dynamics of long-range interacting spin systems

Dynamics of Long-Range Interacting Quantum Spin Systems

Dynamics of Long-Range Interacting Quantum SpinSystems

Mauritz van den Worm and Michael Kastner

Stellenbosch Theory Seminar

Mauritz van den Worm | March 2013 1 / 21

Dynamics of Long-Range Interacting Quantum Spin Systems | Long-Range Interacting Systems

What is a long-range interacting system?

Interaction satisfies:

Ji ,j ∝ r−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

Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model

Our toy model

Emch-Radin Model

Emch’s original work

Radin’s generalization and extension

The Emch-Radin Model

Basic Ingredients

Lattice Λ, dim(Λ) = ν <∞C2i attached at each i ∈ Λ

Dynamics occurs on H = ⊗i∈ΛC2i

Long-Range, α < ν

H = Nα∑

(i ,j)∈Λ×Λ

Ji ,jσzi σ

zj − h

∑i∈Λ

σzi ,

with coupling constant Ji ,j := 1|i−j |α , where α ≥ 0

What makes the Emch-Radin model interesting?

Thermodynamic limit i.e. Large systems∑i ,j

|ri−rj |α →∞

Weight of interactions from far particles nonnegligible

Example:

1 2 3 4 5x0.0

f @xD=

x1�2

f @xD=

What makes the Emch-Radin model interesting?

Thermodynamic limit i.e. Large systems∑i ,j

|ri−rj |α →∞

Weight of interactions from far particles nonnegligible

Example:

1 2 3 4 5x0.0

f @xD=

x1�2

f @xD=

The Emch-Radin Model

The Hamiltonian

H = Nα∑

(i,j)∈Λ×Λ

Ji,jσzi σ

zj − h

∑i∈Λ

What are we interested in?

〈A〉(t) =Tr[e iHtAe−iHtρ(0)

], A =

aiσxi

ρ(0) = initial density matrix

Time Evolution of Emch-Radin model:

Define the time evolution operator:

αΛt : B (H)→ B (H) : O 7→ e iHtOe−iHt

The magnetic terms give

exp[−iht

∑i∈Λ σ

]σak exp

[iht∑

j∈Λ σzj

{σxk cos (2ht) + σy

k sin (2ht) for a = xσyk cos (2ht)− σx

k sin (2ht) for a = y

The interaction terms give

)= σx

k cos (2tPk )− σyk sin (2tPk )

)= σy

k cos (2tPk ) + σxk sin (2tPk )

with H =∑

i,j∈Λ Ji,jσzi σ

zj and Pk :=

∑j∈Λ\k Jk,jσ

Analytic Results:

Expectation values:

〈A〉(t) = 〈αΛt (A)〉 = Tr

[e iHtAe−iHtρ(0)

Initial State ρ(0)

For each A := (A1,A2,A3) with the Ai ⊂ Λ, Ai ∩ Aj = ∅, define

σA :=

(∏i∈A1

)∏j∈A2

(∏k∈A3

Choose ρ(0) such that

Tr[σAρ(0)

for all A such that A3 6= ∅.

Example

ρ(0) prepared diagonal in

σx , or

tensor product eigenbasis.

Analytic Results:

Expectation values:

〈A〉(t) = 〈αΛt (A)〉 = Tr

[e iHtAe−iHtρ(0)

Initial State ρ(0)

For each A := (A1,A2,A3) with the Ai ⊂ Λ, Ai ∩ Aj = ∅, define

σA :=

(∏i∈A1

)∏j∈A2

(∏k∈A3

Choose ρ(0) such that

Tr[σAρ(0)

for all A such that A3 6= ∅.

Example

ρ(0) prepared diagonal in

σx , or

tensor product eigenbasis.

Analytic Results - 1D lattice

Single spin expectation values:

〈σxi 〉(t)

〈σxi 〉(0)

= cos(2ht)

N/2∏j=1

cos2 (2Nαε(j)t)

〈σyi 〉(t)

〈σxi 〉(0)

= sin(2ht)

N/2∏j=1

cos2 (2Nαε(j)t)

Analytic Results - 1D lattice:

Two-spin correlators

〈σxi σxj 〉(t) = P−i ,j + cos(4ht)P+i ,j

〈σyi σyj 〉(t) = P−i ,j − cos(4ht)P+

〈σxi σyj 〉(t) = − sin(4ht)P+

〈σxi σzj 〉(t) = sin(2ht)Pzi ,j

〈σyi σzj 〉(t) = cos(2ht)Pzi ,j

P±i ,j = 12〈σ

xj 〉(0)

∏k 6=i ,j cos [2Nαt (Ji ,k ± Jj ,k)]

Pzi ,j = −〈σxi 〉(0) sin (2NαJi ,j)

∏k 6=i ,j cos (2NαtJi ,k)

Graphical Representation

1 10 100 1000 104t

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 0.4

Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.

Analytic Results - 1D lattice:

Two-spin correlators

〈σxi σxj 〉(t) = P−i ,j + cos(4ht)P+i ,j

〈σyi σyj 〉(t) = P−i ,j − cos(4ht)P+

〈σxi σyj 〉(t) = − sin(4ht)P+

〈σxi σzj 〉(t) = sin(2ht)Pzi ,j

〈σyi σzj 〉(t) = cos(2ht)Pzi ,j

P±i ,j = 12〈σ

xj 〉(0)

∏k 6=i ,j cos [2Nαt (Ji ,k ± Jj ,k)]

Pzi ,j = −〈σxi 〉(0) sin (2NαJi ,j)

∏k 6=i ,j cos (2NαtJi ,k)

1 10 100 1000 104t

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 0.4

Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.

Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization

What is being doneexperimentally?

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

(a) (b) (c)

H = −∑i<j

Ji ,jσzi σ

zj − Bµ ·

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-rangeEmch-Radin model!

(a) (b) (c)

H = −∑i<j

Ji ,jσzi σ

zj − Bµ ·

Tune 0 ≤ α ≤ 3

(a) (b) (c)

H = −∑i<j

Ji ,jσzi σ

zj − Bµ ·

Tune 0 ≤ α ≤ 3

Benchmarking of Penning Ion Trap

Finite Hexagonal lattice:

All results carry over

Previous benchmarking only in mean field limit (α = 0)

Use our exact results for better benchmarking

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 0.25

0.01 0.1 1 10t

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 1.5

0.01 0.1 1 10t

(a) (b)

Figure: Time evolution of the normalized spin-spin correlations. Curves of thesame color correspond to different side lengths L = 4, 8, 16 and 32 (from rightto left) of the hexagonal patches of lattices. In figure (a) α = 1/4, results aresimilar for all 0 ≤ α < ν/2. In figure (b) α = 3/2, with similar results for allα > ν/2.

Benchmarking of Penning Ion Trap

Finite Hexagonal lattice:

All results carry over

Previous benchmarking only in mean field limit (α = 0)

Use our exact results for better benchmarking

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 0.25

0.01 0.1 1 10t

YΣix]

Σ jz]

Σ jy]

Σ jx]

Α = 1.5

0.01 0.1 1 10t

(a) (b)

Figure: Time evolution of the normalized spin-spin correlations. Curves of thesame color correspond to different side lengths L = 4, 8, 16 and 32 (from rightto left) of the hexagonal patches of lattices. In figure (a) α = 1/4, results aresimilar for all 0 ≤ α < ν/2. In figure (b) α = 3/2, with similar results for allα > ν/2.

Mechanism responsible for prethermalization

Relaxation timescales

Dephasing → Prethermalization

Collisions → Dephasing on slower timescale

Which mechanism? (n-spin purity)

γn(t) := Tr [ρn(t)],where ρn(t) is the n-spin reduced density matrix.

Γ2 HtL

Γ1 HtL

Α = 0.5

0.01 0.1 1 10t

ΓnHtL

Figure: Both relaxation steps of spin–spin correlations turn out to be associatedwith a drop in the purity γn. This is an indication that both relaxation steps arecaused by dephasing

Physical Intuition

Γ2 HtL

Γ1 HtL

Α = 0.5

0.01 0.1 1 10t

ΓnHtL

Physical Intuition

Γ2 HtL

Γ1 HtL

Α = 0.5

0.01 0.1 1 10t

ΓnHtL

Physical Intuition

Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities

Two more curiosities...

1 Is the Emch-Radinmodel quantum enough?

2 Lieb-Robinson bounds?

Is the Emch-Radin model quantum enough?

Entropy of entanglement

S [ρn(t)] = −Tr [ρn(t) log2 ρn(t)]

Entropy of Entanglement:

Α = 0.4

2 4 6 8 10t

Α = 0.4

0.1 1 10 100 1000 104t

(a) (b)

Figure: (a) and (b) respectively show the entropy of entanglement of the singleand two spin reduced density matrices of the Emch-Radin model for particlenumbers N = 100, 200 and 300.

Is the Emch-Radin model quantum enough?

Entropy of entanglement

S [ρn(t)] = −Tr [ρn(t) log2 ρn(t)]

Entropy of Entanglement:

Α = 0.4

2 4 6 8 10t

Α = 0.4

0.1 1 10 100 1000 104t

(a) (b)

Figure: (a) and (b) respectively show the entropy of entanglement of the singleand two spin reduced density matrices of the Emch-Radin model for particlenumbers N = 100, 200 and 300.

Lieb-Robinson Bounds

What is it? (short-range)

‖[OA(t),OB(0)]‖ ≤ K‖OA‖‖OB‖ exp[−L−vt

]dist (A,B) = L

Exponential decay of correlators when v < Lt

Extension to short range (power law)

‖[OA(t),OB(0)]‖ ≤ K‖OA‖‖OB‖ ev|t|−1(1+dist(A,B))α

Exponential decay of correlators when v < α ln Lt

Example of short-range Lieb-Robinson

Figure: Notice the exponential decay of correlators.

Behaviour of correlator in Emch-Radin

2 4 6 8 10 12 14 16

Here we expect an exponential decay

t < ln n

20 40 60 80 100n

(a) (b)

Figure: (a) shows “inverted” light-cone behaviour while (b) shows what weexpected to observe.

Lieb, Robinson, Commun. math. Phys. 28, 251—257 (1972)Hastings, Koma, ArXiv:math-ph/0507008 (2005)

]dist (A,B) = L

2 4 6 8 10 12 14 16

t < ln n

20 40 60 80 100n

(a) (b)

]dist (A,B) = L

2 4 6 8 10 12 14 16

t < ln n

20 40 60 80 100n

(a) (b)

Concluding Remarks

The long-range Ising model...

is simple, but exhibits strange quantum behaviour:

Prethermalization plateausDephasingEntanglement increasing in time

can be used to benchmark Penning ion traps

is far from exhausted in terms of research potential

- The End -

dynamics of long-range interacting spin systems

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