supersonic spreading of correlations in long-range quantum lattice models

Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models

Supersonic Spreading of Correlations in Long-RangeQuantum Lattice Models

Jens Eisert2, Mauritz van den Worm1, Salvatorre R Manmana4 andMichael Kastner 1,3

1Institute of Theoretical PhysicsStellenbosch University

2Dahlem Center for Complex Quantum SystemsFreie Universitat Berlin

3National Institute for Theoretical PhysicsStellenbosch

3Institute for Theoretical Physics

Georg-August-Universitat Gottingen

NITHeP Bursars Workshop

Mauritz van den Worm | SU | Oct 2013 1 / 20

Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models | Long-Range Interacting Systems

What is a long-range interacting system?

Interaction satisfies:

Ji ,j ∝ |i − j |−α

0 < α < dim(System)

Example

Gravitating Masses

Coulomb Interactions (no screening)



Why the focus on short-range interacting systems?

The pioneers of statistical physics

Boltzmann Gibbs

Interactions:

Electromagnetic

±q gives rise to screening → effective short-range



What about astrophysics?

Screening?

No negative masses → no screening

Negative heat capacities Nonequivalence of ensembles


Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models | Causality and Lieb-Robinson Bounds



Causality



Lieb-Robinson Bounds

Finite range or exponentially decaying interactions

‖[OA(t),OB(0)]‖ ≤ Cev|t|−d(A,B)

ξ

v |t| > d(A,B) + ξ ln ε; ε > 0

Long-range interactions [α > D = dim(Λ)]

‖[OA(t),OB(0)]‖ ≤ Cev |t| − 1

(d(A,B) + 1)α

v |t| > ln

(1 + ε (d(A,B) + 1)α

)

Schematic of what to expect

Short-Range

Long-Range

Light

2 4 6 8 10dHA,BL

2

4

6

8

10

t



Main Result

α > D necessary andsuffiecient for having causal

region


Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models | Information Propagation

Lower Bounds on Information Propagation

TrΛ\B

(e−itHUAρU

†Ae

itH)

TrΛ\B

(e−itHρe itH

)πBρ

Nt(ρ)

Tt(ρ)



Lower Bounds on Information Propagation

Bound Classical Information Capacity

Ct ≥ pt := |Tr (Tt(ρ)πB)− Tr (Nt(ρ)πB)|

probability of detecting a signal at time t > 0



Toy Model

Ising Type Hamiltonian

HΛ =1

2(1− σzo)

∑j∈B

1

(1 + d(o, j))α(1− σzj )

B := {j ∈ Λ : d(o, j) ≥ δ}A = {o}

POVM and UA

πB = |+〉〈+|⊗|B|

UA = |1〉〈0|



Toy Model - Product Initial State

Product Initial State

ρ = |0〉〈0||Λ|−|B| ⊗ |+〉〈+|⊗|B|

|+〉 =1√2

(|0〉+ |1〉)

Lower bound on information capacity

pt > 1− exp

[−4t2

5

∑j∈B

(1 + d(o, j))−2α

]

Sum converges for

α > D/2 = dim Λ/2



Toy Model - Multi-particle Entangled Initial State

Product Initial State

ρ = |0〉〈0||Λ|−|B| ⊗ |ψ〉〈ψ|

|ψ〉 =1√2

(|0, · · · , 0〉+ |1, · · · , 1

)Lower bound on information capacity

pt > 1− 1

2

1 + cos

t∑j∈B

(1 + d(o, j))−α

Sum converges for

α > D = dim Λ

Causal region looks like

v |t| > εδq with v , q > 0


Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models | Density Plots

Long-Range Ising Exact Resutls

Exact Resutls

H Ising = −∑i>j

1

|i − j |σzi σ

zj

〈σxi 〉(t) = 〈σxi 〉(0)∏j 6=i

cos

(2t

|i − j |α

)〈σxi σxj 〉(t) = P−i ,j + P+

i ,j

P±i ,j =1

2〈σxi σxj 〉(0)

∏k 6=i ,j

cos

[2t

(1

|i − k |α± 1

|j − k|α

)]



Britton et al., Nature 484, 489–492 (26 April 2012)

(a) (b) (c)

H = −∑i<j

Ji ,jσzi σ

zj − Bµ ·

∑i

σi

Coupling Constant Ji ,jExpressed i.t.o. the transverse phonon eigenfunctions

Numerical evaluation shows Ji ,j ∝ D−αi ,j

Tune 0 ≤ α ≤ 3

Exactly the long-rangeIsing model!



Long-Range Ising Exact Results

〈σxoσxδ 〉c = 〈σxoσxδ 〉 − 〈σxo 〉〈σxδ 〉

Α = 1� 4

0 50 100 150

0.00

0.02

0.04

0.06

0.08

0.10

∆

t

Α = 3� 4

0 50 100 150

0.00

0.05

0.10

0.15

0.20

∆

t

Α = 3� 2

20 40 60 80

0.0

0.1

0.2

0.3

0.4

∆

t

Figure: Density contour plots of the connected correlator 〈σxoσ

xδ 〉c 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 = 0 arevanishing.



XXZ -Model DMRG Results

HXXZ =∑

i>j1

|i−j |α

[J⊥2

(σ+i σ−j + σ−i σ

+j

)+ Jzσ

zi σ

zj

]

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

Figure: Density plots of the correlator 〈σzoσ

zδ〉c in the (δ, t)-plane. The results are

for long-range XXZ chains with |Λ| = 40 sites and exponents α = 3/4, 3/2, and 3(from left to right). The left and center plots reveal supersonic spreading ofcorrelations, not bounded by any linear cone, whereas such a cone appears in theright plot for α = 3.



XXZ -Model DMRG Results

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 ∆

Figure: As above, but showing contour plots of ln 〈σzoσ

zδ〉c in the (ln δ, ln t)-plane.

For better visualization, odd/even effects caused by the staggered initial statehave been eliminated. All plots in the bottom row are consistent with a powerlaw-shaped causal region for larger distances δ.



Take Home Message

α > D is necessary and sufficient condition for restricting thespreading of correlations to the interior of a causal region, butallows supersonic (faster than linear) propagation

For models considered causal region has power-law dependence

v |t| > εδq

Starting with product initial states we can already find a causalregion for α > D/2


supersonic spreading of correlations in long-range quantum lattice models

Documents

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theoretical physicsgeorg

mauritz van den worm1

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