Upload
mauritz-van-den-worm
View
194
Download
1
Tags:
Embed Size (px)
DESCRIPTION
This is a talk I gave at the Stellenbosch Theoretical Physics seminar series.
Citation preview
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)
Mauritz van den Worm | March 2013 2 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Long-Range Interacting Systems
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
Mauritz van den Worm | March 2013 3 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Long-Range Interacting Systems
What about astrophysics?
Screening?
No negative masses → no screening
Negative heat capacities Nonequivalence of ensembles
Mauritz van den Worm | March 2013 4 / 21
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
Mauritz van den Worm | March 2013 5 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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
Mauritz van den Worm | March 2013 6 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
What makes the Emch-Radin model interesting?
Thermodynamic limit i.e. Large systems∑i ,j
1
|ri−rj |α →∞
Weight of interactions from far particles nonnegligible
Example:
1 2 3 4 5x0.0
0.5
1.0
1.5
2.0
f HxL
f @xD=
1
x1�2
f @xD=
1
x2
f @xD=
1
x
Mauritz van den Worm | March 2013 7 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
What makes the Emch-Radin model interesting?
Thermodynamic limit i.e. Large systems∑i ,j
1
|ri−rj |α →∞
Weight of interactions from far particles nonnegligible
Example:
1 2 3 4 5x0.0
0.5
1.0
1.5
2.0
f HxL
f @xD=
1
x1�2
f @xD=
1
x2
f @xD=
1
x
Mauritz van den Worm | March 2013 7 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
The Emch-Radin Model
The Hamiltonian
H = Nα∑
(i,j)∈Λ×Λ
Ji,jσzi σ
zj − h
∑i∈Λ
σzi
What are we interested in?
〈A〉(t) =Tr[e iHtAe−iHtρ(0)
], A =
∑i
aiσxi
ρ(0) = initial density matrix
Mauritz van den Worm | March 2013 8 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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
zi
]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
αΛt
(σxk
)= σx
k cos (2tPk )− σyk sin (2tPk )
αΛt
(σyk
)= σy
k cos (2tPk ) + σxk sin (2tPk )
with H =∑
i,j∈Λ Ji,jσzi σ
zj and Pk :=
∑j∈Λ\k Jk,jσ
zj .
Mauritz van den Worm | March 2013 9 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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
σxi
)∏j∈A2
σyj
(∏k∈A3
σzk
).
Choose ρ(0) such that
Tr[σAρ(0)
]= 0
for all A such that A3 6= ∅.
Example
ρ(0) prepared diagonal in
σx , or
σy
tensor product eigenbasis.
Mauritz van den Worm | March 2013 10 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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
σxi
)∏j∈A2
σyj
(∏k∈A3
σzk
).
Choose ρ(0) such that
Tr[σAρ(0)
]= 0
for all A such that A3 6= ∅.
Example
ρ(0) prepared diagonal in
σx , or
σy
tensor product eigenbasis.
Mauritz van den Worm | March 2013 10 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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)
Mauritz van den Worm | March 2013 11 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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+
i ,j
〈σxi σyj 〉(t) = − sin(4ht)P+
i ,j
〈σxi σzj 〉(t) = sin(2ht)Pzi ,j
〈σyi σzj 〉(t) = cos(2ht)Pzi ,j
Where
P±i ,j = 12〈σ
xi 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
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
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.
Mauritz van den Worm | March 2013 12 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Emch-Radin Model
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+
i ,j
〈σxi σyj 〉(t) = − sin(4ht)P+
i ,j
〈σxi σzj 〉(t) = sin(2ht)Pzi ,j
〈σyi σzj 〉(t) = cos(2ht)Pzi ,j
Where
P±i ,j = 12〈σ
xi 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
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
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.
Mauritz van den Worm | March 2013 12 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
What is being doneexperimentally?
Mauritz van den Worm | March 2013 13 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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-rangeEmch-Radin model!
Mauritz van den Worm | March 2013 14 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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-rangeEmch-Radin model!
Mauritz van den Worm | March 2013 14 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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-rangeEmch-Radin model!
Mauritz van den Worm | March 2013 14 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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
Graphical Representation
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 0.25
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 1.5
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
(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.
Mauritz van den Worm | March 2013 15 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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
Graphical Representation
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 0.25
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 1.5
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
(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.
Mauritz van den Worm | March 2013 15 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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.
Graphical Representation
Γ2 HtL
Γ1 HtL
Α = 0.5
0.01 0.1 1 10t
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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
Mauritz van den Worm | March 2013 16 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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.
Graphical Representation
Γ2 HtL
Γ1 HtL
Α = 0.5
0.01 0.1 1 10t
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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
Mauritz van den Worm | March 2013 16 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Experimental Realization
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.
Graphical Representation
Γ2 HtL
Γ1 HtL
Α = 0.5
0.01 0.1 1 10t
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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
Mauritz van den Worm | March 2013 16 / 21
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?
Mauritz van den Worm | March 2013 17 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
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
N=300
N=200
N=100
2 4 6 8 10t
0.2
0.4
0.6
0.8
1.0
SHF1L
N=300
N=200
N=100
Α = 0.4
0.1 1 10 100 1000 104t
0.5
1.0
1.5
2.0
SHF2L
(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.
Mauritz van den Worm | March 2013 18 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
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
N=300
N=200
N=100
2 4 6 8 10t
0.2
0.4
0.6
0.8
1.0
SHF1L
N=300
N=200
N=100
Α = 0.4
0.1 1 10 100 1000 104t
0.5
1.0
1.5
2.0
SHF2L
(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.
Mauritz van den Worm | March 2013 18 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
Lieb-Robinson Bounds
What is it? (short-range)
‖[OA(t),OB(0)]‖ ≤ K‖OA‖‖OB‖ exp[−L−vt
ζ
]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
2
4
6
8
10
0
0.5
i
10 t
Here we expect an exponential decay
t < ln n
20 40 60 80 100n
1
2
3
4
t
(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)
Mauritz van den Worm | March 2013 19 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
Lieb-Robinson Bounds
What is it? (short-range)
‖[OA(t),OB(0)]‖ ≤ K‖OA‖‖OB‖ exp[−L−vt
ζ
]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
2
4
6
8
10
0
0.5
i
10 t
Here we expect an exponential decay
t < ln n
20 40 60 80 100n
1
2
3
4
t
(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)
Mauritz van den Worm | March 2013 19 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
Lieb-Robinson Bounds
What is it? (short-range)
‖[OA(t),OB(0)]‖ ≤ K‖OA‖‖OB‖ exp[−L−vt
ζ
]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
2
4
6
8
10
0
0.5
i
10 t
Here we expect an exponential decay
t < ln n
20 40 60 80 100n
1
2
3
4
t
(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)
Mauritz van den Worm | March 2013 19 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
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
Mauritz van den Worm | March 2013 20 / 21
Dynamics of Long-Range Interacting Quantum Spin Systems | Two more curiosities
- The End -
Mauritz van den Worm | March 2013 21 / 21