Poster presented at the school on non-linear dynamics and dynamical transitions in classical and quantum systems in Trieste Italy.

1. Supersonic Spreading of Correlations in Long-Range Quantum Lattice Models
Jens Eisert2
Mauritz van den Worm1
, Salvatore R. Manmana 3
, and Michael Kastner 1
1
National Institute of Theoretical Physics, Stellenbosch University, South Africa
2
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
3
Institute for Theoretical Physics, Georg-August-Universität Göttingen, 37077 Göttingen, Germany
Introduction
Recent advances in trapped
ions in optical lattices
• Engineering of Ising Hamiltonians
H = −
i<j
Ji,jσz
i σz
j − Bµ ·
i
σi
with hundreds of spins [1]
• Ji,j expressed i.t.o. transverse phonon
eigenfunctions
• Ji,j = J0/|i − j|α
• Tunable 0 ≤ α ≤ 3
Parallel theoretical advances
Analytic expressions for any time dependent
correlation functions with product [3], and
mixed [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 distance
between A and B.
The norm is negligibly small outside the effective causal
cone that is determined for those values of t and d(A, B) for
which the exponential is larger than some > 0, which
happens for
v|t| > d(A, B) + ξ ln .
Long-range case [4]:
The general form of long-range Hamiltonians is
HΛ
X⊂Λ
hX
where hX are local Hamiltonian terms with compact
support on X, assumed to satisfy
X x,y
hX ≤ λ0 [1 + d(x, y)]−α
.
For exponents α > D = dim (Λ) Lieb-Robinson-type
bounds 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 causal
region given by the inequality
v|t| > ln 1 +
[1 + d(A, B)]α
min (|A|, |B|)
.
Nonequilibrium systems as quantum
channels
Ρ ΠB
Tr B e iHt
UA ΡUA
†
eiHt
Tr B e itH
ΡeiHt
0 t
Tt
Nt
At time t = 0 prepare initial state ρ then implement one of
the quantum channels Tt or Nt. At a later time t perform
a positive-operator valued measure πB supported on B only,
with 0 ≤ πB ≤ 1. The classical information capacity Ct is
bounded form below by the probability of detecting a signal
at time t > 0
Ct ≥ pt := |Tr [Tt (ρ) πB] − Tr [Nt (ρ) πB]| .
Lower bounds on information propagation
Hamiltonian, POVM and Local Unitary:
HΛ =
1
2
(1 − σz
o)
j∈B
1
[1 + d(o, j)]α 1 − σz
j .
• 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|⊗|ΛB|
⊗ |+ +|⊗|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 which
d(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)−2α
OΛ,l.
The right hand side converges if
lim
L→∞
L
l=δ
(1 + l)D−1−2α
< ∞
which is when α > D/2.
When α < D/2 signal propagation is therefore
not 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 −
1
2

1 + cos

t
L
l=δ
[1 + l]−α
OΛ,l



 .
From the expansion
1 − [1 + cos x] /2 = x2
/4 + O(x3
)
we see that we have to investigate
f(δ) := lim
L→∞
L
l=δ
(1 + d(o, j))−α
OΛ,l
Exploiting the asymptotic behaviour of the Hurwitz zeta
function we find
f(δ)2
= Θ(δ2(D−α)
).
When α > D it gives rise to a bent causal region
and allows for faster than linear propagation of
information, but slower than permitted by the
long-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
σx
o (t)σx
δ (t) c = σx
o (t)σx
δ (t) − σx
o (t) σx
δ (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 and
initial correlations at time t = 0 are vanishing. For α = 1/4 (left panel) correlations increase in an essentially distance-independent
way. A finite-size scaling analysis confirms that the propagation front indeed becomes flat (δ independent) for 0 ≤ α < D/2, and
hence no effective causal region is present. For α = 3/4 (central panel) the spreading of correlations shows a distance dependence
that 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
3.0
3.5
∆
t
0
0.01
0.02
0.03
0.04
6420246
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
6420246
The figure shows numerical results for the time evolution under the XXZ Hamiltonian
HXXZ
=
i>j
1
d(i, j)α
J⊥
2
σ+
i σ−
j + σ−
i σ+
j + Jz σz
i σz
j .
Top row: Density plots of the correlator σz
0σz
δ c in the (δ, t)-plane. The results are for long-range XXZ chains with |Λ| = 40 sites
and 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 σz
0σz
δ 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, Engineered
two-dimensional Ising interactions in a trapped-ion quantum
simulator 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 Lattice
Models, Phys. Rev. Lett. 111 (2013), 260401.
[3] M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger, A. M. Rey, and
C. W. Clark, Dynamical quantum correlations of Ising models on
an arbitrary lattice and their resilience to decoherence, New J.
Phys. 15 (2013), 113008.
[4] M. B. Hastings and T. Koma, Spectral Gap and Exponential Decay
of Correlations, Commun. Math. Phys. 265 (2006), 781–804.
[5] E. H. Lieb and D. W. Robinson, The Finite Group Velocity of
Quantum 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-range
interacting 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 and
Ana Maria Rey, Quantum correlations and entanglement in
far-from-equilibrium spin systems, arXiv:1406.0937
dB spin squeezing entanglement entropy concurrence
a
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
b
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
c
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
d
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t
e
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t
f
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t