exceptionaly long-range quantum lattice models

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1/6 Exceptionally long-range quantum lattice models Mauritz van den Worm National Institute of Theoretical Physics Stellenbosch University Stellenbosch

Transcript of exceptionaly long-range quantum lattice models

1 / 6

Exceptionally long-range quantum lattice models

Mauritz van den Worm

National Institute of Theoretical Physics

Stellenbosch University

Stellenbosch

| Introductory words 2 / 6

Interaction/hopping satisfies:

Ji,j ∝ |i− j|−α

0 < α < dim(System)

Gravitating Masses Coulomb Interactions (no screening)

| Introductory words 2 / 6

Interaction/hopping satisfies:

Ji,j ∝ |i− j|−α

0 < α < dim(System)

What are we interested in?

Spatio-temporal spreading of

correlators

information

entanglement

| Breakdown of Causality 3 / 6

Linear Cone

‖ [OA(t),OB(0)] ‖ ≤ K exp

[v|t| − d(A,B)

ξ

] ~v t

x

t

Short-Range

Logarithmic Cone, α > D

‖ [OA(t),OB(0)] ‖ ≤ Kev|t| − 1

[d(A,B) + 1]α−D

~ ln x

x

t

Long-Range

New bounds

Bounds of Alexey and co-workers

| Breakdown of Causality 3 / 6

Ρ Π B

TrL� B @e- iHt UA ΡUA†

e iHt D

TrL� B @e- itH ΡeiHt D

0 t

Tt

N t

Toy model

HΛ =1

2(1− σzo)

∑j∈B

1

[1 + dist(o, j)]α(1− σzj )

| Breakdown of Causality 3 / 6

Product initial state

ρ = |0〉〈0||Λ|−|B| ⊗ |+〉〈+|⊗|B|

|+〉 = (|0〉+ |1〉)/√

2

Probability of detecting a signal

pt ≥ 1− exp

{−4t2

5

∑j∈B

[1 + d (A, j)]−2α

}

When α < D/2 we have instantaneousspreading

Well defined causal region down to α = D/2

| Breakdown of Causality 3 / 6

GHZ entangled initial state

ρ =|0〉〈0||Λ|−|B| ⊗ |ψ〉〈ψ||ψ〉 =(|0, . . . , 0〉+ |1, . . . , 1〉)/

√2

Probability of detecting a signal

pt ≥ 1− 1

2

{1 + cos

[t∑j∈B

[1 + d (A, j)]−α]}

When α < D we have instantaneousspreading

Power-law shaped causal region

| Breakdown of Causality 3 / 6

Exact results for Ising

H =1

2

∑i 6=j

1

|i− j|α σzi σ

zj , 〈σxi σxj 〉(t)− 〈σxi 〉(t)〈σxj 〉(t)

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

Flat Power-law Linear

Figure : Density contour plots of the connected correlator 〈σx0σxδ 〉c(t) 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 = 0are vanishing.

| Breakdown of Causality 3 / 6

tDMRG results for XXZ

H =1

2

∑i 6=j

1

|i− j|α

[Jzσzi σ

zj +

J⊥

2

(σ+i σ−j + σ+

j σ−i

)], 〈σz0σzδ 〉c(t)

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 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 ∆

| Lieb-Robinson bounds for α < D 4 / 6

Rescaled time

τ ∝ N−q, q > 0

Connected Correlation Functions for Long-Range Ising

N =102

N =103

N =104

0.1 1 10 100t

0.1

0.2

0.3

0.4

0.5

Y Σix Σ j

x ]c

N =102

N =103

N =104

0.1 1 10 100Τ

0.1

0.2

0.3

0.4

0.5

Y Σix Σ j

x ]c

Physical time Rescaled time

| Lieb-Robinson bounds for α < D 4 / 6

Ingredients

D dimensional lattice Λ

|Λ| = N

Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =

∑X⊂Λ hX

| Lieb-Robinson bounds for α < D 4 / 6

Ingredients

D dimensional lattice Λ

|Λ| = N

Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =

∑X⊂Λ hX

Boundedness∑X3x,y

‖hX‖ =λ

[1 + d(i, j)]α

| Lieb-Robinson bounds for α < D 4 / 6

Ingredients

D dimensional lattice Λ

|Λ| = N

Tensor product Hilbert space, H = ⊗i∈ΛHiGeneric Hamiltonian, H =

∑X⊂Λ hX

Reproducing

∑k∈Λ

1

[1 + d(i, k)]α1

[1 + d(k, j)]α≤ p

[1 + d(i, j)]α

Asymptotics

NΛ ∼

c1N

α/D−1 for 0 6 α < D,

c2/ lnN for α = D,

c3 for α > D,

| Lieb-Robinson bounds for α < D 4 / 6

α > 0 Lieb-Robinson bound

‖ [OA(τ), OB(0)] ‖ ≤ C‖OA‖‖OB‖|A||B|(ev|τ | − 1)

[d(A,B) + 1]α

Rescaled timeτ = t/NΛ

Speed-up in physical time, α = 1/2

N = 10 N = 102 N = 103

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t

| Fermionic Long-Range Hopping 5 / 6

Long-Range fermionic hoppnig model

H = −1

2

N∑j=1

N−1∑l=1

d−αl

(c†jcj+l + c†j+lcj

)

dl =

{l if l ≤ N/2,N − l if l > N/2,

| Fermionic Long-Range Hopping 5 / 6

Long-Range hoppnig model

H = −1

2

N∑j=1

N−1∑l=1

d−αl

(c†jcj+l + c†j+lcj

)

dl =

{l if l ≤ N/2,N − l if l > N/2,

Diagonalize and Dispersion

H =∑k

ε(k)a†kak

ε(k) = −N−1∑l=1

cos (kl)

dαl

| Fermionic Long-Range Hopping 5 / 6

Propagation from staggered initial state

|ψ(0)〉 = |1, 0, 1, 0, . . . , 1, 0〉

Spreading

〈c†j+δ(t)cj(t)〉 =1

2δδ,0 −

(−1)j+δ

2N

∑k

eit[ε(k+π)−ε(k)]e−ikδ

| Fermionic Long-Range Hopping 5 / 6

Spreading of correlations

Linear Linear Linear

Α=0.75

0 10 20 30 40 50

0

5

10

15

20

t

Α=1.5

0 10 20 30 40 50

0

5

10

15

20

tΑ=3

0 10 20 30 40 50

0

5

10

15

20

t

Figure : Contour plots in the (δ, t)-plane, showing correlations between sites 0and δ in the fermionic long-range hopping model for N = 200 lattice sites andvarious values of α, starting from a staggered initial state. Lighter colorsrepresent larger correlations.

| Fermionic Long-Range Hopping 5 / 6

Time dependent number operator

〈nj(t)〉 =1

2− (−1)j

2N

N∑n=1

cos [t∆(k)] , ∆(k) = ε(k + π)− ε(k)

Α=3

Α=0.5

Α=0.25

Α=0.05

Α=0.01

0.1 1 10 100t

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Yn j ]

Figure : Time dependence of the occupation number of site j for different α,starting from a staggered initial state.

| Fermionic Long-Range Hopping 5 / 6

Dispersion Relation

ε(k) = −1

2

[Liα(

eik)

+ Liα(

e−ik)]

| Fermionic Long-Range Hopping 5 / 6

Dispersion relation in large system limit

HaL

-Π - Π2

Π2

Πk

-2

-1

1

Ε

HbL-Π - Π

2

Π2

Πk

-2

-1

1

2

Ε¢Α=1

Α=2

Α=3

Figure : Dispersion relation (a) and its derivative ε′(k) (b) for the long-rangefermionic hopping model with exponents α = 1, 2, and 3.

| Fermionic Long-Range Hopping 5 / 6

Dispersion Relation

ε(k) = −1

2

[Liα(

eik)

+ Liα(

e−ik)]

Density of states

ρ(v) =1

∫ 2π

(v − dε

dk

)dk

ρ(v) =1

π

2

1 + 4v2for α = 1,

Θ (π − 2v) Θ (π + 2v) for α = 2,1√

1− v2for α→∞,

| Fermionic Long-Range Hopping 5 / 6

DOS and Spreading

Α=1

Α=2

Α=¥

-3 -2 -1 1 2 3v

0.2

0.4

0.6

0.8

Ρ

Α=0.75

0 10 20 30 40 50

0

5

10

15

20

t

Α=1.5

0 10 20 30 40 50

0

5

10

15

20

t

Α=3

0 10 20 30 40 50

0

5

10

15

20

t

Figure : Left: Density of states for α = 1, 2 and ∞. Right: Spreading plots fordifferent α.

| Conclusions 6 / 6

Take home message

Even for α < D we can have clear propagation fronts

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

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

1.2

t

In rescaled time τ = t/NΛ we can deriveLieb-Robinson bounds for α < D

‖ [OA(τ), OB(0)] ‖ ≤ C‖OA‖‖OB‖|A||B|(ev|τ| − 1)

[d(A,B) + 1]α

For α < D we can still have cone-like propagation

Α=0.75

0 10 20 30 40 50

0

5

10

15

20

t

Α=1.5

0 10 20 30 40 50

0

5

10

15

20

t

Α=3

0 10 20 30 40 50

0

5

10

15

20

t

| Conclusions 6 / 6

Collaborators

| Conclusions 6 / 6