In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
in a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
. The fractal,
self-similar
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
nature of the spectrum was discovered in the 1976 Ph.D. work of
Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, a ...
and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps
n the graphform a very striking pattern somewhat resembling a butterfly."
The Hofstadter butterfly plays an important role in the theory of the integer
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exh ...
and the theory of
topological quantum number
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
s.
History
The first mathematical description of electrons on a 2D lattice, acted on by a perpendicular homogeneous magnetic field, was studied by
Rudolf Peierls
Sir Rudolf Ernst Peierls, (; ; 5 June 1907 – 19 September 1995) was a German-born British physicist who played a major role in Tube Alloys, Britain's nuclear weapon programme, as well as the subsequent Manhattan Project, the combined Allie ...
and his student R. G. Harper in the 1950s.
Hofstadter first described the structure in 1976 in an article on the
energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
s of
Bloch electron
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
s in perpendicular magnetic fields.
It gives a graphical representation of the spectrum of Harper's equation at different frequencies. One key aspect of the mathematical structure of this spectrum – the splitting of energy bands for a specific value of the magnetic field, along a single dimension (energy) – had been previously mentioned in passing by Soviet physicist
Mark Azbel
Mark Yakovlevich Azbel (russian: Марк Яковлевич Азбель; 12 May 1932 — 31 March 2020) was a Soviet and Israeli physicist. He was a member of the American Physical Society.
Between 1956 and 1958, he experimentally demonstrated ...
in 1964
(in a paper cited by Hofstadter), but Hofstadter greatly expanded upon that work by plotting ''all'' values of the magnetic field against all energy values, creating the two-dimensional plot that first revealed the spectrum's uniquely recursive geometric properties.
Written while Hofstadter was at the
University of Oregon
The University of Oregon (UO, U of O or Oregon) is a public research university in Eugene, Oregon. Founded in 1876, the institution is well known for its strong ties to the sports apparel and marketing firm Nike, Inc
Nike, Inc. ( or ) is a ...
, 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
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by thei ...
, as a function of a magnetic field applied perpendicularly 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
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
repeat
pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...
s seen in the large-scale structure.
"Gplot", as Hofstadter called the figure, was described as a
recursive structure in his 1976 article in ''
Physical Review B
''Physical Review B: Condensed Matter and Materials Physics'' (also known as PRB) is a peer-reviewed, scientific journal, published by the American Physical Society (APS). The Editor of PRB is Laurens W. Molenkamp. It is part of the ''Physical Re ...
'',
written before
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...
's newly coined word "fractal" was introduced in an English text. Hofstadter also discusses the figure in his 1979 book ''
Gödel, Escher, Bach
''Gödel, Escher, Bach: an Eternal Golden Braid'', also known as ''GEB'', is a 1979 book by Douglas Hofstadter.
By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, t ...
''. The structure became generally known as "Hofstadter's butterfly".
David J. Thouless
David James Thouless (; 21 September 1934 – 6 April 2019) was a British condensed-matter physicist. He was the winner of the 1990 Wolf Prize and a laureate of the 2016 Nobel Prize for physics along with F. Duncan M. Haldane and J. Michael ...
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.
Confirmation
In 1997 the Hofstadter butterfly was reproduced in experiments with microwave guide equipped by an array of scatterers. 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
Klaus von Klitzing (, born 28 June 1943, Schroda) is a German physicist, known for discovery of the integer quantum Hall effect, for which he was awarded the 1985 Nobel Prize in Physics.
Education
In 1962, Klitzing passed the Abitur at the A ...
and coworkers realized an experimental setup to test Thouless ''et al.''
's predictions about Hofstadter's butterfly with a
two-dimensional electron gas
A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion ...
in a superlattice potential.
In 2013, three separate groups of researchers independently reported evidence of the Hofstadter butterfly spectrum in
graphene
Graphene () is an allotrope of carbon consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice nanostructure. devices fabricated on hexagonal
boron nitride
Boron nitride is a thermally and chemically resistant refractory compound of boron and nitrogen with the chemical formula BN. It exists in various crystalline forms that are isoelectronic to a similarly structured carbon lattice. The hexagonal ...
substrates. In this instance the butterfly spectrum results from interplay between the applied magnetic field and the large scale
moiré pattern
In mathematics, physics, and art, moiré patterns ( , , ) or moiré fringes are large-scale interference patterns that can be produced when an opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré ...
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 perpendicular magnetic field using interacting photons in 9 superconducting
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
s. The simulation recovered Hofstadter's butterfly, as expected.
In 2021 the butterfly was observed in twisted
bilayer graphene
Bilayer graphene is a material consisting of two layers of graphene. One of the first reports of bilayer graphene was in the seminal 2004 '' Science (journal), Science'' paper by Geim and colleagues, in which they described devices "which containe ...
at the second magic angle.
Theoretical model
In his original paper, Hofstadter considers the following derivation:
a charged quantum particle in a two-dimensional square lattice, with a lattice spacing
, is described by a periodic
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, under a perpendicular static homogeneous magnetic field restricted to a single Bloch band. For a 2D square lattice, the
tight binding
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at eac ...
energy
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
is
:
,
where
is the energy function,
is the
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hbar i ...
, and
is an empirical parameter. The magnetic field
, where
the
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
, can be taken into account by using
Peierls substitution, replacing the crystal momentum with the canonical momentum
, where
is the particle
momentum operator
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensi ...
and
is the charge of the particle (
for the electron,
is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
). For convenience we choose the gauge
.
Using that
is the
translation operator, so that
, where
and
is the particle's two-dimensional
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
. One can use
as an effective
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
to obtain the following time-independent Schrödinger equation:
:
Considering that the particle can only hop between points in the lattice, we write
, where
are integers. Hofstadter makes the following
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
:
, where
depends on the energy, in order to obtain Harper's equation (also known as
almost Mathieu operator
In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by
: ^_\omega un) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \,
acting as a self-adjoint operator on the Hil ...
for
):
:
where
and
,
is proportional to the magnetic flux through a lattice cell and
is the
magnetic flux quantum
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
. The flux ratio
can also be expressed in terms of the magnetic length
, such that
.
Hofstadter's butterfly is the resulting plot of
as a function of the flux ratio
, where
is the set of all possible
that are a solution to Harper's equation.
Solutions to Harper's equation and Wannier treatment
Due to the cosine function's properties, the pattern is periodic on
with period 1 (it repeats for each quantum flux per unit cell). The graph in the region of
between 0 and 1 has
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D the ...
in the lines
and
.
Note that
is necessarily bounded between -4 and 4.
Harper's equation has the particular property that the solutions depend on the rationality of
. By imposing periodicity over
, one can show that if
(a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
), where
and
are distinct
prime numbers
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, there are exactly
energy bands.
For large
, the energy bands converge to thin energy bands corresponding to the
Landau levels.
Gregory Wannier
Gregory Hugh Wannier (1911–1983) was a Swiss physicist.
Biography
Wannier received his physics PhD under Ernst Stueckelberg at the University of Basel in 1935. He worked with Professor Eugene P. Wigner as a post-doc exchange student at Princ ...
showed that by taking into account the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
, one can obtain a
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
that describes the system, as
:
where
:
where
and
are integers, and
is the density of states at a given
. Here
counts the number of states up to the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
, and
corresponds to the levels of the completely filled band (from
to
). This equation characterizes all the solutions of Harper's equation. Most importantly, one can derive that when
is an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, there are infinitely many solution for
.
The union of all
forms a self-similar fractal that is discontinuous between rational and irrational values of
. This discontinuity is nonphysical, and continuity is recovered for a finite uncertainty in
or for lattices of finite size.
The scale at which the butterfly can be resolved in a real experiment depends on the system's specific conditions.
Phase diagram, conductance and topology
The
phase diagram
A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...
of electrons in a two-dimensional square lattice, as a function of a perpendicular magnetic field,
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
and temperature, has infinitely many phases. Thouless and coworkers showed that each phase is characterized by an integral Hall conductance, where all integer values are allowed. These integers are known as
Chern number
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
s.
See also
*
Aubry–André model
References
{{Douglas Hofstadter
Fractals
Condensed matter physics
1976 introductions