This document discusses the spreading of correlations in quantum lattice models with long-range interactions decaying as a power law. It is shown that even when the exponent α is less than the dimension D, Lieb-Robinson bounds can still be derived in rescaled time, indicating cone-like propagation. Exact results are presented for long-range Ising and XXZ models, as well as a fermionic long-range hopping model, demonstrating different types of propagation fronts for varying α. The key conclusion is that while correlations may spread instantaneously in physical time when α < D, rescaling time allows for Lieb-Robinson bounds and conical propagation in certain parameter regimes.

1. 1 / 6
Exceptionally long-range quantum lattice models
Mauritz van den Worm
National Institute of Theoretical Physics
Stellenbosch University
Stellenbosch

2. | Introductory words 2 / 6
Interaction/hopping satisfies:
Ji,j ∝ |i − j|−α
0 < α < dim(System)
Gravitating Masses Coulomb Interactions (no screening)

3. | 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

4. | 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)] ≤ K
ev|t|
− 1
[d(A, B) + 1]α−D
ln x
x
t
Long Range
New bounds
Bounds of Alexey and co-workers

5. | Breakdown of Causality 3 / 6
Ρ ΠB
Tr B e iHt
UA ΡUA
†
eiHt
Tr B e itH
ΡeiHt
0 t
Tt
Nt
Toy model
HΛ =
1
2
(1 − σz
o)
j∈B
1
[1 + dist(o, j)]α
(1 − σz
j )

6. | 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 instantaneous
spreading
Well defined causal region down to α = D/2

7. | 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 instantaneous
spreading
Power-law shaped causal region

8. | Breakdown of Causality 3 / 6
Exact results for Ising
H =
1
2
i=j
1
|i − j|α
σz
i σz
j , σx
i σx
j (t) − σx
i (t) σx
j (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 σx
0 σx
δ c(t) in the
(δ, t)-plane for long-range Ising chains with |Λ| = 1001 sites and three different
values of α. Dark colors indicate small values, and initial correlations at t = 0
are vanishing.

9. | Breakdown of Causality 3 / 6
tDMRG results for XXZ
H =
1
2
i=j
1
|i − j|α
Jz
σz
i σz
j +
J⊥
2
σ+
i σ−
j + σ+
j σ−
i , σz
0 σz
δ 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 ∆

10. | 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 10 100
t
0.1
0.2
0.3
0.4
0.5
Σi
x
Σj
x
c
N 102
N 103
N 104
0.1 1 10 100
Τ
0.1
0.2
0.3
0.4
0.5
Σi
x
Σj
x
c
Physical time Rescaled time

11. | Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHi
Generic Hamiltonian, H = X⊂Λ hX

12. | Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHi
Generic Hamiltonian, H = X⊂Λ hX
Boundedness
X x,y
hX =
λ
[1 + d(i, j)]α

13. | Lieb-Robinson bounds for α < D 4 / 6
Ingredients
D dimensional lattice Λ
|Λ| = N
Tensor product Hilbert space, H = ⊗i∈ΛHi
Generic Hamiltonian, H = X⊂Λ hX
Reproducing
NΛ
k∈Λ
1
[1 + d(i, k)]α
1
[1 + d(k, j)]α
≤
p
[1 + d(i, j)]α
Asymptotics
NΛ ∼



c1Nα/D−1
for 0 α < D,
c2/ ln N for α = D,
c3 for α > D,

14. | 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 time
τ = 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

15. | Fermionic Long-Range Hopping 5 / 6
Long-Range fermionic hoppnig model
H = −
1
2
N
j=1
N−1
l=1
d−α
l c†
jcj+l + c†
j+lcj
dl =
l if l ≤ N/2,
N − l if l > N/2,

16. | Fermionic Long-Range Hopping 5 / 6
Long-Range hoppnig model
H = −
1
2
N
j=1
N−1
l=1
d−α
l c†
jcj+l + c†
j+lcj
dl =
l if l ≤ N/2,
N − l if l > N/2,
Diagonalize and Dispersion
H =
k
(k)a†
kak
(k) = −
N−1
l=1
cos (kl)
dα
l

17. | 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
2
δδ,0 −
(−1)j+δ
2N
k
eit[ (k+π)− (k)]
e−ikδ

18. | 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
∆
t Α 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 0
and δ in the fermionic long-range hopping model for N = 200 lattice sites and
various values of α, starting from a staggered initial state. Lighter colors
represent larger correlations.

19. | Fermionic Long-Range Hopping 5 / 6
Time dependent number operator
nj(t) =
1
2
−
(−1)j
2N
N
n=1
cos [t∆(k)] , ∆(k) = (k + π) − (k)
Α 3
Α 0.5
Α 0.25
Α 0.05
Α 0.01
0.1 1 10 100
t
0.4
0.5
0.6
0.7
0.8
0.9
1.0
n j
Figure : Time dependence of the occupation number of site j for different α,
starting from a staggered initial state.

20. | Fermionic Long-Range Hopping 5 / 6
Dispersion Relation
(k) = −
1
2
Liα eik
+ Liα e−ik

21. | Fermionic Long-Range Hopping 5 / 6
Dispersion relation in large system limit
a
Π Π
2
Π
2
Π
k
2
1
1
Ε
b
Π Π
2
Π
2
Π
k
2
1
1
2
ΕΑ 1
Α 2
Α 3
Figure : Dispersion relation (a) and its derivative (k) (b) for the long-range
fermionic hopping model with exponents α = 1, 2, and 3.

22. | Fermionic Long-Range Hopping 5 / 6
Dispersion Relation
(k) = −
1
2
Liα eik
+ Liα e−ik
Density of states
ρ(v) =
1
2π
2π
0
δ v −
d
dk
dk
ρ(v) =
1
π



2
1 + 4v2
for α = 1,
Θ (π − 2v) Θ (π + 2v) for α = 2,
1
√
1 − v2
for α → ∞,

23. | Fermionic Long-Range Hopping 5 / 6
DOS and Spreading
Α 1
Α 2
Α
3 2 1 1 2 3
v
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 for
different α.

24. | 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 derive
Lieb-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

25. | Conclusions 6 / 6
Collaborators

26. | Conclusions 6 / 6