Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Quantum Statistical Mechanics: Overview

L Triolo, Universita

di Roma ‘‘Tor Vergata’’,

Rome, Italy

Introduction

Quantum theory actually started at the beginning of

the twentieth century as a many-body theory,

attempting to solve problems to which classical

physics gave unsatisfactory answers.

This article aims to follow the developments of

quantum statistical mechanics, hereafter called

QSM, staying close to the underlying physics and

sketching its methods and perspectives. The next

section outlines the historical path, and the first

achievements by Planck (1900) and Debye (1913);

the subsequent free quantum gas theory will be

recalled in the first original insights due to Fermi,

Dirac (1926) and Bose, Einstein (1924–25), when

many open problems began to find a coherent

treatment.

In this framework, an interesting new idea

appeared: the elementary units of the systems could

be ‘‘particles’’, in the usual or in a broad meaning, a

notion which includes photons, phonons, and

quasiparticles of current use in condensed matter

physics. The description of a classical harmonic

system through independent normal modes is an

example of a very fruitful use of collective variable s.

The subsequent section will deal with more recent

achievements, related to the properties of quantum

N-body systems, which are fundamental for the

derivation of their macroscopic behavior. In parti-

cular, the works by Dyson–Lenard and Lieb–

Lebowitz on the stability of matter have to be

recalled: a system made of electrons and ions has a

thermodynamic behavior, thanks to the quantum

nature of its constituents, where the Pauli exclusion

principle plays an essential role .

We will then present relations that arise in

quantum field theory, that is, from the second

quantization methods; related technical and concep-

tual problems will also be presented briefly.

This is necessary for taking into account the

recent works and perspectives, which will be

considered in the last section. Here the new inputs

and challenges from outstanding achievemen ts in

physics laboratories will be taken into account,

referring to some exactly solvable models which

help in understanding and in fixing the boundaries

of approximate methods.

The Crisis of Classical Physics:

The Quantum Free Gas

Let us briefly recall some of what Lord Kelvin called

the ‘‘nineteenth century clouds’’ over the physics

of that time (1884), and the subsequent new ideas,

(Gallavotti 1999).

It is well known that the classical Dulong–Petit law

of specific heat of solid crystals may be derived from

the model of point particles interacting through

harmonic forces; the equipartition of the mean energy

among the degrees of freedom implies, for N

particles, the linear dependence of the internal energy

on absolute temperat ure T, hence a constant heat

capacity C

is the Boltzmann’s consta nt)

¼ 6 

T; C

:¼

¼ 3Nk

½1

Experimentally this is relatively well satisfied at high

temperatures but it is violated for low T: one

observes that U

vanishes faster than linearly as T

goes to zero, so that C

vanishes. Moreover, the

contributions to the heat capacity from the internal

degrees of freedom of the molecular gases or from

the free electrons in conducting solids are negligible,

at room temperature: these degrees of freedom, in

spite of the equipartition principle, seem frozen.

The analysis of the blackbody radiation problem

from the classical point of view, that is, using

equipartition among the normal modes of the

electromagnetic field in the ‘‘black’’ cavity at

temperature T , gives the following dependence,

Rayleigh–Jeans law (1900), of the spectral energy

density u(, T), on frequency  and temperature T

(c is the speed of light in vacuum):

uð;TÞ¼

8

T ½2

The experimental curves for any positive T show a

maximum for a frequency 

max

(T) which increases

linearly with T according to Wien’s displacement

law (1893). The spectral energy density decreases

fast enough to zero as  !1 in such a way that the

overall (integrated) energy is (finite and) propor-

tional to T

, according to Stefan’s law (1879); the

agreement with the classical form holds for low

frequencies. The analytic form of the classical

u(, T)in[2] does not present maxima and the

overall radiated energy is clearly divergent (this bad

behavior for large , present in many formulas for

other models, sometimes in the corresponding

‘‘short-distance’’ form, is called an ‘‘ultraviolet

catastrophe’’).

302 Quantum Statistical Mechanics: Overview

The effort by M Planck (1900) to understand the

right dependence of u from  and T was based on a

thermodynamic argument about the possible

energy–entropy relation, and on an assumption

similar to the discretization rules on which the

‘‘old quantum theory’’ for the atomic structure is

based. The electromagnetic field is represented, via

Fourier analysis, as a set of infinitely many

independent harmonic oscillators, two for every

wave vector k, to take into account the polarization.

The frequency depends linearly on the wave number

k = jkj (linear dispersion law), and the spacing

becomes negligible for macroscopic dimensions of

the cavity. The key idea for computing the partition

function is the discretization of the phase space of

each oscillator (of frequency  = !=2). Putting

there the adimensionalized Lebesgue measure

dp dq=h, where h is a constant with physical

dimensions of an action, we consider the regions

bounded by the constant-energy ellipses and

their areas jR

j, and find

j¼

dpdq

2E

h

If these adimensional areas have integer values, that

is, E = nh, n = 0, 1, 2, ..., the annular region (‘‘cell’’

) between R

and R

nþ1

has unit area and so we

approximate the partition function with the series

( = 1=(k

T), the ubiquitous parameter in statisti cal

mechanics, often called ‘‘inverse temperature’’)

discr

expðnhÞ¼

1  expðhÞ

In this way, the probabilistic weight given to this

cell is

pðC

Þ¼

expðnhÞ

expðjhÞ

¼ expðnhÞð1  expðhÞÞ ½3

A well-defined value for the constant h (i.e., h =

6.626 ... 10

27

erg s, the Planck constant), com-

bined with the usual computation for the density of

states, gives a formula which quantitatively agrees

with experimental data (see Figure 1)

uð;TÞ¼

8

h

expðh=k

TÞ1

½4

Moreover, for a certain range of parameters, that

is, such that h 1, there is agreement with the

classical law.

The ‘‘quantum of light,’’ introduced by Einstein in

1905 in his work on photoelectric effect, was later

(1926) called photon by G N Lewis. The picture for

representing the radiating system was one of a gas

of noninteracting photons, carrying energy and

momentum, and being continuously created and

absorbed.

A sligh tly different approach was used about the

same time, for the proble m of specific heat of

crystalline solids.

The simpler model considers N points on the

nodes of the lattice Z

, in a cubic box of side L, and

interacting through harmonic forces; similarly to the

radiation problem, the system is represented by a

collection of independent harmonic oscillators (nor-

mal modes), which are ‘‘quantized’’ as before: the

corresponding quanta were called phonons (by

Fraenkel, in 1932) for the role of the acoustic band

of frequencies. In this simplified approach (by

Debye, in 1913) the different phonons are deter-

mined by a finite set of wave vectors

k ¼

2

n; n

integer; i ¼ 1; 2; 3; jkjk

where the maximal modulus k

is such that the

total number of different k’s is 3N (degrees of

freedom).

Moreover, the frequency–wave number relation is

simplified too, extrapolating the low-frequency

(acoustic) linear relation  = jkjv

is the sound

speed). In this way, the density of states which is

quadratic in the frequency, has a cutoff to zero at

the maximal frequen cy, 

, corresponding to

j, with an associate temperature 

= h

(Debye’s temperature). The expected energy U

the canonical ensemble, after the computation of the

canonical partition function, is given in term of the

Debye function D():

¼ 3Nk





DðyÞ:¼

exp x  1

½5

u (ν,T

)

Figure 1 Dependence of the electromagnetic energy density

on , for T

< T

Quantum Statistical Mechanics: Overview 303

The agreement with experimental data, for the

specific heat of different materials (i.e., different



), at low and high temperatures, is rather good.

At low temperatures, one recovers the empirical T

behavior (see Figure 2). More careful measurements

at low T put into evidence, for metallic solids, the

role of the conduction electrons: their contribution

to the heat capacity turns out to be linear in T, with

a coefficient such that at room temperature it is

much smaller than the lattice contribution, so that a

satisfactory agreement with the classical law is

found.

Soon after the beginning of qua ntum mechanics in

its modern form (1925–26), physicists considered

many-particle systems, dealing initially with the

simplest situations, with a relatively easy formal

apparatus, yet sufficient enough to understand in the

main lines the ‘‘anomalous,’’ that is, nonclassical,

behavior.

For a system of N free particles in a cubic box of

side L, quantum theory brings the labeling of the

one-particle states with the wave vectors k, recalling

the de Broglie relation for the momentum p =hk,

with a possible additional spin (intrinsic angu lar

moment) label 

k ¼

2

n; n 2 Z

nf0g

and the statistics of the particles: because of

indistinguishability, the wave function of several

identical particles has to be symmetric (B–E, Bose–

Einstein statistics) or antisymmetric (F–D, Fermi–

Dirac statistics) in the exchange of the particl es. This

has the deep im plication that no more than one

fermion shares the same quantum state.

We may here recall the spin–statistics connection,

which, in the framework of a local relativistic

theory, states that integer spin particles are bosons,

while particles with half-odd-integer spin are

fermions.

As the state is completely defined by the knowl-

edge of occupation numbers n

k, 

, we have the

simple and relevant statement on the ground states

for the N spinless bosons and N spin-1/2 fermion

systems are described by the statement:

BE system : n

¼ N

k;k

FD system : n

k;

¼ 1ðjkjk

Þ8

½6

The constant k

(Fermi wave number), or the

equivalent p

=hk

and "

= p

=2m (Fermi momen-

tum and energy, respectively) denotes the higher

occupied level. In the continuum approximation,

this implies the following relation between Fermi

energy "

and density  = N=L

h

ð3

Þ

2=3

½7

Going to the positive-temperature case, the grand

canonical partition function is computed by con-

sidering that occupation numbers are non-negative

integers for the B–E case and just 0 or 1 for the F–D

case. This implies the simple formulas, with obvious

meaning of symbols and leaving more details to the

vast literature (see Figure 3):

k;

;

expðð"

ÞÞ1

; þfor FD; for BE ½8

It is useful to introduce the Fermi temperature

; using some realistic data, that is, for

common metals like copper, T

ranges roughly

between 10

–10

K, that is, well above the ‘‘nor-

mal,’’ room temperatures: the quantum nature (i.e.,

quantum degeneracy) of the conduction electrons,

modeled as free electrons, is macroscopically visible

in normal conditions.

c (T )

Figure 2 The specific heat of crystal solids according to

Debye.

0.2

0.4

0.6

0.8

0 1 2 3

n(ε)

T > 0

= 0

Figure 3 Expected fermionic occupation number, for T = 0,

T > 0, and  = 2.

304 Quantum Statistical Mechanics: Overview

The presence of an external field, like the periodic

one given by the ionic lattice of a crystal, changes

the situation in a relevant way, as the one-particle

spectrum generally gets a band structure, and the

allowed momenta are described in the reciprocal

lattice: the Fermi sphere becomes a surface, and its

structure is central for further developments.

For massive bosons, the strange superfluid fea-

tures of liquid

He at low temperature, that is,

below the critical value 2.17 K, led F London, just

after Kapitza’s discovery in 1937, to speculate that

these were related to a macroscopic occupation of

the ground state (B–E condensation). A more

realistic model has to take into account interaction

between bosons (see last section) as the microscopic

interactions in superfluid liquid

He are not

negligible.

Quantum N -Body Properties:

Second Quantization

The main step in analyzing a quantum N-body

system is its energy spectrum, and in particular its

ground state, as it may represent a good approxima-

tion of the low-tempe rature states: its structure , the

relations with possible symmetries of the Hamilto-

nian, its degeneracy, the dependence of its energy on

the number of pa rticles, are further relevant ques-

tions. The last one is related to the possibility of

defining a thermodynamics for the system (Ruelle

1969). As a physically very interesting example,

consider a system of electrically charged particles, N

electrons with negative unit charge, and K atoms

with positive charge z, say, interacting through

electrostatic forces; the classical Coulomb potential

as a function of distance behaves badly, as it

diverges at zero and decreases slowly at infinity.

The first questio n is about the stability: thanks to

the exclusion principle, for the ground-state energy

N , K

an extensive estimate from below is valid:

N;K

c

ðN þ KzÞ

so that a finite-volume grand partition function

exists, while for the thermodynamic limit, which

involves large distances, we need more, that is,

charge neutrality, which allows for screening, and a

fast-decreasing effective interaction.

Let us see an example (quantum spin, Heisenberg

model) belonging to the class of lattice models,

where the identical microscopic elements are distin-

guishable by their fixed positions, that is, the nodes

of a lattice like Z

. To any site x 2 Z

is associated

acopyH

of a (2s þ 1)-dimensional Hilbert space

H, where an irreducible unitary representation of

SU(2) is given, so that the nonzero values for s are

1=2, 1, 3=2, .... For any x, the generators



(x), ( = 1, 2, 3) satisfy the well-known commuta-

tion relations of the angular momentum; moreover,



(x) = s(s þ 1)1, and operators related to

different sites commute. The ferromagnetic, iso-

tropic, next-neighbors, magnetic field Hamiltonian

for the finite system is



¼J

<x;y>

SðxÞSðyÞh

ðxÞ½9

where J is the positive strength of the next-neighbors

coupling (< x, y> means that x and y are next

neighbors); h is the intensity of the magnetic field

oriented along the third axis. This model is consider-

ably studied even now with several variants regarding

possible anisotropies of the interaction, the possibly

infinite range of the interaction, and the sign of J, for

other (e.g., antiferromagnetic) couplings. Among the

relevant results, the Mermin–Wagner theorem, at

variance with the analogous classical spin model,

states the absence of spontaneous magnetization in

this zero-field model for d = 2 for any positive

temperature; this can also be formulated as absence

of symmetry breaking for this model (Fro¨ hlich and

Pfister in 1981 shed more light on this point).

As mentioned earlier, a useful mathematical tool

for dealing with quantum systems of many particles

or quasiparticles, is the occupation-number repre-

sentation for the state of the system. The vector

space for a system with an indefinite number of

particles is the Fock space: it is the direct sum of all

spaces with any number of particles, starting with

the zero-particle, vacuum state. The operators which

connect these subspaces are the creation and

annihilation operators, very similar to the raising

and lowering operators introduced by Dirac for the

spectral analysis of the harmonic-oscillator Hamil-

tonian and the angular momentum, in the context of

one-particle quantum theory.

It is perhaps worth sketching the action of these

operators on the Fock space.

We consider spinless bosons first, as spin might

easily be taken into account, if necessary. We

suppose that a one-particle Hamiltonian has eigen-

functions labeled by a set of quantum numbers k,

say, as the wave vector for the purely kinetic one-

particle Ham iltonian. Let jn

, n

, ..., n

> denote

a vector state with

i = 1,..., p

particles, where n

denotes the number of particles with wave vector

, i = 1, ..., p; j0 > denotes the no-particle, vacuum

state. We defin e the creation operators a



as follows:



j...n

; ...i¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ 1

j...; n

þ 1; ...i½10

Quantum Statistical Mechanics: Overview 305

Its adjoint a

is called the annihilation operator,

for its action on the vectors

j...n

; ...i¼

ﬃﬃﬃﬃﬃ

j...; n

 1; ...i½11

The operator a



creates a new particle with that

momentum: for any k



j...n

; ...i¼n

j...n

; ...i



:¼

(the occupation-number operator)

The vacuum state belongs to Ker a

for any k , and

the whole space is generated by application of

creation operators on the vacuu m state.

The following basic commutation relations, for

any k, k

, are valid:

½a

; a



¼ðk; k

Þ; ½a

; a

¼½a



; a



¼0 ½12

For fermions, multiple occupancy is forbidden, so

that the analogous annihilation (

) and creation

(



) operators satisfy anticommutation relations:

½

;





¼ ðk; k

Þ; ½

;



¼½



;





¼ 0 ½13

The presence of spin is dealt by an addit ional spin

label  to these symbols, and a (, 

), where

necessary.

The Hamiltonian for a system of particles, say

spinless bosons, in a box , made of its kinetic part

together with a two-body (v(x  y)) interaction, is

written in terms of the ‘‘field operators’’; if {

(x)}

are the one-particle eigenfunctions of the single-

particle purely kinetic Hamiltonian for the spinless

case, and their complex conjugates are {



(x)}, we

define the fields

ðxÞ¼



ðxÞa

; 



ðxÞ¼





ðxÞa



½14

So that the full Hamiltonian is given by



dx 



ðxÞ

h

ðxÞ

þ 



dyvðx  yÞ



ðxÞðxÞ



ðyÞðyÞ

½15

We mention that a theoretical breakthrough in the

analysis of superfluidity was made by Bogoliubov

(1946), who, starting from the Hamiltoni an in [15],

introduced the following Hamiltonian in the

momentum representation:





k;k



kq



þq

½16

where "

is the one-particle kinetic energy and

the Fourier transform of the two-body potential.

To study the excitation spectrum above the ground

state, he introduced an approximation about the

persistency of a macroscopic occupation of the

ground state and a diagonalization procedure

leading to new quasiparticles with a characteristic

energy spectrum, linearly incr easing near jkj= 0,

then presenting a positive minimum before the

subsequent increase (see Figure 4).

Some Mathematical Tools for

Macroscopic Quantum Systems

The formal apparatus of second quantization, born

in the context of the quantum field theory, brought

to statistical mechanics new ideas and techniques

and related difficulties. For instance, the renormali-

zation group was conceived in the 1970s to deal

both with critical phenomena (i.e., power singula-

rities of thermodynamic quantities around the

critical point) and with divergences in quantum

field theory. This subject is currently being devel-

oped and applied in models of quantum statistical

mechanics (QSM) (Benfatto and Gallavotti, 1995).

Another issue, which has again strong relations

with quantum field theory, is the algebraic formula-

tion of QSM. This point of view, which is well

suited for the analysis of infinitely extended quan-

tum systems, uses a unified, synthetic, and rigorous

language. The procedure for passing from a finite

quantum system to its infinitely extended version

deserves some attention.

It is well known that, for finite quantum systems,

say N particles in a box , an observable is represented

by a self-adjoint operator A on a Hilbert space H



,and

the normalized elements {j >}ofthisspacearethe

pure states 

which define the expectations



ðAÞ:¼ < jA >

100

0.5 1 1.5 2 2.5 3 3.5

(ε /k

)

Figure 4 Excitation spectrum for superfluids.

306 Quantum Statistical Mechanics: Overview

The mixed states (mixtures) are defined by convex

combinations of pure states, the coefficients having

an obvious statistical meaning.

Among the observables, the Hamiltonian plays a

special role, as it generates the dynamics of the

system, which evolves the pure states through the

unitary group (Schro¨ dinger picture)

ðt Þexp 

itH



h





To the notion of equilibrium probability measure on

the phase space of a classical system, corresponds

the mixed state 



, 

such that





;

ðAÞ:¼Z

;

1

trðexpðH



ÞAÞ½17

The normalization factor Z

, 

= tr( exp ( H



)) is

the canonical partition function.

Consider now the algebra A() of local observa -

bles; sending  to infini ty, by induction, it is possible

to define the algebra A of quasilocal observables.

The main point is a set of algebraic relations like the

canonical commutation relations (CCRs) and the

canonical anticommutation relation s (CARs) for

the creation/annihilation operators: the observables

of A, through the GNS (Gel’fand, Naimark, and

Segal) construction may be represented as operators

on the appropriate Hilbert spaces, depending on the

chosen state; the representations, at variance with

the finite case, might be inequivalent. It is possible to

define the equilibrium state for the infinite system

and how to insert in a natural way the possible

group invariance of the system (R

or Z

, typically),

ending with characterization of the pure phases of

the system as the ergodic components in the

decomposition of an equilibrium state. These states

have the property that coarse-grained observables

have sharp values (Ruelle 1969, Sewell 2002): if

(A) is the space average on scale l, that is, over

boxes of side l, for an ergodic state ,

lim

l!1

ð½Av

ðAÞðAÞ

Þ¼0

Another issue which is worth mentioning is the

characterization of equilibrium states through

the KMS (Kubo–Martin–Schwinger) condition. The

strong formal similarity between the finite-volume

quantum evolution operator 

:= exp(itH



=h)

and the statistical equilibrium density operator

exp(H



), leads to the identity, valid for any

couple of bounded observables A and B, using the

short symbol <  >

, 

for the expectations with

respect to the statistical operator:

;

¼<BA

tþih

;

½18

This relation is suitably extended for infinite size,

and therefore defines a KMS state; it implies some

physically relevant properties like stability with

respect to local disturbances and dissipativity

(Sewell 2002).

A final issue in this section concerns another

formalism stemming from the Feynman path-integral

formulation of quantum mechanics: here a functional

integral represents the statistical equilibrium density

operator W



= exp(H). For a d-dimensional sys-

tem of N particles in a potential field (X 2 R

)

V = V(X) =

i<j

(x

 x

) and Hamiltonian H =

(1=2)  þ V the Feynman–Kac formula which, for

a test function , may be written as follows:

ðW



ÞðXÞ¼



X;Y

ðd!Þexp 



dsVð!ðsÞÞðYÞdY



where P



X, Y

(d!) is the Wiener measure on the space

of paths {!(s), s 2 [0, ]}. For details on the con-

struction and several other related features on the

treatment of the different statistics, see Glimm and

Jaffe (1981).

New Problems and Challenges

In this final section, we recall some phenomena

which have been observed recently in physics

laboratories, and which presumably deserve con-

siderable efforts to overcome the heuristic level of

explanation. About this last point, it is worth

quoting a method that has been used to get results

even without clear justifications of the underlying

hypotheses, that is, the mean-field procedure. It

started with the Curie–Weiss theory of magnetism

and is based on the following drastic simplification:

the microscopic element of the system feels an

average interaction field due to other elements,

indipendently of the positions of the latter. This

method might provide relatively good results if the

range of the interaction is very large, and in fact, a

clear version with due limiting procedure was

introduced by Kac, and applied by Lebowitz and

Penrose in the 1960s for a microscopic derivation of

van der Waals equation, and soon extended by Lieb

to quantum systems.

We will briefly outline some aspects of three

recent achievements of condensed matter physics for

which modeling is still on the way of further

progress: the B–E condensation, the high-T

super-

conductivity, and the fractional quantum Hall effect.

The first consists in trapping an ultracold (at less

than 50 mK) dilute bosonic gas, for example,

–10

atoms of

Rb, finding experimental evi-

dence for Bos e condensation. To understand the

Quantum Statistical Mechanics: Overview 307

properties of this system, an important tool is the

Gross–Pitaevskii energy functional for the conden-

sate wave function ,

E½¼

h

jrj

þ V

ext

ðxÞjj

jj

where the quartic term represents the reduced

(mean-field) interaction among particles.

The second issue, that is, the high-temperature

superconductivity, certainly deserves much atten-

tion. It has been observed recen tly in some ceramic

materials well above 100 K, and a clear model which

takes into account the formation of pairs and the

peculiar isotropy–anisotropy aspects of the normal

conductivity and superconductivity is still lacking

(Mattis 2003).

Finally, let us consider the fractional quantum

Hall effect; recall that the integer version, that is, a

discretization of the Hall resistivity R

by multiples

of h=(e

), finds an explanation in terms of band

spectra, formation of magnetic Landau levels, and

localization from surface impurities, that is, without

taking into account dire ct interactions among

electrons.

The fractional discretization of R

(Sto¨ rmer 1999)

has a theoretical interpretation, in terms of subtle

collective behavior of the two-dimensional semicon-

ductor electron system: the quasiparticles which

represent the excitations may behave as composite

fermions or bosons, or exhibit a fractional statistics

(see Fractional Quantum Hall Effect).

This brief excursion through these new fascinating

phenomena shows the rich interplay between theory

and experiments: these phenomena are a source of

new ideas and suggest new models for further

progress.

See also: Bose–Einstein Condensates; Dynamical

Systems and Thermodynamics; Exact Renormalization

Group; Falicov–Kimball Model; Fermionic Systems;

Finitely Correlated States; Fractional Quantum Hall

Effect; High T

Superconductor Theory; Hubbard Model;

Quantum Phase Transitions; Quantum Spin Systems;

Stability of Matter.

Further Reading

Benfatto G and Gallavotti G (1995) Renormalization Group.

Princeton: Princeton University Press.

Gallavotti G (1999) Statistical Mechanics: A Short Treatise.

Berlin: Springer.

Glimm J and Jaffe A (1981) Quantum Physics. A Functional

Integral Point of View. New York: Springer.

Landau LD and Lifschitz EN (2000) Statistical Physics, Course of

Theoretical Physics, 3rd edn., vol 5. Parts I and II. Oxford:

Butterworth-Heinemann.

Mattis DC (2003) Statistical Mechanics Made Simple. NJ: World

Scientific.

Ruelle D (1969) Statistical Mechanics. Rigorous Results. New

York: Benjamin.

Sewell GL (2002) Quantum Mechanics and Its Emergent

Macrophysics. Princeton: Princeton University Press.

Sinai YaG (1982) Theory of Phase Transitions: Rigorous Results.

Budapest: Akade´miai Kiado´.

Sto¨ rmer HL (1999) Nobel lecture: the quantum Hall effect.

Reviews of Modern Physics 71: 875–889.

Taylor PL and Heinonen O (2002) A Quantum Approach to

Condensed Matter Physics. Cambridge: Cambridge University

Press.

Quasiperiodic Systems

P Kramer, Universita

tTu

bingen, Tu

bingen, Germany

Introduction: From Periodic

to Quasiperiodic Systems

Periodic systems occur in many branches of physics.

Their mathematical analysis was stimulated in

particular by the analysis of the periodic transla-

tional symmetry of crystals. The systematic study of

the compatibility between translational and crystal-

lographic point or reflection symmetry leads to the

concept of space group symmetry. Mathematical

crystallography in three dimensions (3D) culminated

in 1892 in the complete classification of the 230

space groups due to Fedorov and Schoenflies

(see Schwarzenberger (1980, pp. 132–135). One

characteristic property of periodic systems is that

their Fourier transform has a pure point spectrum.

Since the Fourier spectrum is experimentally acces-

sible through diffraction experiments, it provides

a main tool for the structure determination of

crystals.

With quantum mechanics in the twentieth cen-

tury, it became possible to describe crystal structures

quantitatively as ordered systems of atomic nuclei

and electrons with electromagnetic interactions.

The representation theory of crystallographic

space groups now opened the way to verify the

308 Quasiperiodic Systems

space group symmetry of atomic systems for

example from the band structure of crystals. It

was then believe d that in physics atomic long-

range order is linked to periodicity and hence to

the paradigm of the 230 space groups in 3D.

Mathematical analysis beyond this paradigm

started independently in various directions. Bohr

(1925) studied qua siperiodic functions and their

Fourier transform. He interpreted them as restric-

tions of periodic functions in nD to their value s on a

linear subspace of orientation irrational with respect

to a lattice. Mathematical crystallography in general

dimension n > 3, including point group symmetry,

was started around 1949 in work by Hermann and

by Zassenhaus (see Schwarzenberger (1980)), and

completed in 1978 for n = 4inBrown et al. (1978).

A different route was taken by Penrose (1974).He

constructed an aperiodic tiling (covering without

gaps or overlaps) of the plane. Its tiles in two

rhombus shapes provide global 5-fold point symme-

try and make the tiling incompatible with any

periodic lattice in 2D. The connection between

Penrose’s aperiodic tiling and irrational subspaces

in periodic structures was made by de Bruijn (1981).

He interpre ted the Penrose rhombus tiling as the

intersection of geometric objects from cells of a

hypercubic lattice in 5D with a 2D sub space,

irrational and invariant under 5-fold noncrystallo-

graphic point symmetry. Kramer and Neri (1984)

embedded the icosahedral group as a point group

into the hypercubic lattice in 6D and constructed a

3D irrational subspace invariant under the noncrys-

tallographic icosahedral point group. From intersec-

tions of boundaries of the hypercubic lattice cells

with this subspace, they constructed a 3D tiling of

global icosahedral point symmetry with two rhom-

bohedral tiles.

Shechtman et al. (1984) disco vered in the system

AlMn diffraction patterns of icosahedral point

symmetry. Since icosahedral symmetry is incompa-

tible with a lattice in 3D, they concluded that there

exists atomic long-range order without a lattice. The

new paradigm of quasiperiodic long-range order in

quasicrystals was established and since then stimu-

lated a broad range of theoretical and experimental

research.

The interplay between the notions – (1) of

crystallographic symmetry in nD, n > 3, (2) of

subspaces invariant under a point group but

irrational and hence incompatible with a lattice,

and (3) of discrete geometric periodic objec ts in nD

providing quasiperiodic tilings on these subspaces –

forms the mathematical basis for a new quasiper-

iodic long-range order found in quasicrystals. The

present-day theory of quasicrystals offers the most

elaborate study of quasiperiodic systems. Therefore,

we shall focus in what follows on the concepts

developed in this theory.

In the following section, we briefly review basic

concepts of periodic systems and lat tices in nD, their

classification in terms of point symmetry and space

groups, and their cell structure. In a section on

quasipe riodic point sets and funct ions, a quasiper-

iodic system is taken as a geometric object on an

irrational mD subspace in an n-dimensional space

and lattice. Noncrystallographic point symmetry is

shown to select the irrational subspace. Next,

scaling symmet ry in quasiperiodic systems is demon-

strated. Then, examples of quasiperiodic systems

with point and scaling symmetry are given. The

penultimate section disc usses quasiperiodic tilings

and their windows. Finally, the notion of a funda-

mental domain for quasiperiodic functions compa-

tible with a tiling is illustrated.

Concepts from Periodic Systems

A distribution f

(x) of geometric objects on Eucli-

dean space E

(a real linear space equipped with

standard Euclidean scalar product h, i and metric)

with coordinates x 2 E

is called ‘‘periodic’’ if it is

invariant under translations b

in n linearly indepen-

dent directions,

ðpÞ : f

: f

ðx þb

Þ¼f

ðxÞ; i ¼1; ...; n ½1

The set of all translations on E

forms the discrete

additive abelian translation group

T ¼ b 2 E

: b ¼

; ðm

; ...; m

Þ2Z

()

½2

Any orbit (set of all images of an initial point) under

the action T  E

! E

yields a lattice  on E

Since T acts fixpoint-free, there is a one-to-one

correspondence  $ T. A fundamental domain on

is defined as a subset of points x 2 E

which

contains a representative point from any orbit under

T. Such a fundamental domain can be chosen, for

example, as the unit cell of the lattice  or as the

Voronoi cell (eqn [5]). By eqn [1], the functional

values on E

of a periodic function f

(x) are

completely determined from its values on a funda-

mental domain of E

Given the lattice basis (b

, ..., b

)ofeqn [2]

in E

, the vector components of the basis form the

n  n basis mat rix B of . The most general change

of the basis preserving the lattice is given by acting

with any element h of the general linear group

Quasiperiodic Systems 309

Gl(n, Z), with integral matrix entries and determi-

nant 1, on the lattice basis,

Glðn; ZÞ3h : B !B

¼Bh ½3

The crystallographic classification of inequivalent

lattices in E

starts from Gl(n, Z). In addit ion to

translations, it employs cryst allographic point sym-

metry operations, (Brown et al. 1978, p. 9). A

crystallographic point group operation of a lattice 

is a Euclidean isometry g which belongs to a group

G 3 g with representations D : G ! O(n, R) and

D: G ! Gl(n, Z) such that

G ¼fg : DðgÞB ¼B DðgÞg ½4

The maximal crystallographic point group for given

lattice  is the holohedry of . The group generated

by T, G is a space group which classifies the lattice.

For finer details in the classification of space groups,

we refer to Brown et al. (1978). For crystallography

in E

, t his classification yields 230 space groups.

Crystallography in E

is described in Schwarzenberger

(1980) and in Brown et al. (1978) where it is

elaborated for E

From a lattice  2 E

and from the Euclidean metric,

one constructs a cell structure as follows: the Voronoi

cell V(b), centered at a lattice point b 2 ,knownin

physics as the Wigner–Seitz cell, is the set of points

VðbÞ¼fx 2 E

: jx  bjjx  b

j; b

2g½5

Any Voronoi cell has a hierarchy of boundaries X

of dimension p,0  p  n which we denote as

p-boundaries.

The set of Voronoi cells at all lattice points form

the -periodic Voronoi complex of  2 E

. The

Voronoi cells and complexes associated with a

lattice admit a notion of geometric duality. We

denote dual objects by a star,



. They are built from

convex hulls of sets of lattice points (Kramer and

Schlottmann 1989) as follows. A Voronoi p-boundary

is shared by several Voronoi cells V(b)and

determines a set of lattice points

SðX

Þ: fb 2 :X

2 VðbÞg ½6

The boundary dual to X

is defined as the convex

hull X



(np)

:= conv{b : b 2 S(X

)}. X



(np)

can be

shown to be an (n  p)-boundary of a dual

Delone cell. A Delone cell D is defined as the

convex hull of all lattice points whose Voronoi cells

share a single vertex, called a hole of the lattice.

Since these vertices fall into classes of orbits under

translations, they determine translationally inequi-

valent classes of Delone cells D



, D



, ....

Fourier analysis applied to a periodic function f

(x)

on E

reduces to an n-fold Fourier series. The Fourier

spectrum is a pure point spectrum and the Fourier

coefficients can be referred to the points of a reciprocal

lattice 



(eqn [7]) in Fourier space E

n

.Wedenote

objects belonging to this Fourier space by the index



The basis matrix B



of the reciprocal lattice 



2 E

n

is obtained from B as the inverse transpose,

i

; b

i¼



¼ðB

1

½7

The values of the Fourier coefficients of f

(x) reduce

to integrals over the fundamental domain of the

lattice . From eqns [4] and [7] it follows that the

orthogonal representation of a point group G in

coordinate and in Fourier space coincides. The

Fourier spectrum and its point symmetry in crystals

are observed in diffraction experiments.

Quasiperiodic Point Sets and Functions

Quasiperiodic functions are characterized from their

Fourier spectrum (Bohr 1925)by

(qp



) The Fourier point spectrum of a quasiper-

iodic function forms a Z-module M



of rank

n, n > m on Fourier space E

m

A Z-module of rank n, n > m on E

m

is defined as a set



¼ b



: b



i

; ðm

; ...; m

Þ2Z

()

½8

with the Z-module basis (b

1

, ..., b

n

) linearly

independent with respect to integral linear combina-

tions. The step from a lattice 



to a module M



nontrivial since the set of all module points becomes

dense on E

m

. The Fourier coefficients of a

quasiperiodic function are assigned to the discrete

set of module points (eqn [8]).

Bohr in his analysis of quasiperiodic functions

(Bohr 1925, II, pp. 111–125) shows that a general

Z-module M



of rank n can be taken as the

projection to a subspace E

m

of dimension m of a

(nonunique) lattice 



2 E

n

, n > m. It is convenient

to consider in Fourier space E

n

an orthogonal

splitting which we denote as

n

¼E

m

þE

ðnmÞ

; E

m

? E

ðnmÞ

½9

A characterization of a quasiperiodic function

(x) on coordinate space is obtained as follows.

From 



one can construct with the help of eqn [7]

the lattice  := (



)



reciprocal to 



on a coordi-

nate space E

and associate to it via the Fourier

series a quasiperiodic function on a coordinate

subspace E

of E

= E

þ E

(nm)

, equipped with a

Z-module M (eqn [11]). As a result one finds a

310 Quasiperiodic Systems

characterization of a quasiperiodic function in

coordinate space:

(qp) A quasiperiodic function f

), x

2 E

can

always be interpreted as the restriction to a

subspace E

of a -periodic function f

(x)

on E

¼E

þE

ðnmÞ

; x ¼x

þ x

ðx

þc

Þ¼: f

ðx

½10

In the interpretation (qp)(eqn [10]), the Z-modules

in Fourier space (eqn [8]) and in coordinate space

become projections of reciprocal lattices,



¼

ð



Þ; M ¼

ðÞ½11

The linear independence of the module basis

enforces a splitting (eqn [9]), irrational with respect

to the lattice 



2 E

n

As in the classification of crystal lattices, point

symmetry plays a crucial role in the classification of

Z-modules for quasiperiodic systems like quasicrys-

tals. Noncrystallographic point groups G (with a

representation incompatible with any lattice) give

rise to quasiperiodic systems as follows:

(qp) Given a point group G with orthogonal

representations D

: G !O(m, R), D

:G !O

(n  m, R) such that D

is incompatible with

any lattice in E

, we now require in E

instead

of eqn [10] a lattice  with basis B and a

representation D: G !Gl(n, Z) such that

ðGÞ 0

0 D

ðGÞ



B ¼B DðGÞ½12

Equation [12] requires that the matrix B provides an

irrational reduction of the representation D(G) into

the two representations D

(G), D

(G). Periodic

functions restricted as in the second line of eqn

[10] are quasiperiodic.

For any finite group G, a representation D(G)

allowing for lattice embedding can always be

constructed by the technique of induced representa-

tions. Its reduction into representations D

(G),

(G) contained in this induced representation is

obtained by standard techniques. If D

(G) is non-

crystallographic and inequivalent to D

(G), the

subspace decomposition (eqn [12]) is unique.

Quasiperiodic functions compatible with tilings

and their windows can be constructed from the dual

cell structure (eqns [5] and [6]) of the embedding

lattice (Kramer and Schlottmann 1989). Examples

are given in the sections ‘‘Point symm etry in

quasipe riodic syst ems’’ an d ‘‘Qu asiperi odic tilings

and their windows ’’.

Scaling and Quasiperiodicity

Quasiperiodic systems lack periodicity but can have

scaling symmetries originating from a non-Euclidean

extension of eqn [12].

Example 1: Scaling in the Square Lattice Z

We begin with the Fibonacci scaling on the square

lattice Z

of E

. The symmetric matrix

h ¼



2Glð2; ZÞ½13

has eigenvalues



¼

1

¼ þ1;

¼ :¼ð1þ

ﬃﬃﬃ

Þ=2 ½14

Evaluation of the orthogonal eigenvectors allows us

to define a lattice basis B = (b

, b

) and rewrite the

eigenvalue equation similar to eqn [12] as



1

0 

B ¼B



B ¼



ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ3

q ﬃﬃﬃﬃﬃﬃﬃ

þ2

ﬃﬃﬃﬃﬃﬃﬃ

þ2

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ3

½15

This relation shows that h with respect to the basis

B acts as a non-Euclidean point symmetry of the

square lattice and generates an infinite discrete

group. Equation [15] provides an orthogonal

splitting E

= E

þ E

. The element h acts on the

two subspaces as a discrete linear scaling by



1

, , r espectively. It maps points of Z

in E

hence also their projections to E

, into one

another.

Figure 1 shows the lattice basis from eqn [15].

We choose as fundamental domain of Z

two

squares A, B whose boundaries are parallel or

perpendicular to E

. A horizontal line E

intersects

these two squares at vertical distances varying with

respect to their horizontal boundaries. The quasi-

periodic restriction f

) = f

þ c

)ofa

-periodic function f

(x)toalinex = x

þ c

picks up varying functional values on these sec-

tions. Clearly, one needs all the values of f

on its

fundamental domain i n E

to obtain all the values

taken by f

Scaling symmetry appears in conjunction with

noncrystallographic point symmetry (cf . the follow-

ing section). Combined with quasiperiodic tilings, it

gives rise to a hier archy of self-similar tilings whose

tiles scale with .

Quasiperiodic Systems 311