Hofstadter's butterfly

The first mathematical description of electrons on a 2D lattice, acted on by a homogeneous magnetic field, was studied by Rudolf Peierls and his student R. G. Harper in the 1950s.[2][3]

Hofstadter described the structure in 1976 in an article on the energy levels of Bloch electrons in magnetic fields.[1] It gives a graphical representation of the spectrum of Harper's equation at different frequencies. The intricate mathematical structure of this spectrum was independently discovered by Soviet physicist Mark Azbel in 1964 (the Azbel-Hofstadter model),[4] but Azbel did not plot the structure as a geometrical object.

Written while Hofstadter was at the University of Oregon, his paper was influential in directing further research. It predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional square lattice, as a function of a magnetic field applied to the system, formed what is now known as a fractal set. That is, the distribution of energy levels for small scale changes in the applied magnetic field recursively repeat patterns seen in the large-scale structure.[1] "Gplot", as Hofstadter called the figure, was described as a recursive structure in his 1976 article in Physical Review B,[1] written before Benoit Mandelbrot's newly coined word "fractal" was introduced in an English text. Hofstadter also discusses the figure in his 1979 book Gödel, Escher, Bach. The structure became generally known as "Hofstadter's butterfly".

David J. Thouless and his team discovered that the butterfly's wings are characterized by Chern integers, which provide a way to calculate the Hall conductance in Hofstadter's model.[5]

Confirmation

A simulation of electrons via superconducting qubits yields Hofstadter's butterfly

In 1997 the Hofstadter butterfly was reproduced in experiments with microwave guide equipped by an array of scatterers.[6] Similarity between the mathematical description of the microwave guide with scatterers and Bloch's waves in magnetic field allowed the reproduction of the Hofstadter butterfly for periodic sequences of the scatterers.

In 2001, Christian Albrecht, Klaus von Klitzing and coworkers realized an experimental setup to test Thouless et al.'s predictions about Hofstadter's butterfly with a two-dimensional electron gas in a supperlattice potential.[7][2]

In 2013, three separate groups of researchers independently reported evidence of the Hofstadter butterfly spectrum in graphene devices fabricated on hexagonal boron nitride substrates.[8][9][10] In this instance the butterfly spectrum results from interplay between the applied magnetic field and the large scale moiré pattern that develops when the graphene lattice is oriented with near zero-angle mismatch to the boron nitride.

In September 2017, John Martinis’s group at Google, in collaboration with the Angelakis group at CQT Singapore, published results from a simulation of 2D electrons in a magnetic field using interacting photons in 9 superconducting qubits. The simulation recovered Hofstadter's butterfly, as expected.[11]

Hofstadter butterfly is the graphical solution to Harper's equation, where the energy ratio ${\displaystyle \epsilon }$ is plotted as a function of the flux ratio ${\displaystyle 2\pi \alpha }$.

In his original paper, Hofstadter considers the following derivation:[1] a charged quantum particle in a two-dimensional square lattice, with a lattice spacing ${\displaystyle a}$, is described by a periodic Schrödinger equation, under a static homogeneous magnetic field restricted to a single Bloch band. For a 2D square lattice, the tight binding energy dispersion relation is

${\displaystyle W(\mathbf {k} )=E_{0}(\cos k_{x}a+\cos k_{y}a)={\frac {E_{0}}{2}}(e^{ik_{x}a}+e^{-ik_{x}a}+e^{ik_{y}a}+e^{-ik_{y}a})}$,

where ${\displaystyle W(\mathbf {k} )}$ is the energy function, ${\displaystyle \mathbf {k} =(k_{x},k_{y})}$ is the crystal momentum, and ${\displaystyle E_{0}}$ is an empirical parameter. The magnetic field ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$, where ${\displaystyle \mathbf {A} }$ the magnetic vector potential, can be taken into account by using Peierls substitution, replacing the crystal momentum with the canonical momentum ${\displaystyle \hbar \mathbf {k} \to \mathbf {p} -q\mathbf {A} }$, where ${\displaystyle \mathbf {p} =(p_{x},p_{y})}$ is the particle momentum operator and ${\displaystyle q}$ is the charge of the particle (${\displaystyle q=-e}$ for the electron, ${\displaystyle e}$ is the elementary charge). For convenience we choose the gauge ${\displaystyle \mathbf {A} =(0,Bx,0)}$.

Using that ${\displaystyle e^{ip_{j}a}}$ is the translation operator, so that ${\displaystyle e^{ip_{j}a}\psi (x,y)=\psi (x+a,y)}$, where ${\displaystyle j=x,y,z}$ and ${\displaystyle \psi (\mathbf {r} )=\psi (x,y)}$ is the particle's two-dimensional wave function. One can use ${\displaystyle W(\mathbf {p} -q\mathbf {A} )}$ as an effective Hamiltonian to obtain the following time-independent Schrödinger equation:

${\displaystyle E\psi (x,y)={\frac {E_{0}}{2}}\left[\psi (x+a,y)+\psi (x-a,y)+\psi (x,y+a)e^{-iqBxa/\hbar }+\psi (x,y-a)e^{+iqBxa/\hbar }\right].}$

Considering that the particle can only hop between points in the lattice, we write ${\displaystyle x=na,y=ma}$, where ${\displaystyle n,m}$ are integers. Hofstadter makes the following ansatz: ${\displaystyle \psi (x,y)=g_{n}e^{i\nu m}}$, where ${\displaystyle \nu }$ depends on the energy, in order to obtain Harper's equation (also known as almost Mathieu operator for ${\displaystyle \lambda =1}$):

${\displaystyle g_{n+1}+g_{n-1}+2\cos(2\pi n\alpha -\nu )g_{n}=\epsilon g_{n},}$

where ${\displaystyle \epsilon =2E/E_{0}}$ and ${\displaystyle \alpha =\phi (B)/\phi _{0}}$, ${\displaystyle \phi (B)=Ba^{2}}$ is proportional to the magnetic flux through a lattice cell and ${\displaystyle \phi _{0}=2\pi \hbar /q}$ is the magnetic flux quantum. The flux ratio ${\displaystyle \alpha }$ can also be expressed in terms of the magnetic length ${\textstyle l_{\rm {m}}={\sqrt {\hbar /eB}}}$, such that ${\textstyle \alpha =(2\pi )^{-1}(a/l_{\rm {m}})^{2}}$.[1]

Hofstadter's butterfly is the resulting plot of ${\displaystyle \epsilon _{\alpha }}$ as a function of the flux ratio ${\displaystyle \alpha }$, where ${\displaystyle \epsilon _{\alpha }}$ is the set of all possible ${\displaystyle \epsilon }$ that are a solution to Harper's equation.

Solutions to Harper's equation and Wannier treatment

Hofstadter's butterfly phase diagram at zero temperature. The horizontal axis indicates electron density, starting with no electrons from the left. The vertical axis indicates the strength of the magnetic flux, starting from zero at the bottom, the pattern repeats periodically for higher fields. The colors represent the Chern numbers of the gaps in the spectrum, also known as the TKNN (Thouless, Kohmoto, Nightingale and Nijs) integers. Blueish cold colors indicate negative Chern numbers, warm red colors indicate positive Chern numbers, white indicates zero.[2]

Due to the cosine function's properties, the pattern is periodic on ${\displaystyle \alpha }$ with period 1 (it repeats for each quantum flux per unit cell). The graph in the region of ${\displaystyle \alpha }$ between 0 and 1 has reflection symmetry in the lines ${\textstyle \alpha ={\frac {1}{2}}}$ and ${\displaystyle \epsilon =0}$.[1] Note that ${\displaystyle \epsilon }$ is necessarily bounded between -4 and 4.[1]

Harper's equation has the particular property that the solutions depend on the rationality of ${\displaystyle \alpha }$. By imposing periodicity over ${\displaystyle n}$, one can show that if ${\displaystyle \alpha =P/Q}$ (a rational number), where ${\displaystyle P}$ and ${\displaystyle Q}$ are distinct prime numbers, there are exactly ${\displaystyle Q}$ energy bands.[1] For large ${\displaystyle Q\gg P}$, the energy bands converge to thin energy bands corresponding to the Landau levels.

Gregory Wannier showed that by taking into account the density of states, one can obtain a Diophantine equation that describes the system,[12] as

${\displaystyle {\frac {n}{n_{0}}}=S+T\alpha }$

where

${\displaystyle n=\int _{-4}^{\epsilon _{\rm {F}}}\rho (\epsilon )\mathrm {d} \epsilon \;;\;n_{0}=\int _{-4}^{4}\rho (\epsilon )\mathrm {d} \epsilon }$

where ${\displaystyle S}$ and ${\displaystyle T}$ are integers, and ${\displaystyle \rho (\epsilon )}$ is the density of states at a given ${\displaystyle \alpha }$. Here ${\displaystyle n}$ counts the number of states up to the Fermi energy, and ${\displaystyle n_{0}}$ corresponds to the levels of the completely filled band (from ${\displaystyle \epsilon =-4}$ to ${\displaystyle \epsilon =4}$). This equation characterizes all the solutions of Harper's equation. Most importantly, one can derive that when ${\displaystyle \alpha }$ is an irrational number, there are infinitely many solution for ${\displaystyle \epsilon _{\alpha }}$.

The union of all ${\displaystyle \epsilon _{\alpha }}$ forms a self-similar fractal that is discontinuous between rational and irrational values of ${\displaystyle \alpha }$. This discontinuity is nonphysical, and continuity is recovered for a finite uncertainty in ${\displaystyle B}$[1] or for lattices of finite size.[13] The scale at which the butterfly can be resolved in a real experiment depends on the system's specific conditions.[2]