Simulation of dynamic abelian and non-abelian gauge theories with ...

Simulation of dynamic abelian and
non-abelian gauge theories
with ultracold atoms
Benni Reznik
Tel Aviv University
Joint work with
Erez Zohar, Tel Aviv University
J. Ignacio Cirac, MPQ
MPQ, January 29, 2013
1

Talk outline
• HEP and Lattice gauge theory – overview
• Confinement in 1+1 – a simple realization
• Further examples: non-abelian and higher
dimensional gauge theories.
• Outlook and Summary
• List of relevant publications

Simulating systems
Using one controllable quantum system in
order to simulate another system,
for example by:
3
• BECs
• Atoms in optical lattices
• Rydberg Atoms
• Trapped Ions
• Superconducting devices
• …
• Water flow tanks
•…

Simulated physics
• Condensed Matter Models ( high TC?)
Artificial potentials (AB phases, Hall effect, etc)
• Gravity: Black hole Hawking radiation,
quantum radiation in cosmological expansion.
• QFT: Scalar and fermion relativistic fields.

Simulated physics
• Condensed Matter Models ( high TC?)
Artificial potentials (AB phases, Hall effect, etc)
• Gravity: Black hole Hawking radiation,
quantum effects of cosmological expansion.
• QFT: Scalar and fermion relativistic fields,
• .
• High Energy physics (HEP)
• Quantum Chromodynamics (QCD) effects?
• Confinement of Quarks?

HEP vs. cond. mat.
• Experimentally more challenging compared with cond. mat. simulations.
(For reasons to be explained).
• Could be used to explore otherwise experimentally and computationally
inaccessible effects.
• There are different proposals with different levels of difficulty.
Starting with relatively simple models, that can
be developed to study complex challenging problems.
Note also recent works by :
P. Zoller group, and M. Lewenstein group.

Requirements: HEP models
Fermion fields : = Matter
Bosonic, Gauge fields:= Interaction mediators
Constraints:
• local gauge invariance = “charge” conservation
(exact, or low energy, effective )
• Relativistic invariance = causal structure,
( In the continuum limit.)

Structure of HEP models
• Matter particles:= fermions (+Higgs)
(Charges: electric, color , flavor. Spin. Mass)
• Gauge fields:= mediate the Interactions
Electromagnetic: massless photon, (1), U(1)
Weak interaction : massive Z, W’s , (3), SU(2)
Strong interaction : massless Gluons , (8), SU(3)

Structure of HEP models
The abelian/non-abelian local symmetries much
determines the Gauge field dynamics.
It also constrains the interactions between
Gauge fields and Matter-field.
U(1) => linear Maxwell’s theory
SU(N) => non-linear Yang-Mills theory

Abelian U(1): QED
α QQQ ≪ 1,
V QQQ r ∝ 1 r
We (ordinarily) don’t need second quantization to understand
the structure of atoms. m e c 2 2
≫ E RRRRRRR ≃ α QQQ m e c 2
In HEP scattering, perturbation theory ( Feynman diagrams)

Abelian U(1): QED
α QQQ ≪ 1,
V QQQ r ∝ 1 r
We (ordinarily) don’t need second quantization to understand
the structure of atoms. m e c 2 2
≫ E RRRRRRR ≃ α QQQ m e c 2
In HEP scattering, perturbation theory ( Feynman diagrams)
Works well.

Non-abelian: QCD
α QQQ > 1 , V QQQ r ∝ r
non-perturbative confinement effect!
=> structure of Hadrons: quark pairs form Mesons ,
quark triplets form Baryons.
Color Electric flux-tubes:
“a non-abelian Meissner effect”.

Non-abelian: QCD
α QQQ > 1 , V QQQ r ∝ r
non-perturbative confinement effect!
=> structure of Hadrons: quark pairs form Mesons ,
quark triplets form Baryons.
Color Electric flux-tubes:
“a non-abelian Meissner effect”.

Gauge fields
Abelian Fields
Massless
Long-range forces
Chargeless
Linear dynamics
Non-Abelian fields
Massless
Confinement
Carry charge
Self interacting & NL

Lattice gauge theory
• The standard avenue: Z lllllll − computed using
Monte Carlo “sampling”
(Other methods exist: non-abelian bosonization (1+1); instanton semiclassical
summation, large N perturbation, variational methods… )
------------------
… but:
• Limited applicability with too many quarks .
Computationally hard “sign problem”.
color superconductivity, Quark-Gluon plasma. )
• Quark confinement with dynamic matter never
proven.

First step:
• Confinement in Abelian lattice models!
• Toy models with “QCD-like properties” that
capture the essential physics mechanism
of confinement.

Confinement in abelian
compact QED lattice models
• 1+1D: “Schwinger’s model” manifests confinement.
(analytic and lattice results available).
• 2+1D: confinement for all values of coupling constant,
a non-perturbative mechanism (Polyakov).
• 3+1D: phase transition between strong coupling
confinement phase, and weak coupling coulomb
phase.
• For T > 0: there is a phase transition also in 2+1 D.

QED in 1+1 : Schwinger’s model
• No magnetic fields: EM has no dynamics of its own.
Non trivial dynamics obtained by coupling to
dynamical charge sources.
• Schwinger: e + e − form bound states. (analytic and
lattice results available.)
(Non-abelian extension: same results holds in 1+1:
QCD 2 version, but are not completely solved. Only in
the large-N limit.)

1+1, U(1) gauge theory
Start with a hopping fermionic Hamiltonian, in 1
spatial direction
H = M n ψ n † ψ n
+ α n ψ n † ψ n+1 + H. c.
n
This Hamiltonian is invariant to global gauge
transformations,
ψ n ⟶ e −iΛ ψ n ; ψ n † ⟶ e iΛ ψ n
†

1+1, U(1) gauge theory
Promote the gauge transformation to be local:
ψ n ⟶ e −iΛ nψ n ; ψ † n ⟶ e iΛ † nψ n
Then, in order to make the Hamiltonian gauge
invariant, add unitary operators, U n ,
H = M n ψ n † ψ n
+ α n ψ n † U n ψ n+1 + H. c.
n
U n = e iθ n ; θ n ⟶ θ n + Λ n+1 - Λ n

Dynamic 1+1 U(1) gauge theory
Add dynamics to the gauge field:
H E = g2
2 L n 2
Where L n is the angular momentum operator
conjugate to θ n , representing the (integer) electric
field.
n

Gauge fields: A → θ , E → L
Matter fields: ψ, ψ †
Figure taken from ref. 4

Realization of a link

Realization of a link
B 1, B 2
F L
F R

Realization of a link
L → L − 1
B 1, B 2
F L
F R

Realization of a link
L → L + 1
B 1, B 2
F L
F R

Angular Momentum conservation ↔
Local gauge invariance
B 1, B 2
F L
F R
ψ L † b 1 † b 2 ψ R + ψ R † b 2 † b 1 ψ L
m F (B 1 )
m F (F R )
m F (B 1 )
m F (F L )

Angular Momentum conservation ↔
Local gauge invariance
B 1, B 2
F L
F R
ψ L † b 1 † b 2 ψ R + ψ R † b 2 † b 1 ψ L
m F (B 1 )
m F (F R )
m F (B 1 )
m F (F L )

Angular Momentum conservation ↔
Local gauge invariance
B 1, B 2
F L
F R
ψ L † b 1 † b 2 ψ R + ψ R † b 2 † b 1 ψ L
m F (B 1 )
m F (F R )
m F (B 1 )
m F (F L )

Gauge bosons: Schwinger algebra
L + = b 1 † b 2 ; L − = b 2 † b 1
L z = 1 2 N 1 − N 2 ;l = 1 2 N 1 + N 2

Gauge bosons: Schwinger algebra
L + = b † 1 b 2 ; L − = b † 2 b 1
L z = 1 N 2
1 − N 2 ;l = 1 N 2
1 + N 2
and thus what we have is
ψ † L L + ψ R + ψ † R L − ψ L

Gauge bosons: Schwinger algebra
ψ † L L + ψ R + ψ † R L − ψ L
For large l , m ≪ l
L + = b † 1 b 2 ~e i φ 1−φ 2 ≡ e ii = U
ψ L † Uψ R + ψ R † U † ψ L
Qualitatively similar results can be obtained with two bosons on the link.

The electric dynamic term
L z 2 = 1 4 N 1 − N 2
2
= 1 4 b † 2 1
1 b 1 +
4 b † 2 1
2b 2 −
2 b † 1 b 1 b † 2 b 2
H E = g2
2 L n 2
n
E = N 1 − N 2 ccccccccc tt θ = φ 1 − φ 2

cQED: U(1) in 1+1
Scattering fermion-boson
H = M −1 n ψ n † ψ n
n
+ α ψ n † U n ψ n+1 + H. c.
+ g2
2 L n 2
n
Scattering boson-boson

Mass and Charge
• Even n (particles): Q = N = ψ † ψ
0 atoms: zero mass, zero charge
1 atom: M, Q=1
• Odd n (anti-particles): Q = N − 1
(note: mass measured relative to –M),
1 atom : zero mass, zero charge (“Dirac sea”)
0 atoms: mass M (relative to –M), charge Q=-1

Quark confinement
N =0 1 0 1
Q =0 0 0 0
L z = 0 0 0
N =1 0 1 0
Q =1 -1 1 -1
L z = 1 0 1
N =1 1 0 0
Q =1 0 0 -1
L z = 1 1 1

SU(2) in 1+1-d
• Confinement of “color” charges.
• Non-abelian Schwinger “QCD_2”
• Hamiltonian:
on each link – an SU(2) element ;
conjugate to the group’s left and right
generators, and , satisfying
(separately) SU(2) algebras
37

SU(2) 1+1 simulation
• Gauge field SU(2): 4 bosons on each link
Figure taken from ref. 5
38

d+1 (d>1) : plaquette terms
In more spatial dimensions, one should
consider also gauge-gauge interactions, which
involve plaquette terms:
− 1 1 2 1 2
cos θ
g 2 m,n + θ m+1,n − θ m,n+1 − θ m,n
m,n
(see ref. 1)
In the continuum limit, this corresponds to
∇ × A 2 - gauge invariant magnetic energy
term.

Non-Abelian plaquettes
In this case, the plaquette terms take the
form
− 1
2g 2 Tr □ UUU † U † + H. c.
- This gives the nonlinear Yang-Mills
model.

Realization of plaquette terms
Unlike in 1+1 dimensional case, here one has to
obtain “un-natural” plaquette terms using
effective methods.
For example – imposing Gauss’s law as a
constraint.
The abelian Kogut-Susskind
model.
Figure taken from ref. 1
PRL 2011.

Possible Measurements
Electric flux tubes
Flux loops deforming and breaking effects
Figure taken from ref. 4
Measurement of Wilson Loop’s area law.
Figure taken from ref. 4

Our current situation
Theory 1+1
Pure
U(1) -
cQED
SU(2) –
Yang Mills
Z N
Trivial
1+1 with
matter
K.S. and
truncated
d+1 Pure d+1 with
matter
K.S. and
truncated
Trivial Full Simulation Strong limit
Simulation
Trivial
K.S. and
truncated
Strong limit
Simulation
Published
In preparation
K.S. = Kogut Susskind Hamiltonian LGT

Summary
• We have discussed a simple realization of lattice gauge
theory that manifests confinement: the Abelian Schwinger
model.
• Higher dimensional extension are theoretically more
interesting problems, experimentally harder, but reachable
in the next step by extending upon 1+1-d models.
• This may then allow for insights into non-trivial high energy
physics problems.
• Improved simplified realization are possibly still awaiting.

Summary
• We have discussed a simple realization of lattice gauge
theory that manifests confinement: the Abelian Schwinger
model.
• Higher dimensional extension are theoretically more
interesting problems, experimentally harder, but reachable
in the next step by extending upon 1+1-d models.
• This may then allow for insights into non-trivial high energy
physics problems.
• Improved simplified realization are possibly still awaiting.
Thank you for your attention !

List of relevant publications
1. Confinement and Lattice Quantum-Electrodynamic Electric Flux Tubes Simulated with Ultracold Atoms
Erez Zohar, Benni Reznik
Phys. Rev. Lett. 107, 275301 (2011) . Preprint: arXiv: 1108.1562v2 [quant-ph]
2. Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and
nonperturbative effects
Erez Zohar, J. Ignacio Cirac, Benni Reznik
Phys. Rev. Lett. 109, 125302 (2012) Preprint: arXiv: 1204.6574 [quant-ph]
3. Topological Wilson-loop area law manifested using a superposition of loops
Erez Zohar, Benni Reznik To appear in New J. Phys.. Preprint: arXiv:1208.1012 [quant-ph]
4. Simulating 2+1d Lattice QED with dynamical matter using ultracold atoms
Erez Zohar, J. Ignacio Cirac, Benni Reznik
Phys. Rev. Lett. 110, 055302 (2013), Preprint: arXiv:1208.4299 [quant-ph]
5. A cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory
Erez Zohar, J. Ignacio Cirac, Benni Reznik
Phys. Rev. Lett. 110, 125304 (2013) , Preprint: arXiv:1211.2241 [quant-ph]
6. Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular
momentum conservation,
Erez Zohar, J. Ignacio Cirac, Benni Reznik preprint: arXiv:1303.5040