Nick Bonesteel

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Bond Fluctuations in Random Singlet Phases
NHMFL Dept. of Physics, Florida State University
Nick Bonesteel
Work primarily with Huan Tran (FSU)
Thanks also to Kun Yang, Gil Refael, Lukasz
Fidkowski, Joel Moore, and many others.
Support US DOE
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Valence Bonds
Non-crossing valence bond basis
Complete, linearly independent basis for the
space of all singlet states.
Unique representation of any singlet state
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Valence Bonds
Non-crossing valence bond basis
Complete, linearly independent basis for the
space of all singlet states.
Unique representation of any singlet state
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
Strongest bond
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
Strongest bond
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
Strongest bond
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
Strongest bond
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
,
Bond strength distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
flows to fixed point distribution
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Random Spin ½ AFM Heisenberg Chains
(D. Fisher 94)
Random Singlet State
flows to fixed point distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Strongest field
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Strongest field
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Strongest field
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Strongest bond
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
Bond strength distribution
Field strength distribution
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Random Transverse Field Ising Model
(D. Fisher 95)
0
1
0
1
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Heisenberg Chain
Singlet Projection Operator

2

i1
i




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Quantum Q d2 State Potts Models


d
i1
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Interacting Chains of SU(2)k Particles
Quantum Dimension
Hilbert space dimensionality for N particles


d
i1
i




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Random Transverse Field Ising Model
(d )
Standard method for solving the 1D TFIM
represent spins using pairs of Majorana fermions,
or equivalently SU(2)2 particles.


1
0
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0
0
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Random critical TFIM has an SU(2)2 random singlet
inside it, special case of random singlet
phases for SU(2)k particles.
(NEB, Kun
Yang 07)
More interesting phases if FM bonds are included.
(Fidkowski, Refael, NEB, Moore,08)
(Fidkowski, Lin, Titum, Refael,09)
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Valence-Bond Monte Carlo
(A. Sandvik PRL 06)
Idea Project out ground state of H by
repeatedly applying H to some initial
valence-bond state S0gt
(
)
å
n
P
P

-
0
0
L
L
S
J
J
H
0
i
i
1
i
i
n
1
n
i
i
L
1
n
Sum over non-crossing valence-bond states.
Initial valence-bond state
Weight factors w(a) are easy to compute and
update for efficient Monte Carlo sampling.
Straightforward to generalize to SU(2)k particles.
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8
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2
3
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Uniform Heisenberg Chain
0
1
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Valence Bond Entanglement Entropy
(Alet et al. PRL07)
L 7
Number of bonds leaving block of size L
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Valence Bond Entanglement Entropy
(Alet et al. PRL07)
L 7
Number of bonds leaving block of size L
For a single valence bond state
entanglement entropy
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Valence Bond Entanglement Entropy
(Alet et al. PRL07)
L 7
Number of bonds leaving block of size L
For a single valence bond state
entanglement entropy
Valence bond entanglement entropy for state
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Exact Result for Uniform Chains
(Jacobsen Saleur. PRL08)
ltnLgt scales logarithmically with L
Valence-Bond entanglement entropy for uniform
Heisenberg chain
1
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Exact Result for Uniform Chains
(Jacobsen Saleur. PRL08)
ltnLgt scales logarithmically with L
Valence-Bond entanglement entropy for uniform
Heisenberg chain
1
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Valence Bond Entanglement Uniform Case (d2)
Block Size L
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Valence Bond Entanglement Uniform Case (d2)
Exact Result (Jacobsen Saleur)
Block Size L
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SU(2)k Singlet Bond Entanglement
(NEB K.Yang PRL07)
N gtgt 1 singlet bonds
A
B


N particles
Dimensionality of Hilbert space d N
Entropy per bond
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Exact Result for Uniform Chains
(Jaconsen Saleur. PRL07)
ltnLgt scales logarithmically with L
Valence-Bond entanglement entropy for uniform
SU(2)k chain
log2d
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Exact Result for Uniform Chains
(Jacobsen Saleur. PRL08)
ltnLgt scales logarithmically with L
Valence-Bond entanglement entropy for uniform
SU(2)k chain
Valence-bond central charge cVB close to, but
not equal to true central charge.
log2d
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Valence Bond Entanglement Entropy Uniform Case
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Valence Bond Entanglement Entropy Uniform Case
Exact result (Jacobsen Saleur, PRL 08)
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Real and Valence Bond Central Charge

- true central charge

d
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Real and Valence Bond Central Charge


- true central charge


- VBMC results for cVB
d
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Real and Valence Bond Central Charge


- true central charge


- VBMC results for cVB
- Exact result for cvb (JS)
d
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Random Singlet Phase Near Fixed Point
0
1
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Random Singlet Phase Near Fixed Point
0
1
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ltnLgt in Random Singlet Phase
(Refael Moore. PRL04)
In the random singlet phase ltnLgt also scales
logarithmically with L
Average over disorder
Entanglement entropy
1
effective central charge
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Entanglement in Random SU(2)k Chains
(NEB Yang.07)
In the random singlet phase ltnLgt also scales
logarithmically with L
Average over disorder
Entanglement entropy
log2d
effective central charge
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Valence Bond Entanglement Entropy Random Case
case first studied by Alet et al.
PRL 07
H. Tran, NEB, unpublished
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Valence Bond Entanglement Entropy Random Case
case first studied by Alet et al.
PRL 07
H. Tran, NEB, unpublished
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Random Singlet Phase Far From Fixed Point
1-u
0
1u
u 0.5
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Random Singlet Phase Far From Fixed Point
1-u
0
1u
u 0.5
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How do we know bonds are freezing?
Look at fluctuations in ltnLgt
L
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How do we know bonds are freezing?
Look at fluctuations in ltnLgt
L
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How do we know bonds are freezing?
Look at fluctuations in ltnLgt
L
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How do we know bonds are freezing?
Look at fluctuations in ltnLgt
L
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How do we know bonds are freezing?
Look at fluctuations in ltnLgt
L
If bonds are frozen, only fluctuations near
boundary of region change the number of bonds
leaving that region.
Expect sn2 to be independent of L for large L if
bonds freeze.
Average over disorder
Bond fluctuations for particular realization of
disorder
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Random Singlet Phase Formation (d2)
Uniform chain
Block Size L
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Random Singlet Phase Formation (d2)
Uniform chain
Exact Result (Jacobsen Saleur)
Uniform chain
Block Size L
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Random Singlet Phase Formation (d2)
Uniform chain
Exact Result (Jacobsen Saleur)
u 1.0
Block Size L
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Random Singlet Phase Formation (d2)
Uniform chain
Exact Result (Jacobsen Saleur)
u 0.5
u 1.0
Block Size L
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Random Singlet Phase Formation (d2)
Uniform chain
Exact Result (Jacobsen Saleur)
u 0.1
u 0.5
u 1.0
Block Size L
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Random Singlet Phase Formation (d )
Uniform chain
u 0.1
u 0.5
u 1.0
Block Size L
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Conclusions
nL and its fluctuations easy to compute
quantities which can be used to study random
singlet formation.
Valence bond basis a natural and intuitive
basis for visualizing singlet states.