An Institute of Physics booklet | September 2014
Mathematical
physics
What is it and
why do we need it?

uantum states, conceptual artwork.
Q
In physics, a quantum state is a set
of mathematical variables that fully
describes a quantum system, such
as the state of an electron within an
hydrogen atom at any given time
(Richard Kail/Science Photo Library)
Front cover image
Supersymmetry, conceptual artwork.
Superstring theory is an attempt
to explain all of the particles and
fundamental forces of nature in one
theory by modelling them as vibrations
of tiny supersymmetric strings
(David Parker/Science Photo Library)
2
Mathematical Physics What it is and why do we need it?
Contents
04
Mathematical physics. What is
it and why do we need it?
6 The synergy between
mathematics and physics
9 Mathematical physics – its
purpose and applications
10
Entanglement entropy –
characterising quantum
entanglement of many-body systems
11 Entanglement entropy
12 Quantum applications
14
From gravity to fluids
and back
15 Fluid/gravity correspondence
17 The nature of gravity
18
Holography, black holes and
superconductors
19 Superconducting materials
20 Black holes
21
Random matrix theory – linking
quantum chaos with the
Riemann hypothesis
22 The Riemann zeta function
23 Random matrix theory
24
Skyrmions
25 A mathematical model of
atomic nuclei
26 The application of Skyrme’s
model to larger nuclei
28
Topological insulators
– a new phase of matter
29 Topological insulators
30 Applications of a
not-so-new material
3
Mathematical
What is it and why do
“Don’t worry about your difficulties in
mathematics. I assure you mine are greater.”
Albert Einstein
p
4
A lbert Einstein (1879-1955),
German-born physicist. Famous for
his theories of relativity, Einstein has
become a cultural icon, his name
synonymous with genius. In 1905,
whilst working as a patent clerk, he
wrote four papers, including one
on special relativity. From this the
idea of a space-time continuum
followed and is represented here as
space warped into the shape of an
hourglass (Bill Sanderson/Science
Photo Library)
This quote attributed to Albert
Einstein rarely fails to raise a
smile. It is reassuring for those who
struggle with mathematics, but it
is also surprising. Even a cursory
brush with physics confirms that
this science is highly mathematical.
Whether it is describing the force
of gravity, as was Einstein’s goal,
understanding a simple light switch,
or designing sophisticated GPS
satellites, physicists need numbers
to measure and to quantify, and
they need mathematical equations
to describe the relationships
between physical objects and
the forces that act on them.
Mathematics is the indispensable
language of physics — which is why
Einstein’s admission strikes many
as amusing.

ravitational lensing. Optical CCD
G
(charge coupled device) image of a
large "luminous arc" associated with
Galaxy Cluster 2242-02. Several
theories have been put forward to
explain such arcs. Most popular of
these is that the arc is part of an
Einstein ring: the smeared-out image
of a distant galaxy or quasar whose
light has been "lensed" by the gravity
of the cluster of galaxies. Einstein
proposed the effect of gravitational
lensing in 1936 (NOAO/Science
Photo Library)
Mathematical Physics What it is and why do we need it?
physics
we need it?
5
The synergy
between mathematics
and physics
If all of physics is mathematical then
what is meant by “mathematical
physics”? The boundaries are
not clearly defined. Despite his
misgivings, Einstein himself could
be counted as a mathematical
physicist. His general theory
of relativity was not a result of
extensive experimentation, but
of theoretical and mathematical
considerations. Because it differs
from Newton’s classical theory of
gravity only when high energies and
massive bodies are involved, the
theory was difficult to test at the
time. But it has stood up admirably
to the experimental tests that have
been developed in the near century
since its inception.
Einstein’s work provides one of
many examples of the synergy
between mathematics and physics.
In the middle of the 19th century,
the pure mathematician Bernhard
Riemann developed a range of
new geometrical concepts. His
aim was to drive geometry to
abstract perfection; to render it
independent of how we happen
to perceive the physical space
around us. Decades later, Einstein
found a very physical application
for Riemann’s mathematics: it was
exactly what he needed to describe
the geometry of space and time.
Riemann’s geometry has now
become an essential prerequisite
for anyone wishing to understand
Einstein’s physics.
Mathematics does not just provide
tools for physics. It can also
drive physical insight. A striking
example is Paul Dirac’s prediction
of antimatter. Dirac was searching
for an equation to describe the
behaviour of electrons, tiny building
blocks of matter, taking account of
insights from special relativity and
the other great success story of early
twentieth-century physics, quantum
mechanics. When he found a
suitable mathematical expression he
realised it contained twice as many
pieces of information as necessary,
leading him to suggest that each
electron comes with an anti-particle,
Mathematics does not just provide the tools for
physics – it can drive physical insight. When
Paul Dirac found a suitable mathematical
expression to describe electrons, he discovered
the existence of positrons
6
the positron. Its existence, and that
of many other anti-particles, was
later confirmed in the laboratory.
Dirac’s efforts contributed vital
pieces to a jigsaw puzzle physicists
are still trying to complete
today, aiming to describe all the
fundamental forces and particles of
nature in one theoretical framework,
known as the Standard Model of
particle physics. The language of
this framework is mathematics,
and mathematical considerations
have led to the discovery of other
particles. A recent addition is the
Higgs boson, traces of which were
glimpsed at CERN in July 2012.
Its existence had been predicted
back in the 1960s, resulting from
a mathematical model devised
to explain events in the very early
universe that led to matter acquiring
mass. Two physicists, Peter Higgs
from the UK and François Englert
from Belgium, were awarded the
Nobel Prize in Physics 2013 for their
role in developing the theory that led
to the recent discovery.
positron
decays via two
gamma rays
Mathematical Physics What it is and why do we need it?
Quarks
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Leptons
The greatest missing piece of
the puzzle is the gravitational
force, which cannot yet be
accommodated in the Standard
Model. The nature of the
mathematical problems involved
has spawned string theory, a major
contender for a unified theory
of everything. It appeals through
its coherence and mathematical
elegance, but it will be many
years before experiments will be
available to test whether it provides
a correct model. Despite this, string
theory is promising to shed light
on the physics of real materials,
such as superconductors, that find
applications in a variety of contexts.
However, it is not just mathematics
that fertilises physics. Physical
considerations can pose difficult
mathematical problems which then
turn into active research areas in
their own right. For instance, in
1834 the Scottish engineer John
Scott Russell observed a curious
solitary wave on the Union Canal
near Edinburgh that could not be
captured by the existing theory of
hydrodynamics. It took 60 years for
physicists to develop an equation
describing the wave and nearly
another century to find a general
method for solving it. Their efforts
have spawned a vibrant area of
mathematics concerned with
 The

Standard Model
A48 experiment apparatus.
N
Engineers working on the
north area 48 (NA48) particle
detector at CERN (the European
particle physics laboratory)
near Geneva, Switzerland.
This apparatus was used for
the NA48 experiment to study
the direct effect of CP (charge
parity) violation, which may
be able to explain matter/
antimatter imbalance
(CERN/Science Photo Library)
6
4
I?(z)
2
0
-1
0
Re(z)
10
1
2
7
Im
Mathematical physics is best described as
consisting of two parts: physical research that
proceeds primarily through mathematical means
and areas of mathematics that work to solve the
problems posed by physics
certain types of equations and
their wave-like solutions known
as solitons. The results feed back
into physics, where solitons have
become ubiquitous. The skyrmions
explored in one of the case studies
in this booklet are examples of
solitons (page 24). What may
appear surprising is that they
describe the protons and neutrons
found in the nuclei of atoms.
Particle physics has also
influenced Riemann’s geometry,
which provided Einstein’s tools.
The equations used to describe
fundamental particles have a
geometrical interpretation and
come with a particular algebraic
structure, which captures the
symmetries physicists believe
govern the laws of nature. This has
resulted in vigorous research efforts
in an area of pure mathematics that
combines algebra and geometry.
Insights from quantum physics
have also nudged mathematicians
nearer to the solution of one of
8
the greatest unsolved problems in
pure mathematics, the Riemann
Hypothesis, first posed by
Riemann in 1859. It concerns the
prime numbers, considered the
fundamental building blocks of
number theory.
Considering this two-way
interaction, the field of
mathematical physics is best
described as consisting of two
parts: physical research that
proceeds primarily through
mathematical means and areas
of mathematics that work to solve
the problems posed by physics.
As such, mathematical physics
does not pertain to specific areas
of either of the two disciplines.
It is impossible to predict which
mathematical methods will find
applications in physics and what
kind of mathematical problems
will arise from physical research.
Mathematical physics is a dynamic
field full of surprises.

Hydrostatics. A page of illustrations
showing various aspects of
hydrostatic theory. At top left is a
diagram illustrating the calculation
of forces in uneven water surfaces,
at bottom left are two methods
of using weights to raise a water
column. The remainder are
illustrations of how water finds a
common level in joined vessels
open to the air irrespective of
their shape. This page was first
published in Daniel Bernoulli's
'Hydrodynamica' of 1738 (Royal
Institution of Great Britain/Science
Photo Library)
Mathematical Physics What it is and why do we need it?
Mathematical
physics – its purpose
and applications
But do we really need it? Its main
goals are to expand the boundaries
of knowledge and provide a
fundamental understanding of
physical reality. This is a fascinating
pursuit in its own right, but it is also
a prerequisite for discoveries and
applications that directly impact
on our lives. GPS technology would
not be possible without insights
from general relativity, which in
particular describes how the Earth’s
gravity distorts space and time.
brain research. The discovery of
topological insulators, one of the first
truly new materials to be discovered
in decades, was only possible due
to the light shed by a mathematical
understanding of materials.
A common feature of many such
applications is that they could not
have been predicted before the
underlying theory was developed
— they are the results of openended research conducted over
The discovery of topological insulators, one of the
first truly new materials to be discovered in
decades, was only possible due to the light shed
by a mathematical understanding of materials
Quantum physics is paving the
way towards superfast quantum
computers, opening up a new era for
information technology. The medical
sciences will continue to benefit
from progress in particle physics.
For example, the development of
high Tesla superconducting magnets
for CERN’s Large Hadron Collider
allows a better understanding of
the superconducting materials
used in MRI scanners. There are
now a small number of 7-Tesla MRI
scanners in clinical use worldwide,
and a tiny number of scanners with
extremely high field strengths have
been developed for state-of-the-art
a sustained period of time. In the
current economic climate this
type of research is under threat as
government and funding agencies
hope for more immediate returns.
This booklet presents some
examples of recent successes in
mathematical physics and their
potential impacts, highlighting
the UK’s role as a world leader
in this research area. If we want
to pursue fundamental research
in physics and benefit from the
economic impacts the resulting
technological innovations will bring,
we must ensure the UK maintains
its position as a world leader in
mathematical physics.
9
Entanglement
entropy –
characterising
quantum
entanglement of
many-body systems
Quantum entanglement is a very
counterintuitive idea. It is possible
to entangle two or more particles,
such as electrons or photons, so
that it seems that the particles are
connected; for example, a spin
measurement of a particle in an
experiment instantaneously affects
the others, whether they are in the
next room or the next galaxy. When
Albert Einstein first encountered the
concept of quantum entanglement,
he famously thought this “spooky
action at a distance” was proof that
the theory of quantum mechanics
was wrong. Despite Einstein’s
concerns, this effect has been
observed in experiments and
quantum mechanics has gone on
to be one of the most successful
scientific theories ever proposed.

Quantum entanglement is a key
resource in quantum information
processing and forms the bedrock
of any design of quantum
algorithms. However, to make the
most efficient use of this resource
we need to be able to quantify it
exactly and develop methods for
measuring entanglement in real
quantum systems.
uantum entanglement, computer artwork.
Q
One of the strangest consequences of the
quantum theories is that some quantum
events can become entangled. Two
particles will appear to be linked across
space and time, with changes to one
of the particles affecting the other one
(Harald Ritsch/Science Photo Library)
Quantum entanglement is a key resource in
quantum information processing and forms the
bedrock of any design of quantum algorithms

10
ehaviour of the entanglement entropy near the
B
transition of a one-dimensional quantum magnet.
One can see a peak in the entanglement entropy
(S) at the quantum critical point where the applied
magnetic field, lambda, equals 1
Mathematical Physics What it is and why do we need it?
Entanglement
entropy
In a simple quantum system of just
two particles for example, it is easy
to quantify how entangled the system
is. But it is far harder to measure
the amount of entanglement in a
system of many particles. Over the
last decade John Cardy (University of
Oxford) and his colleagues, including
Pasquale Calabrese (now at the
University of Pisa), have contributed
to the systematic study of exact
predictions for entanglement entropy,
a way of quantifying the degree of
entanglement of such many-body
systems. Their elegant approach uses
all the tools of modern mathematical
physics, including those developed
in quantum field theory, integrable
systems and string theory.
The spin of an electron can have one
of two values: it can point either up or
down. For a system of two electrons,
there are 22= 4 possible states
(shown in the figure, left). A quantum
system can occupy all these
states simultaneously.

The
spin of an
electron, a system
of two electrons,
there are 22= 4
possible states
Now, if two particles are not
entangled you can treat them
independently, ignoring the one when
making calculations for the other.
But the more entangled things are,
the harder the system is to simulate.
You have to take all the entangled
variables into account and solve
the relevant equations for all these
particles at the same time. One can
think of the entanglement entropy of
a system as a measure of how much
information is needed to simulate the
quantum system in a computer.
Numerical analysis had already
revealed that certain large quantum
systems were easier to simulate
than others, though it was not
known why. Entanglement entropy
explains this, with the calculated
value of entanglement entropy closely
agreeing with the numerical results for
the difficulty of simulating equivalent
quantum systems.
Any simulation of this quantum
system will need to allow for all these
possibilities at the same time. For
a system of N electrons, there are
2N possible states, a number that
increases rapidly with N, and all of
these must be encoded into any
simulation of the system.
11
Quantum
applications
This mathematical tool can also be
used to characterise the quantum
critical point of quantum systems,
for example, the point at which a
material becomes superconducting.
As the material nears this point the
electrons become more and more
entangled, something described
most clearly by the increase in the
superconductor’s entanglement
entropy. This has shed new light
on the processes involved in these
transitions. Another example is that
of a quantum magnet, composed
of many interacting spins. As the
applied magnetic field approaches
the critical point where the phase
transition occurs, a sharp peak
in the value of the entanglement
entropy can be seen (illustrated
in the figure, right). Measuring
the entanglement entropy could
provide a way to track the physical
process. Deeper understanding
of these materials, both
superconductors and quantum
magnets, will aid the development
of novel high-tech devices.
Entanglement entropy now informs
the numerical simulations of these
large quantum systems. Such
simulations are traditionally very
hard as you have to simultaneously
solve Schrödinger’s equations
describing their quantum state
for many particles. Understanding
the degree of entanglement of a
large quantum system indicates
how many particles need to
be considered: the smaller the
entanglement entropy the fewer
the particles that are needed to
simulate the system numerically.
There are hopes to test the
predictions of entanglement
entropy experimentally in the
future. Proposals are being
developed to measure the degree
of quantum entanglement of a
system, comparing it with the
predicted value of entanglement
entropy. Some of these
experiments involve the design
of quantum switches and other
components that potentially
could be used in the development
of quantum computing. The
measurement of entanglement
Quantum entanglement will help to characterise
a deeper understanding of superconductors and
quantum magnets, which will aid the development
of novel high-tech devices
12
Mathematical Physics What it is and why do we need it?
entropy in many-body systems
will be vital in understanding how
to scale up these components
to eventually build functioning
quantum computers.
Quantum computing will
revolutionise how we store,
process and use information. A
full-scale quantum computer has
the potential to perform certain
calculations significantly faster
than any silicon-based computer.
While a normal computer can only
work on one calculation at any
time, a quantum computer can
work on millions simultaneously.
Such a technology could have a
tremendous effect on financial
asset movement, potentially
leading to a more efficient financial
market and, perhaps, a more
predictable one.

uantum computer, conceptual artwork. Quantum
Q
computers, which are under development, are
based on quantum mechanics and the principle
of representing information using quantum
properties. Quantum computing has the potential
to massively increase computing power (Harald
Ritsch/Science Photo Library)
Entanglement entropy is an
example of how the abstract
tools of mathematical physics,
often developed for theoretical
mathematics with no hint of their
future application, are now integral
to the research and development of
essential future technologies.
13
From gravity to
fluids and back
If, as Einstein postulated, gravity is
the result of massive objects such
as planets and stars warping space
and time, then what can be said
about the nature of this malleable
spacetime? What can be said
about those ultimate gravitational
objects, black holes, which warp
spacetime to the extreme?
Over decades, mathematical
scrutiny of Einstein’s equations
has led to an interesting idea:
that aspects of spacetime might
resemble a fluid. If this is true,
then can they be captured by the
central equations of fluid dynamics,
developed nearly 200 years ago by
the French engineer and physicist,
Claude-Louis Navier, and the
Cambridge-based mathematician
14
and physicist, George Gabriel
Stokes? Conversely, can we address
open questions about fluids using
insights from gravity?
One of the most exciting recent
achievements in mathematical
physics has been to provide an
explicit correspondence between
Einstein’s equations and the
Navier-Stokes equations, which
works both ways. On the fluid side
it gives a new tool for studying one
of the last remaining questions of
classical physics, that of how to
describe turbulent fluid flow, and
also for studying substances that
seem to defy a description in terms
of classical physics. On the gravity
side it may give deep insights into
the nature of spacetime.
s
lack hole. Artwork of the
B
spherical region where light
is trapped around a black
hole, with surrounding
interstellar material (blue)
being pulled inwards (Victor
De Schwanberg/Science
Photo Library)
Mathematical Physics What it is and why do we need it?
Fluid/gravity
correspondence
The fluid/gravity correspondence
was formulated in 2008 by
Veronika Hubeny and Mukund
Rangamani (both at Durham
University), Shiraz Minwalla
(Tata Institute of Fundamental
Research in India) and Sayantani
Bhattacharya (now at the Indian
Institute of Technology in Kanpur).
The fluids to which it currently
applies are not ordinary ones.
They are nearly perfect fluids
Superfluids and the quark-gluon
plasma are also examples of socalled strongly coupled systems.
These are made of particles
interacting so strongly that they
can no longer be considered
individually. Such systems
lack a coherent theoretical
description, but they may find
analogues on the gravity side of
the correspondence, represented
by mathematical descriptions of
black holes.
Superfluids to which the fluid/gravity correspondence
applies are not ordinary, some seeming to defy the
laws of physics by climbing up walls
which exhibit little viscosity and
heat conduction. Examples are
superfluids, such as liquid helium
cooled to very low temperatures,
which seems to defy the laws of
physics, appearing to self-propel
and climb up walls. Another
example is the mysterious quarkgluon plasma, a particle soup
thought to have last existed in
nature right after the Big Bang
13.7 billion years ago, and which
has recently been produced in tiny
quantities in the laboratory.
The equations here can be more
tractable and the hope is that the
correspondence will illuminate how
these systems work.
Over recent years, hundreds of
research projects around the
world have exploited the fluid/
gravity correspondence. On the
topic of fluid motion, researchers
have used it to tackle a regime in
which the traditional macroscopic
approach breaks down. In this
non-hydrodynamic regime, which
includes turbulence, the velocity or
temperature of the fluid shows large
15
variations on very small scales, so
a microscopic description would, in
principle, be needed. In 2012 Pau
Figueras (University of Cambridge)
and Toby Wiseman (Imperial
College London) used solutions
to Einstein’s equations, which
describe a new type of theoretical
black hole, to derive for the first
time a description of both the
hydrodynamic and part of the nonhydrodynamic regime for a certain
class of fluids. In 2013, Paul
Chesler, Hong Liu and Allan Adams
(all at MIT) used gravity to derive,
from first principles, a description of
turbulence in a class of superfluids
flowing on a surface. Surprisingly,
their work predicts that in this
16
turbulent motion large structures
tend to break down into smaller
ones, just like a uniform stream of
smoke breaks down into smaller
eddies. That is the exact opposite
of what happens to ordinary fluids
flowing on surfaces. Here smaller
structures tend to merge to form
larger ones.
a circle to its circumference. It has
been suggested that 1/(4pi) is a
universal bound for this characteristic
in all fluids, although this idea has
been under active debate. Using
the fluid/gravity correspondence
Bhattacharya, Hubeny, Minwalla and
Rangamani have derived the same
value in a simpler way than had
previously been possible, providing
Gravitational insights may also
a new angle on the problem. They
give results in connection with the hope that the correspondence will
quark-gluon plasma. A physical
help to shed more light on this poorly
characteristic of the viscosity of the understood quark-gluon plasma.
plasma (the viscosity to entropy
density ratio) can be described in
terms of the number 1/(4pi), where
pi is the famous mathematical
constant relating the diameter of
Mathematical Physics What it is and why do we need it?
The nature of gravity
Understanding strongly coupled
systems is a major aim researchers
are hoping to achieve, but the
fluid/gravity correspondence also
raises questions about the nature
of gravity.
s Turbulence
is a common
phenomenon in nature.
Hurricane Katrina
seen from NASA’s Terra
satellite (NASA)
How should we interpret concepts
such as turbulence in gravitational
systems? Physicists are now taking
the first steps towards answering
this question by identifying
properties of black holes that
correspond to turbulence in fluids.
An example is work published
in 2013 by Adams, Chesler and
Liu, which uses the fluid/gravity
correspondence to construct
“turbulent” black holes. The
identifying feature uncovered by the
researchers was an approximately
self-similar structure in the horizons
of their black holes: they look
the same at several scales of
magnification.
The black holes that are involved
in this kind of research are not
real astrophysical objects. They
are mathematical descriptions of
black holes that exist in unusual
mathematical spaces. But the hope
is that results like these will shed
light on aspects of gravity in our own
world that we do not yet understand.
At the same time researchers are
working to extend the fluid/gravity
correspondence to capture less
exotic fluids and spacetimes.
The overarching goal of this research
is to understand the physics that
governs our universe, from the
smallest to the largest scales. What
applications will eventually arise
from it cannot be foretold. A better
understanding of strange metals and
superfluids may lead to materials
that become superconducting
at high temperatures, but it may
also open up uses we cannot yet
imagine. A theoretical description
of turbulence may ultimately find
applications in the full range of
contexts in which this motion occurs,
from aircraft design to meteorology.
The most fascinating question
remains the fundamental one: the
fluid/gravity correspondence hints
towards a type of universality in
nature we do not yet understand.
Further research is needed to
illuminate its true meaning.
A better understanding of strange metals and
superfluids may lead to materials that become
superconducting at high temperatures, but it may
also open up uses we cannot yet imagine
17
Holography,
black holes and
superconductors
Superconductors have intriguing
properties: these materials
exhibit zero electrical resistance
when cooled below a critical
temperature and they expel
magnetic fields. The applications
are broad; superconductors are
used in MRI scanners, particle
accelerators like those at CERN,
and to levitate trains, to name
just a few. However, the critical
temperatures are very low and the
cost of the required cooling limits
their use. An important aim is to
find a material that superconducts
at room temperature. A step in this
direction is to try to understand
the quantum properties of
a class of known, but still
mysterious, "high temperature
superconductors". This is
where recent research using an
interesting principle that emerged
from the mathematical physics of
string theory is showing promise.
Demonstration of magnetic
levitation of one of the new hightemperature superconductors – yttrium
barium copper oxide. Discovered
in 1986, the new superconducting
ceramic materials are expected to
lead to a technological revolution and
are the subject of intensive worldwide
research. The photograph shows a
small, cylindrical magnet floating freely
above a nitrogen-cooled, cylindrical
specimen of a superconducting
ceramic (made by IMI Ltd). The glowing
vapour is from liquid nitrogen, which
maintains the ceramic within its
superconducting temperature range.
Photographed at the University of
Birmingham (David Parker/Science
Photo Library)
t
18
Mathematical Physics What it is and why do we need it?
Superconducting
materials
 The
Japanese JR–Maglev MLX01-1 uses superconductors (Daylight9899)
Superconductivity was first
discovered in 1911 by the Dutch
physicist Heike Kamerlingh Onnes,
who observed it in mercury. By the
1960s physicists had found many
more examples of superconducting
materials and had developed a
comprehensive understanding
of the sub-atomic quantum
processes that made these
“conventional” superconductors
work. In 1986, however, new
types of superconductors were
discovered with much higher
critical temperatures, which could
not be described by the existing
theory. Since then physicists
have proposed a range of
different approaches to explain
the fundamental principles
underlying these “high temperature
superconductors”. However,
despite an enormous amount of
work, there is no consensus as to
which approach, if any, is correct.
Radically new paradigms may be
needed, one of them potentially
emerging from string theory,
which provides a mathematical
description of the world aiming at
unifying gravity with the other three
An important aim in superconductor research is to
find a material that superconducts at room
temperature. A step in this direction is to try to
understand the quantum properties of a class of
known, but still mysterious, "high temperature
superconductors"
fundamental forces of nature: the
electromagnetic, the weak nuclear
and the strong nuclear forces. While
the elegance of string theory is
highly appealing, technology has
not yet advanced far enough to test
it in experiments.
Nevertheless, in 1997 a remarkable
observation within the mathematics
of string theory was made by the
physicist Juan Maldacena (now
at the Institute for Advanced
Study in Princeton), which led
him to conjecture a far-reaching
correspondence between two
apparently very different theories.
He developed a mathematical
toy universe which, by very loose
analogy, can be thought of as an
inflated balloon. The physics on
the boundary of that universe,
corresponding to the surface of the
balloon, is described by a quantum
theory with no gravity. The physics in
the interior is described by a string
theory, which does incorporate
gravity. Remarkably, these two
physical systems are equivalent:
the quantum theory defined on the
boundary is sufficient to describe
all the physics in the interior and
vice-versa. One direction of this
two-way correspondence has given
Maldacena’s discovery its name:
it is known as the holographic
principle because the boundary
physics captures the interior physics
just as a 2D hologram captures a
3D image.
19
Black holes
It is this principle that physicists
are now using to try to understand
superconductors. A beautiful feature
is that the thermal properties of the
boundary world can be described
in terms of the mathematical
properties of theoretical black holes
in the interior of the model, as we
have seen already in the fluid/gravity
context. The results are intriguing:
the phase transition that turns a
material into a superconductor at
the critical temperature, represented
in the boundary of the model,
corresponds to the existence of a
new kind of theoretical black hole
in the interior of the model, which
comes with a halo of charged matter
around it. It is the properties of this
halo that characterise the black
hole, via the holographic principle,
as a superconducting phase.
The first breakthroughs regarding
superconducting black holes
were made in 2008 by Steven
Gubser (Princeton University)
and by Sean Hartnoll (Stanford
University), Chris Herzog (now at
Stony Brook University) and Gary
Horowitz (University of California,
Santa Barbara). Shortly afterwards,
in 2009, Jerome Gauntlett and
Toby Wiseman (both at Imperial
College London), and their former
colleague Julian Sonner, now at
MIT, demonstrated that such black
holes exist within the mathematical
framework of string theory. These
findings, and others, have set off
a flurry of activity in the area, with
encouraging results. Research has
revealed theoretical black holes
whose properties share similarities
with properties seen in real
materials. For example, Gauntlett
and Aristomenis Donos (University
of Cambridge) have recently found
theoretical black holes with a
halo of matter exhibiting a striped
pattern. A similar striped phase,
where the electric charge density
has a periodic behaviour, has
been observed in several materials
including high temperature
superconductors.
Research has revealed theoretical black holes
whose properties share similarities with properties
seen in real materials
20

lack hole, artwork. Technically, a
B
black hole is a region of spacetime
where, by nature of its great
mass, gravity prevents anything
from escaping; this includes light
(Henning Dalhoff/Science Photo
Library)
Further research is necessary to
ascertain whether this approach will
lead to the ultimate insight into the
physics behind high temperature
superconductors, but it may provide
the basis for a new paradigm. What
is more, many other types of exotic
black holes are being discovered
in string theory, and via the
holographic principle these point to
an untold variety of new materials
with exotic properties yet to be
discovered. This demonstrates
the power of mathematical
physics: its sophisticated
mathematical techniques find
applications in physical contexts
far removed from the ones in which
they were developed.
Mathematical Physics What it is and why do we need it?
 Computer generated
image of a mathematical
saddle (Riemann
surface). It is a way of
representing a form
of space where the
conventional rules of
geometry no longer apply
(Alfred Pasieka/Science
Photo Library)
Random matrix
theory –
linking quantum chaos with
the Riemann hypothesis
In 1859 the mathematician
Bernhard Riemann posed a
hypothesis that remains one of the
most important unsolved problems
in mathematics. Mathematicians
have been studying prime
numbers for millennia as they
are the atoms of arithmetic – any
integer can be written as a unique
product of prime numbers. Much
is known about primes and they
have become a vital part of the
economy due to their role in the
encryption of digital information
used in e-commerce and secure
communication. Yet despite our
reliance on prime numbers there is
much we still do not understand.
In particular, a fundamental
question remains open: how are
prime numbers distributed along
the number line? In the 1970s
a surprising connection was
discovered between this purest
area of mathematics and quantum
physics. This analogy has enabled
mathematical physicists to
give new insights into the
Riemann hypothesis.
Much is known about primes and they have
become a vital part of the economy due to their
role in the encryption of digital information used
in e-commerce and secure communication
21
 A
wave of the type occurring
in quantum chaos,
associated with a classically
chaotic system with a
downward force – quantum
chaos of Galileo's falling
body (Sir Michael Berry/
University of Bristol)
The Riemann zeta function
Riemann considered a particular
function, the Riemann zeta function,
which describes the distribution of
the prime numbers. The points at
which this function is zero can be
thought of as the sea level points
in a landscape defined by the
function. The Riemann hypothesis
states that these points, rather than
being scattered over the landscape,
all lie in a straight line (called the
critical line). If this is true, it gives
important information about prime
numbers; Riemann himself wrote a
formula connecting the zeroes with
the primes.
In the 1960s a similar formula
was developed in an entirely
different area. Dynamic physical
systems (for example, electrons
moving in atoms or molecules,
or light moving in a laser) can be
described in two ways: using the
classical physics developed by
Newton or using the newer theory
of quantum mechanics. The formula
linked the energy levels found in
such systems when described
by quantum mechanics to the
periodic trajectories one finds
when describing the systems using
Newtonian physics.
Random matrix theory has crossed the boundary of
mathematical physics and made a huge impact on
a completely different field, connecting deep ideas
between mathematics and physics giving novel
insights in both areas
22
Mathematical Physics What it is and why do we need it?
Random
matrix
theory
The two formulae looked remarkably
similar: the Riemann zeroes were
like the quantum energy levels and
the primes were like the classical
trajectories. Furthermore, these
formulae suggested the classical
orbits in question should be chaotic.
This connection deepened in the
1980s when Sir Michael Berry
(University of Bristol) discovered
that the distributions of the
quantum energy levels of all chaotic
systems were statistically the same.
Remarkably, the Riemann zeroes
had the same statistical distribution.
This universality meant that all
these systems look essentially
the same and could be described
using random matrix theory. This
statistical universality applies at
short range. Over longer ranges, the
same 1980s theory predicted nonuniversal deviations from random
matrix theory, also for the Riemann
zeroes, where quantum chaos theory
gave a detailed description, new to
mathematicians, of their statistics.
In 2000 Jon Keating and Nina
Snaith (both at the University of
Bristol) used this connection to
shed new light on the Riemann
zeta function. Any function can be
characterised by its moments – a
sequence of numbers that describe
its shape to greater and greater
degrees of accuracy. The first
moment can be thought of as the
mean, the average height of the
function; the second moment the
variance, the range of fluctuation in
the height; and higher moments give
more detailed information about the
fluctuations themselves.
Number theorists had been
struggling for nearly a hundred years
to calculate the higher moments of
the Riemann zeta function along the
critical line. The first two moments
had been calculated by the mid1920s but there was then little
progress until the 1990s. At this
point number theorists had guessed
the next two moments, giving the
sequence: 1, 2, 42 and 24024.
However, the methods they had used
failed beyond this point and they
could not generate any sensible
answers for the next numbers in
the sequence.
Using the analogy with quantum
chaos, Keating and Snaith expressed
this problem in terms of random
matrices. Not only were they able
to calculate the moments already
known, they developed a general
formula for the whole sequence.
Their solution to this long-standing
problem in number theory revealed
that the sequence of moments
had surprising mathematical
properties, which have deepened
p
iemann zeta function in the critical
R
strip, showing the zeroes where the
colours meet (Sir Michael Berry/
University of Bristol)
our understanding of the Riemann
zeta function and the Riemann
hypothesis.
This result has led to a new and
stimulating collaboration between
number theorists and mathematical
physicists. After making these
calculations of the moments, Snaith
and Keating have gone on to answer
more general questions in number
theory, such as how quickly the
Riemann zeta function grows in
height, a question that is similar in
spirit to the Riemann hypothesis.
New questions about the Riemann
zeta function are now being studied
as a result of known properties of
the analogous quantum chaotic
systems. Similarly, known results
about the Riemann zeta function
have prompted new directions in the
study of quantum chaos. Random
matrix theory has crossed the
boundary of mathematical physics
and made a huge impact on a
completely different field, connecting
deep ideas between mathematics
and physics giving novel insights in
both areas.
23
Skyrmions
What does the nucleus of a carbon
atom look like? Given the success
of nuclear physics, be it in power
generation or medical applications,
it may come as a surprise that we
still do not have a detailed answer
to this question. Atomic nuclei are
too small to observe directly and
the fundamental theory describing
the smallest building blocks of
matter, quantum chromodynamics
24
(QCD), does not allow concrete
calculations of nuclear properties.
It gives us the theory of matter, but
the equations are so difficult to
solve that even simple questions,
such as how protons and neutrons
(collectively called nucleons) are
configured inside a nucleus, remain
unanswered. To study nuclei directly
from QCD would require immense
computing resources that are
beyond current capabilities, and
are likely to remain so for some
time. Applications, such as nuclear
power, hinge on approximate
models that are disconnected from
the fundamental theory. These work
well, but much can be gained from
a true understanding of the nature
of nuclei and sub-atomic particles.
Mathematical Physics What it is and why do we need it?
t
Simulation
results showing a Skyrmion
in the magnetisation vector field in a
chiral ferromagnetic material at the
nanoscale (Mark Vousden
and Hans Fangohr/University
of Southampton)
A mathematical
model of
atomic nuclei
Sometimes the path to such an
understanding winds in surprising
ways. In the early 1960s the
British physicist Tony Skyrme, then
Senior Principal Scientific Officer
in the Theoretical Physics Division
of the Atomic Energy Research
Establishment (AERE) at Harwell,
devised a mathematical model
of atomic nuclei, based on a
novel idea that provided a unified
description of matter and force. The
nuclear force is described by wave-
analogue of the nuclear force. The
elastic band can also be twisted,
which is the analogue of the
particle-like solutions describing
nucleons. Skyrme discovered the
particle-like solution with a single
twist (now known as a skyrmion)
and identified the number of twists
(the skyrmion number) with the
number of nucleons. Studying
nuclei requires finding solutions
containing several skyrmions. This
is a difficult problem that is of
In an attempt to make explicit
calculations more feasible,
scientists tried to simplify QCD
using novel approximations.
Remarkably, Edward Witten
(Institute for Advanced Study in
Princeton) found that the Skyrme
model emerged from these
approximations, together with an
understanding, directly from
QCD, of Skyrme’s identification
of the particle-like solution with
the nucleon.
We still do not know what the nucleus of a carbon
atom looks like. Applications such as nuclear power
hinge on approximate models that are disconnected
from the fundamental theory
like solutions of the theory, but
Skyrme recognised that the theory
also has particle-like solutions
that describe nucleons. A useful
analogy is to consider a piece of
elastic band held at both ends.
If one end is wiggled then waves
carrying energy move along the
elastic band: these waves are the
mathematical interest in its
own right, with fascinating
geometrical aspects.
Despite its ingenuity, Skyrme’s
model was quickly superseded
by the newly-fledged theory of
QCD, but in the 1980s it made
an unexpected reappearance.
25
The application
of Skyrme’s model
to larger nuclei
In the 1990s researchers started
applying the Skyrme model to
larger nuclei, beyond the single
proton and neutron. Among these
researchers were the mathematical
physicists Richard Battye (University
of Manchester), Nicholas Manton
(University of Cambridge) and
Paul Sutcliffe (Durham University),
and the pure mathematician
Sir Michael Atiyah (University of
Edinburgh). According to classical
physics protons and neutrons can
be thought of as rigid spheres that
cluster together to form the nucleus
of an atom. In Skyrme’s model the
relevant question was how several
skyrmions, representing protons
and neutrons, would combine to
form a larger nucleus.
Although Skyrme’s equations
are simpler than those of QCD,
significant computing power
was still needed to answer this
question. Battye and Sutcliffe set
the Cosmos supercomputer at the
University of Cambridge to work for
weeks at a time. The pictures they
saw emerging came as a pleasant
surprise. Skyrmions merged
to form beautiful symmetric
structures, shapes related to the
famous Platonic solids described
over 2,000 years ago.
Why these particular shapes
should form was initially a mystery.
Each shape corresponds to a
nucleus with a given number of
basic skyrmions (representing
protons and neutrons) but there
was no obvious explanation for
the relation between the number
of skyrmions and the shape that
forms. For example, when four
skyrmions merge you might expect

26
kyrmions with skyrmion
S
numbers 1, 4, 12, 32,108
(Dankrad Feist and Chris Lau/
University of Cambridge)
Mathematical Physics What it is and why do we need it?
p A
system of 22 skyrmions
in a 2D magnetic film with
periodic boundary conditions
(Mark Vousden, Marijan Beg,
Hans Fangohr/University
of Southampton)
the number four to correspond to
a defining feature of the resulting
shape. Yet the shape that formed
was a cube, whose defining
features, its faces, vertices and
edges, come in the numbers six,
eight and twelve, respectively, but
not four. Battye and Sutcliffe would
play a game trying to guess what
shape was going to emerge after
starting with a certain number of
skyrmions. Often the result was a
surprise and more interesting than
they had imagined.
The explanation was found by
revealing a hidden mathematical
structure. The shapes correspond
to the symmetries of a class
of mathematical objects that
lie concealed within Skyrme’s
model. There is also a physical
consequence of these symmetries:
they predict the possible spin
states of the nucleus that is being
modelled. These are nuclear
properties that can be measured
in the laboratory. It is gratifying that
the sophisticated mathematics at
the heart of Skyrme’s model, with
the aid of geometry, can link up real
nuclear data.
Skyrme’s model remains a
simplification of QCD and we
cannot be sure that real nuclei
always have the predicted
structure. But the agreement with
experimental data is encouraging.
Recent results provide new insights
into some of the most abundant
elements in nature, such as helium,
lithium and carbon, and how these
vital elements formed from the
primordial soup of particles in the
very early universe.
But this is not all. The description
of nuclei in terms of skyrmions has
recently been obtained within string
theory, so different approaches are
converging to provide a growing body
of evidence in support of Skyrme’s
idea. Related skyrmions also appear
in the completely different context of
magnetic materials at the nanoscale.
These have potential applications
in data storage: the skyrmions’ twist
can be used to represent the ‘0’ and
‘1’ bits that underlie information
technology; the presence of a
skyrmion in a medium denotes
‘1’ and the absence denotes ‘0’.
Since a skyrmion’s twist cannot be
undone, skyrmions may provide a
robust alternative to conventional
storage devices; the latter become
unstable as devices get smaller and
their capacity increases. Whether
it will be possible to manipulate
large numbers of skyrmions in the
necessary manner is still unclear,
but a team of researchers led by
Roland Wiesendanger (University of
Hamburg) have recently taken the
first steps by “writing” and “deleting”
single magnetic skyrmions on an
ultrathin magnetic film.
27
Topological
insulators –
a new phase of matter
The discovery of a new type of
material is a rare and precious
thing. As well as deepening our
understanding, new materials
spur research, and innovative
applications drive the development
of future technologies.
Mathematical physics has led the
way in discovering a new class of
materials with novel properties
that promise to revolutionise
the area of electronics and
quantum computation.
For more than 80 years physicists
believed they had a solid
understanding of how insulators
worked. The Swiss physicist
Felix Bloch solved Schrödinger's
equations for the wave functions
of electrons in a solid in 1928,
revealing that the wave functions of
the electrons form discrete energy
bands. The lower energy bands in
an insulator are completely filled
with electrons and there is a large
gap to the next band: insulators do
not conduct electricity as there is
no way for the electrons to flow.

28
uantum waves in topological
Q
insulators. Computer model
showing interference patterns
formed by quantum waves in a
type of new material known as a
topological insulator. This is an
example of computational and
quantum models being used to
predict the properties of new
materials (Ali Yazdani/Science
Photo Library)
Mathematical Physics What it is and why do we need it?
Topological
insulators
There is, however, a subtly different
solution to Bloch's equations
than this ordinary band insulator.
In a flurry of papers appearing
within just a few weeks in 2006,
several groups of mathematical
physicists independently predicted
the existence of a new type of
insulator – a topological insulator.
Such insulators have surprising
properties: while the interior looks
like the ordinary band insulator,
the surfaces conduct electricity.
These surface states are also
incredibly robust: it does not matter
where or how the material is cut,
the resulting surface will always
be conductive. Moreover, unlike
the surfaces of ordinary metals
that lose conductivity as they
oxidise or corrode, the surface of
a topological insulator will always
conduct electricity, even in the
presence of impurities and defects.
This is because the conductive
states at the surface of these
materials are topologically
protected. Topology is the area of
mathematics that is concerned
with how things are connected:
two shapes are topologically
equivalent if you can bend or
stretch one into the other, without
cutting or tearing. The most
famous example is that a coffee
cup is topologically the same as
a donut. Topological insulators,
however, are not concerned with
the shape of the physical material
but instead the shapes of the
mathematical forms that describe
the states of the electrons. The
mathematical description of the
electron states in a band insulator
can be transformed (in an exactly
analogous way to bending and
p Topological
insulators are a new
state of matter, topologically
distinct from ordinary insulators
(Yulin Chen/University of Oxford)
stretching a coffee cup into a
donut) to one that essentially ties
each electron to a specific atom.
However, this cannot be done for
the mathematical descriptions of
the electron states on the surface
of a topological insulator. These
electrons are fundamentally mobile
as their mathematical forms
are topologically different to the
electrons in a band insulator.
Unlike the surfaces of ordinary metals that lose
conductivity as they oxidise or corrode, the surface
of a topological insulator will always conduct
electricity, even in the presence of impurities
and defects
29
Applications of a
not-so-new material
What is surprising is that these
topological insulators turned out
not to be new materials after all,
although the properties inferred
from revisiting Bloch’s equations
were definitely novel. Indeed, a
few years after the theoretical
prediction of these properties came
their experimental confirmation,
and the materials found with
these properties were available
on the shelf of any chemistry
laboratory and were commonly
used in industrial processes. A
deeper understanding of how
these materials work will impact on
their existing use in academic and
industrial settings.
p Artist's
The novel properties of topological
materials also promise to
revolutionise electronics. Over
the last decades electronic
components have shrunk in
size, enabling them to run at
increasingly faster speeds.
However, there is a limit to how far
this miniaturisation can go: the
electric current passing through
these components generates heat
and they must be of a certain size
in order to dissipate that heat
energy. However, spin currents,
which transfer the spin state of
an electron rather than its electric
charge, do not produce heat.
Spintronics, electronics based on
30
impression of an electron wave being transmitted through
a step edge (Courtesy: Ali Yazdani/Princeton University)
these spin currents, should lead
to decreased chip size and, hence,
faster computer processors.
Topological insulators are an ideal
material to build such spintronic
components. On the surface of these
materials the spin of an electron is
coupled to its electric charge: the
direction of the spin of the electrons
is always perpendicular to the
direction of the electric current. This
property allows the manipulation of
spin currents and these materials
could form part of the first
functioning spintronic components.
Mathematical Physics What it is and why do we need it?
a
It is highly likely that topology
plays an important part in many
materials. Steve Simon (University
of Oxford) and Rahul Roy (now at
the University of California, Los
Angeles) have been working on
classifying topological materials
using mathematical techniques
first developed in string theory
in the 1980s. Roy is one of the
leading theorists in the field and
received the 2010 McMillan Award
for his prediction of 3D topological
insulators. Simon is part of the
UK wide collaboration, TOPNES
b
One of the greatest challenges in
quantum computing is isolating a
system so that it can maintain a
particular quantum state. Simon
and Fiona Burnell (the latter now
at the University of Minnesota)
are examining if the topological
properties of these materials
can be harnessed to create a
topological quantum computer.
Such quantum systems promise
to be far more stable due to the
topological protection inherent in
these materials.
c

( a) The spin of electrons on
the surface is correlated
with their direction of motion
(b) The lattice structure
of bismuth telluride and
the predicted relativistic
"Dirac cone-like" electronic
structure formed by the
surface electrons
(c) The electronic structure
measured by angleresolved photoemission
that confirmed the
theoretical prediction and
the topological nature of
bismuth telluride (Yulin
Chen/University of Oxford)
Mathematical physics will lead the way to new
materials, guiding the work of experimentalists
in identifying the most promising areas for
future discoveries
(Topological Protection and NonEquilibrium States in Strongly
Correlated Electron Systems)
that aims to provide a theoretical
and experimental foundation for
quantum mechanical electronics.
As well as providing promising
avenues for producing smaller
and faster computer components
using spintronics, this collaboration
is also investigating new
materials and components for
quantum computing.
The prediction and discovery of
topological insulators has been one
of the most important advances
in condensed matter physics in
the last 20 years and promises to
transform this field. It is also an
illustration of a growing trend, in
which mathematical physics will
lead the way to new materials,
guiding the work of experimentalists
in identifying the most promising
areas for future discoveries.
31
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Published: September 2014
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