<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
is	O
an	O
approximate	O
model	O
used	O
to	O
describe	O
the	O
transition	O
between	O
conducting	O
and	O
insulating	O
systems	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
states	O
that	O
each	O
electron	O
experiences	O
competing	O
forces	O
:	O
one	O
pushes	O
it	O
to	O
tunnel	O
to	O
neighboring	O
atoms	O
,	O
while	O
the	O
other	O
pushes	O
it	O
away	O
from	O
its	O
neighbors	O
.	O
</s>
<s>
The	O
particles	O
can	O
either	O
be	O
fermions	O
,	O
as	O
in	O
Hubbard	O
's	O
original	O
work	O
,	O
or	O
bosons	O
,	O
in	O
which	O
case	O
the	O
model	O
is	O
referred	O
to	O
as	O
the	O
"	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
"	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
is	O
a	O
useful	O
approximation	O
for	O
particles	O
in	O
a	O
periodic	O
potential	O
at	O
sufficiently	O
low	O
temperatures	O
,	O
where	O
all	O
the	O
particles	O
may	O
be	O
assumed	O
to	O
be	O
in	O
the	O
lowest	O
Bloch	O
band	O
,	O
and	O
long-range	O
interactions	O
between	O
the	O
particles	O
can	O
be	O
ignored	O
.	O
</s>
<s>
If	O
interactions	O
between	O
particles	O
at	O
different	O
sites	O
of	O
the	O
lattice	O
are	O
included	O
,	O
the	O
model	O
is	O
often	O
referred	O
to	O
as	O
the	O
"	O
extended	O
Hubbard	B-Algorithm
model	I-Algorithm
"	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
introduces	O
short-range	O
interactions	O
between	O
electrons	O
to	O
the	O
tight-binding	O
model	O
,	O
which	O
only	O
includes	O
kinetic	O
energy	O
(	O
a	O
"	O
hopping	O
"	O
term	O
)	O
and	O
interactions	O
with	O
the	O
atoms	O
of	O
the	O
lattice	O
(	O
an	O
"	O
atomic	O
"	O
potential	O
)	O
.	O
</s>
<s>
When	O
the	O
interaction	O
between	O
electrons	O
is	O
strong	O
,	O
the	O
behavior	O
of	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
can	O
be	O
qualitatively	O
different	O
from	O
a	O
tight-binding	O
model	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
correctly	O
predicts	O
the	O
existence	O
of	O
Mott	O
insulators	O
:	O
materials	O
that	O
are	O
insulating	O
due	O
to	O
the	O
strong	O
repulsion	O
between	O
electrons	O
,	O
even	O
though	O
they	O
satisfy	O
the	O
usual	O
criteria	O
for	O
conductors	O
,	O
such	O
as	O
having	O
an	O
odd	O
number	O
of	O
electrons	O
per	O
unit	O
cell	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
is	O
based	O
on	O
the	O
tight-binding	O
approximation	O
from	O
solid-state	O
physics	O
,	O
which	O
describes	O
particles	O
moving	O
in	O
a	O
periodic	O
potential	O
,	O
typically	O
referred	O
to	O
as	O
a	O
lattice	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
introduces	O
a	O
contact	O
interaction	O
between	O
particles	O
of	O
opposite	O
spin	O
on	O
each	O
site	O
of	O
the	O
lattice	O
.	O
</s>
<s>
When	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
is	O
used	O
to	O
describe	O
electron	O
systems	O
,	O
these	O
interactions	O
are	O
expected	O
to	O
be	O
repulsive	O
,	O
stemming	O
from	O
the	O
screened	O
Coulomb	O
interaction	O
.	O
</s>
<s>
The	O
physics	O
of	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
is	O
determined	O
by	O
competition	O
between	O
the	O
strength	O
of	O
the	O
hopping	O
integral	O
,	O
which	O
characterizes	O
the	O
system	O
's	O
kinetic	O
energy	O
,	O
and	O
the	O
strength	O
of	O
the	O
interaction	O
term	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
can	O
therefore	O
explain	O
the	O
transition	O
from	O
metal	O
to	O
insulator	O
in	O
certain	O
interacting	O
systems	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
can	O
explain	O
the	O
transition	O
from	O
conductor	O
to	O
insulator	O
in	O
systems	O
such	O
as	O
rare-earth	O
pyrochlores	O
as	O
the	O
atomic	O
number	O
of	O
the	O
rare-earth	O
metal	O
increases	O
,	O
because	O
the	O
lattice	O
parameter	O
increases	O
(	O
or	O
the	O
angle	O
between	O
atoms	O
can	O
also	O
change	O
)	O
as	O
the	O
rare-earth	O
element	O
atomic	O
number	O
increases	O
,	O
thus	O
changing	O
the	O
relative	O
importance	O
of	O
the	O
hopping	O
integral	O
compared	O
to	O
the	O
on-site	O
repulsion	O
.	O
</s>
<s>
This	O
orbital	O
can	O
be	O
occupied	O
by	O
at	O
most	O
two	O
electrons	O
,	O
one	O
with	O
spin	O
up	O
and	O
one	O
down	O
(	O
see	O
Pauli	B-General_Concept
exclusion	I-General_Concept
principle	I-General_Concept
)	O
.	O
</s>
<s>
Expressed	O
using	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
,	O
the	O
Hamiltonian	O
is	O
made	O
up	O
of	O
two	O
terms	O
.	O
</s>
<s>
If	O
is	O
not	O
too	O
large	O
,	O
the	O
overlap	O
integral	O
provides	O
for	O
superexchange	B-General_Concept
interactions	O
between	O
neighboring	O
magnetic	O
moments	O
,	O
which	O
may	O
lead	O
to	O
a	O
variety	O
of	O
interesting	O
magnetic	O
correlations	O
,	O
such	O
as	O
ferromagnetic	O
,	O
antiferromagnetic	O
,	O
etc	O
.	O
</s>
<s>
The	O
one-dimensional	O
Hubbard	B-Algorithm
model	I-Algorithm
was	O
solved	O
by	O
Lieb	O
and	O
Wu	O
using	O
the	O
Bethe	O
ansatz	O
.	O
</s>
<s>
Although	O
Hubbard	O
is	O
useful	O
in	O
describing	O
systems	O
such	O
as	O
a	O
1D	O
chain	O
of	O
hydrogen	O
atoms	O
,	O
it	O
is	O
important	O
to	O
note	O
that	O
more	O
complex	O
systems	O
may	O
experience	O
other	O
effects	O
that	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
does	O
not	O
consider	O
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
as	O
analogous	O
to	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
for	O
hydrogen	O
chains	O
,	O
where	O
conduction	O
between	O
unit	O
cells	O
can	O
be	O
described	O
by	O
a	O
transfer	O
integral	O
.	O
</s>
<s>
The	O
fact	O
that	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
has	O
not	O
been	O
solved	O
analytically	O
in	O
arbitrary	O
dimensions	O
has	O
led	O
to	O
intense	O
research	O
into	O
numerical	O
methods	O
for	O
these	O
strongly	O
correlated	O
electron	O
systems	O
.	O
</s>
<s>
Approximate	O
numerical	O
treatment	O
of	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
on	O
finite	O
systems	O
is	O
possible	O
via	O
various	O
methods	O
.	O
</s>
<s>
With	O
projector	O
and	O
finite-temperature	O
auxiliary-field	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
,	O
two	O
statistical	O
methods	O
exist	O
that	O
can	O
obtain	O
certain	O
properties	O
of	O
the	O
system	O
.	O
</s>
<s>
The	O
Hubbard	B-Algorithm
model	I-Algorithm
can	O
be	O
studied	O
within	O
dynamical	O
mean-field	O
theory	O
(	O
DMFT	O
)	O
.	O
</s>
<s>
This	O
scheme	O
maps	O
the	O
Hubbard	O
Hamiltonian	O
onto	O
a	O
single-site	B-Algorithm
impurity	I-Algorithm
model	I-Algorithm
,	O
a	O
mapping	O
that	O
is	O
formally	O
exact	O
only	O
in	O
infinite	O
dimensions	O
and	O
in	O
finite	O
dimensions	O
corresponds	O
to	O
the	O
exact	O
treatment	O
of	O
all	O
purely	O
local	O
correlations	O
only	O
.	O
</s>
<s>
DMFT	O
allows	O
one	O
to	O
compute	O
the	O
local	O
Green	O
's	O
function	O
of	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
for	O
a	O
given	O
and	O
a	O
given	O
temperature	O
.	O
</s>
<s>
Within	O
DMFT	O
,	O
the	O
evolution	O
of	O
the	O
spectral	B-Algorithm
function	I-Algorithm
can	O
be	O
computed	O
and	O
the	O
appearance	O
of	O
the	O
upper	O
and	O
lower	O
Hubbard	O
bands	O
can	O
be	O
observed	O
as	O
correlations	O
increase	O
.	O
</s>
<s>
The	O
device	O
functioned	O
at	O
a	O
temperature	O
of	O
5	O
Kelvins	B-Operating_System
,	O
far	O
above	O
the	O
temperature	O
at	O
which	O
the	O
effect	O
had	O
first	O
been	O
observed	O
.	O
</s>
