(Non-)Abelian Lattice Gauge Theories

45 Pages·2013·6.82 MB·English

Atomic Quantum Simulation of
(Non-)Abelian Lattice Gauge Theories:
Quantum Link Models
UNIVERSITY OF INNSBRUCK
Peter Zoller
IQOQI
Innsbruck ITP Bern AUSTRIAN ACADEMY OF SCIENCES
€U AQUTE
M. Dalmonte D. Banerjee €U COHERENCE
E. Rico→ Ulm M. Bögli
M. Müller→ Madrid P. Stebler
UQUAM
U. J. Wiese
D. Marcos
K. Stannigel
(quantum optics) (high-energy physics) AFOSR
Atomic Quantum Simulation of
(Non-)Abelian Lattice Gauge Theories:
Quantum Link Models
UNIVERSITY OF INNSBRUCK
Peter Zoller
… possible future developments at the interface between
IQOQI
AMO and cond mat & particle physics AUSTRIAN ACADEMY OF SCIENCES
€U AQUTE
•
Abelian U(1) Schwinger model, D. Banerjee et al., PRL 2012 €U COHERENCE
•
U(N), SU(N) Non-Abelian QLM, D. Banerjee et al., PRL in print
UQUAM
•
Other
- dissipative techniques K. Stannigel et al.
- other implementations: superconducting qubits, D. Marcos et al.
AFOSR
Related work: MPQ-Tel Aviv, ICFO, Cornell, ...
… lattice gauge theories [in particle physics]
•
Gauge theories on a discrete lattice structure
K. Wilson, Phys. Rev. D (1974).
•
Fundamental gauge symmetries: standard model (every force has a
gauge boson)
non-perturbative approach to
fundamental theories of matter
(e.g. QCD)
a → classical statistical mechanics
quarks and gluons
lattice
Classical Monte Carlo simulations:
achievements issues
Sign problem in its various flavors:
• first evidence of quark-gluon plasma
• finite density QCD (=fermions)
• ab initio estimate of proton mass
• real time evolution
• entire hadronic spectrum
Quantum simulation (with atoms)? … toy models & simple phenomena
3
Quantum Simulation of Lattice Gauge Theories with Atoms
Lattice Gauge Theories Atomic Quantum Simulation
gauge
field
matter field optical lattice
Hamiltonian formulation
Kogut & Susskind
Hubbard models with bosonic and
fermionic atoms in optical lattices
(Non-)Abelian LGT:
✓QED U(1)
✓QCD SU(N), U(N)
?
gauge symmetry as
“emergent lattice gauge theory”
local symmetry
Quantum Simulation of Lattice Gauge Theories with Atoms
Lattice Gauge Theories Quantum Link Models (QLM)
Example:
gauge
U(1) / QED
field
fermionic atoms spin S
matter field
as matter field as quantum link: “spin ice”
(Non-)Abelian LGT: QLM
✓QED U(1) ✓gauge fields in finite dim spaces
✓QCD SU(N), U(N) ✓Abelian & Non-Abelian
particle physics: U.J. Wiese et al., Horn, Orland & Rohrlich, …
cond mat: … [spin ice, quantum spin liquids]
QLM have “natural” implementations
with cold atoms in optical lattices
5
•
Quantum Link Models (as Hubbard Models for Atoms)
U(1) Abelian LGT
D. Banerjee et al., PRL 2012
“spin ice” QED + matter
L R
fermionic atoms spin S
Schwinger boson
as matter field as quantum link
atomic boson-fermi mixtures in optical lattices
U(N),SU(N) Non-Abelian LGT
D. Banerjee et al., PRL in print
fermions
L R
i
(i = 1, 2, 3)
x
color index
fermionic rishons
multi-species fermi gases
6
H
gauge
invariant
subspace
⇢ E = 0
� r ·
Gauss law as a constraint
“spin ice” QED + matter
Hamiltonians, (local) gauge symmetries, Gauss law
etc.
an AMO perspective
7
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
c-number / static gauge fields
particles hopping around a plaquette acquire a finite phase
x y
2 ~ ~ ~
�= d f A
ˆ · r ⇥
flux
i'
H = t e xy + h.c.
x† y
�
y
~ ~
matter ' = dl A
phase xy
ˆ ·
field x
U(1) (abelian)
theory & AMO experiments
on “synthetic gauge fields”
- Hofstadter butterfly
- Fractional Quantum Hall,
Fractional Chern insulators
Review: J. Dalibard et al. Rev. Mod. Phys. (2011)
8
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
dynamical gauge fields
particles hopping around a plaquette assisted by link degrees of freedom
gauge
x y
field
H = t U + h.c. + . . .
x† xy y
�
link operator
matter field
9
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
dynamical gauge fields
particles hopping around a plaquette assisted by link degrees of freedom
G G
x x y y
H = t U + h.c. + . . .
x† xy y
�
gauge (local) symmetry U(1) local conserved quantity
V
i↵ i↵ [H, G ] = 0 x
U e xU e y,
gauge bosons xy xy � x
�! 8
V
fermions ei↵x generator of gauge transformation
x x
�!
i↵ G
V = e x x
unitary trafo:
generator
x
Y