7.

S t a t i s t i c a l M e c h a n i c s o f A n y o n s

In this chapter we are going to study some statistical mechanical properties of

systems of many anyons, and in particular we will discuss the partition functions,
the virial coefficients, the magnetization and the magnetic moment. However only
very partial results can be obtained in this context, because the exact solution
of a gas of anyons is not known. In fact, in contrast to the bosonic or fermionic
case where the statistics is implemented by hand on the many body Hilbert space
by constructing completely symmetric or antisymmetric products of s~ngle particle
wavefunctions, for anyons the complicated boundary conditions for the interchange
of any two particles require the knowledge of the complete many-body configura-
tions. As we have pointed out in the previous chapter, only the two-body problem
is exactly soluble for anyons, and hence only the two-body partition function can
be computed exactly. Since the thermodynamic limit cannot be performed, one
has to resort to approximate or alternative methods to study the statistical me-
chanics of anyons. For example if the thermodynamic functions are analytic in
the particle density, it is well-known that the low density, or equivalently the high
temperature limit, of a (free) gas can be investigated using the virial expansion
(see for instance (Huang 1987; Reichl 1980)) in which the equation of state t h a t
relates the pressure P , the temperature T and the density p, can be expressed as
follows
P=pkBT (l+a2p+a3p2+...) (7.1)
where kB is the Boltzmann constant and a2, C a , . . . are the so-called virial coef-
ficients. For bosons and fermions, (7.1) is known to be consistent and one can
assume that a similar expansion holds also in the case of particles with fractional
statistics.
When a2, a 3 , . . , are all vanishing, the equation of state is that of a classical
ideal gas ( P = p kB T). The second virial coefficient a2 depends only on the two-
body quantum correlations, and it is positive for bosons and negative for fermions,
signaling the presence of a statistical attraction or repulsion respectively. Since the
two-body problem is completely soluble for anyons, the second virial coefficient can
be computed exactly also for intermediate statistics (Arovas et al. 1985), and it
turns out that it continuously interpolates between the negative bosonic value and
the positive fermionic one. The third virial coefficient a3 depends on the q u a n t u m
two- and three-body correlations. It is known for bosons and fermions, but not for
anyons since the three-anyon problem is not solved. However, even in the case of
intermediate statistics, it is possible to derive some qualitative features of a3 using
semiclassical arguments (Bhaduri et al. 1991)'or, as we will see, using the Type-I
 92 Statistical Mechanics of Anyons

and Type-II states we constructed in the previous chapter (Dunne et al. 1992b).
For the higher virial coefficients instead, only very preliminary and perturbative
results are available at the moment (Comtet et al. 1991).
Remarkably enough, the exact solution of the two body problem allows to
extract a few interesting features of the anyon system, like the presence or absence
of spontaneous magnetization, or the presence or absence of parity violating effects,
or the relation between the canonical magnetic moment and the fractional orbital
angular m o m e n t u m that we introduced in Chapter 5. However it would be even
more remarkable and interesting to extend these results to the thermodynamic
limit and deduce macroscopic effects that, at least in principle, could be detected.
Lacking for the time being such possibility, we will content ourselves to compute
the canonical partition functions and derive some consequences using standard
techniques of statistical mechanics.

7.1 Partition Functions

The canonical N - b o d y partition function, Zt¢, for a system of free particles is

clearly divergent because each energy level is infinitely degenerate. Thus, in or-
der to get a finite Z~r, one has to break the degeneracy of the energy spectrum.
This can be done, for instance, by confining the particles in a finite volume or,
as we will do, by introducing a confining interaction, like a harmonic potential
with frequency w, which serves as a regulator. Since we will be interested in the
magnetic properties of the anyon gas, we introduce also an external magnetic field
B as we did in the previous chapter. The coupling to a magnetic field alone is not
enough to regularize the theory and remove the degeneracy of the energy levels.
Indeed, even though the presence of the magnetic field organizes the spectrum in
discrete Landau levels spaced by the cyclotron frequency wc = e B / m , each level
remains infinitely degenerate in an infinite volume, thus yielding divergent parti-
tion functions. Therefore hereinafter we will consider a system of N particles in
the infinite plane interacting both with a harmonic potential and with a magnetic
field; sometimes the latter will be switched off to simplify the situation and render
the formulas more manageable.
For bosons and fermions it is well-known how to compute the N-body par-
tition functions ZN. First one solves the single-particle problem and then forms
all N - b o d y wavefunctions by constructing all Slater permanents (for bosons) or
determinants (for fermions) with the single-particle wavefunctions. In this way
the energy levels and their degeneracies follow immediately, and one can easily
compute Z~r. For example, for particles of mass m interacting in the plane only
with a harmonic potential of frequency w, the single-particle wavefunctions are (cf
(6.1.12))
¢~ ~ z¢ L~(mwlz[ 2) e-½ m'~l~l~ (7.1.1)
where n = 0, 1, 2 , . . . , j = - n , - n + 1,... and L~ is the generalized polynomial of
degree n (see Appendix 6.A). The energy corresponding to (7.1.1) is
E~-w (2n+j+l) , (z.z.2)
 93

while the canonical one-body partition function is

(x)

Za--= ~ e -#E'~ (7.1.3)

n'-O
j"- --n

where, as usual,/3 = 1/(kB T). If we define j ' = j + n so t h a t j ' = O, 1 , . . . and

E~' = w(n + j' + 1), we can easily evaluate the summations in (7.1.3) and get

Z1 = y ~ e -/3E~' -- Z e-/3w(nTJ"k'l)
n=0 n=0
j'=o ./'=o
(7.1.4)
----
e-~° _._
1
( 1 - e-O~) 2 4 sinh 2 (2~-~--)

Obviously Z1 is the same for bosons, for fermions and for anyons because no
statistical effect is present in the one-body problem.
Let us now compute the two-body partition function Z2. For simplicity we
first consider the case of bosons. The two-body bosonic wavefunctions are totally
symmetric products of the single-particle wavefunctions (7.1.1). If ( n i , j ~ ) and
(n2, j~) are respectively the labels of the two component factors, the total energy
is
•I .$
E31,32
n,,-2 = w (hi + j~ + n2 + j~ + 2) , (7.1.5)

and the partition function is

~ii,4
Z~°" -= Z e-t~-"''2 (7.1.6)

where the symbol ~ denotes the sum over all s y m m e t r i c states meaning
{~ ,Jl ; ~ ,j'~}~
that the state characterized by the ordered set { n i , j ~ ; n 2 , j ~ } must be identified
with the one characterized by {n2,j~; ni,j~ }. To calculate Z2 it is convenient to
make the following observation: If we sum independently over all labels h i , j ~ , n 2
and j~, we clearly overcount the bosonic states because we do not implement the
identifications due to s y m m e t r y ; however if we add to this sum the contribution
of the diagonal states for which nl -- n2 and j~ -- j~, we get exactly twice the
wanted bosonic states, t h a t is

~+ Z - 2 ~ . (7.1.7)
11,1 Irt 2 I~,1~n 2 {1%1 "t. -t
•I -I -I .I 'J1 , n 2 ' J 2 } S
J1 J2 J 1 --"~J2

The bosonic partition function, Z b°8 , then follows immediately from (7.1.7) and
(7.1.5), and we get
94 Statistical Mechanics of Anyons

Z~O~ - 1 Z
e--/3w(n1+J1+n2+J2+2) .~_ Z
.i .i
e--~w(2nl+2j~+2)
11,1= 0 n2--O nl--O
j; =o

= 2
1[ (1 - e - ~ ' ) 4 + (1 - e--2f~w) 2
] (7.1.8)

cosh(fl w)
8sinh 2 ( 2~ - ) sinh2(flw) "

The fermionic two-body partition function

Z2f e r - e -f~-"1,"2 , (7.1.9)

where the symbol ~ denotes the sum over all antisymmetric states, can
• .i

be computed in an analogous way. A moment thought reveals that

ZZ- Z --2 Z " (7.1.10)

"1., "~., "1.,=~., 1,1,~; ;,~,~ }~
J1 J2 J1--J2

The diagonal states for which two quantum numbers are the same, are clearly
impossible according to the Pauli exclusion principle, and thus must be removed.
From (7.1.10), (7.1.9) and (7.1.5), it is easy to get

z:erlln~llO:~Oe-~W(rtl+Jll+n2+J2+2):
= = --~e I"L 1 = 0 lf~ w (2n1"~'2Ji "~'2)

j~ =0 j; =0
_ e1- 2[ f ~ w _ _ e-2f~w ] (7.1.11)
-- 2 (1 - e - ~ ) 4 (I - e - 2 ~ ) 2
I

8sinh' (2g-~-) sinh'(/~w) "

Let us now turn to the three-body partition functions for bosons and fermions.
The bosonic three-body wavefunctions are totally symmetric products of three
single-particle wavefunctions (7.1.1), and hence can be labeled by (na,j:), (n2,j:),
(n3, j~). By taking into account the symmetry imposed by Bose statistics, it is not
difficult to prove that, in obvious notations,

~-'~y~-~+3 Z Z +2 Z --6 ~ . (7.1.12)

111
•!
n 2
.I
n3
"1
n i ~-~lrb2
"! "1
f'b 3
"I
11,1 " " 1~2 ~_~1'/,3
.I -I .I {,,1 ,j; ;"~,j'}s
3x J2 33 31=32 3a J1=32=Ja

Therefore, since
 95

.,,..,.~ = ~ ( ~ + j~ + ~ + j~ + ~3 + j~ + 3) , (7.1.13)
the partition function Z b°" is given by

Z b ° S = 61 I ~ ~~' ~ "
e - f ~ tO ( " 1 d ' j ~ "]- n 2 " { - f f ~ ~ - " 3 " ~ ' ' 3 " 1 " 3 )

nl--O n2--O na--O

L.~i=o j~=o s:,=" o

oo oo oo

+ 3 "' 2 + 3x +3)
na=O na=O nx=O
j~=0 j;=0 ji=0
1 [ e-3~ ~ ~-3~, ~-3~,~, ]
= 6 ( 1 - e - a ) 6" +3 (1 - e -f3'' )2 (1 - - e -2f3 ~)2 + 2 (1 - - e - 3 f ~ " )3

cosh(3flw) + 2cosh 2 (2P--~)

32 sinh 2 (2P~-) sinh2(flw)sinh 2 (32 - ~ ) "
(7.1.14)
In the case of fermions we must exclude all states in which at least two quantum
numbrs of the constituent factors are the same; simple combinatorial considerations
lead to the following result
~~-3 ~ ~+2 Z =6 ~ . (7.1.15)
nl
•I
n2
"I
na
.I
nl=n2
"I .I
n3
"l
nl=n2=n3
"l .I .I
{nl,Jll ;n2, 2
3z 32 3a 31=32 3a 31"--32=33

Using (7.1.13), we get

•t 7"1, -t ~,~ .e
e-f~w(nl+3x+ 2+32+ 3+.23+3)
nl=O n:~=O na=O
•l .I .l
31=0 32-'0 3a=0

c~

-3~ Z e - ~ ' (2.,+2jl+n3+j~+3) + 2 Z e - ~ (3n1+3jl +3)

nl----O na--O ni--O
•I .? "t
3x=O 3a=0 31=0
1 [ e -3fl " e -3fl " e-3~w ]
- 6 [ (1 - ~_~)6 - 3(1 _ e _ ~ ) 2 (I - e-2~)2 + 2 (1 _ e _ a ~ . , ) 3

l + 2cosh 2 (2~-~-)

(7.1.16)
This method can be easily applied to compute the bosonic and fermionic parti-
tion functions for any number N of particles, but the resulting formulas are more
96 Statistical Mechanics of Anyons

and more cumbersome when N increases. However, such a procedure cannot be

applied to anyons of intermediate statistics. In fact, in this case, the N - b o d y wave-
functions cannot be constructed from products of single-particle wave functions,
because one has to represent the braid group and not simply the permutation
group. An alternative approach is therefore needed. This is provided by the step
operators introduced in Chapter 6 (Dunne et al. 1992b). The idea is to work in the
anyon gauge in which case the Hamiltonian is identical for bosonic, fermionic or
anyonic systems, but the many-body wavefunctions have different symmetry prop-
erties for different statistics. The Hamiltonian for N particles interacting with an
external uniform magnetic field B and a harmonic force of frequency w, is (in
complex coordinates)

N ( e2B2 ]2 1 )eB N
g= ~ 20ibi+ Izi + mw21zlI2 ~ ( z , 0 i - 2iOi) ,
1=1 - m Sm -2 2m 1=1
(7.1.17)
which is a simple generalization of (6.1.9). We are interested in the eigenvalue
problem H ¢ -- E ¢ for multi-valued functions ¢. For convenience let us define

e2B 2
j~2 _ 40)2 .~ m 2 = 40)2 + wc ,
- ~ 2
(7.1.18)

and

= exp m/2 ~ I~iI 2 ¢, (7.1.19)

I----1

so t h a t the "reduced" Hamiltonian acting on ¢ is

~I_- NF2
2 + ~ ( -2°~&m 1
+ ~_z~O~ 1 2iOi)
+ -~.,+ (7.1.20)
I=1

where w+ = F2 4- we. In analogy with (6.2.3a) the Hamiltonian H can be written

as follows
/?/__ N~2 1 N ( )
-- 2 + -2 Z w+ a~ai + w _ b?Ibi , (7.1.21)
I--1

where the operators

a?i -- 21 ~..0 0i , aI -- (91

2 -
(7.1.22)
mJl

r
satisfy < , i ° - - ° , + e n , > e r , + : : + , . , , wit,+, o<,,°," ,+ommu,,,+-
I- ,,,I I,,, ,,,II

tors being zero. This problem can be analyzed in complete analogy with the one
discussed in Section 6.2. The symmetric step operators
 97

N
c.. : 7: o*;b~" , (n -t- m < N) (7.1.23)
I--1

act on the on the base states ~0) _ YI (z,j)", and ~,,a~'(°)"- H (~,ij)(2-z,), which
I<J l<J
entirely encode the anyonic statistics (here we take 0 _< v < 2 ) , and the following
two families of regular anyonic wavefunctions can be constructed
1 N--n

~,: H H (c-) ~"" ~0) (7.1.24a)

n--O m--O
for Type-I, and
1 N--m,
'~II -- H H (Chin)An" "7'(0)
'vii (7.1.24b)
m=0 n=0

for Type-II. Here Anm are non-negative integers. For N - 2 the states in (7.1.24),
completely exhaust the space of wavefunctions for anyons of arbitrary statistics.
The energy corresponding to (7.1.24a) and (7.1.245) is respectively

N +
EI ({An,~}) =-~-~2 1N(N - 1)w_v
(7.1.25a)
1 1 N--m
-t--~ E E A,~,n(nw++mw-)
m=0 n=0

and
N 4-
EII ({A,,m}) --~-~2 1N(N- 1 ) w + ( 2 - v)
1 N-,, (7.1.25b)
1
-t--~ E E Anm(nw++row_) .
m=0 n=0

With the anyonic states represented as in (7.1.24), it is immediate to compute the

corresponding partition functions. In fact, the contribution of the Type-I states
(7.1.25a) to the partition function is

Z(~)= E e-/~E'({X"m})
{~,,}
I N-n oo (7.1.26)

n = 0 m = 0 Xn,~ = 0

The summations over A,~m can be easily done and, with trivial manipulations, one
gets
1 N 1
Z(I) = e--}/3f2--~N(N--1)w_v__
N
~inh . ~ (l- 1
(V.l.2Va)
98 Statistical Mechanics of Anyons
7(II) of the Type-II states (7.1.24b) to the parti-
The corresponding contribution "-'N
tion function can be computed in the same way or simply obtained from (7.1.27a)
with the substitutions w+ ~ w_, v ~ 2 - v, yielding

N
Z(II) -~-fl~--~N(N--1)w+(2--u) 1 1
t=l sinh ( 4~w+l) sinh
(w+ + ( l - 1)w_) ].
(7.1.27b)
Hence the total partition function for our boson-based anyons is given by

7.(I) ~(II)
ZN(v, B) = "~N + "~N

= 2- ~e l ~N(N--1)w_(l--v)H 1
~g----1 sinh ( ~ w _ g ) s i n h [4~ (w_ + ( g - 1)w+)]

+ e-~N(N-1)~+(1-~) H g) (~
1 " 1)w_
}
t=l sinh (~w+ sinh [4~ (w+ + - )]
(7.1.28)
Note that this partition function enjoys the property Z N ( v , B ) -- Z N ( 2 - v , - B )
for any N and Z2(0, B) - Z2(2, B) for the special case of two anyons. Moreover,
we observe that Z1 is obviously independent of v, and that Z2 in (7.1.28) is the
complete partition function for two anyons of arbitrary statistics, and smoothly
tends to the bosonic and fermionic ones for v --~ 0, 2 and v --, 1, respectively. For
N >_ 3, ZN(v,,B) does not reduce to the known bosonic and fermionic partition
functions for v -- 0 and v - 1, since the states (7.1.24) do not cover the entire
spectrum.
Let us exemplify this for the special case of B -- 0. In this case w+ - w_ --- 2w
and (7.1.28) gives

cosh [(1 - v)flw]

z ~ ( ~ , 0) =
8 sinh 2 ( 2~ - ) sinh2(flw) '
(7.1.29)
cosh [3(1 - v)flw]
z 3 ( . , 0) =
32 sinh 2 (2P-~-) sinh2(flw)sinh 2 (32 - ~ )

By comparing these with (7.1.8) and (7.1.11), we see that Z2(0,0) = Z2b°" and
Z2(1,0) = Z2fer. These equalities do not hold for Za(v, 0). However, let us note
that the difference with respect to the exact results (7.1.14) and (7.1.16) is the
same for u = 0, 1. Indeed,

z ~ °" - z 3 ( o , o) = = z~er - Za(1, 0) .

16 sinh 2 (2P-~-) sinh 2 (flw)sinh 2 (32 - ~ )
(7.1.30)
 99

Furthermore, we note that for N = 3, the coefficient of the contribution at first

order in v, i.e. ° Z 3 ( v , 0)1~=0, coincides exactly with the one computed with n u -
m e r i c a l methods in perturbation theory around bosonic statistics (McCabe and
Ouvry 1991).
Having established that (7.1.28) is the complete partition function for two
anyons in a magnetic field B and a confining harmonic potential of frequency w,
in the next sections we will compute the virial coefficients and the magnetization
properties.

7.2 V i r i a l C o e f f i c i e n t s

The low density or equivalently the high temperature limit of a gas of particles can
be described using the virial expansion, in which the equation of state is expressed
as a series in the particle density p (see (7.1)). The coefficients in this expansion
are known as virial coefficients. To set up the notations, let us first recall the
basic definition of the first few of these. In the low density approximation, the
gran canonical partition function Q can be written as a cluster expansion (see for
instance (Huang 1987; Reichl 1980))

{2 - exp A~ b~ z ~ (7.2.1)

where bl are the so-called cluster integrals (Ursell 1927; Pais and Uhlenbeck 1959),
z is the fugacity and A is the area of the system (we obviously limit our consid-
erations to the case of two dimensions; in higher dimensions A would be replaced
by the volume V). On the other hand, using standard techniques of statistical
mechanics, the pressure P and the density p are given by

P = kBT
(1) ~lnQ ,

p--Z-~z
0(1 ) lnQ .
(7.2.2)

A,T

Upon inserting the cluster expansion (7.2.1) into (7.2.2), we get

oo

P = kBTZb~zt ,

(x)
£--1 (7.2.3)
p= ZIb~zt ,
l--1

so that the virial expansion of the equation of state (7.1) becomes

Z
t=l
bl z t --
t=l
I bt z t 1 + a2
£=1
~ bt z t + a3
t=l
I bt z t +... ] (7.2.4)
100 Statistical Mechanics of Anyons

If we now expand both sides of (7.2.4) and equate the coefficients of equal powers
of z, we easily obtain the expressions for the virial coefficients in terms of the
cluster integrals; the first two are

b2
a2 - b2 ,
(7.2.5a)

a3 = 4a 2 - 2 b_.33 (7.2.5b)
b~
We can make further progress if we use the definition of the gran canonical partition
function
Q= ~ z~ z~ (7.2.6)
N--0

to express the cluster integrals be in t e r m s of the N - b o d y canonical partition

functions ZN. From the comparison of (7.2.1) with (7.2.6), we obtain

Zl
bl = ~ , (7.2.7a)
A
2z2 - z~
b2 -- , (7.2.7b)
2A
3 z 3 - 3 z 2 z~ + z 3
53 -- . (7.2.7c)
3A
A c t u a l l y all these expressions are meaningful only in the infinite area limit, so t h a t
p a r t i c u l a r care must be used in their evaluation.
In two spatial dimensions, one has
1
bl = ~ (7.2.8)

where AT is the t h e r m a l wavelength of a particle of mass m and is given by AT =

(27rfl/m) 1/2. Therefore from (7.2.7a) we have
1 Zl (7.2.9)
~ = W
Using the expression of Z1 given in (7.1.4) and taking the high t e m p e r a t u r e limit
(i.e./~ --, 0), we are lead to the identification
w2,, ~ 27r 1 . (7.2.10)
BmA
T h u s the harmonic frequency w can be t h o u g h t as being related to the inverse of
the area A, and hence the infinite area limit (A ~ 0) becomes equivalent to the
zero h a r m o n i c frequency limit (w --, 0). This is of course to be expected because
p u t t i n g the system in a harmonic potential confines the particles to move in a
finite region; removing the harmonic force (i.e. letting w --, 0) a m o u n t s to remove
any restriction on the space (i.e. A ---, 0).
 101

Let us now e v a l u a t e t h e cluster integrals and the first virial coefficients in t h e

w --, 0 limit. For s i m p l i c i t y we will use the anyon p a r t i t i o n functions ZN in (7.1.28)
with B - 0; t h e case w i t h a non-vanishing m a g n e t i c field is left as an exercise for
the reader.
Recalling t h a t t h e h a r m o n i c regularization requires to m u l t i p l y b~ by a n o r m a l -
ization factor ~.d/2 w h e r e d is t h e n u m b e r of space dimensions 22, the second virial
coefficient t u r n s o u t to be

b2 - A~ lim [Z1_ 2 Z2(u' O) ]

a2(u) = bl2 ~-,0 Zl
= ~
(1 - ~ + ~,_ :~,~
1) ,
(7.2.11)

in a g r e e m e n t w i t h t h e original derivation (Arovas et al. 1985) which is b a s e d on

different m e t h o d s (cf also ( C o m t e t et al. 1989; B l u m et al. 1990; Sen 1991a)). We
r e m a r k t h a t b o t h Z1 a n d 2 Z2(u, O)/Z1 are divergent in t h e limit w --, 0 - in fact
t h e y are O(1/w 2) - b u t t h e divergence cancels in t h e c o m b i n a t i o n

z~(~,0)
Zl - 2 ~
z~
a p p e a r i n g in (7.2.11). T h e second virial coefficient a2(v) i n t e r p o l a t e s c o n t i n u o u s l y
between t h e bosonic value

a~o. _ a ~ ( 0 ) = -~1 ~¢ , (7.2.12)

and the fermionic one

1
a~er ---- a2(1) -- ~ A2 . (7.2.13)

This is an obvious c o n s e q u e n c e of the fact t h a t the t w o - b o d y p a r t i t i o n f u n c t i o n s

is exact for a r b i t r a r y s t a t i s t i c s u.
Let us now t u r n to t h e t h i r d virial coefficient which is given by

53
aa(u) = 4 a 2 ( u ) 2 - 2 b--~
(~.2.14)
= 4~(~)~-2~ ~i%[3z~(~,z~0)-3 z~(~,o)+z~]
22For two particles the harmonic potential is
o)2
~= (=~ + =~) - 2~ = z = + T (zI - zI)

where Z - ( 1 / v f 2 ) ( z l ~- z2) is the center of mass coordinate. Thus, in the two-body

2
problem O)¢.m. -- 2 O)2 , and in general in the Lbody problem O)c.m.2 -- i We. Since it is
2 which is actually related to the area according to (7.2.10) the two-dimensional
o)c.m.
cluster integral be must be multiplied by I.
102 Statistical Mechanics of Anyons

If we substitute in (7.2.14) the explicit expressions of the partition functions given

in (7.1.29) and compute the w --, • limit, we find a divergent result. However,
remarkably enough, the coefficients of the divergences turn out to be independent
oj' u, and thus the difference a 3 ( u ) - aa(u') is well-defined and finite. Furthermore,
a simple calculation shows that

- = 0 .

This result is the consequence of highly non-trivial cancellations and was first
pointed out in (Dunne et al. 1992b).
We remark t h a t the third virial coefficient is finite for bosons and fermions;
indeed one has 23
abo. = a~r = 1
~--~A~ . (7.2.16)

This result can be easily obtained if we use the complete bosonic or fermionic par-
tition functions given in (7.1.14) and (7.1.16) respectively. As previously noted,
these are not the smooth limit of Za(u, 0) for u --~ 0, 1, and in fact the differ-
ence given in (7.1.30) precisely cancels the divergences in the cluster integrals and
renders the third virial coefficient finite.
The presence of divergences in a3 for fractional statistics could signal either
the failure of the virial expansion due to long-range statistical interactions, or the
presence of "missing states" which are not included in (7.1.24). This last possibility
seems to be suggested by recent numerical (Sporre et al. 1991a; Murphy et al.
1991) and perturbative (McCabe and Ouvry 1991; Comtet et al. 1991; Khare and
McCabe 1991) calculations. To shed some light on this problem, it is therefore
necessary to investigate in more detail the third virial coefficient ca(u). To this
purpose, let us turn back to the three-anyon problem, described by the reduced
Hamiltonian (see (7.1.20) for B = 0)

/-'1 m

As we have seen in Chapter 6,/-t can be nicely split into a center of mass part and
a relative part by introducing the Jacobi coordinates (of (6.3.8))
1
z= (Zl+Z2+Z3) ,

1
Wl ---- ~ (Zl -- Z2) , (7.2.18)

1
W2 -- ~ ( Z l +Z2 --2Z3) •

Indeed we have
f/-- 2~rc.m.-~-f/rel

23It is well-known that all odd virial coefficients, a2k+l, are equal for bosons and fermions
(Huang 1987).
 103

where

[-I¢.m. = W +
(2 - - - - Oz Oz + W Z Oz + o., 2 0 z
) ,

2(m

) (7.2.19)

m
i--a

From now on, we focus only on the relative problem and neglect the trivial center
of mass dynamics. In analogy with (7.1.22) we introduce the operators

A~ = ~ , - 1--~-Ow, , A, = Ow,
mo.)
(7.2.20)
1 -
BJ = wi - 0,,,, , Bi = 0,,,,
mw
for i = 1, 2, so that ~rre I becomes

[_irel=w[2+~(A~Ai+BtBi)]i=l (7.2.21)

The operators (7.2.20) satisfy the Heisenberg algebra

[Ai, A~] = [B,, B~] = 6ij , (7.2.22)

with all other c o m m u t a t o r s being zero, and are energy step operators. In fact, using
(7.2.21) and (7.2.22), it is easy to prove that A~ and B~ lower the eigenvalues of
Hrel by w, i.e.

[f/rel, A i ] - - w Ai , [~rrel, Bi] -- - w Bi , (7.2.23a)

whereas A~ and Bit raise the eigenvalues by w, i.e.

[/~,¢,, A~] - w A~ , [/-Ir~,, B~*I - w B t . (7.2.23b)

Following (Sen 1991b,c), we define the operator

Q=AIB2-A~BI (7.2.24)

which is completely antisymmetric under particle exchanges, and commutes with

//re1. Being antisymmetric, Q acts on symmetric wavefunctions to produce anti-
symmetric ones with the same energy, and vice versa. Thus Q changes the statis-
tics mapping bosons into fermions or vice versa, and for this reason it has been
called a s u p e r s y m m e t r y operator (Sen 1991b,c). More generally, Q connects staes
of equal energy in two anyonic theories with statistics v and v + 1. In order to keep
the statistics within the range [0, 1], we can follow the action of Q with a parity
transformation P which makes the replacements wi *-~ wi in the wavefunctions,
corresponding to ( x a , y i ) ~ ( x i , - y i ) in terms of the cartesian coordinates. Since
P changes the sign of the statistical parameter, the combined operator
104 Statistical Mechanics of Anyons

Q - P Q (7.2.25)

maps anyonic states of statistics v to anyonic states of statistics 1 - v without

changing their energy.
Let us now consider a wavefunction ¢ ( v ) describing the relative motion of three
anyons of statistics v with a given energy E ( ¢ ) . Under the action of Q, in principle
there are three possibilities:
i) The wavefunction (~ ¢(u) is a regular wavefunction for three anyons of statis-
tics 1 - v and energy E ( ¢ ) ;

ii) The wavefunction (~ ¢ ( v ) i d e n t i c a l l y vanishes;

iii) The wavefunction (~ ¢ ( v ) is singular.
A detailed analysis is therefore necessary to see which one of these possibilities
actually occurs. We refer the reader to the original literature (Sen 1991b,c) for a
complete discussion; here we limit ourselves to summarize the results and see their
consequences on the third virial coefficient. It turns out t h a t if ¢ ( v ) is a regular
wavefunction, then case iii) never occurs, that is (~ ¢ ( v ) i s always an acceptable
wavefunction (eventually vanishing) of a new anyonic theory with statistics 1 - v.
Furthermore, the operator Q annihilates only all the Type-I and Type-II states
which we constructed in Chapter 6 and which contributed to the partition function
Z3(v, 0) in (7.1.29). The missing states (if they exist) are instead mirror symmetric,
t h a t is for any regular missing wavefunction ¢~(v) there is a corresponding regular
missing wavefunction ¢ ' ( 1 - v). This property is confirmed also by recent numerical
calculations (Sporre et al. 1991a; Murphy et al. 1991).
Let us now assume that the missing states exist and construct the full three-
body partition function according to

Z3(y) - ~ e-f~ E(~)

{all states ~}
(7.2.26)
= z3(., 0) +
{missing states ~l}

By construction we have
23( = o) = o" ,

and
,~a(v = 1 ) - Z3fer , (7.2.27b)
since the missing states are defined to make the bosonic and fermionic limits
smooth. Taking into account the mirror s y m m e t r y of the missing states, we easily
obtain
Z a ( v ) - 23(1 v) Za(v, 0 ) - Z3(1 v, 0) ,
- - - (7.2.28)
-

which shows t h a t only the regular Type-I and Type-II states contribute to the
difference Z a ( v ) - Z 3 ( 1 - v). Let us now define
 105

ha(v) -- 4 a 2 ( u ) 2 - 2A~ lim 3 Za(v) 3 0) + (7.2.29)

w--*0 Z1

which is the same expression as in (7.2.14) with Za(u, 0) replaced by the .full
partition function Za(u), and then compute

53(v)-~ta(1-u) .

Using (7.2.28), we have

a 3 ( v ) -- a 3 ( 1 -- v ) - - 4 a 2 ( / / ) 2 - 4 a 2 ( 1 -/])2
{[ Zl l (7.2.30)
- 3 o)- z (1- ,,o11 }
-- a3(u) -- a3(1 -- v)

From (7.2.15) with v' = 1 - v, it immediately follows t h a t

a3(v) -- 53(1 -- u) . (7.2.31)

The mirror s y m m e t r y of the third virial coefficient expressed by (7.2.31) is a rig-

orous result which was first pointed out in (Sen 1991c). In particular for v - 0,
we have the well-known property
1 ~4
a b°' -- h3(0) = h3(1) = af3er (7.2.32)

Unfortunately, the mirror symmetry (7.2.31) does not allow to deduce the value of
ha(v) for any v; however, using (7.2.31) together with the perturbative calculations
of (McCabe and Ouvry 1991), it can be conjectured t h a t actualy the third virial
coefficient does not depend on the statistics v. A proof (or a disproof) of this
conjecture is still missing.
For the higher virial coefficients, only partial pertubative results are available
at the moment but no definite conclusions can be drawn (McCabe et al. 1991).

7.3 Magnetic Moment

In this section we are going to analyze some magnetic properties of the anyon gas.
Let us consider a two-body system and define the magnetization according to the
standard formula
1 0
M2 = fl OB log Z2 . (7.3.1)

As a consequence of the properties of Z2 given after (7.1.28), it is easy to see t h a t

M2 satisfies
106 Statistical Mechanics of Anyons

M 2 ( v , B ) = -21//2(2- v , - B ) . (7.3.2)
There are two regimes in which one can extract simple and interesting results.
The first is the regime of high magnetic fields, in which B sets the biggest scale in
the problem, n a m e l y
m w~Tz
<< 1 << 1 . (7.3:3)
eflB ' eB
The first inequality tells us that the thermal excitations whose energies are ,,~ 1//3
are negligible with respect to the magnetic excitations whose energies are ,-, we =
e B/re. Instead the second equality in (7.3.3) implies t h a t the excitations induced
by the harmonic force whose energies are ,,~ w are small compared to the magnetic
excitations. In this regime M2 reduces to a double series in the small parameters
(7.3.3), whose first terms are given by

-2----* [1 ~¢3B
2,, ¢v + ~ - 28~,,2)¢~-)2]
~,m , B > 0
2m
M2(B, v)
e [1- e~lB' ] + ( v - 5+ 2¢,0) ,,,m , B <0 . (7.3.4)
For w = 0 the magnetic moment 21//2 is independent of statistics; the second term
on the right-hand side of (7.3.4) represents a finite t e m p e r a t u r e correction. As
expected, the statistical dependence appears in the t e r m proportional to w; it is
in fact the harmonic potential that breaks the infinite degeneracy of the Landau
levels of two anyons. The last term in (7.3.4) can also be reinterpreted as a "finite
area correction" if we take into account the identification of w 2 with the inverse
of the area as shown in (7.2.10). With such identification the last term of (7.3.4)
represents a correction inversely proportional to the total flux B A through the
sample.
We remark t h a t 21//2 in (7.3.4) satisfies

2142 ( - B , v = 1) - -2142 (B, v - 1) (7.3.5)

M2 ( - B , v = 0, 2) = - M 2 (B, v -- 0, 2) .

which show t h a t parity is not broken for bosons and fermions; instead for all other
values of v parity is broken.
The second regime which leads to an interesting result is that of low magnetic
field
Wc eB
- - = << 1 . (7.3.6)
6J m0)

This is the regime in which one can take the limit B --, 0 for finite values of w and
fl, and is the relevant one to study spontaneous magnetization effects. Indeed for
low magnetic field, M2 is given by a series in the small p a r a m e t e r (7.3.69) with
functions of flw as coefficients; more precisely we have

M2(B, v) = ~ e { t h [ f l w ( 1 - v ) ] c o t h ( f l w ) - ( 1 - v)} + O ( em---~

B) " (7.3.7)

For intermediate statistics (i.e. for v # 0, 1, 2) there is a parity breaking sponta-

neous magnetic moment # given by
 107

# - M2(B = O, v ) . (7.3.8)

This spontaneous magnetic moment vanishes in the limit of high temperature,

lim # = 0 (7.3.9)

whereas at zero temperature it reduces to

e
#L -- ~-,oolim# = ~mL~ (7.3.10)

where L~ is the orbital fractional angular momentum defined in Chapter 5 (see

in particular (5.85) and (5.86) for N = 2). Notice that the gyromagnetic factor
for #L is 1, as usual for an o r b i t a l magnetic moment. This is to be contrasted
with the spin magnetic moment # s computed in (Kogan 1991; Stern 1991) for
which the gyromagnetic factor is 2, as should be expected. Finally, we remark
t h a t the spontaneous magnetic moment # vanishes for non-confined anyons ( i . e .
when ~v = 0) at any finite temperature. It would be interesting and important
to extend these results to systems of many anyons, and see if the spontaneous
magnetization we have evidentiated for two anyons, survives in the thermodynamic
limit. Another interesting open problem is the study of the possible existence of
a Curie-like critical t e m p e r a t u r e and a corresponding phase transition; however
to make progress in this direction, it is necessary first to solve the many-anyon
problem.