Hofstadter-Toda spectral duality and quantum groups

Among the diversity of natural phenomena, intriguing connections sometimes emerge, where different physical systems share similar mathematical equations or structures. These unexpected parallels may seem coincidental or, conversely, hint at a deeper unity within the physical and mathematical worlds. One such example is the relationship between the relativistic Toda lattice model^1,2 in high energy physics and the Hofstadter^3–5 model in condensed matter physics.

The Toda lattice model describes a one-dimensional lattice of N particles with exponentially decaying interactions.¹ Initially conceived as a simple toy model for one-dimensional crystals, it is a prominent example of a nonlinear system in mathematical physics, which stands out for the remarkable property of being exactly solvable and integrable. Moreover, the model exhibits the emergence of solitonic excitations, providing insights into the complex dynamics of nonlinear systems. Furthermore, its relativistic generalization² is studied in high-energy physics for its connection to supersymmetric Yang-Mills theories in five dimensions⁶ and topological string theories on Calabi-Yau manifolds.^7–11 

On the other hand, the Hofstadter model describes noninteracting charged fermions (e.g., electrons) on a two-dimensional lattice in a magnetic field (more generally, a gauge field) perpendicular to the plane or, alternatively, noninteracting fermions in a one-dimensional quasiperiodic lattice.^3–5 The most distinctive feature of the model is the presence of a fractal energy spectrum, which is one of the few examples of fractals in quantum physics, and the very rich phase diagram¹² with topological gapped phases indexed by a topological invariant, the Chern number of the occupied bands.¹³ The Hofstadter model is the discrete counterpart of a quantum Hall system, in the sense that it describes the Landau levels of free electrons in a magnetic field regularized on a discrete lattice.¹³ Indeed, in the weak field limit, i.e., when the magnetic cyclotron radius becomes much larger than the lattice constant, the quantum Hall system and the Hofstadter model become physically equivalent. It is however when the magnetic cyclotron radius is comparable with the lattice constant that the fractal properties emerge, as a consequence of the incommensuration between these two characteristic lengths.

The Hofstadter Hamiltonian is moreover identical to the model introduced by Aubry and André,⁵ describing fermions on a one-dimensional discrete lattice in the presence of a harmonic potential not necessarily commensurate with the lattice parameter. In the Aubry-André model, the role of the magnetic flux per unit cell in the two-dimensional system is played by the angular wavenumber of the harmonic potential (in units of the lattice parameter), and the role of the second dimension is played by the phase shift of the harmonic potential with respect to the discrete lattice, giving rise to a so-called second synthetic dimension. This also leads to the emergence of topological charge pumping,¹⁴ which shares the same topological origin as the quantum Hall effect. The fractal nature of the spectra emerges here as a consequence of the incommensuration between the periodicity of the harmonic potential and the periodicity of the underlying lattice. This model can describe the effects of disorder, including Anderson localization⁵ and quasiperiodicity.^15–17 

Physical realizations of the Hofstadter model include microwave photonic systems,¹⁸ Moiré superlattices in graphene,^19–21 ultracold atoms in incommensurate optical lattices,²² and interacting phonons in superconducting qubits.²³ 

Although originating from distinct branches of physics, the N = 2 relativistic Toda lattice and Hofstadter models are described by Hamiltonians which transform one into the other by a formal substitution of the position and momenta operators x, p ↦ ix, ip. Mathematically, this corresponds to substituting Mathieu (cos) and the modified Mathieu (cosh) potentials or operators. Recently, a deep relationship between the energy spectra of the Hofstadter model, the relativistic Toda lattice model, and their modular dual was found by Hatsuda et al.¹⁰ Building on these results, it has been conjectured that the Hofstadter model is intimately related to the Langlands duality of quantum groups,²⁴ suggesting a deep connection between the Langlands program, quantum geometry, and quasiperiodicity.

Here, using the representation theory of the elementary quantum group, and the fundamental properties of tridiagonal matrices, we study the properties of a polynomial which appears to capture spectral information for both models. More precisely, the polynomial is implicitly defined by the fundamental self-recurring property of the Hofstadter butterfly.

We notice that both the Hofstadter and Toda lattice models are known to be related to the root systems A_N−1. Hence it should be no surprise that $Uq(sl2)$ (the simplest quantum group) plays a central role in the study of these models. More generally, we stress the fact that noncommutative geometry is the most natural language to investigate the properties of the Hofstadter model and the related Aubry and André model, as well as other quasiperiodic and aperiodic structures emerging in condensed matter physics, such as quasicrystals, Penrose tilings, and disordered systems.^25–28 

It should be emphasized that the first result below (Theorem II.2) has already been understood¹⁰ in the context of the relativistic Toda model, and our contribution is a clarification of the relationship with the characteristic polynomials associated with the Hofstadter model, making use of a “quantum group-adjusted” gauge introduced by Wiegmann and Zabrodin,²⁹ and further studied by Hatsugai et al.,³⁰ and the spectral duality developed by Molinari (see Ref. 31, where general Hamiltonian matrices similar to the one considered here are investigated).

The formula given in Theorem III.9 and the parametrization in Theorem IV.2 are new, and they are obtained by using the representation theory of $Uq(sl2)$ and standard formulas involving Chebyshev polynomials.

Let us briefly summarize our results.

The polynomial relationship of Eq. (1) was first proved by Hatsuda et al.¹⁰ by comparing the eigenvalue equation of the Hofstadter Hamiltonian H, the Toda Hamiltonian H_Toda, and its modular dual.

Here, we clarify the argument by using the spectral duality of tridiagonal matrices with corners developed in Ref. 31 and proving that the polynomials involved are indeed the characteristic polynomials determining the spectrum of the Hofstadter model [as observed in Ref. 10 (p. 10)].

For more details on the result above, see Theorem IV.2 and related discussion above it. This result should be interpreted as follows: in order to gain more insight into the mapping $E↦Ẽ$⁠, we can try to obtain explicit formulas for f for all values of α = P/Q and then invert one side of Eq. (1) (at least locally) to get an analytic expression $Ẽ=Ẽ(E)$⁠. This theorem is an attempt toward this goal. It should be noted that a formula for the Hofstadter spectrum at the mid-band point appeared previously in Ref. 32, by using a strategy related to the one employed in this paper, but without involving the notion of quantum group.

The layout of the paper is as follows. We first briefly review the relevant mathematical structures (generalized Clifford algebras, rotation algebra, the elementary quantum group) and physical models from a unified perspective based on the principal series representation of $Uq(sl2)$⁠. The second section starts with a brief recall of the spectral duality of tridiagonal matrices and continues on proving Theorem A above. In the third section, we exploit the irreducible representations of the quantum group to turn $Ĥ$ into a bidiagonal matrix without corners and with one constant diagonal. This allows us to establish a formula for its characteristic polynomial (Theorem B). In the fourth section, we employ Chebyshev polynomials in order to give a different parametrization of $Ĥ$ over the Brillouin torus (equivalent to the Chambers relation). We then finish with a conclusion and a brief outlook.

In this section, we will introduce the Hofstadter and Toda models in a somewhat unified way by emphasizing the key mathematical notions which underlie both systems.

Let us start by introducing the generalized Clifford algebra of order n on three generators, i.e., elements {e₁, e₂, e₃} satisfying the Weyl braiding relations e_ie_j = ωe_je_i (i < j) and $ein=ωn=1$ for a primitive n-th root of unity ω. As the classical case (n = 2, ω = −1) is generated by the Pauli matrices, the matrix representations of the e_i’s are also known as generalizations of the Pauli matrices.

There are several such constructions, most notably the Gell-Mann matrices and Sylvester’s shift and clock matrices.³³ The former are Hermitian and traceless (just like the Pauli matrices), while the latter are unitary and traceless, which makes them preferable for our goals.

The connection with quantum mechanics is evident from the explicit form of V and U: the clock matrix U amounts to the exponential of position in a periodic space of n “hours” (e.g., discrete lattice sites), and the circular shift matrix V is just the translation operator in this cyclic space, i.e., the exponential of the momentum.

Equation (3) provides our second contact point with the Hofstadter model, as those two transformations also appear as the fundamental self-similarity relations of the Hofstadter butterfly.^4,10,24 In addition, the modular duality discussed above has inspired Faddeev’s notion of modular double of a quantum group.³⁸ The modular double of $Uq(sl2)$ plays a central role in the study of the relativistic Toda lattice,³⁹ particularly in the argument of Ref. 10 where the polynomial relation which inspired this paper is established. To understand this, let us first recall the definition of the elementary quantum group.

In this paper, we will not need the coproduct which makes $Uq(sl2)$ a bialgebra. When α is real, the commutation relations above are compatible with a certain involution, giving a real form for $Uq(sl2)$⁠. The key point is that the principal series representations of this quantum group factors through the rotation algebra $Aq2$⁠. Indeed, the following proposition is readily verified.

The general definition of this object involves the Langlands dual Lie algebra.³⁹ However, we do not need to worry about such details here as $sl2$ is self-dual. We can anticipate that the Toda Hamiltonian is H ∝ ρϕ(E + F).

This motivates us to find a similar expression for the Hofstadter model.

The Hofstadter Hamiltonian describes the Hamiltonian of a charged particle in a magnetic field and in a periodic potential V(x, y) describing a two-dimensional lattice. There are two opposite ways to derive this Hamiltonian: In the limit of strong fields, one takes the Hamiltonian of a charged particle in a field (describing the quantum Hall effect), which is diagonalized in terms of Landau levels, and then treats the periodic potential V(x, y) as a perturbation on each Landau level (as done by Thouless-Kohmoto-Nightingale-den Nijs¹³). On the other hand, in the limit of weak fields, one starts with the so-called “tight-binding” Hamiltonian describing the lowest energy sector of a charged particle in a periodic potential, and treats the magnetic field as a perturbation (as done by Harper³ and Hofstadter⁴). These two approaches lead to the same Hamiltonian, with one remarkable exception, as already noted in Ref. 13: the value of the magnetic flux is replaced as Φ → 1/Φ. As noted in Ref. 10, this formal operation coincides precisely with taking the modular dual of the Hamiltonian.

One can introduce an anisotropy parameter R in the Hamiltonian, by taking $H=Tx+Tx*+R(Ty+Ty*)$⁠. The parameter R can be understood as a measure of the degree of anisotropy of the system, with R = 1 corresponding to x and y directions being perfectly symmetric. To simplify notations, we will assume R = 1 hereafter, except when explicitly noted.

Let us point out that the measure-theoretic properties of the spectrum of the almost Mathieu operator crucially depend on R. When α is irrational, the spectrum is a Cantor set of Lebesgue measure |4 − 4R|. Hence, the case R = 1 has spectrum of measure zero, and it can be shown to be surely purely singular continuous spectrum. The case R > 1 is notable for having almost surely pure point spectrum and exhibiting Anderson localization (see Refs. 5, 42, and 43).

Let us assume α is rational now. Imposing Q-periodicity, and treating g_n as the n-th coordinate of g in $CQ$⁠, the equation above becomes the eigenvalue problem for the matrix below, which we record as a separate definition as it will play a major role in the rest of the paper.

It will be convenient to set $Tx=eiνxV,Ty=eiνyU$⁠. The group spanned by T_x, T_y is referred to as the group of magnetic translations.⁴⁴ 

As already mentioned, Hatsuda et al.¹⁰ found a relationship between the characteristic polynomial of the Hofstadter Hamiltonian H, the self-similarity relation defining the fractal properties of the Hofstadter butterfly, and the spectral transformation between the Toda Hamiltonian H_Toda and its modular dual.

Let us denote the polynomial induced by the modular duality between the Toda Hamiltonian H_Toda and its modular dual defined in Ref. 10 by f = f_P/Q(E), where E is the energy. We are going to prove that f is the characteristic polynomial $f(E)=det(Ĥ−E)$ of the finite-dimensional Hofstadter Hamiltonian (note that $Ĥ$ depends on P/Q, although we suppressed this dependence from the notation).

Note that Eq. (17) is in fact a proof of the Chambers relation for the x-coordinate. We will see how to obtain the y-dependence later after using the representation theory of the quantum group. We will refer to the points $ν⃗=(0,0)$⁠, $ν⃗=(π2Q,π2Q)$⁠, and $ν⃗=(πQ,πQ)$ as the center, mid-band, and corner points, and restrict ourselves to the reduced Brillouin zone ν_x, ν_y ∈ [0, 2π/Q], except when explicitly noted. This can be done without loss of generality, since the Hamiltonian is periodic in ν_x, ν_y with a period 2π/Q up to unitary transformations.⁴⁶ 

It is well-known that the characteristic polynomial in f(E, ν_x, ν_y) (of order Q) contains only terms with powers E^Q−2i. This is because odd powers in E necessarily contains terms in cos(2πnα − ν_y), which cancel each other when summed over n = 0, Q − 1 at the mid-band point ν_y = 2π/Q. Consequently, f(E, ν_x, ν_y) is an even function of E and the spectrum is symmetric under E ↦ −E if Q is even, while f(E, ν_x, ν_y) is an odd function and the spectrum at the mid-band point contains E = 0 if Q is odd, since f(0, π/2Q, π/2Q) = 0. It is also well-known that f(0, π/2Q, π/2Q) = 4(−1)^Q/2 if Q is even, which mandates that f(0, 0, 0) = 0 if Q/2 is even and f(0, π/Q, π/Q) = 0 if Q/2 is odd. Consequently, the spectrum contains E = 0 at the center point if Q is doubly even, at the corner point if Q is singly even, and at the mid-band point if Q is odd, respectively. In particular, the zeros E = 0 in the spectra when Q is even are doubly degenerate and form Dirac cones with a linear dependence on the momentum.⁴⁷ 

We are now able to apply the spectral duality discussed above (coupled with the Chambers relation) and prove the first main result of this section.

From the representation theory of $Uq(sl2)$⁠, when q² is a primitive root of unity, we have essentially three families of representations. The first two are low-dimensional, and hence they must be discarded in favor of the last one, which we denote Z_a,b(λ). This family is topologically parametrized by a three-dimensional complex space. However, our system naturally comes with a single parameter R, which we have seen in Sec. II together with the almost Mathieu operator.

Heuristically, this is the reason why the “correct” representation for our purposes is Z_λ = Z_0,0(λ). A more formal argument goes like this: the representation in Eq. (21) has nonzero entries only along the secondary diagonals with zero corner elements. The parameters a, b do appear in the corners of E and F, thus we need to set them to zero. The following theorem illustrates the situation precisely.

Set $ζ=Zq−1$ and $Ĥ′=i(q−q−1)ζ(F−E)$⁠. We have the following proposition.

The advantage of ζ over ζ₀ is twofold: the freedom in choosing the parameter allows us to write an expression that is independent of the parity of P, and ζ(F) is represented through a matrix with a constant (secondary) diagonal, which makes the formula for $det(Ĥ′−E)$ more easily guessed.

The following lemma is easily verified.

Let us introduce a simple formula for the determinant of tridiagonal matrices.

To calculate the determinants of tridiagonal matrices with corners, we will need the following result.

Combining the previous theorem with Proposition III.2, we get the main result on the polynomial f. Recall that α = P/Q is a rational number in reduced form.