[PPT] - Saarland University Simulating non-equilibrium quantum relaxation PowerPoint Presentation

SLIDE 1

Simulating non-equilibrium quantum relaxation in closed interacting quantum many body systems

H. Rieger, B. Blass

Theoretical physics, Saarland University, Saarbrücken, Germany

Simulating quantum processes and devices

624. Heraeus Seminar, Bad Godesberg, 19.-22.9.2019

Saarland University

SLIDE 2

Quantum thermalization – 2008 (numerics)

H = -Σ<ij> Jij(ai

+aj + aj +ai), hard core bosons

SLIDE 3

Quantum Thermalization – 2016 (experimental)

,

t STATISTICAL PHYSICS

Quantum thermalization through entanglement in an isolated many-body system

Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Markus Greiner*

4

19 AUGUST 2016 • VOL 353 ISSUE 6301

rg SCI E N CE

H = -Σ<ij> Jij(ai

+aj + aj +ai) + U Σini(ni-1)

SLIDE 4

Thermalization in closed systems

SLIDE 5

Outline

Focus on transverse Ising model
1d : integrable: Generalized Gibbs ensemble (GGE)
2d: non-integrable:

numerical studies of thermalizaIon aJer quantum quenches

SLIDE 6

The transverse Ising model

σx, σz Pauli matrices, n.n. = nearest neighbors in d-dimensions, p.b.c.

Order parameter: Magne&za&on (not conserved!) m = hσx

i i

h T

hc (=3.04 J)

QC hσx

i i = 0

2d

Tc (=2.25 J) FM PM

hσx

i i > 0

1d

h T

hc (=J)

FM PM QC

hσx

i i > 0

hσx

i i = 0

hσx

i i = 0

H = J X

(ij) n.n.

σx

i σx i h

X

i

σz

i

✓ ◆ ✓ ◆

Equilibrium phase diagram:

hσx

i i = 0

SLIDE 7

Non-equilibrium relaxation with heat bath

Quench from T=0, h=0 to T>0, h>0; heat bath dynamics, thermalizaIon è ρ~e-H/T

1d

h T

hc

log m(t) t ~ e-t/τ

è

2d

h T

hc

Tc

t 1 m(t)

~mrem + ce-t/τ è

SLIDE 8

Non-equilibrium dynamics in closed system

Quench from h=0 to h>0; Schrödinger dynamics (conserved energy E) è Thermaliza&on ?? Temperature T(E) ?

1d

h T

hc

log m(t) t ~ e-t/τ

è ? ? ? if thermalizing ?

2d

h T

hc

Tc

t 1 m(t)

~mrem + ce-t/τ è ? ? ? if thermalizing ?

SLIDE 9

Quantum Quench in the 1d TIM (at T=0)

h T

hc

log m(t) t

~ e-t/τ(h)

è

L=∞

Quench: |ψ(t=0> GS of H with e.g. h=0, dynamics for H with h>0 (exactly calculable)

H =

L

X

i=1

(Jσx

i σx i+1 + hσz i )

( X

X = X

p

εpη+

p ηp

|ψ(t)i = e−itH|ψ0i

ηp Fermion operators

mi(t) = hψ(t)|σx

i |ψ(t)i / exp(t/τ(h))

1/τ(h) = 2 π Z π dp vp · fp(h0, h) ( with

vp = ∂εp ∂p

Does the exponenIal relaxaIon mean that the system is thermalized? (no, because fp ≠ e-βε(p))

and

fp(h, h0) = hψ0|η+

p (h)ηp(h)|ψ0i

[Rieger, Iglói 2011] [Calabrese, Essler, Fagon 2012]

SLIDE 10

0→0.5 ~const

Exponential relaxation
Quasi-stationary regime
Exponential recovery

Look at finite system – what‘s going on?

ml(t) = <σl(t)> magne&za&on at site l

SLIDE 11

Kinks = non-interacting fermions are created in pairs (+p,-p) move with velocity ±vp

E.g. C(r1t1;r2t2) = <σr1(t1) σr2(t2)>: Reflection at the boundaries at i=0 and i=L!

Finite system: … and will flip spins upon arrival!

Quasi-particles = kinks (in FM phase: h<J)

SLIDE 12

log ml(t)

exact semi classical

0→0.2 0→0.1, L=256

log ml(t)

Period: Tperiod = L/vmax ~ L/h

1d TIM does not thermalize

QP occupation probability: thermal True fp QPs non-interacIng, è fp conserved è no thermalizaIon towards fp~e-βε(p)

SLIDE 13

The 2d transverse Ising model: equil. & quenches

ˆ H = J 2 X

<R,R0>

ˆ x

Rˆ

x

R0 h

2 X

R

ˆ z

R

ˆ µx = 1 N X

R

ˆ x

R .

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

T/J h/J

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

T/J h/J

−1.8 −1.6 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4

Equilibrium phase diagram

<μx> <H>

(Ji; hi) ! (Jf; hf)

h Ef = P

λ Ef,λ| hΨi,0|Ψf,λi |2

energy in the system after th

Eexc = Ef Ef,0 ,

h ˆ Hfi

Teff CGE = Ef

Quenches: Final energy: Excess energy: Effec&ve temperature: Interac&on quenches: (0,h) -> (J,h)

T/J h/J

1 2 3 4 1 2 3 4 5 6 7 8 9 10

L = 4
L = 8
L = 16
L = 32

T/J h/J

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5

ferromagnetic paramagnetic

|Ψi,0i = |"" . . . ""iz = 1 p 2N X

x

|xi

field quenches: (J,0) -> (J,h)

5 2 0 2

T/J h/J

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5

ferromagnetic paramagnetic |Ψi,0i = 1 p 2 n |"" . . . ""ix + |## . . . ##ix

SLIDE 14

Time evolution

|Ψ(t = 0)i = |Ψi,0i

|Ψi,0i = |"" . . . ""iz = 1 p 2N X

x

|xi

|Ψi,0i = 1 p 2 n |"" . . . ""ix + |## . . . ##ix

|Ψ(t)i = eı ˆ

Hft |Ψ(t = 0)i

O h ˆ Oit = X

λ

|cf,λ|2Oλλ + X

λ6=λ0

c⇤

f,λcf,λ0eı(Ef,λEf,λ0)tOλλ0

(15)

lim

T !1

1 T Z T dt h ˆ Oit = X

λ

|cf,λ|2Oλλ .

h ˆ Oidiag = Tr[ ˆ O ˆ ρdiag]

h ˆ ρdiag = P

λ pf,λ |Ψf,λihΨf,λ| an

r the distributions of the (pos

d pf,λ = |cf,λ|2.

SLIDE 15

Techniques for non-integrable systems

|ψ(t)i = e−itH|ψ0i

2d TFIM Non-integrable (in parIcular not a free fermion model as in 1d) ! L x L square lance, p.b.c. What can one do to study the (n.eq.) dynamics ? 1) Mean field theory (or truncated hierarchy of correlaIons) 2) Exact diaginalizaIon of snall systems (up L=4 or 5) 3) Time series expansion 4) PerturbaIon theory (e.g. in h) 5) Time dependent variaIonal calculaIons 6) Real Ime Quantum Monte Carlo 7) Quantum Boltzmann equaIon (?) 8) ... ˆ H = J 2 X

<R,R0>

ˆ x

Rˆ

x

R0 h

2 X

R

ˆ z

R

SLIDE 16

Real Time Dependent Variational Ansatz Principle:

Choose set of „physically relevant“ subspace V = span{|Φ1>,..., |ΦM>}, M<<2N
IniIal state

, Ansatz:

At each Ime t minimize distance between

and

ResulIng differenIal equaIon for variaIonal parameters αk(t):
Solve and calculate observales

|ψ0i =

M

X

k=1

αk|φki X |ψ(t)i =

M

X

k=1

αk(t)|φki

X i ∂ ∂t X

k

αk(t)|φki X

X H X

k

αk(t)|φki

i ∂ ∂tαk(t) = X

k0

αk0(t)hφk|H|φk0i

reads in the x-basis D(t) = X

x

˙

Ψexact(x, t) ˙ Ψvar(x, t)

2 .

SLIDE 17

Interaction quenches (paramagnetic phase)

|Ψ(t)i = exp ⇣ X

r

αr(t) ˆ Cxx

r

⌘ |"" . . . ""iz

ˆ Cxx

r

= 1 Nr X

R

ˆ σx

Rˆ

σx

R+r

X

r0

hδ ˆ Cxx

r δ ˆ

Cxx

r0 it ˙

αr0(t) = ı hElocal

f

(t)δ ˆ Cxx

r it

th δ ˆ O = ˆ O h ˆ Oit ˆ

gy Elocal

f

(x, t) =

with δO = O hOit hx| ˆ Hf|Ψ(t)i / hx|Ψ(t)i. values h ˆ Oit = P

x |Ψ(x, t)|2O(x)

P

x |Ψ(x, t)|2

αr(t = 0) = 0 . Jastrow Ansatz as variaIonal funcIons EquaIons of MoIon for αr ExpectaIon values IniIal condiIon CalculaIon via Monte Carlo | i | i A (x ! x0, t) = min ⇥ 1, Q (x ! x0, t) ⇤

Q (x ! x0, t) = exp n 2 X

r

αR

r (t)

Cxx

r (x0) Cxx r (x)

(A5)

c.f. Carleo etal PRA 2014

SLIDE 18

Field quenches (ferromagnetic phase)

|Ψ(t)i = X

m,n

αm,n(t) |Ψm,ni

|Ψm,ni = 1 pNm,n

Nm,n

X

k=1

|Ψk

m,ni

Symmetric superposiIon of all states with n spins up and m kinks

L 4 8 12 16 number of αm,n 45 848 4551 14834

ı ˙ αm,n(t) = J (N n) αm,n(t) h X

h

2 X

m0,n0

tm0,n0;m,n αm0,n0(t)

h tm0,n0;m,n = Tm0,n0;m,n/pNm0,n0 Nm,n. T er m0 and n0 runs over all subspaces th

sitions bet Tm,n;m0,n0. T = number of transiIons between Hn,m and Hn‘,m‘ via 1 spin flip. CalculaIon with rare event sampling Monte Carlo

Eq. Of

moIon:

αm,n(t = 0) = (

1 √ 2

if (m, n) = (0, 0) or (N, 0) else

IniIal valuse:

SLIDE 19

Observables

h ˆ Oit = 1 ∆t Z t0+∆t

t0

dt Tr[ ˆ O ˆ ρ(t)]

pt(Oj) = 1 ∆t Z t0+∆t

t0

dt Tr[δ(Oj ˆ O) ˆ ρ(t)]

h ˆ Oi

Teff CGE =

1 ZTeff

CGE

Tr[ ˆ Oe− ˆ

H/Teff]

Time average: Thermal average:

Z th ˆ ρ(t) = |Ψ(t)ihΨ(t)| an ˆ

pTeff

CGE(Oj) =

1 ZCGE Tr[δ(Oj ˆ O)e− ˆ

H/Teff] .

Calculated with conInuous imaginary Ime cluster Monte Carlo

SLIDE 20

Comparison (4x4): thermal, rt-VMC, exact

Interaction quenches (0; h) → (J; h)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

(0; 10) ! (1; 10)

µx

0.00 0.05 0.10 0.15 0.20 1 0.750.50.25 0 0.25 0.5 0.75 1

(0; 10) ! (1; 10)

pTeff

CGE(µx)

pt(µx) rt-VMC
pt(µx) ED

ˆx

0.00 0.02 0.04 0.06 0.08 0.10 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

(0; 7.5) ! (1; 7.5)

µx

0.00 0.05 0.10 0.15 0.20 1 0.750.50.25 0 0.25 0.5 0.75 1

(0; 7.5) ! (1; 7.5)

0.00 0.05 0.10 0.15 0.20 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

(0; 5) ! (1; 5)

µx

0.00 0.05 0.10 0.15 0.20 1 0.750.50.25 0 0.25 0.5 0.75 1

(0; 5) ! (1; 5)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

(0; 4) ! (1; 4)

µx

0.00 0.05 0.10 0.15 0.20 1 0.750.50.25 0 0.25 0.5 0.75 1

(0; 4) ! (1; 4)

0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

(0; 3.5) ! (1; 3.5)

µx

0.00 0.05 0.10 0.15 0.20 1 0.750.50.25 0 0.25 0.5 0.75 1

(0; 3.5) ! (1; 3.5)

InteracIon quench (PM)

Field quenches (J; 0) → (J; h)

1 0.990 0.992 0.994 0.996 0.998 1.000 5 10 15 20 25

hˆ ˜ µxit t

(1; 0) ! (1; 0.25)

µx

0.0 0.1 0.2 0.3 0.4 0.5 1 0.750.50.25 0 0.25 0.5 0.75 1

(1; 0) ! (1; 0.25)

1 0.95 0.96 0.97 0.98 0.99 1.00 5 10 15 20 25

hˆ ˜ µxit t

(1; 0) ! (1; 0.5)

µx

0.0 0.1 0.2 0.3 0.4 0.5 1 0.750.50.25 0 0.25 0.5 0.75 1

(1; 0) ! (1; 0.5)

1 0.90 0.92 0.94 0.96 0.98 1.00 5 10 15 20 25

hˆ ˜ µxit t

(1; 0) ! (1; 0.75)

µx

0.0 0.1 0.2 0.3 0.4 0.5 1 0.750.50.25 0 0.25 0.5 0.75 1

(1; 0) ! (1; 0.75)

1 0.80 0.84 0.88 0.92 0.96 1.00 5 10 15 20 25

hˆ ˜ µxit t

(1; 0) ! (1; 1)

µx

0.0 0.1 0.2 0.3 0.4 0.5 1 0.750.50.25 0 0.25 0.5 0.75 1

(1; 0) ! (1; 1)

1 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

hˆ ˜ µxit t

(1; 0) ! (1; 1.25)

µx

0.0 0.1 0.2 0.3 0.4 0.5 1 0.750.50.25 0 0.25 0.5 0.75 1

(1; 0) ! (1; 1.25)

Field quench (FM)

<μx> P(<μx>)

[Blass, Rieger: arXiv 2016]

SLIDE 21

Comparison (4x4) - magnetization

Interaction quenches Field quenches (0; h) → (J; h) (J; 0) → (J; h)

0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit rt-VMC

hˆ

˜ µxit ED hc/J 0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit rt-VMC

hˆ

˜ µxit ED hc/2J 0.15 0.10 0.05 0.00 3 4 5 6 7 8 9 10

Relative error h/J

0.005 0.000 0.005 0.010 0.015 0.020 0.25 0.5 0.75 1 1.25 1.5

Relative error h/J

SLIDE 22

Comparison time (rt-VMC) / thermal average (L=8,12,16)

Interaction quenches (0; h) → (J; h) ˜ µx Cxx

nn

L = 8

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 2 3 4 5 6 7 8 9 10

h ˆ Cxx

nnit

t

0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/J 0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/J

L = 12

0.00 0.02 0.04 0.06 0.08 0.10 0.12 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 2 3 4 5 6 7 8 9 10

h ˆ Cxx

nnit

t

0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/J 0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/J

L = 16

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1 2 3 4 5 6 7 8 9 10

hˆ ˜ µxit t

0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 2 3 4 5 6 7 8 9 10

h ˆ Cxx

nnit

t

0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/J 0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/J

InteracIon quench (PM)

Field quenches (J; 0) → (J; h) ˜ µx Cxx

nn

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

hˆ ˜ µxit t

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

h ˆ Cxx

nnit

t

0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/2J 0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/2J 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

hˆ ˜ µxit t

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

h ˆ Cxx

nnit

t

0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/2J 0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/2J 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

hˆ ˜ µxit t

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 5 10 15 20 25

h ˆ Cxx

nnit

t

0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

hˆ

˜ µxi

Teff CGE

hˆ

˜ µxit hc/2J 0.5 0.6 0.7 0.8 0.9 1.0 0.25 0.5 0.75 1 1.25 1.5 1.75

h/J

h ˆ

Cxx

nni Teff CGE

h ˆ

Cxx

nnit

hc/2J

Field quench (FM)

[Blass, Rieger: arXiv 2016]

SLIDE 23

Deviations time - thermal: finite size effects

∆(ˆ ˜ µ) 1/N

1 10−3 10−2 10−1 256−1 144−1 64−1 16−1

∆( ˆ Cxx

nn)

1/N

1 10−3 10−2 10−1 256−1 144−1 64−1 16−1

(a) Interaction quenches (0,h) → (J,h)

i. Magnetization
ii. Correlation function

∆(ˆ ˜ µ) 1/N

1 10−5 10−4 10−3 10−2 10−1 256−1 144−1 64−1 16−1

∆( ˆ Cxx

nn)

1/N

1 10−5 10−4 10−3 10−2 10−1 256−1 144−1 64−1 16−1

(b) Field quenches (J,0) → (J,h)

i. Magnetization
ii. Correlation function

SLIDE 24

Time / thermal – increasing quench strength (L=16)

Interaction quenches (0; h) → (J; h) ˜ µx Cxx

nn

L = 16

µx

0.00 0.01 0.02 0.03 0.04 0.05 −0.5 −0.25 0.25 0.5

(0; 10) → (1; 10)

pTeff

CGE(µx)

pt(µx)

εx

0.000 0.025 0.050 0.075 0.100 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

(0; 10) → (1; 10)

pTeff

CGE(εx)

pt(εx)

µx

0.00 0.01 0.02 0.03 0.04 0.05 −0.5 −0.25 0.25 0.5

(0; 7.5) → (1; 7.5)

pTeff

CGE(µx)

pt(µx)

εx

0.000 0.025 0.050 0.075 0.100 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

(0; 7.5) → (1; 7.5)

pTeff

CGE(εx)

pt(εx)

µx

0.00 0.01 0.02 0.03 0.04 0.05 −0.5 −0.25 0.25 0.5

(0; 5) → (1; 5)

pTeff

CGE(µx)

pt(µx)

εx

0.000 0.025 0.050 0.075 0.100 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

(0; 5) → (1; 5)

pTeff

CGE(εx)

pt(εx)

µx

0.00 0.01 0.02 0.03 0.04 0.05 −0.5 −0.25 0.25 0.5

(0; 4) → (1; 4)

pTeff

CGE(µx)

pt(µx)

εx

0.000 0.025 0.050 0.075 0.100 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

(0; 4) → (1; 4)

pTeff

CGE(εx)

pt(εx)

µx

0.00 0.01 0.02 0.03 0.04 0.05 −0.5 −0.25 0.25 0.5

(0; 3.5) → (1; 3.5)

pTeff

CGE(µx)

pt(µx)

εx

0.000 0.025 0.050 0.075 0.100 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

(0; 3.5) → (1; 3.5)

pTeff

CGE(εx)

pt(εx)

InteracIon quench (PM)

Field quenches (J; 0) → (J; h) ˜ µx Cxx

nn

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 −1 −0.98 −0.96 0.96 0.98 1

µx

(1; 0) → (1; 0.25)

pTeff

CGE(µx)

pt(µx)

εx

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.92 0.94 0.96 0.98 1

(1; 0) → (1; 0.25)

pTeff

CGE(εx)

pt(εx)

0.000 0.025 0.050 0.075 0.100 0.125 0.150 −1 −0.96 −0.92 0.92 0.96 1

µx

(1; 0) → (1; 0.5)

pTeff

CGE(µx)

pt(µx)

εx

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.84 0.88 0.92 0.96 1

(1; 0) → (1; 0.5)

pTeff

CGE(εx)

pt(εx)

0.000 0.025 0.050 0.075 0.100 −1 −0.92 −0.84 0.84 0.92 1

µx

(1; 0) → (1; 0.75)

pTeff

CGE(µx)

pt(µx)

εx

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.75 0.8 0.85 0.9 0.95 1

(1; 0) → (1; 0.75)

pTeff

CGE(εx)

pt(εx)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 −1 −0.84 −0.68 0.68 0.84 1

µx

(1; 0) → (1; 1)

pTeff

CGE(µx)

pt(µx)

εx

0.00 0.02 0.04 0.06 0.08 0.6 0.7 0.8 0.9 1

(1; 0) → (1; 1)

pTeff

CGE(εx)

pt(εx)

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 −1 −0.75 −0.50 0.50 0.75 1

µx

(1; 0) → (1; 1.25)

pTeff

CGE(µx)

pt(µx)

εx

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.4 0.5 0.6 0.7 0.8 0.9 1

(1; 0) → (1; 1.25)

pTeff

CGE(εx)

pt(εx)

Field quench (FM)

[Blass, Rieger: arXiv 2016]

SLIDE 25

Correlation functions

Interaction quenches Field quenches (0; h) → (J; h) (J; 0) → (J; h) L = 4 L = 12 L = 4 L = 12

d

0.00 0.01 0.02 0.03 0.04 0.05 0.06 1 2 3 4

(0; 10) ! (1; 10)

h ˆ

Cxx

r i Teff CGE

h ˆ

Cxx

r it rt-VMC

h ˆ

Cxx

r it ED

d

0.00 0.01 0.02 0.03 0.04 0.05 0.06 1 2 3 4 5 6 7 8 9 10 11 12

(0; 10) ! (1; 10)

d

0.9916 0.9918 0.9920 0.9922 1 2 3 4

(1; 0) ! (1; 0.25)

d

0.9918 0.9919 0.9920 0.9921 0.9922 0.9923 1 2 3 4 5 6 7 8 9 10 11 12

(1; 0) ! (1; 0.25)

d

0.00 0.02 0.04 0.06 0.08 1 2 3 4

(0; 7.5) ! (1; 7.5)

d

0.00 0.02 0.04 0.06 0.08 1 2 3 4 5 6 7 8 9 10 11 12

(0; 7.5) ! (1; 7.5)

d

0.966 0.967 0.968 0.969 1 2 3 4

(1; 0) ! (1; 0.5)

d

0.965 0.966 0.967 0.968 0.969 1 2 3 4 5 6 7 8 9 10 11 12

(1; 0) ! (1; 0.5)

d

0.00 0.05 0.10 0.15 1 2 3 4

(0; 5) ! (1; 5)

d

0.00 0.05 0.10 0.15 1 2 3 4 5 6 7 8 9 10 11 12

(0; 5) ! (1; 5)

d

0.915 0.920 0.925 0.930 1 2 3 4

(1; 0) ! (1; 0.75)

d

0.910 0.915 0.920 0.925 0.930 1 2 3 4 5 6 7 8 9 10 11 12

(1; 0) ! (1; 0.75)

d

0.05 0.10 0.15 0.20 1 2 3 4

(0; 4) ! (1; 4)

d

0.00 0.05 0.10 0.15 0.20 1 2 3 4 5 6 7 8 9 10 11 12

(0; 4) ! (1; 4)

d

0.835 0.840 0.845 0.850 0.855 0.860 0.865 0.870 1 2 3 4

(1; 0) ! (1; 1)

d

0.830 0.840 0.850 0.860 0.870 0.880 1 2 3 4 5 6 7 8 9 10 11 12

(1; 0) ! (1; 1)

d

0.05 0.10 0.15 0.20 0.25 1 2 3 4

(0; 3.5) ! (1; 3.5)

d

0.00 0.05 0.10 0.15 0.20 0.25 1 2 3 4 5 6 7 8 9 10 11 12

(0; 3.5) ! (1; 3.5)

d

0.70 0.72 0.74 0.76 0.78 0.80 1 2 3 4

(1; 0) ! (1; 1.25)

d

0.70 0.72 0.74 0.76 0.78 0.80 0.82 1 2 3 4 5 6 7 8 9 10 11 12

(1; 0) ! (1; 1.25)

[Blass, Rieger: arXiv 2016]

SLIDE 26

Conclusion

Quantum relaxaIon aJer field quench in 1d TIM:
No thermalizaIon, quasi parIcles (kinks) do not interact,
fp conserved, reconstrucIon of magneIzaIon in finite systems
Quantum relaxaIon aJer quenches in 2d TIM
Time dependent variaIonal calculaIon (rt-VMC):
Comparison of Ime averages with thermal expectaIon values
Good agreement for interacIon quenches (in the PM phase)
Absence of thermalizaIon for field quenches (in the FM phase)
MagneIc correlaIons do not decay in the FM phase
Note: FM phase is gapless (-> lack of clustering property?)