<s>
The	O
classical	B-Algorithm
XY	I-Algorithm
model	I-Algorithm
(	O
sometimes	O
also	O
called	O
classical	O
rotor	O
(	O
rotator	O
)	O
model	O
or	O
O(2 )	O
model	O
)	O
is	O
a	O
lattice	B-Algorithm
model	I-Algorithm
of	O
statistical	O
mechanics	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
XY	B-Algorithm
model	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
specialization	O
of	O
Stanley	O
's	O
n-vector	B-Algorithm
model	I-Algorithm
for	O
.	O
</s>
<s>
The	O
notation	O
indicates	O
the	O
expectation	O
of	O
the	O
random	O
variable	O
in	O
the	O
infinite	O
volume	O
limit	O
,	O
after	O
periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
have	O
been	O
imposed	O
.	O
</s>
<s>
Using	O
the	O
Griffiths	O
inequality	O
in	O
the	O
formulation	O
of	O
Ginibre	O
,	O
Aizenman	O
and	O
Simon	O
proved	O
that	O
the	O
two	O
point	O
spin	O
correlation	O
of	O
the	O
ferromagnetics	O
XY	B-Algorithm
model	I-Algorithm
in	O
dimension	O
,	O
coupling	O
and	O
inverse	O
temperature	O
is	O
dominated	O
by	O
(	O
i.e.	O
</s>
<s>
As	O
in	O
any	O
'	O
nearest-neighbor	O
'	O
n-vector	B-Algorithm
model	I-Algorithm
with	O
free	O
(	O
non-periodic	O
)	O
boundary	O
conditions	O
,	O
if	O
the	O
external	O
field	O
is	O
zero	O
,	O
there	O
exists	O
a	O
simple	O
exact	O
solution	O
.	O
</s>
<s>
Indeed	O
,	O
like	O
the	O
one-dimensional	O
Ising	O
model	O
,	O
the	O
one-dimensional	O
XY	B-Algorithm
model	I-Algorithm
has	O
no	O
phase	O
transitions	O
at	O
finite	O
temperature	O
.	O
</s>
<s>
The	O
same	O
computation	O
for	O
periodic	B-Algorithm
boundary	I-Algorithm
condition	I-Algorithm
(	O
and	O
still	O
)	O
requires	O
the	O
transfer	O
matrix	O
formalism	O
,	O
though	O
the	O
result	O
is	O
the	O
same	O
.	O
</s>
<s>
The	O
two-dimensional	O
XY	B-Algorithm
model	I-Algorithm
with	O
nearest-neighbor	O
interactions	O
is	O
an	O
example	O
of	O
a	O
two-dimensional	O
system	O
with	O
continuous	O
symmetry	O
that	O
does	O
not	O
have	O
long-range	O
order	O
as	O
required	O
by	O
the	O
Mermin	O
–	O
Wagner	O
theorem	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
discrete	O
lattice	B-Algorithm
of	O
spins	O
,	O
the	O
two-dimensional	O
XY	B-Algorithm
model	I-Algorithm
can	O
be	O
evaluated	O
using	O
the	O
transfer	O
matrix	O
approach	O
,	O
reducing	O
the	O
model	O
to	O
an	O
eigenvalue	O
problem	O
and	O
utilizing	O
the	O
largest	O
eigenvalue	O
from	O
the	O
transfer	O
matrix	O
.	O
</s>
<s>
The	O
2D	O
XY	B-Algorithm
model	I-Algorithm
has	O
also	O
been	O
studied	O
in	O
great	O
detail	O
using	O
Monte	B-Application
Carlo	I-Application
simulations	O
,	O
for	O
example	O
with	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
Monte	B-Application
Carlo	I-Application
simulation	O
,	O
each	O
spin	O
is	O
associated	O
to	O
a	O
continuously-varying	O
angle	O
(	O
often	O
,	O
it	O
can	O
be	O
discretized	O
into	O
finitely-many	O
angles	O
,	O
like	O
in	O
the	O
related	O
Potts	O
model	O
,	O
for	O
ease	O
of	O
computation	O
.	O
</s>
<s>
At	O
each	O
time	O
step	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
chooses	O
one	O
spin	O
at	O
random	O
and	O
rotates	O
its	O
angle	O
by	O
some	O
random	O
increment	O
.	O
</s>
<s>
The	O
Monte	B-Application
Carlo	I-Application
method	O
has	O
been	O
used	O
to	O
verify	O
,	O
with	O
various	O
methods	O
,	O
the	O
critical	O
temperature	O
of	O
the	O
system	O
,	O
and	O
is	O
estimated	O
to	O
be	O
.	O
</s>
<s>
The	O
Monte	B-Application
Carlo	I-Application
method	O
can	O
also	O
compute	O
average	O
values	O
that	O
are	O
used	O
to	O
compute	O
thermodynamic	O
quantities	O
like	O
magnetization	O
,	O
spin-spin	O
correlation	O
,	O
correlation	O
lengths	O
,	O
and	O
specific	O
heat	O
.	O
</s>
<s>
Similarly	O
the	O
mean-squared	O
magnetization	O
characterizes	O
the	O
average	O
of	O
the	O
square	O
of	O
net	O
components	O
of	O
the	O
spins	O
across	O
the	O
lattice	B-Algorithm
.	O
</s>
<s>
Rigorous	O
analysis	O
of	O
the	O
XY	B-Algorithm
model	I-Algorithm
shows	O
the	O
magnetization	O
in	O
the	O
thermodynamic	O
limit	O
is	O
zero	O
,	O
and	O
that	O
the	O
square	O
magnetization	O
approximately	O
follows	O
,	O
which	O
vanishes	O
in	O
the	O
thermodynamic	O
limit	O
.	O
</s>
<s>
The	O
magnetization	O
shown	O
(	O
for	O
a	O
25x25	O
lattice	B-Algorithm
)	O
is	O
one	O
example	O
of	O
this	O
,	O
that	O
appears	O
to	O
suggest	O
a	O
phase	O
transition	O
,	O
while	O
no	O
such	O
transition	O
exists	O
in	O
the	O
thermodynamic	O
limit	O
.	O
</s>
<s>
This	O
peak	O
position	O
and	O
height	O
have	O
been	O
shown	O
not	O
to	O
depend	O
on	O
system	O
size	O
,	O
for	O
lattices	O
of	O
linear	O
size	O
greater	O
than	O
256	O
;	O
indeed	O
,	O
the	O
specific	O
heat	O
anomaly	O
remains	O
rounded	O
and	O
finite	O
for	O
increasing	O
lattice	B-Algorithm
size	O
,	O
with	O
no	O
divergent	O
peak	O
.	O
</s>
<s>
The	O
nature	O
of	O
the	O
critical	O
transitions	O
and	O
vortex	O
formation	O
can	O
be	O
elucidated	O
by	O
considering	O
a	O
continuous	O
version	O
of	O
the	O
XY	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
continuous	O
version	O
of	O
the	O
XY	B-Algorithm
model	I-Algorithm
is	O
often	O
used	O
to	O
model	O
systems	O
that	O
possess	O
order	O
parameters	O
with	O
the	O
same	O
kinds	O
of	O
symmetry	O
,	O
e.g.	O
</s>
<s>
Topological	O
defects	O
in	O
the	O
XY	B-Algorithm
model	I-Algorithm
lead	O
to	O
a	O
vortex-unbinding	O
transition	O
from	O
the	O
low-temperature	O
phase	O
to	O
the	O
high-temperature	O
disordered	O
phase	O
.	O
</s>
<s>
The	O
presence	O
of	O
these	O
contributes	O
an	O
entropy	O
of	O
roughly	O
,	O
where	O
is	O
an	O
effective	O
length	O
scale	O
(	O
for	O
example	O
,	O
the	O
lattice	B-Algorithm
size	O
for	O
a	O
discrete	O
lattice	B-Algorithm
)	O
Meanwhile	O
,	O
the	O
energy	O
of	O
the	O
system	O
increases	O
due	O
to	O
the	O
vortex	O
,	O
by	O
an	O
amount	O
.	O
</s>
<s>
One	O
can	O
then	O
study	O
snapshots	O
of	O
the	O
spin	O
configurations	O
at	O
different	O
temperatures	O
to	O
elucidate	O
what	O
happens	O
above	O
and	O
below	O
the	O
critical	O
temperature	O
of	O
the	O
XY	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
To	O
identify	O
vortices	O
(	O
or	O
antivortices	O
)	O
present	O
as	O
a	O
result	O
of	O
the	O
Kosterlitz	O
–	O
Thouless	O
transition	O
,	O
one	O
can	O
determine	O
the	O
signed	O
change	O
in	O
angle	O
by	O
traversing	O
a	O
circle	O
of	O
lattice	B-Algorithm
points	O
counterclockwise	O
.	O
</s>
<s>
In	O
the	O
continuous	O
XY	B-Algorithm
model	I-Algorithm
,	O
the	O
high-temperature	O
spontaneous	O
magnetization	O
vanishes	O
:	O
</s>
<s>
As	O
mentioned	O
above	O
in	O
one	O
dimension	O
the	O
XY	B-Algorithm
model	I-Algorithm
does	O
not	O
have	O
a	O
phase	O
transition	O
,	O
while	O
in	O
two	O
dimensions	O
it	O
has	O
the	O
Berezinski-Kosterlitz-Thouless	O
transition	O
between	O
the	O
phases	O
with	O
exponentially	O
and	O
powerlaw	O
decaying	O
correlation	O
functions	O
.	O
</s>
<s>
In	O
three	O
and	O
higher	O
dimensions	O
the	O
XY	B-Algorithm
model	I-Algorithm
has	O
a	O
ferromagnet-paramagnet	O
phase	O
transition	O
.	O
</s>
<s>
Many	O
three-dimensional	O
physical	O
systems	O
belong	O
to	O
the	O
same	O
universality	O
class	O
as	O
the	O
three	O
dimensional	O
XY	B-Algorithm
model	I-Algorithm
and	O
share	O
the	O
same	O
critical	O
exponents	O
,	O
most	O
notably	O
easy-plane	O
magnets	O
and	O
liquid	O
Helium-4	O
.	O
</s>
<s>
The	O
values	O
of	O
these	O
critical	O
exponents	O
are	O
measured	O
by	O
experiments	O
,	O
Monte	B-Application
Carlo	I-Application
simulations	O
,	O
and	O
can	O
also	O
be	O
computed	O
by	O
theoretical	O
methods	O
of	O
quantum	O
field	O
theory	O
,	O
such	O
as	O
the	O
renormalization	O
group	O
and	O
the	O
conformal	O
bootstrap	O
.	O
</s>
<s>
Renormalization	O
group	O
methods	O
are	O
applicable	O
because	O
the	O
critical	O
point	O
of	O
the	O
XY	B-Algorithm
model	I-Algorithm
is	O
believed	O
to	O
be	O
described	O
by	O
a	O
renormalization	O
group	O
fixed	O
point	O
.	O
</s>
<s>
Most	O
important	O
critical	O
exponents	O
of	O
the	O
three	O
dimensional	O
XY	B-Algorithm
model	I-Algorithm
are	O
.	O
</s>
<s>
Monte	B-Application
Carlo	I-Application
methods	O
give	O
compatible	O
determinations	O
:	O
.	O
</s>
