# Hofstadter quasicrystal problem and irrational quantum oscillations

Kirill Kozlov  Moscow Institute of Physics and Technology, Dolgoprudny, 141700
Russia  Grigor Adamyan  Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA  Mariia Kryvoruchko  Department of
Biophysics, Johns Hopkins University, Baltimore, MD 21218, USA  Yelizaveta
Kulynych  Department of Physics, Taras Shevchenko National University of Kyiv,
Kyiv 03680, Ukraine  Department of Physics, The École polytechnique fédérale
de Lausanne, Lausanne 1015, Switzerland  Leonid Levitov  Physics Department,
Massachusetts Institute of Technology, Cambridge MA02139, USA

###### Abstract

Landau levels perturbed by a periodic potential have long been known broaden
into fractals. Motivated by recent advances in twistronics, we introduce a
‘Hofstadter quasicrystal’ problem describing Landau levels perturbed by
incommensurate cosine waves. We consider a quasiperiodic potential
representing a pair of $45^{\circ}{}$-rotated Hofstadter problems, finding a
complex spectrum completely distinct from the Hofstadter spectrum, and yet
sharing with it many common features. Surprisingly, due to the eight-fold
symmetry of the incommensurate potential, the problem exhibits hidden symmetry
arising at special ‘magic’ values of the magnetic field. The periodicity in
the magic field values explains striking wide-range oscillations in the
spectrum width that have irrational periodicity distinct from the Aharonov-
Bohm and Brown-Zak periodicities. The prominent character of these
oscillations makes them readily accessible in state-of-the-art moiré graphene
systems.

Perturbing Landau levels by periodic potentials offers seminal examples of
quantum fractals [1, 2, 4, 3, 5]. The intricacy, richness, and general
character of the fractal behavior originate from the unique symmetries
governed by magnetic translations. These are symmetry operations that commute
with the Hamiltonian but not with each other, and, together, span a large
symmetry group [6, 7]. Magnetic translations make the phases of particle
states depend on the order in which translations are performed. This results
in the disappearance and recurrence of Bloch bands, governed by
commensurability of the magnetic flux through the lattice period, generating
self-similar fractal bands resembling a Cantor set. This intriguing behavior
has garnered significant interest over the years and has been extensively
studied for the case of Landau levels of Bloch electrons [4, 8, 9, 3, 5, 12,
10, 13, 11],

$\displaystyle
H=\epsilon({\boldsymbol{p}}-e{\boldsymbol{A}}(r))+V({\boldsymbol{r}}),$ (1)

where $V({\boldsymbol{r}})$ is a periodic potential,
$\epsilon({\boldsymbol{p}})$ is particle kinetic energy and
${\boldsymbol{A}}(r)$ describes a uniform magnetic field. Exquisite number-
theoretic and topological structures discovered for this problem suggest
extensions to a wider family of non-periodic potentials. Recently, a number of
generalizations to hierarchical and aperiodic lattices have been considered
[14, 15, 16]. At the same time, magnetotransport measurements unveiled fractal
behavior in graphene superlattices [17, 18, 19, 20, 21, 22].

Figure 1:  a), b) Periodic two-cosine and quasiperiodic four-cosine potentials
representing Hamiltonians $H_{N=2}$ and $H_{N=4}$ [Eqs. (4) and (7)],
exhibiting 4-fold and 8-fold point symmetry group comprising rotations and
mirrors, respectively. The extremal points form both a periodic and a
quasiperiodic lattice, respectively. The range shown:
$-8\pi/\kappa<x,y<8\pi/\kappa$.  Figure 2: Energy spectrum of a quasicrystal
$H_{N=4}$ vs. $\beta$ (the inverse $B$ field) obtained from a Harper-like 1D
problem. The factor $\tilde{V}$ describing the oscillations decay with
$\beta$, Eq. (5), is factored out and included in the energy units. Note pairs
of positive and negative pyramids filled with Landau-like fans of energy
states and complex deep-level structure of spectral gaps detailed in Fig. 3,
where zoom-ins of the regions marked by c), d), e) and f) are shown. The
modulation in the spectrum width seen on a large scale is explained by
recurring hidden symmetry originating from magnetic translations, see Eq. (9).
Red lines mark the ’magic’ $\beta$ values defined in Eqs. (3), at which the
$N=4$ quasicrystal Hamiltonian, Eq. (7), decouples into a pair of commuting
Hofstadter $N=2$ Hamiltonians, Eq. (4). This irrational periodicity, appearing
over a wide range of fields and not explained by the Aharonov-Bohm or Brown-
Zak oscillations represents a readily observable signature of the Hofstadter
quasicrystal.

However, relatively little is known about the more general problem of Landau
levels perturbed by potentials with incommensurate periods. Here we consider
moiré quasicrystal potentials [23], comprised of harmonics defined by stars of
vectors with non-crystallographic symmetries:

$\displaystyle
V_{N}({\boldsymbol{r}})=2V\cos{\boldsymbol{k}}_{1}{\boldsymbol{r}}+...+2V\cos{\boldsymbol{k}}_{N}{\boldsymbol{r}},$
(2)
$\displaystyle{\boldsymbol{k}}_{s}=(\kappa\cos\theta_{s},\kappa\sin\theta_{s}),\quad\theta_{s=1...N}=\pi
s/N,$

which defines a crystal for $N=2,\,3$ and a quasicrystal for $N>3$ (see Fig.
1). On a Landau level, this defines a quantum problem for noncommuting
coordinates of a cyclotron orbit center, see below. Besides an obvious
fundamental interest, this problem is timely due to the advent of twistronics
— a field that produces and probes incommensurate 2D systems comprising two or
more periodic lattices rotated relative to one another by a fixed angle and
strongly coupled at the atomic scale [24, 25, 26]. This triggered interest in
dodecagonal graphene quasicrystals [27, 28, 29, 30, 31, 32] and in topological
superconductivity in eight-fold symmetric quasicrystals obtained from
$45^{\circ}$-twisted stacks of high-$T_{c}$ superconductors [33, 34, 35, 36,
37, 38, 39], leading to structures closely related to the discussion below.

Figure 3: a), b) A comparison of the periodic and quasiperiodic problems.
Energy spectra of $H_{N=2}$ and $H_{N=4}$ vs. $\beta$, obtained from a 1D
Harper-like problem. For $H_{N=4}$ the incommensurate translations $T_{\pm 1}$
and $T_{\pm 1/\sqrt{2}}$ are tackled using rational approximant
$\frac{1}{\sqrt{2}}\approx\frac{p}{q}=\frac{5}{7}$ (higher-order approximants
were checked to yield similar results, see text). The spectrum is the familiar
Hofstadter’s fractal for $H_{N=2}$ and has a complex multiscale character for
$H_{N=4}$. A double arrow between a) and b) highlights similar fans of Landau-
like levels. As in Fig. 2, the factor $\tilde{V}$ describing the decay with
$\beta$, Eq. (5), is factored out and included in the energy units. Zoom-ins
c), d), e), and f) illustrate specific motifs at the locations marked in Fig.
2. In c), d) the spectrum displays pairs of pyramids filled with fans of
Landau-like energy levels, multiple band gaps and complex internal structure.
e), f) Large butterfly-shaped bandgaps with smaller, nested butterfly-shaped
gaps. Note the web-like patterns discernible at the center of the large
butterflies and a fractal web near the right edge.

Here we consider a Landau level perturbed by a $N=4$ quasiperiodic potential,
Eq. (2), finding the spectrum shown in Figs. 2 and 3. While the spectrum is
clearly far more complex than the Hofstadter fractal, it shares with it common
features such as pyramids filled with Landau-like fans of energy levels and
spectral gaps, some of which resemble butterflies. However, the most prominent
behavior seen in Fig. 2 are strong oscillations in the spectrum width
occurring over a wide range of $\beta$. The periodicity of these oscillations
is incommensurate with the periods expected for the Aharonov-Bohm (AB) or
Brown-Zak (BZ) oscillations.

As we will see, these oscillations reflect a surprising behavior originating
from non-crystallographic symmetry of the quasicrystal problem. Namely, the
problem possesses hidden symmetry arising at an infinite family of ‘magic’
field values highlighted by red lines in Fig. 2, forming a $1/B$-periodic
family with an irrational period:

$\displaystyle B_{m}a^{2}=\frac{\phi_{0}}{\sqrt{2}m},\quad m=\pm 1,\pm 2,\pm
3...$ (3)

where $\phi_{0}=h/e$ is the flux quantum and $a=2\pi/\kappa$ is the
periodicity of the cosine modulation in Eq. (2). This counterintuitive
behavior will be explained below

The problem of a particle in a periodic potential, Eq. (1), describes several
distinct regimes in which fractal spectra can emerge. The best-studied regime
is a tight-binding problem on a lattice in direct space [2, 44, 45]. In this
case, the AB interference leads to a periodic dependence on magnetic flux
through the lattice unit cell, $\omega=Ba^{2}/\phi_{0}$, where $a^{2}$ is the
unit cell area, with Bloch states disappearing at irrational values $\omega$
and recurring at rational $\omega$, producing Hofstadter fractal.

A less widely known occurrence of fractal spectra is when the cyclotron energy
greatly exceeds the periodic potential amplitude. In this case, oscillations
in the spectrum, known as the BZ oscillations [20, 21], arise as a function of
an inverse magnetic field. The relation to the Hofstadter problem can be
illustrated for a Landau level perturbed by a weak square lattice potential
[5]

$\displaystyle H_{N=2}=2\tilde{V}\cos\kappa x+2\tilde{V}\cos\kappa
y,\quad[x,y]=i\frac{\hbar}{eB},$ (4)

where $x$ and $y$ are operators of the electron orbit center
${\boldsymbol{r}}_{c}=(x,y)$ and $\kappa$ describes the modulation
periodicity. The commutator in Eq. (4) follows from the relation between
electron coordinates and the orbit center radius vector,
${\boldsymbol{r}}_{c}={\boldsymbol{r}}-{\boldsymbol{v}}\times{\boldsymbol{z}}/\omega_{c}$,
$\omega_{c}=eB/m$. The potential effective strength $\tilde{V}$ seen by an
electron equals the bare potential strength in Eq. (2) reduced by a “Debye-
Waller factor” reflecting averaging of $V({\boldsymbol{r}})$ over electron
orbitals. For $n=0$ Landau level this factor equals[5, 40, 41]

$\tilde{V}/V=e^{-\kappa^{2}\ell_{B}^{2}/2}=e^{-\beta/2},\quad\beta=\kappa^{2}\ell_{B}^{2}=2\pi\phi_{0}/Ba^{2}$
(5)

with the magnetic length $\ell_{B}=\sqrt{\phi_{0}/2\pi B}$ representing the
characteristic radius of an electron orbital. This behavior was well
understood and documented in the early work on fractal spectra in
nanostructures [5, 42, 43].

As a brief detour, we solve the problem in Eq. (4) by a method that reveals
its relation with the Hofstadter problem and explains periodicity in $1/B$.
This is done by transforming it to Harper’s equation, a discrete one-
dimensional wave equation that can be tackled numerically or analytically
[44]. As a first step, we nondimensionalize the problem by using the magnetic
length. By rescaling coordinates as $x\to x/\ell_{B}$, $y\to y/\ell_{B}$, we
absorb $\ell_{B}$ into a dimensionless wavenumber
$\tilde{\kappa}=\kappa\ell_{B}$.

Next, using the commutator $[x,y]=i$, we identify $y$ with $-i\partial_{x}$
and introduce the one-dimensional translation operators
$T_{\lambda}f(x)=e^{-\lambda\partial_{x}}f(x)=f(x-\lambda)$. This transforms
Eq. (4) to a 1D problem
$H=2\tilde{V}\cos\tilde{\kappa}x+\tilde{V}T_{\tilde{\kappa}}+\tilde{V}T_{-\tilde{\kappa}}$.
After rescaling $x\to\tilde{\kappa}x$, translation operators become $T_{\pm
1}$, leading to the Harper problem

$H=2\tilde{V}\cos\beta x+\tilde{V}T_{+1}+\tilde{V}T_{-1}.$ (6)

The spectrum vs. $\beta$, shown in Fig. 3 a), is nothing but the Hofstadter’s
fractal, periodic under $\beta\to\beta\pm 2\pi$. Periodicity in the inverse
$B$ field is in agreement with the properties of the BZ oscillations in a
periodic potential observed in strong fields [20, 21].

In this context, we extend this approach to describe the new regimes arising
due to the quasiperiodic potential. We will focus on the $N=4$ quasicrystal
case and consider a pair of identical square-lattice potentials given in Eq.
(4) rotated by $45^{\circ}{}$. This provides a representation

$\displaystyle H_{N=4}=H_{N=2}+H^{45^{\circ}}_{N=2},$ (7)

where $H_{N=2}$ is given by Eq. (4) and $H^{45^{\circ}}_{N=2}$ takes the form

$\displaystyle
H^{45^{\circ}}_{N=2}=2\tilde{V}\cos\frac{\kappa(x+y)}{\sqrt{2}}+2\tilde{V}\cos\frac{\kappa(x-y)}{\sqrt{2}}$
(8)

Upon rescaling $x$ and $y$ by $\ell_{B}$ as above, $H_{N=4}$ preserves its
form and maintains the $8$-fold rotation symmetry, whereas the commutation
relations become $[x,y]=i$.

To understand the origin of the ‘magic’ magnetic field values in Eq. (3) we
consider the commutator of $H_{N=2}$ and $H^{45^{\circ}}_{N=2}$. Its
properties can be understood from the commutators of individual cosine terms
in Eq. (2):

$\displaystyle[\cos{\boldsymbol{k}}_{1}{\boldsymbol{r}},\cos{\boldsymbol{k}}_{2}{\boldsymbol{r}}]=\sum_{s_{1,2}=\pm
1}\frac{1}{4}[e^{is_{1}{\boldsymbol{k}}_{1}{\boldsymbol{r}}},e^{is_{2}{\boldsymbol{k}}_{2}{\boldsymbol{r}}}]$
(9) $\displaystyle=\sum_{s_{1,2}=\pm
1}\frac{i}{2}\sin\left(\frac{s_{1}s_{2}}{2}({\boldsymbol{k}}_{1}\times{\boldsymbol{k}}_{2})_{z}\right)e^{is_{1}{\boldsymbol{k}}_{1}{\boldsymbol{r}}+is_{2}{\boldsymbol{k}}_{2}{\boldsymbol{r}}}$

obtained with the help of the Baker-Campbell-Hausdorff (BCH) formula
$e^{Z}=e^{X}e^{Y}$, $Z=X+Y+\frac{1}{2}[X,Y]+...$, where the higher-order terms
vanish when $[X,Y]$ is a c-number. For the $N=4$ quasicrystal, possible angles
between ${\boldsymbol{k}}_{1}$ and ${\boldsymbol{k}}_{2}$ are $\pm 45^{\circ}$
and $\pm 135^{\circ}$ degrees, whereas
$|{\boldsymbol{k}}_{1,2}|=\kappa\ell_{B}=\sqrt{\beta}$. Therefore, the
commutator $[H_{N=2},H^{45^{\circ}}_{N=2}]$ vanishes whenever
$\beta\frac{1}{2\sqrt{2}}$ equals $\pi$ times an integer, which gives the
condition in Eq. (3). At these ’magic field’ values, the spectrum is a sum of
two independent Hofstadter spectra. Away from these values the Hamiltonians
$H_{N=2}$ and $H^{45^{\circ}}_{N=2}$ do not commute. As a result, the
eigenstates are hybridized and the eigenvalues are reduced. Therefore, the
oscillations in the spectrum width vs. $\beta$ is a signature due to a hidden
symmetry appearing at magic $\beta$ values. The strong amplitude and
irrational period of these oscillations makes them easy to delineate from the
conventional AB and BZ oscillations.

Next, we discuss the approach used to obtain the spectrum of $H_{N}$, focusing
on the $N=4$ quasicrystal Hamiltonian. Following the steps that have led to
Eq. (6), we arrive at a generalized 1D Harper-like problem involving
incommensurate translations by $\pm 1$ and $\pm 1/\sqrt{2}$

$\displaystyle H_{4}=H_{2}+H_{2}^{45^{\circ}}=2\tilde{V}\cos\beta
x+\tilde{V}T_{-1}+\tilde{V}T_{+1}$ (10)
$\displaystyle+2\tilde{V}\cos\left(\frac{\beta}{4}+\frac{\beta
x}{\sqrt{2}}\right)T_{\frac{1}{\sqrt{2}}}+2\tilde{V}\cos\left(\frac{\beta}{4}-\frac{\beta
x}{\sqrt{2}}\right)T_{-\frac{1}{\sqrt{2}}}.$

A problem involving incommensurate translations cannot be reduced, as is, to a
problem on a periodic lattice. To transform it into a Harper-like problem, we
use rational approximants of $1/\sqrt{2}\approx p/q$ found from a step-by-step
expansion of a continued fraction:

$\displaystyle\frac{1}{\sqrt{2}}=\frac{1}{1+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}}\to\frac{p}{q}=\frac{2}{3},\quad\frac{5}{7},\quad\frac{12}{17}...$
(11)

As is well known, infinite continued fractions representing irrational
numbers, when truncated at a certain place, yield the best rational
approximants with not-too-large denominators. E.g., already $5/7$ approximates
$1/\sqrt{2}$ with $1\%$ accuracy, which proves sufficient for our purposes.
The results found for the next approximant, $12/17$, support this conclusion.

A rational approximation $\frac{1}{\sqrt{2}}\to\frac{p}{q}$ yields a
Hamiltonian with commensurate translations $T_{\pm 1}$, $T_{\pm p/q}$. This is
nothing but a 1D tight-binding problem on a lattice with periodicity $1/q$ and
non-nearest-neighbor hoppings, a problem that can be readily solved. It is
convenient to move to an integer lattice by replacing
$x\rightarrow\frac{x}{q}$. Upon factoring $\tilde{V}$, this gives a hermitian
Harper-like operator

$\displaystyle H_{p/q}=2\cos\frac{\beta x}{q}+\hat{T}_{q}+\hat{T}_{-q}$ (12)
$\displaystyle+2\cos\left(\frac{\beta
px}{{q^{2}}}+\frac{\beta}{4}\right)\hat{T}_{p}+2\cos\left(\frac{\beta
px}{q^{2}}-\frac{\beta}{4}\right)\hat{T}_{-p}.$

Studying the approximant Hamiltonian $H_{p/q}$ starts with defining a Hilbert
space. In that, one must exercise caution in choosing the system length and
selecting suitable parameter values. First, we consider $H_{p/q}$ on an
integer lattice $\\{x\\}$ using as a basis the Kronecker delta functions,
$\braket{x}{n}=\delta_{x,n}$, which is an infinite-dimensional Hilbert space.
To render it finite-dimensional, ${\rm dim}({\cal{V}})=M<\infty$, we focus on
the states $\ket{n}$, $1\leq n\leq M$ and apply periodic boundary conditions
$\psi(x+M)=\psi(x)$.

The periodicity condition, applied to the cosine terms in Eq. (12), restricts
possible $\beta$ values. In the Harper problem, a periodicity condition for
the cosine term generates one constraint for the allowed $\beta$ values. Here,
however, we have cosine terms with two different periods, which generates two
constraints:

${\rm(i)}\quad\beta=2\pi\frac{qk}{M};\qquad{\rm(ii)}\quad\beta=2\pi\frac{q^{2}k^{\prime}}{pM},$
(13)

with integer $k$ and $k^{\prime}$. Combined, these conditions are more
stringent than in the Harper problem case, restricting the allowed values of
$\beta$ as:

$\beta_{m}=2\pi m\frac{q^{2}}{M}$ (14)

with integer $m$. This gives the minimal allowed step $\Delta\beta=2\pi
q^{2}/M$ which is $q^{2}$ times larger than the minimal step found in a
similar manner for the Harper problem. This implies that for a fixed step in
$\beta$, if we aim to enhance the accuracy of approximating $1/\sqrt{2}$ by
increasing the value of $q$, we must also expand the Hilbert space dimension
by increasing $M$. In practice, this can prove costly. A better sampling of
the spectrum vs. $\beta$ can be achieved by relaxing the constraints imposed
by periodic boundary conditions. This yields a spectrum that exhibits a
certain level of ‘noise’, which can be traced to the appearance of the
boundary states of the Hamiltonian. The resulting spectrum, which is depicted
in Figs. 2 and 3, was obtained by optimizing $\Delta\beta$ in order to achieve
high resolution and, at the same time, minimize the added ‘noise’ (parameters
used: $M=1960$, $\Delta\beta=2\pi\cdot 10^{-3}$).

The resulting spectrum, shown in Fig. 3 alongside the Hofstadter butterfly
(HB) spectrum obtained by the same method, exhibits pyramids filled with
Landau-like families of energy levels, and a multiscale hierarchy of energy
gaps. Both the large-scale and the fine structure seen in this spectrum differ
significantly from those of HB. Nevertheless, closer inspection reveals
fragments resembling HB-like patterns, as illustrated in the zoom-ins shown in
Fig. 3 c), d), e) and f).

However, the most prominent aspect unique to this problem is oscillations vs.
$\beta$ with an irrational period occurring on a large scale. Unlike the
periodic $\beta$ dependence for the HB spectrum in Fig. 3 a), it is a
signature that has no known counterpart in the HB problem and its
generalizations. Inspecting other quasicrystal problems, including dodecagonal
Hamiltonians $H_{N=6}$, confirms the general character of this behavior and
its relation to a hidden symmetry appearing at isolated $\beta$ values, in
general forming an irrational periodic sequence. Occurring at large scales and
having a large amplitude, these irrational-period oscillations, like other BZ-
type effects [20, 21, 46], are expected to persist to high temperatures and be
readily observable experimentally. The hierarchy of gapped states seen in the
spectrum points to a variety of interesting questions, from Brown-Zak band
formation in quasiperiodic potentials to moiré quasicrystal Quantum Hall
effect.

We are grateful to Joseph Avron, Leonid Glazman, Philip Kim and Klaus Ensslin
for useful discussions. This work was performed in part at Aspen Center for
Physics, which is supported by NSF grant PHY-2210452.

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