Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
Seemingly Unrelated Mathematical Question
?
=
Knot Invariant:
Picture of a Knot (or Link)
Rules
Such that topologically
equivalent pictures give the
same output
Some Mathematical Quantity
For two knots to be the same, it is necessary but not sufficient
that they have the same output.
Rules for Kauffman Invariant ( ≈ Jones Polynomial )
=
q1/2
+ q-1/2
= (-q-q-1) = d
Rules for Kauffman Invariant ( ≈ Jones Polynomial )
=
q1/2
+ q-1/2
=
q1/2
+ q -1/2
= (-q-q-1) = d
Ex:
= q1/2
q1/2
= q1/2 q1/2
+ q-1/2
+ q-1/2
+ q-1/2
q1/2
+ q-1/2 q1/2
+ q-1/2
+ q-1/2
= qd2 + d +d3 +q-1d2 = d + d3 + (q+q-1)d2 = d
Seemingly Unrelated: Topological Quantum Field Theory
•
TQFT = QFT where amplitudes depend only on the topology of the process.
(Witten, Moore, Seiberg, Froelich, … 1980s)
•
Amplitudes
are link invariants!
For Chern-Simons TQFT, amplitude of a process is given by the Jones
Polynomial of the knot (Witten, 1989).
time
≈
Kauffman Invariant + Spin Statistics
=
q1/2
+ q -1/2
= (-q-q-1) = d
=
=
q1/2
(q1/2 d + q-1/2)
+q-1/2
=
-q3/2
Not a true topological
Knot Invariant?
World Ribbons not World Lines
• Pulling tight rotates particle
• Rotation gives a phase (spin physics)
q is a phase only !
• Removing this phase gives Jones polynomial
Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
Topological Phase of Matter :
• Def: A phase of matter which, at long distance and low energy is described
by a Topological Quantum Field Theory.
Topological Phase of Matter
Time
TQFT = QFT where amplitudes depend only on the topology of the process
≈
Multiple Quasiparticles
Time
≠
• In Nontrivial Topological Theories, the ground state in the presence of
quasiparticles/quasiholes is degenerate (multiple ground states).
Multiple Quasiparticles
Time
=
• In Nontrivial Topological Theories, the ground state in the presence of
quasiparticles/quasiholes is degenerate (multiple ground states).
Multiple Quasiparticles
How do we know
≠
?
Time
=
- The ONLY “dynamics” in bulk is that braiding quasiparticles makes
transitions between degenerate ground states
• In Nontrivial Topological Theories, the ground state in the presence of
quasiparticles/quasiholes is degenerate (multiple ground states).
Preparing quasiparticle “ket” states
≠
?
Time
How do we know
Also “bra” states
Matrix Elements:
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
= d2
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
=d
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
= q -9 d
• Can deduce overlaps
between any two
states
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
• Can deduce overlaps
between any two
states
• Can take matrix elements
of braids.
Matrix Elements:
For TQFT of Jones type
= ( -q-q-1 ) = d
Where a given TQFT
corresponds to some
particular value of q
• Can deduce overlaps
between any two
states
• Can take matrix elements
of braids.
BRAID
=…
Topological Phase of Matter :
• Def: A phase of matter which, at long distance and low energy is described
by a Topological Quantum Field Theory.
Topological Phase of Matter
• In Nontrivial Topological Theories, the ground state in the presence of
quasiparticles/quasiholes is degenerate (multiple ground states).
- The ONLY “dynamics” in bulk is that braiding quasiparticles makes
transitions between degenerate ground states
- No Local operator can mix the multiple ground states
= Quasiparticles Exhibit NonAbelian Statistics
Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
Topological Phase of Matter :
• Def: A phase of matter which, at long distance and low energy is described
by a Topological Quantum Field Theory.
Topological Phase of Matter
• In Nontrivial Topological Theories, the ground state in the presence of
quasiparticles/quasiholes is degenerate (multiple ground states).
- The ONLY “dynamics” in bulk is that braiding quasiparticles makes
transitions between degenerate ground states
- No Local operator can mix the multiple ground states
Almost all noise processes are local !
“Qubits” are HIGHLY protected from decoherence/error
Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Because it
really exists!
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
• Current theoretical work
strongly suggests that…
Mobility 31 million cm2/ V-sec
T = 9 mK (Xia et al, PRL 2004)
NonTrivial Topological Phases of Matter
= NonTrivial TQFTs = NonAbelian Statistics0.50
Really Exist !!!
T = 9 mK
5/2 ≈ SU(2)2 →
• Many experiments under way
to try to prove this!
0.45
7/3
2
Rxy (h/e )
12/5 ≈ SU(2)3 →
2
0.40
5/2
A
8/3
2
0.35
3
5/2
- Other quantum Hall states
- Chiral p-wave superconductors (Sr2RuO4),
- Superfluid He-3 films
- Cold Atomic Gases (Rotating or not)
- Atom/Ion Lattices
- Josephson Junction Arrays,
- Bismuth-Antimonide-Super Junctions..
- others?
Rxx (kΩ)
• Proposals to realize similar physics in:
7/3
2+3/8
8/3
14/5
1
12/5
0
4.6
4.8
5.0
5.2
B (T)
5.4
5.6
5.8
6.0
Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
1. Degenerate ground states (in presence of quasiparticles) act as the qubits
2. Unitary operations (gates) are performed on ground state by braiding
quasiparticles around each other.
Particular braids correspond to particular computations.
Reference:
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008
Bonesteel, Hormozi, Simon, … ; PRL 2005, 2006; PRB 2007
=
U
U
Quantum Circuit
time
Braid
1. Degenerate ground states (in presence of quasiparticles) act as the qubits
2. Unitary operations (gates) are performed on ground state by braiding
quasiparticles around each other.
Particular braids correspond to particular computations.
3. State can be initialized by “pulling” pairs from vacuum
State can be measured by trying to return pairs to vacuum
Ref: Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. DasSarma
Rev Mod Phys, September 2008
“Artist’s conception” of device
Theorem (Simon, Bonesteel, Freedman… PRL05): In any topological
quantum computer, all computations can be performed by moving
only a single quasiparticle!
1. Degenerate ground states (in presence of quasiparticles) act as the qubits
2. Unitary operations (gates) are performed on ground state by braiding
quasiparticles around each other.
Particular braids correspond to particular computations.
3. State can be initialized by “pulling” pairs from vacuum
State can be measured by trying to return pairs to vacuum
This idea due to: Freedman and Kitaev
TOPOLOGICAL QUANTUM COMPUTATION
Advantages:
• Topological Quantum “memory” highly protected from noise
• The operations (gates) are also topologically robust
Ref: Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. DasSarma
Rev Mod Phys, September 2008
Topological Phases of Matter
And Why You Should Be Interested
Steven H. Simon
φ
xford
hysics
Non-Abelian Anyons and Topological Quantum Computation
C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma
arXiv:0707.1889, Rev Mod Phys Sept 2008