Random Chains of Interacting NonAbelian Quasiparticles

1
Random Chains of Interacting Non-Abelian
Quasiparticles
Nick Bonesteel Kun Yang Huan Tran
Florida State University

• NEB, K. Yang, Phys. Rev. Lett. , in Press.
  quant-ph/0612503
• some unpublished numerical results by H. Tran

Support US DOE
2
Non-Abelian FQH States (Moore, Read 91)
Fractionally charged quasiparticles
Essential features
A degenerate Hilbert space whose dimensionality
is exponentially large in the number of
quasiparticles.
States in this space can only be distinguished by
global measurements provided quasiparticles are
far apart.
3
Possible Non-Abelian FQH States
J.S. Xia et al., PRL (2004).
4
Possible Non-Abelian FQH States
J.S. Xia et al., PRL (2004).

• 5/2 Probable Moore-Read Pfaffian state.
  Charge e/4 quasiparticles described by SU(2)2
  Chern-Simons Theory.
• Nayak and Wilczek, 96

5
Possible Non-Abelian FQH States
J.S. Xia et al., PRL (2004).

• 5/2 Probable Moore-Read Pfaffian state.
  Charge e/4 quasiparticles described by SU(2)2
  Chern-Simons Theory.
• Nayak and Wilczek, 96

• 12/5 Possible Read-Rezayi Parafermion
  state.
• Read and Rezyai, PRB 99.
• Charge e/5 quasiparticles described by SU(2)3
  Chern-Simons Theory
• Slingerland and Bais 01

6
SU(2)k Nonabelian Particles
Describe quasiparticle excitations of the
Read-Rezayi Parafermion FQH states at level k
(up to Abelian phases).
Read and Rezyai, 99
Slingerland and Bais, 01
Read-Rezayi states may be realizable in spinning
Bose gases.
Cooper, Wilkin, Gunn, 02
Rezayi, Read, Cooper, 05
Related non-Abelian states may also be realizable
as string-net condensates.
Levin and Wen 04
Fendley and Fradkin 05
Freedman, Nayak, Shtengel, Walker and Wang 03
But before SU(2)k, there was just plain old
SU(2).
7
Particles with Ordinary Spin SU(2)

• Particles have spin s 0, 1/2, 1, 3/2 ,

spin ½
2. Triangle Rule for adding angular momentum
For example
Two particles can have total spin 0 or 1.

a
b
0
1
Numbers label total spin of particles inside ovals
8
Hilbert Space
9
Hilbert Space
S
7/2
Paths are basis states
3
5/2
2
3/2
1
1/2
N
11
12
0
1
2
3
4
5
6
7
8
10
9
1
1/2
1
3/2
10
Valence Bonds Basis

0
Non-crossing valence bond basis
Any two particles connected by a bond form a
singlet
Complete, linearly independent basis for the
space of all singlet states.
11
Valence Bonds Basis
Nonorthogonal basis, but easy to compute with
12
S
44
1
8
7/2
35
7
1
3
110
6
27
1
5/2
75
20
5
1
2
165
14
48
4
1
3/2
90
28
9
3
1
1
132
5
14
42
2
1
1/2
5
1
2
132
14
42
N
11
12
0
1
2
3
4
5
6
7
8
10
9
Singlet Space Dim(N) N!/((N/21)!(N/2)!)
2N
13
Particles with Ordinary Spin SU(2)

• Particles have spin s 0, 1/2, 1, 3/2 ,

spin ½
2. Triangle Rule for adding angular momentum
For example
Two particles can have total spin 0 or 1.

a
b
0
1
14
Nonabelian Particles SU(2)k

• Particles have topological charge s 0, 1/2, 1,
  3/2 , , k/2

topological charge ½
2. Fusion Rule for adding topological charge
For example
Two particles can have total topological
charge 0 or 1.
a
b
0
1
15
S
44
1
8
7/2
35
7
1
3
110
6
27
1
5/2
75
20
5
1
2
165
14
48
4
1
3/2
90
28
9
3
1
1
132
5
14
42
2
1
1/2
5
1
132
2
14
42
N
11
12
0
1
2
3
4
5
6
7
8
10
9
Dim(N) 2N
16
k 4
Dim(N) 3N/2
17
k 3
Dim(N) Fib(N1) fN
18
(No Transcript)
19
Quantum Dimension
Hilbert space of N particles with topological
charge ½ grows asymptotically as d N where d is
the quantum dimension of the particles.
k
d
2
3
4
2
20
Valence Bonds Basis for SU(2)k

0
Non-crossing valence bond basis
Any two particles connected by a bond fuse to
trivial topological charge 0 if brought together.
A complete, but linearly dependent basis for the
space of all singlet states.
21
Valence Bonds Basis for SU(2)k
Again, nonorthogonal, but still easy to compute
with
Quantum Dimension
22
Interacting Non-Abelian Anyons
Localized quasiparticles
Topological degeneracy is lifted when
quasiparticles are close together (for FQHE
states, this means within a few magnetic lengths).
Assume trivial topological charge is
energetically favored
å
P
-

J
0
J
H

j
i
,
j
i
j
i
,
,
energy
j
i
,
1
Projection onto state of particle i and j with
total topological charge 0.
J
0
23
Simplifed Model Anyon Chain
(Feiguin et al., PRL 98, 160409 (2007).)
Ji
Ji-1
Ji-2
Ji1
Ji2
Assume trivial topological charge is
energetically favored
energy
1
J
(
)
0
24
Properties of Projection Operator
For ordinary spin-1/2 particles
Action on valence-bond states for SU(2)k
particles

1
2
3
4
1

1
1
d
2
3
4
25
Uniform SU(2)k Chains
Ordinary spin-1/2 AFM Heisenberg model
Conformally invariant quantum critical model with
central charge c1
Uniform SU(2)k chains are all conformally
invariant with central charge
(Feiguin et al., PRL 98, 160409 (2007).)
Heisenberg Model
Golden Chain
Critical TFIM
26
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
27
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
J2
J1
J3
Strongest Bond
28
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
J2
J1
J3
Strongest Bond
29
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
Effective interaction from 2nd order perturbation
theory
Spin-1/2 particles (Ma, DasGupta, Hu 79)
J2
J1
J3
Strongest Bond
30
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
Effective interaction from 2nd order perturbation
theory
SU(2)k particles (NEB, K.Yang 06)
J2
J1
J3
Strongest Bond
31
Random SU(2)k Chains
Given the similarity between ordinary spin and
SU(2)k particles we can apply the real space RG.
(Ma, DasGupta, Hu 79, D. Fisher 94)
Random Singlet Phase for SU(2)k particles
Bonds freeze into a particular non-crossing
valence-bond state.
32
Random Singlet Phase
(D. Fisher 94)
Strongest remaining bond
Logarithmic bond strength
1
0
33
Random Singlet Phase
(D. Fisher 94)
Strongest remaining bond
Logarithmic bond strength
1
0
34
Random Singlet Phase
(D. Fisher 94)
Strongest remaining bond
Logarithmic bond strength
Distribution of b s broadens and flattens as
bonds are decimated.
0
35
Random Singlet Phase
(D. Fisher 94)
Strongest remaining bond
Logarithmic bond strength
Distribution of b s broadens and flattens as
bonds are decimated.
0
36
Random Singlet Phase
(D. Fisher 94)
Flow parameter
strongest bond at start of RG
Flow equation
Fixed point solution
G
Independent of details of initial bond
distribution.
0
37
Random Singlet Phase
(D. Fisher 94)
Number of unpaired particles as a function of
flow parameter
At temperature T assume all particles with bond
energies W gtkT are frozen, and all particles with
bond energies W gt kT are free.
of free particles
Entropy
Should hold for random SU(2)k chains.
Specific Heat
38
Random SU(2)k chains all flow to random singlet
phases with bond distribution
Random AFM spin-1/2 Heisenberg Model
Critical Transverse Field Ising Model
39
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
40
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
41
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
42
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
43
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
44
SU(2)2 Hilbert Space
S
1
1/2
N
0
11
12
0
1
2
3
4
5
6
7
8
10
9
Map interacting anyon model onto spin model
å

P
0
J
H
i

i
i

Random TFIM at Critical Point
45
Entanglement Entropy
A quantum system composed of two parts A and B
Reduced density matrix
Entanglement entropy
Simple example An SU(2) singlet bond
A
B
46
Entanglement Entropy
At 11 dimensional conformally invariant quantum
critical points, the entanglement entropy scales
logarithmically with the size of region A with a
universal coefficient
A
L
c central charge
(Holzhey et al. 94, Calabrese Cardy 04)
For uniform Heisenberg model (c1)
For uniform critical TFIM (c1/2)
47
Entanglement Entropy of Random Chains
(Refael Moore PRL 93, 260602 (2004))
In the random singlet phase the entanglement
entropy also scales logarithmically with L
A
L
Avg. of bonds leaving region of length L
1
effective central charge
48
Entanglement Entropy of Random Chains
(Refael Moore PRL 93, 260602 (2004))
Uniform
Random
Heisenberg Model
Critical TFIM
Effective central charge decreases from uniform
to random models, consistent with a generalized
c-theorem.
(But see Santachiara cond-mat/0602527 for
counterexamples)
49
Entanglement Entropy of Random SU(2)k Chains
For SU(2)k random chains the only thing that is
different is the entanglement per bond.
Imagine N gtgt 1 singlet pairs
A


N particles
Dimensionality of Hilbert space d N
Entropy per bond
(NEB, K.Yang 06)
50
Entanglement Entropy of Random SU(2)k Chains
A
L
log2d
Effective central charge
(NEB, K.Yang 06)
51
Entanglement Entropy of Random Chains
Quantum Dimension
Random
Uniform
SU(2)k Chain
Heisenberg Model
Critical TFIM
Golden Chain
52
Valence-Bond Monte Carlo
(Sandvik, PRL 95, 207203 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.
53
Properties of Projection Operator
For ordinary spin-1/2 particles
Action on valence-bond states for SU(2)k
particles

1
2
3
4
1

1
1
d
2
3
4
54
Valence-Bond Entanglement
(Alet, Capponi, Laflorencie, Matthieu,
cond-mat/0703027)
For the ground state wavefunction
the valence-bond entanglement is defined to be
Entanglement entropy in the valence-bond state a
gt computed a la Refael and Moore.
If bonds freeze on long length scales then
SVB(L) should show the same scaling as S(L) for
large L.
55
Valence-Bond Entanglement

Random Heisenberg spin chain
(First computed by Alet et al. cond-mat/0703027)
56
Valence-Bond Entanglement

Random critical TFIM
57
Valence-Bond Entanglement

Random golden chain
58
Conclusions
SU(2)k particles can enter random singlet phases.
Random AFM Heisenberg Chain
Random Critical TFIM
Universal entanglement scaling.