Semiclassical approach for interacting fermionic systems: - - PowerPoint PPT Presentation

Semiclassical approach for interacting fermionic systems: interference and echos in the Hubbard model

Thomas Engl1, Peter Schlagheck2, Juan Diego Urbina1 and Klaus Richter1

1Universit¨

at Regensburg

2Universit´

e de Li` ege

March 18, 2015

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 1 / 10

Introduction & Motivation

Coherent backscattering

[cf. F. Jendrzejewski, K. M¨ uller, J. Richard, A. Date,

• T. Plisson, P. Bouyer, A. Aspect and V. Josse, PRL 109

195302 (2012)]

0.004 223334 232334 233243 233432 242333 322334 323243 323432 332234 332342 333242 334332 343223 423233 432323 433322 probability final state ϕ=0 ϕ=0.125π ϕ=0.25π ϕ=0.5π ϕ=0, classical

[TE, J. Dujardin, A. Arg¨ uelles, P. Schlagheck, K. Richter and J. D. Urbina, PRL 112 140403 (2014)] → talk by Peter Schlagheck (Fr. 11h) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 2 / 10

Introduction & Motivation

Coherent backscattering

[cf. F. Jendrzejewski, K. M¨ uller, J. Richard, A. Date,

• T. Plisson, P. Bouyer, A. Aspect and V. Josse, PRL 109

195302 (2012)]

0.004 223334 232334 233243 233432 242333 322334 323243 323432 332234 332342 333242 334332 343223 423233 432323 433322 probability final state ϕ=0 ϕ=0.125π ϕ=0.25π ϕ=0.5π ϕ=0, classical

[TE, J. Dujardin, A. Arg¨ uelles, P. Schlagheck, K. Richter and J. D. Urbina, PRL 112 140403 (2014)] → talk by Peter Schlagheck (Fr. 11h)

Echoes:

[cf. http://en.wikipedia.org/wiki/Spin echo]

?

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 2 / 10

Fermionic Hubbard model

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

|n = ˆ c†

1↓ |0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

|n = ˆ c†

2↑ˆ

c†

2↓ |0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

|n = − ˆ c†

3↑ |0, 1, 1, 1, 0, 0, 0, 0, 0, 0

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

|n = |0, 1, 1, 1, 1, 0, 1, 1, 1, 0

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

Hamiltonian

ˆ H =

• j

σ=↑,↓

• ǫjˆ

c†

jσˆ

cjσ − J

• ˆ

c†

jσˆ

cj+1σ + ˆ c†

j+1σˆ

cj+1σ

• + Uˆ

c†

j↑ˆ

c†

j↓ˆ

cj↓ˆ cj↑ + κ

• ˆ

c†

j,↓ˆ

cj+1,↑ − ˆ c†

j+1,↓ˆ

cj,↑

• + κ∗

ˆ c†

j+1,↑ˆ

cj,↓ − ˆ c†

j,↑ˆ

cj+1,↓

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

J J

Hamiltonian

ˆ H =

• j

σ=↑,↓

• ǫjˆ

c†

jσˆ

cjσ − J

• ˆ

c†

jσˆ

cj+1σ + ˆ c†

j+1σˆ

cj+1σ

• + Uˆ

c†

j↑ˆ

c†

j↓ˆ

cj↓ˆ cj↑ + κ

• ˆ

c†

j,↓ˆ

cj+1,↑ − ˆ c†

j+1,↓ˆ

cj,↑

• + κ∗

ˆ c†

j+1,↑ˆ

cj,↓ − ˆ c†

j,↑ˆ

cj+1,↓

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

J J −κ κ

Hamiltonian

ˆ H =

• j

σ=↑,↓

• ǫjˆ

c†

jσˆ

cjσ − J

• ˆ

c†

jσˆ

cj+1σ + ˆ c†

j+1σˆ

cj+1σ

• + Uˆ

c†

j↑ˆ

c†

j↓ˆ

cj↓ˆ cj↑ + κ

• ˆ

c†

j,↓ˆ

cj+1,↑ − ˆ c†

j+1,↓ˆ

cj,↑

• + κ∗

ˆ c†

j+1,↑ˆ

cj,↓ − ˆ c†

j,↑ˆ

cj+1,↓

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model

J J −κ κ U

Hamiltonian

ˆ H =

• j

σ=↑,↓

• ǫjˆ

c†

jσˆ

cjσ − J

• ˆ

c†

jσˆ

cj+1σ + ˆ c†

j+1σˆ

cj+1σ

• + Uˆ

c†

j↑ˆ

c†

j↓ˆ

cj↓ˆ cj↑ + κ

• ˆ

c†

j,↓ˆ

cj+1,↑ − ˆ c†

j+1,↓ˆ

cj,↑

• + κ∗

ˆ c†

j+1,↑ˆ

cj,↓ − ˆ c†

j,↑ˆ

cj+1,↓

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Semiclassical theory

Propagator in Fock basis: K(n, m, t) =

• n
• e− i

ˆ

Ht

• m
• Construct Path integral

→ talk by Juan Diego Urbina (Fr. 11h30) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory

Propagator in Fock basis: K(n, m, t) =

• n
• e− i

ˆ

Ht

• m
• Construct Path integral

→ talk by Juan Diego Urbina (Fr. 11h30)

⇒ Classical limit:

ˆ c†

jσˆ

cjσ →

• φjσ(t)
• 2 ,

ˆ c†

jσˆ

ckσ′ → φ∗

jσ(t)φkσ′(t)e−|φjσ(t)|

2−|φkσ′(t)|2

(k,σ′)

• (l,σ′′)=(j,σ)
• 1 − 2 |φlσ′′(t)|2

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory

Propagator in Fock basis: K(n, m, t) =

• n
• e− i

ˆ

Ht

• m
• Construct Path integral

→ talk by Juan Diego Urbina (Fr. 11h30)

⇒ Classical limit:

ˆ c†

jσˆ

cjσ →

• φjσ(t)
• 2 ,

ˆ c†

jσˆ

ckσ′ → φ∗

jσ(t)φkσ′(t)e−|φjσ(t)|

2−|φkσ′(t)|2

(k,σ′)

• (l,σ′′)=(j,σ)
• 1 − 2 |φlσ′′(t)|2

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory

Propagator in Fock basis: K(n, m, t) =

• n
• e− i

ˆ

Ht

• m
• Construct Path integral

→ talk by Juan Diego Urbina (Fr. 11h30)

⇒ Classical limit:

ˆ c†

jσˆ

cjσ →

• φjσ(t)
• 2 ,

ˆ c†

jσˆ

ckσ′ → φ∗

jσ(t)φkσ′(t)e−|φjσ(t)|

2−|φkσ′(t)|2

(k,σ′)

• (l,σ′′)=(j,σ)
• 1 − 2 |φlσ′′(t)|2

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory

Propagator in Fock basis: K(n, m, t) =

• n
• e− i

ˆ

Ht

• m
• Construct Path integral

→ talk by Juan Diego Urbina (Fr. 11h30)

⇒ Classical limit:

ˆ c†

jσˆ

cjσ →

• φjσ(t)
• 2 ,

ˆ c†

jσˆ

ckσ′ → φ∗

jσ(t)φkσ′(t)e−|φjσ(t)|

2−|φkσ′(t)|2

(k,σ′)

• (l,σ′′)=(j,σ)
• 1 − 2 |φlσ′′(t)|2

Stationary phase analysis ⇒ Sum over classical trajectories

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical propagator1

K(m, n, t) =

• m
• e− i

ˆ

Ht

• n
• ≈
• γ:n→m

Aγe

i Rγ+iµγ π 2

Classical action: Rγ =

t

• dt′

θ (t′) ˙ J (t′) − H(cl) (ψ∗ (t′) , ψ (t′))

• 1TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133, 1563;

arXiv:1409.4196

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator1

K(m, n, t) =

• m
• e− i

ˆ

Ht

• n
• ≈
• γ:n→m

Aγe

i Rγ+iµγ π 2

Classical trajectory γ : n → m: |ψjσ(0)|2 = njσ |ψjσ(t)|2 = mjσ Classical action: Rγ =

t

• dt′

θ (t′) ˙ J (t′) − H(cl) (ψ∗ (t′) , ψ (t′))

• 1TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133, 1563;

arXiv:1409.4196

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator1

K(m, n, t) =

• m
• e− i

ˆ

Ht

• n
• ≈
• γ:n→m

Aγe

i Rγ+iµγ π 2

Classical trajectory γ : n → m: |ψjσ(0)|2 = njσ i ˙ ψ = ∂H(cl)

∂ψ∗

|ψjσ(t)|2 = mjσ Classical action: Rγ =

t

• dt′

θ (t′) ˙ J (t′) − H(cl) (ψ∗ (t′) , ψ (t′))

• 1TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133, 1563;

arXiv:1409.4196

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator1

K(m, n, t) =

• m
• e− i

ˆ

Ht

• n
• ≈
• γ:n→m

Aγe

i Rγ+iµγ π 2

Classical trajectory γ : n → m: θ(γ=1)(0) θ(γ=2)(0) θ(γ=3)(0) |ψjσ(0)|2 = njσ i ˙ ψ = ∂H(cl)

∂ψ∗

|ψjσ(t)|2 = mjσ Classical action: Rγ =

t

• dt′

θ (t′) ˙ J (t′) − H(cl) (ψ∗ (t′) , ψ (t′))

• 1TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133, 1563;

arXiv:1409.4196

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t)

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t) interference between time-reverse paths γ′ = T γ

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t) interference between time-reverse paths γ′ = T γ Rγ − RT γ = 0 n = m

ψ

T

→ ψ∗

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t) interference between time-reverse paths γ′ = T γ δnmPcl (n, m, t) n = m

ψ

T

→ ψ∗

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t) interference between time-reverse paths γ′ = T γ Rγ − RT γ = π

j

(nj↑ − mj↑) δnmPcl (n, m, t) n = m

ψ

T

→ ψ∗

n m

ψ↑ ψ↓

• T

→ −ψ↓ ψ↑ ∗

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

P (n, m, t) = |K (n, m, t)|2 =

• γ,γ′:m→n

AγA∗

γ′e

i (Rγ−Rγ′)+i π 2 (µγ−µγ′)

diagonal approximation γ′ = γ: =: Pcl (n, m, t) interference between time-reverse paths γ′ = T γ δnmPcl (n, m, t) (−1)Nδn↑m↓δn↓m↑Pcl (n, m, t) n = m

ψ

T

→ ψ∗

n m

ψ↑ ψ↓

• T

→ −ψ↓ ψ↑ ∗

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 6 / 10

Transition probability

0.0001 0.0002 0.0001 0.0002

N = 8 N = 7

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 7 / 10

Transition probability

0.0001 0.0002 0.0001 0.0002

N = 8 N = 7

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 7 / 10

Transition probability

0.0001 0.0002 0.0001 0.0002

N = 8 N = 7 ˆ H =

j σ=↑,↓

• ǫjˆ

c†

jσˆ

cjσ − J

• ˆ

c†

jσˆ

cj+1σ + ˆ c†

j+1σˆ

cj+1σ

• + Uˆ

c†

j↑ˆ

c†

j↓ˆ

cj↓ˆ cj↑ +κ

• ˆ

c†

j,↓ˆ

cj+1,↑ − ˆ c†

j+1,↓ˆ

cj,↑

• + κ∗

ˆ c†

j+1,↑ˆ

cj,↓ − ˆ c†

j,↑ˆ

cj+1,↓

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 7 / 10

Transition probability

0.0001 0.0002 0.0001 0.0002

N = 8 N = 7 symmetry under ψ↑ ψ↓

• →

ψ↓eiarg(κ) ψ↑e−iarg(κ)

• Thomas Engl (Uni Regensburg)

Interference in the Fermi-Hubbard model March 18, 2015 7 / 10

Transition probability

0.0001 0.0002 0.0001 0.0002

N = 8 N = 7 symmetry under ψ↑ ψ↓

• →

ψ↓eiarg(κ) ψ↑e−iarg(κ)

• Every discrete (antiunitary) symmetry gives another peak/dip!

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 7 / 10

Many-Body Spin Echo

propagate: t flip spins propagate: t + τ

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 8 / 10

Many-Body Spin Echo

propagate: t flip spins propagate: t + τ → γ1, γ3 → γ2, γ4 4 trajectories:

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 8 / 10

Many-Body Spin Echo

1 2 Pmbse/Pmbse

(incoh)

τ τN

1 2 0 1 2 3 4 5 6 F N=6 1 2 0 1 2 3 4 5 6 F N=7 1 2 0 1 2 3 4 5 6 F N=9 1 2 0 1 2 3 4 5 6 F N=10

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 9 / 10

Conclusion

[TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133, 1563 (2014);

arXiv:1409.4196]

Semiclassical theory for interacting Fermions in Fock space Successful prediction of interference phenomena in transition probability [TE, J. D. Urbina and K. Richter, arXiv:1409.5684] Prediction of Many-Body Spin Echo

Outlook

Additional symmetries in Spin Echo Trace formula

Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 10 / 10