581 Final: Due May 15, 2009
•
Instructions:
Work independently and turn in exam by 5:00pm May 15.
Turn in
directly to me in my office or place in envelope outside my office.
•
Problem 1:
Quantum phase transitions occur at zero temperature and arise funda
mentally from the uncertainty principle. Hence, present in any Hamil
tonian that admits a quantum phase transition must be two non
commuting operators. You are to consider the following Hamiltonian
H
=
−
J
summationdisplay
i
parenleftBig
gσ
x
i
+
σ
z
i
σ
z
i
+1
parenrightBig
(1)
where
σ
μ
i
is a Pauli matrix,
i
refers to site
i
and
J
and
g
are coupling
constants.
(a) Identify the two ordering tendencies of this Hamiltonian. That is,
identify the ground state for
g
= 0 and
g
=
∞
.
(b) For
g
≫
1, a fairly accurate solution for the wavefunction is

i
)
=
−)
i
productdisplay
j
negationslash
=
i

+
)
j
±)
i
= (
 ↑)
i
±  ↓)
i
)
/
√
2
.
(2)
Show that the energy of the state

k
)
=
1
√
N
summationdisplay
i
e
ikx
i

i
)
,
(3)
is given by
ǫ
k
=
Jg
parenleftBigg
2
−
2
g
cos
k
+
O
(1
/g
2
) +
...
parenrightBigg
.
(4)
(c) This problem can be solved exactly using the JordanWigner trans
formation:
σ
x
i
= 1
−
2
n
i
,
σ
z
i
=
−
productdisplay
j<i
(1
−
2
n
j
)(
c
i
+
c
†
i
)
,
σ
y
i
=
i
productdisplay
l<i
(1
−
2
n
l
)(
c
†
i
−
c
i
) (5)
1
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where the
c
i
(
c
†
i
) are fermionic annihilation (creation) operators and
n
i
=
c
†
i
c
i
. You are to apply this transformation to the Hamiltonian for
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 Spring '08
 Phillips
 mechanics, Work, Uncertainty Principle, Phase transition, Quantum phase transition, canonical partition function

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