Supersonic Spreading of Correlations in Long-Range Quantum Lattice ModelsJens Eisert2 Mauritz van den Worm1, Salvatore R. Manmana 3, and Michael Kastner 1

1National Institute of Theoretical Physics, Stellenbosch University, South Africa2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

3Institute for Theoretical Physics, Georg-August-Universität Göttingen, 37077 Göttingen, Germany

Introduction

Recent advances in trappedions in optical lattices• Engineering of Ising Hamiltonians

H = −∑i<j

Ji,jσzi σ

zj −Bµ ·

∑i

σi

with hundreds of spins [1]• Ji,j expressed i.t.o. transverse phononeigenfunctions

• Ji,j = J0/|i− j|α• Tunable 0 ≤ α ≤ 3

Parallel theoretical advancesAnalytic expressions for any time dependentcorrelation functions with product [3], andmixed [6] initial states.

How do correlations spread in long-range interacting quantum systems?

Lieb-Robinson Bounds

Short-range case [5]:Observables OA and OB with support

A,B ⊂ Λ,with A ∩B = ∅evolving in the Heisenberg picture satisfy‖[OA(t), OB(0)]‖ = C‖OA‖‖OB‖min (|A|, |B|) e[v|t|−d(A,b)]/ξ

with C, v, ξ > 0 and d(A,B) the graph theoretical distancebetween A and B.The norm is negligibly small outside the effective causalcone that is determined for those values of t and d(A,B) forwhich the exponential is larger than some ε > 0, whichhappens for

v|t| > d(A,B) + ξ ln ε.

Long-range case [4]:The general form of long-range Hamiltonians is

HΛ∑X⊂Λ

hX

where hX are local Hamiltonian terms with compactsupport on X , assumed to satisfy∑

X3x,y‖hX‖ ≤ λ0 [1 + d(x, y)]−α .

For exponents α > D = dim (Λ) Lieb-Robinson-typebounds have been proved to be of the form

‖[OA(t), OB(0)]‖ = C‖OA‖‖OB‖min (|A|, |B|)

(ev|t| − 1

)[1 + d(A,B)]α

.

Similar to the short-range case we find an effective causalregion given by the inequality

v|t| > ln[1 + ε [1 + d(A,B)]α

min (|A|, |B|)

].

Nonequilibrium systems as quantumchannels

Ρ Π B

TrL� B @e- iHt UA ΡUA†

e iHt D

TrL� B @e- itH ΡeiHt D

0 t

Tt

N t

At time t = 0 prepare initial state ρ then implement one ofthe quantum channels Tt or Nt. At a later time t performa positive-operator valued measure πB supported on B only,with 0 ≤ πB ≤ 1. The classical information capacity Ct isbounded form below by the probability of detecting a signalat time t > 0

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

Lower bounds on information propagation

Hamiltonian, POVM and Local Unitary:

HΛ = 12

(1− σzo)∑j∈B

1[1 + d(o, j)]α

(1− σzj

).

•A = {o} and B = {j ∈ Λ : d(o, j) ≥ δ} for δ ∈ N

• πB = |+〉〈+||B| with |+〉 = (|0〉 + |1〉) /√

2•UA = |1〉〈0|

Product Initial State:As initial state, we choose

ρ = |0〉〈0|⊗|Λ\B| ⊗ |+〉〈+|⊗|B|.For times 2t < (1 + δ)α we can bound

pt ≥ 1− exp

−4t2

5∑j∈B

[1 + d (o, j)]−2α

.Let OΛ,l denote the number of sites j ∈ Λ for whichd(o, j) = l. By definition we have OΛ,l = Θ

(lD−1

). The

sum in the exponential can then be written as∑j∈B

[1 + d(o, j)]−2α =L∑l=δ

(1 + l)−2αOΛ,l.

The right hand side converges if

limL→∞

L∑l=δ

(1 + l)D−1−2α <∞

which is when α > D/2.

When α < D/2 signal propagation is thereforenot restricted to any causal region.

Multipartite Entangled Initial States:As initial state choose

ρ = |0〉〈0||Λ|−|B| ⊗ |GHZ〉〈GHZ|with

|GHZ〉 = 1√2

(|0, . . . , 0〉 + |1, . . . , 1〉) .

Here we find

pt = 1− 12

1 + cos

t L∑l=δ

[1 + l]−αOΛ,l

.From the expansion

1− [1 + cos x] /2 = x2/4 + O(x3)we see that we have to investigate

f (δ) := limL→∞

L∑l=δ

(1 + d(o, j))−αOΛ,l

Exploiting the asymptotic behaviour of the Hurwitz zetafunction we find

f (δ)2 = Θ(δ2(D−α)).

When α > D it gives rise to a bent causal regionand allows for faster than linear propagation ofinformation, but slower than permitted by thelong-range Lieb-Robinson bound.

Spreading of correlations in long-range Ising model [2]

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

∆

Density contour plots of the connected correlator〈σxo (t)σxδ (t)〉c = 〈σxo (t)σxδ (t)〉 − 〈σxo (t)〉〈σxδ (t)〉

in the (δ, t)-plane for long-range Ising chains with |Λ| = 1001 and three different values of α. Dark colors indicate small values andinitial correlations at time t = 0 are vanishing. For α = 1/4 (left panel) correlations increase in an essentially distance-independentway. A finite-size scaling analysis confirms that the propagation front indeed becomes flat (δ independent) for 0 ≤ α < D/2, andhence no effective causal region is present. For α = 3/4 (central panel) the spreading of correlations shows a distance dependencethat is consistent with a power-law-shaped causal region; plots for other D/2 < α < D are similar. For α = 3/2 (right panel)correlations initially seem to spread linearly, but not further than a few tens of lattice sites; plots for other α > D are similar.

Spreading of correlations in long-range XXZ model

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

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3.5

∆

t

0

0.01

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0.03

0.04

- 6- 4- 20246

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 ∆

- 16

- 12

- 8

- 4

- 6- 4- 20246

The figure shows numerical results for the time evolution under the XXZ Hamiltonian

HXXZ =∑i>j

1d(i, j)α

[J⊥2(σ+i σ−j + σ−i σ

+j

)+ Jz σ

zi σ

zj

].

Top row: Density plots of the correlator 〈σz0σzδ〉c in the (δ, t)-plane. The results are for long-range XXZ chains with |Λ| = 40 sitesand exponents as indicated. The left and center plots reveal supersonic spreading of correlations, not bounded by any linear cone,whereas such a cone appears in the right plot for α = 3. Bottom row: As above, but showing contour plots of ln〈σz0σzδ〉c in the(ln δ, ln t)-plane. All plots in the bottom row are consistent with a power-law-shaped causal region for larger distances δ.

References

[1] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K.Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Engineeredtwo-dimensional Ising interactions in a trapped-ion quantumsimulator with hundreds of spins, Nature 484 (2012), 489–492.

[2] Jens Eisert, MVDW, Salvatore R. Manmana, and Michael Kastner,Breakdown of Quasilolcality in Lon-Range Quantum LatticeModels, Phys. Rev. Lett. 111 (2013), 260401.

[3] M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger, A. M. Rey, andC. W. Clark, Dynamical quantum correlations of Ising models onan arbitrary lattice and their resilience to decoherence, New J.Phys. 15 (2013), 113008.

[4] M. B. Hastings and T. Koma, Spectral Gap and Exponential Decayof Correlations, Commun. Math. Phys. 265 (2006), 781–804.

[5] E. H. Lieb and D. W. Robinson, The Finite Group Velocity ofQuantum Spin Systems, Commun. Math. Phys. 28 (1972), 251–257.

[6] MVDW., B. C. Sawyer, J. J. Bollinger, and M. Kastner,Relaxation timescales and decay of correlations in a long-rangeinteracting quantum simulator, New J. Phys. 15 (2013), 083007.

Some related work...

• Kaden R. A. Hazzard, MVDW, Michael Foss-Feig, Salvatore R.Manmana, Emanuele Dalla Torre, Tilman Pfau, Michael Kastner andAna Maria Rey, Quantum correlations and entanglement infar-from-equilibrium spin systems, arXiv:1406.0937

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