exceptionaly long-range quantum lattice models
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Transcript of exceptionaly long-range quantum lattice models
1 / 6
Exceptionally long-range quantum lattice models
Mauritz van den Worm
National Institute of Theoretical Physics
Stellenbosch University
Stellenbosch
| Introductory words 2 / 6
Interaction/hopping satisfies:
Ji,j ∝ |i− j|−α
0 < α < dim(System)
Gravitating Masses Coulomb Interactions (no screening)
| Introductory words 2 / 6
Interaction/hopping satisfies:
Ji,j ∝ |i− j|−α
0 < α < dim(System)
What are we interested in?
Spatio-temporal spreading of
correlators
information
entanglement
| Breakdown of Causality 3 / 6
Linear Cone
‖ [OA(t),OB(0)] ‖ ≤ K exp
[v|t| − d(A,B)
ξ
] ~v t
x
t
Short-Range
Logarithmic Cone, α > D
‖ [OA(t),OB(0)] ‖ ≤ Kev|t| − 1
[d(A,B) + 1]α−D
~ ln x
x
t
Long-Range
New bounds
Bounds of Alexey and co-workers
| Breakdown of Causality 3 / 6
Ρ Π B
TrL� B @e- iHt UA ΡUA†
e iHt D
TrL� B @e- itH ΡeiHt D
0 t
Tt
N t
Toy model
HΛ =1
2(1− σzo)
∑j∈B
1
[1 + dist(o, j)]α(1− σzj )
| Breakdown of Causality 3 / 6
Product initial state
ρ = |0〉〈0||Λ|−|B| ⊗ |+〉〈+|⊗|B|
|+〉 = (|0〉+ |1〉)/√
2
Probability of detecting a signal
pt ≥ 1− exp
{−4t2
5
∑j∈B
[1 + d (A, j)]−2α
}
When α < D/2 we have instantaneousspreading
Well defined causal region down to α = D/2
| Breakdown of Causality 3 / 6
GHZ entangled initial state
ρ =|0〉〈0||Λ|−|B| ⊗ |ψ〉〈ψ||ψ〉 =(|0, . . . , 0〉+ |1, . . . , 1〉)/
√2
Probability of detecting a signal
pt ≥ 1− 1
2
{1 + cos
[t∑j∈B
[1 + d (A, j)]−α]}
When α < D we have instantaneousspreading
Power-law shaped causal region
| Breakdown of Causality 3 / 6
Exact results for Ising
H =1
2
∑i 6=j
1
|i− j|α σzi σ
zj , 〈σxi σxj 〉(t)− 〈σxi 〉(t)〈σxj 〉(t)
α = 1/4 α = 3/4 α = 3/2
0 50 100 150
0.00
0.02
0.04
0.06
0.08
0.10
∆
t
0 50 100 150
0.00
0.05
0.10
0.15
0.20
∆
20 40 60 80
0.0
0.1
0.2
0.3
0.4
∆
Flat Power-law Linear
Figure : Density contour plots of the connected correlator 〈σx0σxδ 〉c(t) in the(δ, t)-plane for long-range Ising chains with |Λ| = 1001 sites and three differentvalues of α. Dark colors indicate small values, and initial correlations at t = 0are vanishing.
| Breakdown of Causality 3 / 6
tDMRG results for XXZ
H =1
2
∑i 6=j
1
|i− j|α
[Jzσzi σ
zj +
J⊥
2
(σ+i σ−j + σ+
j σ−i
)], 〈σz0σzδ 〉c(t)
α = 3/4 α = 3/2 α = 3
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
∆
t
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
∆
t
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
∆
t
0.0 0.5 1.0 1.5 2.0 2.5
- 5
- 4
- 3
- 2
- 1
0
ln ∆
lnt
0.0 0.5 1.0 1.5 2.0 2.5
- 5
- 4
- 3
- 2
- 1
0
ln ∆
0.0 0.5 1.0 1.5 2.0 2.5
- 5
- 4
- 3
- 2
- 1
0
1
ln ∆
| Lieb-Robinson bounds for α < D 4 / 6
Rescaled time
τ ∝ N−q, q > 0
Connected Correlation Functions for Long-Range Ising
N =102
N =103
N =104
0.1 1 10 100t
0.1
0.2
0.3
0.4
0.5
Y Σix Σ j
x ]c
N =102
N =103
N =104
0.1 1 10 100Τ
0.1
0.2
0.3
0.4
0.5
Y Σix Σ j
x ]c
Physical time Rescaled time
| Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =
∑X⊂Λ hX
| Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =
∑X⊂Λ hX
Boundedness∑X3x,y
‖hX‖ =λ
[1 + d(i, j)]α
| Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =
∑X⊂Λ hX
Reproducing
NΛ
∑k∈Λ
1
[1 + d(i, k)]α1
[1 + d(k, j)]α≤ p
[1 + d(i, j)]α
Asymptotics
NΛ ∼
c1N
α/D−1 for 0 6 α < D,
c2/ lnN for α = D,
c3 for α > D,
| Lieb-Robinson bounds for α < D 4 / 6
α > 0 Lieb-Robinson bound
‖ [OA(τ), OB(0)] ‖ ≤ C‖OA‖‖OB‖|A||B|(ev|τ | − 1)
[d(A,B) + 1]α
Rescaled timeτ = t/NΛ
Speed-up in physical time, α = 1/2
N = 10 N = 102 N = 103
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
0.6
∆
t
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
0.6
∆
t
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
0.6
∆
t
| Fermionic Long-Range Hopping 5 / 6
Long-Range fermionic hoppnig model
H = −1
2
N∑j=1
N−1∑l=1
d−αl
(c†jcj+l + c†j+lcj
)
dl =
{l if l ≤ N/2,N − l if l > N/2,
| Fermionic Long-Range Hopping 5 / 6
Long-Range hoppnig model
H = −1
2
N∑j=1
N−1∑l=1
d−αl
(c†jcj+l + c†j+lcj
)
dl =
{l if l ≤ N/2,N − l if l > N/2,
Diagonalize and Dispersion
H =∑k
ε(k)a†kak
ε(k) = −N−1∑l=1
cos (kl)
dαl
| Fermionic Long-Range Hopping 5 / 6
Propagation from staggered initial state
|ψ(0)〉 = |1, 0, 1, 0, . . . , 1, 0〉
Spreading
〈c†j+δ(t)cj(t)〉 =1
2δδ,0 −
(−1)j+δ
2N
∑k
eit[ε(k+π)−ε(k)]e−ikδ
| Fermionic Long-Range Hopping 5 / 6
Spreading of correlations
Linear Linear Linear
Α=0.75
0 10 20 30 40 50
0
5
10
15
20
∆
t
Α=1.5
0 10 20 30 40 50
0
5
10
15
20
∆
tΑ=3
0 10 20 30 40 50
0
5
10
15
20
∆
t
Figure : Contour plots in the (δ, t)-plane, showing correlations between sites 0and δ in the fermionic long-range hopping model for N = 200 lattice sites andvarious values of α, starting from a staggered initial state. Lighter colorsrepresent larger correlations.
| Fermionic Long-Range Hopping 5 / 6
Time dependent number operator
〈nj(t)〉 =1
2− (−1)j
2N
N∑n=1
cos [t∆(k)] , ∆(k) = ε(k + π)− ε(k)
Α=3
Α=0.5
Α=0.25
Α=0.05
Α=0.01
0.1 1 10 100t
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Yn j ]
Figure : Time dependence of the occupation number of site j for different α,starting from a staggered initial state.
| Fermionic Long-Range Hopping 5 / 6
Dispersion relation in large system limit
HaL
-Π - Π2
Π2
Πk
-2
-1
1
Ε
HbL-Π - Π
2
Π2
Πk
-2
-1
1
2
Ε¢Α=1
Α=2
Α=3
Figure : Dispersion relation (a) and its derivative ε′(k) (b) for the long-rangefermionic hopping model with exponents α = 1, 2, and 3.
| Fermionic Long-Range Hopping 5 / 6
Dispersion Relation
ε(k) = −1
2
[Liα(
eik)
+ Liα(
e−ik)]
Density of states
ρ(v) =1
2π
∫ 2π
0δ
(v − dε
dk
)dk
ρ(v) =1
π
2
1 + 4v2for α = 1,
Θ (π − 2v) Θ (π + 2v) for α = 2,1√
1− v2for α→∞,
| Fermionic Long-Range Hopping 5 / 6
DOS and Spreading
Α=1
Α=2
Α=¥
-3 -2 -1 1 2 3v
0.2
0.4
0.6
0.8
Ρ
Α=0.75
0 10 20 30 40 50
0
5
10
15
20
∆
t
Α=1.5
0 10 20 30 40 50
0
5
10
15
20
∆
t
Α=3
0 10 20 30 40 50
0
5
10
15
20
∆
t
Figure : Left: Density of states for α = 1, 2 and ∞. Right: Spreading plots fordifferent α.
| Conclusions 6 / 6
Take home message
Even for α < D we can have clear propagation fronts
0 50 100 150
0.00
0.02
0.04
0.06
0.08
0.10
∆
t
0 50 100 150
0.00
0.05
0.10
0.15
0.20
∆
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
∆
t
In rescaled time τ = t/NΛ we can deriveLieb-Robinson bounds for α < D
‖ [OA(τ), OB(0)] ‖ ≤ C‖OA‖‖OB‖|A||B|(ev|τ| − 1)
[d(A,B) + 1]α
For α < D we can still have cone-like propagation
Α=0.75
0 10 20 30 40 50
0
5
10
15
20
∆
t
Α=1.5
0 10 20 30 40 50
0
5
10
15
20
∆
t
Α=3
0 10 20 30 40 50
0
5
10
15
20
∆
t