Quantum Kibble-Zurek mechanism: scaling hypothesis in the Ising - - PowerPoint PPT Presentation
Quantum Kibble-Zurek mechanism: scaling hypothesis in the Ising and Bose-Hubbard models Anna Francuz & Jacek Dziarmaga @ Jagiellonian U. Bartek Gardas @ U. of Silesia & Los Alamos Wojciech H. urek @ Los Alamos coming soon
review on quantum KZM: JD, Adv. in Phys. 59, 1063 (2010)
Quantum Kibble-Zurek mechanism: scaling hypothesis in the Ising and Bose-Hubbard models
coming soon
Anna Francuz & Jacek Dziarmaga @ Jagiellonian U. Bartek Gardas @ U. of Silesia & Los Alamos Wojciech H. Żurek @ Los Alamos
= +
N n x n x n z n
1 1
Quantum Ising Chain Quantum phase transition at g=1 Strong transverse field g>>1
... →→→→→ ...
... ↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑
Ferromagnetic states at g=0
∞ → → Δ ξ length n Correlatio gap Energy
Ideal Adiabatic Quantum State Preparation (or Adiabatic Quantum Computation) i
f
Simple Interesting Adiabatic
Real Adiabatic Quantum State Preparation i
f
Simple Interesting
Transition Phase Quantum g Interestin Simple ⇓ ≠
Non-adiabatic Mott BEC
= +
+ − =
N n x n x n z n
g H
1 1
σ σ σ
Quantum Ising Chain ... →→→→→
...
... ↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑
``Simple’’ ``Interesting’’
... ↑↑ ↓↑↑↑↑↓↓↓↓↓ ↑↑↑↑↑↓↓↓↓↓
Excited
Adiabatic Non-adiabatic
? ˆ = ξ
Quantum Kibble-Zurek mechanism (KZM)
c c
g g g − = ε
distance from the critical point
ν
ε ξ
−
∝
ν
ε
z
∝ Δ
energy gap correlation length
Q
t τ ε =
linear(ized) quench
t dt d 1 / = ε ε
transition rate
ν ν ν ν
τ ξ τ
z z z
Q Q
t t t
+ +
∝ ∝ − = =
1 1
ˆ and ˆ where ˆ at GAP RATE ... ↑↑ ↓↑↑↑↑↓↓↓↓↓ ↑↑↑↑↑↓↓↓↓↓
ξ ˆ
Q Q
t τ τ ξ ∝ ∝ ˆ and ˆ
Quantum Ising chain Final excited state
t ˆ −
GS ) ˆ ( t GS − Adiabatic Adiabatic Non-adiabatic (impulse)
ξ ˆ ξ ˆ
) ˆ ( t GS −
t ˆ +
K-Z scaling hypothesis
O
O
Δ −
rescaled time rescaled distance scaling dimension scaling function
Q Q
t τ τ ξ = = ˆ , ˆ
n z n x n
c → σ σ ,
+ +
=
n R n R
c c α
) ˆ / , ˆ / ( ˆ
1
t t R F
R
ξ ξ α
α −
=
Jordan-Wigner transformation quadratic correlator
before: after: at:
Q Q
t τ τ ξ = = ˆ , ˆ
n z n x n
c → σ σ ,
n R n R
c c + = β
Jordan-Wigner transformation anomalous correlator
) ˆ / , ˆ / ( ˆ1 t t R F
R
ξ β ξ
β
=
before at after SAME
Q Q
t τ τ ξ = = ˆ , ˆ
x n x R n x n x R n xx R
C σ σ σ σ
+ +
− = ) (
before: at: after:
ferromagnetic correlator
) ˆ / , ˆ / ( ) ( ˆ
4 / 1
t t R F R C
xx xx
ξ ξ =
see also M. Kolodrubetz
SAME
b n a R n b n a R n ab R
C σ σ σ σ
+ +
− = ) (
more correlation functions
Q Q
t τ τ ξ = = ˆ , ˆ
) ˆ / , ˆ / ( ) ( ˆ
4 / 5
t t R F R C
xy xy
ξ ξ = ) ˆ / , ˆ / ( ) ( ˆ
4 / 9
t t R F R C
yy yy
ξ ξ = ) ˆ / , ˆ / ( ) ( ˆ1 t t R F R C
zz zz
ξ ξ =
mutual information & quantum discord
SAME
Q Q
t τ τ ξ = = ˆ , ˆ
L L L
entanglement entropy: block of L spins
before: at: after:
) ˆ / , ˆ / ( ˆ log ) ˆ / ( ) (
3 1
t t L F t t S t S
S L
ξ ξ = +
SAME
2 1
entanglement gap: block of L spins
before: at: after:
Q Q
t τ τ ξ = = ˆ , ˆ
) ˆ / , ˆ / ( ˆ
8 / 1
t t L F ξ λ ξ
λ Δ
= Δ
1 , 1 ,...., 1 , 1 , 1 , 1 , 1 ) ( = ψ
1D Bose-Hubbard model: Mott -> superfluid transition
s s M s s s M s s s s s
a a a a a a a a J H
= + + = + + + +
+ + − =
1 1 1 1
2 1
Q cr
t J t J τ = ) (
MPS
10 <<
Q
τ
1D Bose-Hubbard model: correlation function
s R s R
a a C
+ +
= ξ ˆ / R
R
C ˆ
4 / 1
ξ
soon coming is 100 50 = = L L
7 .
ˆ
Q
τ ξ ∝
Kosterlitz-Thouless
ˆ / = t t
previous simulations & an experiment
classical supercomputer
1) For theorists: 2) For experimentalists:
quantum simulator
) ˆ / , ˆ / ( ) ( ˆ
4 / 1
t t R F R C
xx xx
ξ ξ =
KZ scaling hypothesis renormalization group in both space and time
2
2
CO