Quench, Hydro and Floquet dynamics in integrable systems RAQIS, - - PowerPoint PPT Presentation

Jean-Sébastien Caux Universiteit van Amsterdam RAQIS, Annecy, 12 September 2018

Quench, Hydro and Floquet dynamics in integrable systems

Plan of the talk

Out-of-equilibrium dynamics Quenches Quasisolitons and Generalized Hydrodynamics Equilibrium dynamics Floquet dynamics

Applications of integrability

Ultracold atoms Quantum magnetism Atomic nuclei Quantum dots, NV centers

|{λ}i

The general idea, simply stated:

Start with your favourite quantum state (expressed in terms of Bethe states)

O

Apply some operator on it Reexpress the result in the basis of Bethe states:

O|{λ}i = X

{µ}

F O

{µ},{λ}|{µ}i

F O

{µ},{λ} = h{µ}|O|{λ}i

using ‘matrix elements’

Equilibrium Dynamics from Integrability

S(k, ω), ∆ = 1, h = 0

Heisenberg spin chain

Integrability ‘probed’ in the lab

Ultracold atoms Quantum magnetism

Hope: also Quantum dots, NV centers, Atomic nuclei

• 1

ΔE [a.u.]

0.0 0.5 1.0 1.5 2.0 2.5 1 2 3 Bragg detuning HkHzL <p2> Harb. u.L

HaL

2nK 4nK

0.0 0.5 1.0 1.5 2.0 2.5 1 2 Bragg detuning HkHzL <p2> Harb. u.L

HbL

1nK 5nK 10nK

0.0 0.5 1.0 1.5 2.0 2.5 1 2 Bragg detuning HkHzL <p2> Harb. u.L

HcL

1nK 10nK 15nK

0.0 0.5 1.0 1.5 2.0 2.5 1 2 Bragg detuning HkHzL <p2> Harb. u.L

HdL

1nK 10nK 20nK

Out-of- Equilibrium Dynamics from Integrability

The super Tonks-Girardeau gas Split Fermi seas (Moses states) Spin echo in quantum dots Quasisolitons Interaction quench in Richardson Domain wall release in Heisenberg Geometric quench Interaction cutoff in Lieb-Liniger Release of trapped Lieb-Liniger BEC to Lieb-Liniger quench Quantum Newton’s Cradle in TG Néel to XXZ quench Generalized hydrodynamics Floquet driving central spin Floquet driving spin chains

Pulsed: Quenched: Driven:

Out-of-equilibrium using integrability

Quantum Quenches

Progress on quenches

Quasisoliton Dynamics in Spin Chains

In memoriam

Ludvig Dmitrievich Faddeev 23/3/1934 - 26/2/2017

Spinon dynamics in real space/time

Vlijm, Caux, PRB 2016

Spinon dynamics in real space/time

Vlijm, Caux, PRB 2016

Solitons (classical)

John Scott Russell: solitary wave of translation (1834)

(Herriot-Watt University)

Solitons (classical)

Further simulations: Zabusky & Kruskal 1965

concept of a soliton

(Boussinesq) Korteweg-de Vries equation

∂tu + u∂xu + δ2∂3

xu = 0

Classical inverse scattering First simulations: Fermi-Pasta-Ulam- (Tsingou) absence of ergodicity

Quasisoliton scattering (quantum)

tJ j j 20 40 25 50 75 50 100 150 200 25 50 75 0.2 0.3 0.4 0.5 hSz

j (t)i

∆ = 2

100 200 20 40 60 80 tJ j 2 4 6 1 2 3 4 Displacement ∆ Measured av. Linear fits Single magnons Bound magnons −χ(1,1) −χ(2,2)

Displacement as a function of anisotropy

(fixed incoming momenta) 1str-1str 2str-2str

‘Worldlines’

• f colliding

wavepackets:

Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, 2015

Generalized Hydrodynamics

Generalized Hydrodynamics (GHD)

• B. Bertini, M. Collura, J. De Nardis and M. Fagotti, PRL 117, 207201 (2016)
• O. A. Castro-Alvaredo, B. Doyon and T.

Yoshimura, PRX 6, 041065 (2016)

• B. Doyon and T.

Yoshimura, SciPost Phys. 2, 014 (2017) 


∂tρ(λ) + ∂x(v

eff(λ)ρ(λ)) = 0

v

eff(λ) = v gr(λ) +

Z dλ0 ϕ(λ, λ0) p0(λ) ρ(λ0) (v

eff(λ0) − v eff(λ))

Local continuity equation

(in terms of Bethe root densities)

Local effective velocity Quench from spatially inhomogeneous state After initial dephasings: ‘hydrodynamic’ evolution described by local GGE

• Fig. 3 from B. Bertini, M. Collura, J. De Nardis and
• M. Fagotti, PRL 117, 207201 (2016)

Two-equation summary:

v

gr

v

eff

GHD as ‘molecular dynamics’:

Encode initial state as a gas of quasisolitons Loop: evolve, collide and scatter

Attractive interactions Repulsive interactions

• B. Doyon, T.

Yoshimura and JSC, PRL 2018

the flea gas

As simple as it gets in the integrability business!

(as if quasisolitons were classical particles, using displacement calculated from quantum phase shifts)

Ergodicity (or lack thereof) in interacting quantum systems close to an integrable model

David Weiss’s quantum Newton’s cradle experiment

‘Oscillation’-like dynamics at short time scales

JSC, B. Doyon, J Dubail, R. Konik and T. Yoshimura, arXiv1711.00873

The flea gas in a force field: simulating the quantum Newton’s cradle

The flea gas in a force field: simulating the quantum Newton’s cradle

‘Relaxation’-like dynamics at long time scales

JSC, B. Doyon, J Dubail, R. Konik, and T. Yoshimura, arXiv:1711.00873

Floquet Dynamics

Pyotr L. Kapitza (8/7/1894-8/4/1984)

The simple pendulum on its head

Floquet basics

“fast motion”

• perator

Floquet Hamiltonian Unitary time evolution operator under action of periodic Hamiltonian

ˆ U(t) = T(e−i

R t

0 dt0H(t0)) = ˆ

P(t)e−i ˆ

HF t

ˆ P(t + T) = ˆ P(t) ˆ P(nT) = 1 (n ∈ Z)

|ψn(t)i = e−i✏nt|φn(t)i, |φn(t)i = ˆ P(t)|φni

t evolution:

ˆ UF ≡ ˆ U(T) = e−i ˆ

HF T

Stroboscopic Floquet operator

ˆ HF = X

n

✏n|nihn|

Diag’n:

ˆ UF = X

n

e−iθn|φnihφn|

ˆ H(t) = ˆ H(t + T)

Integrable manifold

H1 H2

Havg = ηH1 + (1 − η)H2

η = 1 η = 0

Floquet’ing integrable models

take a one-parameter family of integrable models do quench/dequench sequences on this manifold Idea:

P . Claeys and JSC, arXiv:1708.07324

Time-avg Hamiltonian also integrable t

{

T

ηT

(1 − η)T

H1 H2

ˆ H(t) = ( ˆ H1 for 0 < t < ηT, ˆ H2 for ηT < t < T,

“Quench-dequench” Floquet protocol

Let’s consider the simple case of periodically “switching” between two Hamiltonians: so

ˆ UF ≡ e−i ˆ

HF T = e−i(1−η)T ˆ H2e−iηT ˆ H1

One then finds the nice identities

ˆ HAvg = η ˆ H1 + (1 − η) ˆ H2

with t-avg Hamiltonian

∂θn ∂T = hφn| ˆ Havg|φni

✏n = ✓n T = hn| ˆ HF |ni

Floquet Dynamics: XXZ chain

ˆ H(t) = −J X

i

⇥ Sx

i Sx i+1 + Sy i Sy i+1 + ∆(t)Sz i Sz i+1

⇤

Focus on sector with k = 0 and

mz = 1/3

XXZ Hamiltonian:

ˆ UF = e−i(1−η)T ˆ

H2e−iηT ˆ H1

Floquet’ing integrable models

P . Claeys and JSC, arXiv:1708.07324

Floquet protocol: binary switch between ∆1

∆2

and No analytical solution ☹, must rely on numerics Stroboscopic Floquet operator:

P . Claeys and JSC, arXiv:1708.07324

Floquet’ing integrable models

−π π

θn

∆1,2 = −2, −3 −4 −2 2 4

θn/T

1 2 3

Period T

−4 −2 2 4

∂T θn

−π π ∆1,2 = −2, 3 −2 −1 1 2 2 4 6

Period T

−2 −1 1 2

Floquet phases, quasienergies and avg quasienergies as a function of T All crossings are avoided Possible to achieve targeted state preparation

Eigenvalue statistics

r = min(sn, sn+1) max(sn, sn+1) ∈ [0, 1], sn = En+1 − En

hriGOE ⇡ 0.535989 hriP OI ⇡ 0.386295

T1 T2 1.5 T3 2.5 3 0.35 0.386 0.4 0.45 0.5 0.536 0.55

hri L = 18 L = 21

T1 1 T2 3 T3 5 6

Period T

0.35 0.386 0.4 0.45 0.5 0.536 0.55

hri

hr(θn/T)i (symbols)

hr(∂T θn)i (dashed lines)

Expect Quasienergies mark transition from Poisson (high frequency) to GOE (low freq)

Δ1,2 = − 2, − 3 Δ1,2 = − 2,3

Floquet Dynamics: central spin

Floquet’ing the central spin model

P . Claeys, S. De Baerdemacker, O. El Araby and JSC, PRL 2018

−π π

θn

−2 −1 1

θn/T

0.5 1 1.5 2 2.5

T/Tc

−2 −1 1

∂T θn

H = BzSz

0 + L

X

j=1

Aj ~ S0 · ~ Sj

Floquet protocol: binary switch between two values of field on central spin Hamiltonian: Example with L=5 (bigger is possible, but ugly)

Floquet’ing the central spin model

P . Claeys, S. De Baerdemacker, O. El Araby and JSC, PRL 2018

−1 1

hHAvgi

0.9 1 1.1

T(n)/Tc

−0.5 0.5

hSz

0i

−1 1 1.9 2 2.1

T(n)/Tc

−0.5 0.5

Modulation of driving frequency (cf Landau-Zener) Red line: effective model using integrability

Ti+1 = Ti + 10−4

using old results on quenches in central spin A. Faribault, P . Calabrese and JSC (2009)

To improve: simply use more states in effective basis

Blue line: exact “Adiabatic” change:

Floquet Dynamics: XY chain

Floquet’ing the XY model

HXY = J

N

X

j=1

⇥ (1 + γ)Sx

j Sx j+1 + (1 − γ)Sy j Sy j+1

⇤ − h(t)

N

X

j=1

Sz

j

HXY = −J X

k>0

h (cos(k) − h(t)/J)(c†

kck + c† −kc−k) + iγ sin(k)(c† kc† −k − c−kck)

i − h(t)N/2

Model: old friend For calculations: as usual, map to free fermions Floquet operator/Hamiltonian: exactly calculable Fun bit: Floquet, switch value of external field

Uk(t0 + T, t0) = e−iH[t0,T ]

F

nT = uk,01k + i

X

a=x,y,z

uk,aσa

k

Pauli matrices spanning { |0i, c†

kc† −k|0i }

S.E. Tapias Arze, JSC, P . Claeys and I. Perez Castillo (arXiv:1804.10226)

Avoided Crossings

Tc=π/2

• 0.0

0.5 1.0 1.5 2.0

• 4
• 2

2 4 T T ϵk

avoided crossings in red regions no crossings in blue regions

Floquet’ing the XY model

Floquet quasienergies as a function of driving period T

J = 1, γ = 0.8, h1 = 15, h2 = 0.5, α = 0.5, N = 10001

• 2

• J/2

Floquet’ing the XY model

Nature of the Floquet Hamiltonian

A` = 2 N X

k>0

uk,z q 1 − u2

k,0

cos(`k)✏T (k)

H[t0,T ]

F

=

N

X

j=1

X

`≥1

h A`c†

jcj+` + B`c† jc† j+` + h.c.

i + A0

N

X

j=1

c†

jcj

B` = − 2i N X

k>0

(uk,x − iuk,y) q 1 − u2

k,0

sin(`k)✏T (k)

“local” “non-local”

ρ =

L

X

`=2

[|A`| + p ||B`||2]/2(L − 1)

with extended-range couplings

Floquet’ing the XY model

Stroboscopic expectation value of observables Magnetization along z NN xx correlator

• range α = 0.5, γ = 0.8, h1 = 0.2, h2 = 1.2, T = 10

green blue

α = 0.5, γ = 0.551, h1 = 0.2, h2 = 1.2, T = 0.75

α = 0.5, γ = 0.8, h1 = 2.0, h2 = 0.5, T = 0.75

Relaxation behaviour displays robustness/universality

Floquet’ing the XY model

Time decay of observables

O(decay)(t) = N 2π Z π dk h f(k)e−2inθT (k) + h.c. i

✓T (k) = ✏T (k)T

f(k) = hGSk

• Φ[0,T ]

k,− (0)ihΦ[0,T ] k,− (0)

• Ok
• Φ[0,T ]

k,+ (0)ihΦ[0,T ] k,+ (0)

• GSki

hO(t)i = O(ss)(δt) + O(decay)(t)

Splits into synch’d/steady-state value + decaying part, Synch’d part: “Periodic GGE” or “Periodic Quench Action” Decaying part: depends on “universal” dispersion and observable- specific ME

100 200

n

0.49892 0.499

Mz(nT)

t−3/2

200 400 600

n

0.35 0.5

t−1/2

200 400 600 800

n

0.2 0.5

t−3/4

−π −π/2 π/2 π

k

−6 −7 −8

θT(k)

−π −π/2 π/2 π

k

−6.0 −6.4 −6.8 −π −π/2 π/2 π

k

−0.6 −1.0 −1.4

Floquet’ing the XY model

Time decay of observables: exponent of envelope related to nature of quasienergy spectrum local non-local local (special)

Z 1

1

dk 1 2f 00(k0)(k − k0)2einθ00

T (k0)(kk0)2 ∝ n3/2

for θ0

T (k0) = 0 and k0 = 0, π

Z ∞

−∞

dkf(k0)e−inθ00

T (k0)(k−k0)2 ∝ n−1/2

for θ0

T (k0) = 0 and k0 6= 0, π

Z 1

1

dk 1 2f 00(k0)(k − k0)2einθ0000

T (k0)(kk0)4/12 ∝ n3/4

for θ0

T (k0) = θ00 T (k0) = θ000 T (k0) = 0 and k0 = 0, π

0.0 0.5 1.0 1.5 2.0 0.20 0.25 0.30 0.35 0.40 0.45 T M z(T )

• 0.10
• 0.05

0.00 0.05 Cxx(T )

Floquet’ing the XY model

Sroboscopic steady-state expectation values Steady state value of Mz and NNxx correlator plotted along this line Shaded: “local”

Observables display non-analyticities at local/non-local boundaries

Conclusions

Quench dynamics in integrable systems Surprisingly healthy recent history!

Quench Action yields exact solutions for nontrivial cases Generalized hydrodynamics: from quasisolitons to the flea gas

Floquet dynamics in integrable systems Prototypical examples: XXZ, central spin, XY

New “universalities” displayed in stroboscopic behaviour Much work to be done, many opportunities for integrability

Equilibrium dynamics in integrable systems Very healthy recent history

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