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]

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)

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

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

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

◆

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

pinch points in Ho2Ti2O7

experiment theory

vanishes for kz=0

T. Fennell et al, 2009

S(K002 + k) ⇠ k2zk2

kz

Quality of singularity

pinch point sharpens with lower T

“Correlation length” for rounding of pinch point

Roughly ξ~ e1.8K/T

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

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

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

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

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

Monopoles for dumbbells

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