structure factor - uc santa barbara
TRANSCRIPT
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Structure Factor
• In spin ice, there is no incipient ordered state: feature in correlations is more subtle than a peak
• Coarse-graining argument: correlations are governed by effective free energy
• Need to calculate He� =
�d3r
c
2|⌃b|2
hbµ(r)b⌫(r0)i =1
Z
Z[d~b(r)] �[~r ·~b] bµ(r)b⌫(r0) e��Heff [~b]
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Structure factor
• Fourier
• Constraint
• Structure factor
~r ·~b = 0
He↵ =X
k
c
2|~bk|2
bz
= �(kx
bx
+ ky
by
)/kz
He↵ =X
k
c
2B†
k
0
@1 + k2x
k2z
kx
ky
k2z
kx
ky
k2z
1 +k2y
k2z
1
ABk Bk
=
✓bx
by
◆,
hbµ(r)b⌫(r0)i =1
N
X
k
hb⇤µ(k)b⌫(k)ieik·(r�r0)
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Gaussian integrals
• General rule
• Proved in many many references...
�H =1
2
X
ij
Kijxixj
hxixji =1
Z
Z[Y
k
dxk]xixje��H
=⇥K�1
⇤ij
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Proof
• Generating function
• Shift
• Result
• Differentiating twice gives
heP
i qixii = 1
Z
Z[Y
k
dx
k
]e�12
Pij Kijxixj+
Pi qixi
xi ! xi +X
j
[K�1]ijqj
heP
i qixii = e12
Pij [K
�1]ijqiqj
hxixji =⇥K
�1⇤ij
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Gaussian integrals
• General rule: invert the quadratic form
• With some algebra
• We could have guessed this!X
µ
kµhb⇤µ(k)b⌫(k)i = 0
hb⇤µ(k)b⌫(k)i =kBT
c
0
@1 + k2x
k2z
kx
ky
k2z
kx
ky
k2z
1 +k2y
k2z
1
A�1
hb⇤µ(k)b⌫(k)i =kBT
c
✓�µ⌫ � kµk⌫
k2
◆
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Power law correlations
• Neutrons
• Measured near a reciprocal lattice vector
• Not a peak but a singularity
S(k) =X
µ⌫
(�µ⌫ � kµk⌫k2
)Sµ⌫(k)
S(K002 + k) = Sxx
(K + k) + Syy
(K + k)
⇡ 2�k2x
+ k2y
k2= 1 +
k2z
k2
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pinch points in Ho2Ti2O7
experiment theory
vanishes for kz=0
T. Fennell et al, 2009
S(K002 + k) ⇠ k2zk2
kz
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Quality of singularity
pinch point sharpens with lower T
“Correlation length” for rounding of pinch point
Roughly ξ~ e1.8K/T
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Defects
• The ice rules constraint is not perfectly enforced at T>0
• Primitive defect is a “charged” tetrahedron with ∑i σi = ±1.
costs energy 2Jeff
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What to call it?
• Consider Ising “spin”
• Single flipped tetrahedron has SzTOT=±1/2
• “spinon”? (M. Hermele et al, 2004)
• But Sz is not very meaningful in spin ice
• Use magnetic analogy: magnetic monopole
SzTOT =
X
i
�i =1
2
X
t
Szt
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Magnetic monopoles• Defect tetrahedra are sources and sinks of
“magnetic” flux
• It is a somewhat non-local object
• Must flip a semi-infinite string of spins to create a single monopole
• Note similarity to 1d domain wall
div b = 1
Castelnovo et al, 2008
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String
stolen (by somebody else on youtube) from Steve Bramwell
• Note that the string is tensionless because the energy depends only on ∑i σi on each tetrahedra
• In an ordered phase, this would cost energy
• Once created, the monopole can move by single spin flips
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Monopoles are “real”
• Monopoles actually are sources for (internal) magnetic field
• Magnetization M ∝ b
• hence div M ~ div H ~ q δ(r)
• Actual magnetic charge is small
Castelnovo et al, 2008
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Monopoles for dumbbells
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Dumbbell modelad magnetic charge ±q q = µ/ad
potential Vqq =µ0
4⇡
qaqbrab
µ ⇡ 10µBDy, Ho
=µ0
4⇡
µ2
a2d
1
rab
Vee =1
4⇡✏0
e2
rCoulomb