q2 ) . (A.8) x y t L R 2 R L 2 R L

For C-cell butterflies whose center do not reside at E = 0, qc is odd. Such odd-parity butterflies can be described by a ‘dual’ Pythagorean tree—a tree where the legs of the right triangle (that is nx and ny) are interchanged. The Pythagorean triplets for these butterflies where qR and qL have opposite parity are obtained by multiplying by a factor of 2 in equation (A.8).

30 J. Phys. A: Math. Theor. 54 (2021) 025701 I I Satija

We note that the integers nx and κ0 determine the horizontal size Δφ of a butterfly: Δφ = | pR − pL | = 1 = 1 . qR qL qLqR nx In view of the fact that the Pythagorean tree or its dual preserve the parity, the hierarchical character of the E-cell butterflies cannot be described by the Pythagorean tree. Perhaps a quadruplet tree (a tree of Lorentz quadruplets) can be constructed as a generalization of the Pythagorean triplet tree. However, to best of our knowledge, there is no proper generalization of the Pythagorean tree to the quadruplet tree although the topic has been the subject of some discussion in the literature [27]. We note that some of the Lorentz quadruplets that describe the E-cell butterflies are non-primitive and therefore the existence of any tree structure that describes the entire butterfly graph seems rather unlikely.

ORCID iDs

Indubala I Satija https://orcid.org/0000-0001-8694-6283

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