Anyon Physics of Ultracold Atomic Gases

86 Pages·2014·12.42 MB·English

Majorana fermions,
Pfaffian states,
& Ising anyons
Matteo Rizzi
Johannes Gutenberg-Universität Mainz
Lectures at the International School & Workshop
“Anyon Physics of Ultracold Atomic Gases”
TU Kaiserslautern, 12-15 Dec. 2014
Lectures’ Plan
•
1D: Kitaev-Majorana Chain & Fractionalization
•
Fermionic Gaussian States
•
2D: Chiral BCS Pairing (p + i p ) & Vortices
x y
•
Ideas for Experiments
•
Lowest Landau Level & 3-body interactions (Pfaffian)
•
3-hardcore Bosons on a Lattice: FQH states?
•
Ideas for Experiments
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
Lectures’ Plan
•
1D: Kitaev-Majorana Chain & Fractionalization
•
Fermionic Gaussian States
•
2D: Chiral BCS Pairing (p + i p ) & Vortices
x y
•
Ideas for Experiments
•
Lowest Landau Level & 3-body interactions (Pfaffian)
•
3-hardcore Bosons on a Lattice: FQH states?
•
Ideas for Experiments
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
1D: Kitaev chain
1/6
• spinless fermions
Kitaev, arXiv:cond-mat/0010440 (2001)
• BCS-like pairing (i.e. Cooper pairs tunnelling in/out)
• dual to Ising / XY model via non-local Jordan-Wigner mapping
➡
what happens to local order parameter ?
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
Majorana Fermions
2/6
• split fermionic operators into Majorana ones
• focus on the “simple” case
Somehow analog
to spin-1 SU(2) chain:
AKLT & Haldane phase
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
Fractional Edge Modes
3/6
• two uncoupled Majorana’s at : &
• one non-local Dirac fermion:
• easy Bogolubov eigenmodes
• gapped system with degenerate grounds: &
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
Fractional Edge Modes
3/6
• two uncoupled Majorana’s at : &
• one non-local Dirac fermion:
• easy Bogolubov eigenmodes
• gapped system with degenerate grounds: &
• exponential corrections for : etc.
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
Topological order & simmetry
4/6
• indistinguishability by local measures
& stability to local perturbations
==> topological quantum order (TQO)
Kitaev, Annals Phys. 303, 2-30 (‘03) // Bravyi et al, J Math Phys 51 093512 (‘10)
Dennis, et al., J. Math. Phys. 43 4452 (2002) // Nayak, et al., RMP 80 1083 (2008)
• fermionic parity as non-local order parameter
==> symmetry protected topological order (SPTO)
Pollmann, et al., PRB 81, 064439 (2010); Turner, et al. , PRB 83, 075102 (2011); Pollmann, et al., PRB 85, 075125 (2012).
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
A quantum memory !?
5/6
• robust(?) storing of qubits in fractional edges (2 chains) …
too many Refs :)
• single particle poisoning is detrimental Budich, Walter, Trauzettel, PRB 85, 121405 (2012)
Rainis, Loss, Phys. Rev. B 85, 174533 (2012)
• parity preserving noise does not affect, even above gap !
1
2 N = 8
(t) || / tr00..9968 NN == 1126
−
− (cid:108) 0.94 N = 20
) N = 24
(t+0.92
(cid:108)
|| 0.9 (a) (b)

0 50 100 150 2000 50 100 150 200
−1 −1
time (J ) time (J )
200
(c)
1) 100
−
J
( 50
0
t 25
8 12 16 20 24
L.Mazza, M.Rizzi, M. Lukin, J.I. Cirac, PRB 88, 205142 (2013) N
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons
1D: Kitaev chain
Physical implementation?
6/6
• proximity effect between • optical lattices with spin-orbit
supercond. & metallic wires coupled to a BEC cloud
Fu, Kane, PRL 102, 216403 (2009) L. Jiang et al., PRL 106 220402 (2011)
Stanescu, et al, PRB 84 144522 (2011) Kraus, et al., NJP 14 113036 (2012)
Pachos, Alba, et al. PRA 88, 013622 (2013)
• recent claims of observation !
• dissipative engineering
Mourik, et al., Science 336, 1003 (2012)
Diehl, Rico, Baranov, Zoller, Nat. Phys. 7, 971 (2011)
Nadj-Perge, et al., Science 346, 602(2014).
Majorana Fermions,
“Anyon Physics of
[email protected] Pfaffian states,
Ultracold Atomic Gases”
Ising anyons