Slow heating, effective Hamiltonians, prethermalization€¦ · Dynamics is described by a quasi-conserved effective Hamiltonian . Driven many-body systems: idea of the approach 1)“Gauge

Slow heating, effective Hamiltonians, prethermalization

Kapitza pendulum

Rapidly oscillating suspension point: “inverted” pendulum

Periodic driving à new states, not possible in static systems

Realization of topological states by driving

Driving à topological Bloch bands Bands with non-zero Chern number (recently measured) Many theory works: Floquet topological insulators, fractional Chern insulators, SPTs Oka, Aoki’09, Lindner, Refael, Galitski’10, Kitagawa, Rudner, Berg, Demler’10, Caysoll, Moessner’12,

Rudner et al’13, Neupert, Grushin’14, many others

Chern number

BUT: Theory mostly limited to single-particle physics This talk: Driven many-body systems

Jotzu et al’14, Aidelsburger et al’14

Driven many-body systems

NO! Driven MBL systems do not heat up (at high enough driving frequency) Is this the fate of all driven systems?

-Many-body systems heat up to infinite T -- a challenge for “Floquet engineering”

How rapid is heating in driven lattice ergodic systems? -At fast driving, heating is exponentially slow (theorem for spins and fermions)

Floquet operator F = T exp− i H (t)dt0

T

∫

H (t +T ) = H (t)

F = exp[−iHeffT ]Conventional approach: (low order) Magnus expansion in

Conventional approach: Magnus expansion

T = 1ν

Floquet operator

Magnus expansion à effective, time-independent Hamiltonian

F = T exp− i H (t)dt0

T

∫

H (t +T ) = H (t)

F = exp[−iHeffT ]

Heff = Heff(0) +Heff

(1) +Heff(2) +...

Heff(0) =

1T

H (t)dt0

T

∫ Heff(1) =

12T

dt1 dt2[H (t1),H (t2 )]0

t1

∫0

T

∫ ,.....

Conventional approach: (low order) Magnus expansion in


T = 1ν

Floquet operator

Magnus expansion à effective, time-independent Hamiltonian BUT: known to converge only for bounded Hamiltonians PROBLEM: BREAKS DOWN IN MANY-BODY SYSTEMS


T

∫

H (t +T ) = H (t)

F = exp[−iHeffT ]


(1) +Heff(2) +...

Heff(0) =

1T

H (t)dt0

T

∫ Heff(1) =

12T

dt1 dt2[H (t1),H (t2 )]0

t1

∫0

T

∫ ,.....

H (t) T < π



T = 1ν

Floquet operator

Magnus expansion à effective, time-independent Hamiltonian BUT: known to converge only for bounded Hamiltonians PROBLEM: BREAKS DOWN IN MANY-BODY SYSTEMS


T

∫

H (t +T ) = H (t)

F = exp[−iHeffT ]


(1) +Heff(2) +...

Heff(0) =

1T

H (t)dt0

T

∫ Heff(1) =

12T

dt1 dt2[H (t1),H (t2 )]0

t1

∫0

T

∫ ,.....

H (t) T < π



T = 1νValidity of Magnus expansion in many-body

systems? Asymptotic expansion? Optimal order of expansion? Rigorous results?

Heating in lattice systems: qualitative considerations H (t) = H + gV cosωtDriven many-body lattice system

ω >> t,UH = t ci+ci+1 + h.c.+U

i∑ nini+1

i∑

ε(k)

k


1 photon absorbed à create e-h pairs

ω >> t,U

N ~ ωt>>1

H = t ci+ci+1 + h.c.+U

i∑ nini+1

i∑

ε(k)

k

ε(k)

k

ω


1 photon absorbed à create e-h pairs Seems difficult, starting from ground state at weak interaction (V is local!)

ω >> t,U

N ~ ωt>>1


i∑ nini+1

i∑

ε(k)

k

ε(k)

k

ω


1 photon absorbed à create e-h pairs Seems difficult, starting from ground state at weak interaction (V is local!)

ω >> t,U

N ~ ωt>>1


i∑ nini+1

i∑

ε(k)

k

ε(k)

k

ω

E.g.: Doublon decay in large-U Hubbard model is slow (exp+perturbation th.)

Strohmaier et al’10; Sensarma et al’10

General bound for the heating rate

Heating rate (golden rule) Γβ (ω)∝ g

2 e−βEη η ' V ηη,η '∑

2δ(ω − (Eη ' −Eη ))

H = hii∑ V = Vi

i∑

Driven many-body lattice system H (t) = H + gV cosωtLocality:


Heating rate (golden rule)

Theorem:

Γβ (ω)∝ g2 e−βEη η ' V ηη,η '∑

2δ(ω − (Eη ' −Eη ))

Γβ (ω) ≤C1 exp −ω

C2 hi

$

%&&

'

())

H = hii∑ V = Vi

i∑

Fundamental reason: locality of quantum dynamics




Theorem:


2δ(ω − (Eη ' −Eη ))


C2 hi

$

%&&

'

())

H = hii∑

EXPONENTIALLY SLOW HEATING AT FAST DRIVING

V = Vii∑





Theorem:


2δ(ω − (Eη ' −Eη ))


C2 hi

$

%&&

'

())

H = hii∑

EXPONENTIALLY SLOW HEATING AT FAST DRIVING General: holds for any initial state and for strong interactions

V = Vii∑



The idea of the proof 1) Consider the case of a local operator V. Rewrite

A(ω) ~ µ V ηη,µ∑

2δ(ω − (Eµ −Eη )) =

µ [V,H ] ηω 2

η,µ∑

2

δ(ω − (Eµ −Eη )) =

=µ [V,H,H,...,H ] η

ω 2kη,µ∑

2

δ(ω − (Eµ −Eη ))

k commutators

2) Use locality of H to bound the commutators

[V,H,H,...,H ] <Ckk!3) Choose optimal 4) Bound: 5) For global driving, use Lieb-Robinson bounds to estimate “cross-terms” involving with large

k* ≈ωCe

A(ω)< e−κω κ =2ce

Vi,Vj | i− j |



Theorem:


2δ(ω − (Eη ' −Eη ))


C2 hi

$

%&&

'

())

H = hii∑

EXPONENTIALLY SLOW HEATING AT FAST DRIVING General: holds for any initial state and for strong interactions

V = Vii∑



DA, Huveneers, De Roeck, arXiv:1507.01474

Result can be made fully non-perturbative (in driving strength and not only in interaction!) à Controlled version of high-frequency expansion

Driven many-body systems: non-perturbative results

Very long heating times What governs dynamics at intermediate times?

τ * ~ eCω

t ≤ τ *



τ * ~ eCω

t ≤ τ *

Dynamics is described by a quasi-conserved effective Hamiltonian

H* = H0 +1ωH1 +

1ω 2 H2 +...+

1ω n Hn n ~ Cω >>1

Conservation breaks down only at exponentially long times t ~ τ *



τ * ~ eCω

t ≤ τ *

H* = H0 +1ωH1 +

1ω 2 H2 +...+

1ω n Hn n ~ Cω >>1


t

Prethermalization

Heating to

0 τ *

T =∞




τ * ~ eCω

t ≤ τ *

H* = H0 +1ωH1 +

1ω 2 H2 +...+

1ω n Hn n ~ Cω >>1


t

Prethermalization

Heating to

0 τ *

T =∞

Applies to: driven systems of spins and fermions with local interactions

Rapidly driven many-body systems show a long prethermalization regime


Driven many-body systems: idea of the approach

ψ(t) = Q̂(t) ϕ(t)1)“Gauge transformation”: Q̂+Q̂ = IQ̂(t +T ) = Q̂(t)

Stroboscopic evolution not affected, but Hamiltonian changed:

H '(t) = Q̂+H (t)Q̂− iQ̂+ dQ̂dt

(1)

Driven many-body systems: idea of the approach

ψ(t) = Q̂(t) ϕ(t)1)“Gauge transformation”: Q̂+Q̂ = IQ̂(t +T ) = Q̂(t)

Stroboscopic evolution not affected, but Hamiltonian changed:

H '(t) = Q̂+H (t)Q̂− iQ̂+ dQ̂dt

2) Goal: choose to minimize the driving term Result: Using locality, can find for that decreases driving term exponentially, by 3) Obtain the quasi-conserved quantity using (1)

Q̂

Q̂ = eTΩ1+T2Ω2+..+T

nΩn

e−Cω

H*

n ~ Cω

(1)

Numerics on finite-size systems

Kick protocol: UF = e−iH0te−iH1t

Err

or, e

xpan

sion

ord

er n

H0 = hi∑ si

ziz + Jsi

zizsi+1

ziz

H1 = Jsizixsi+1

x

i∑t -driving period

-Confirmed that this is an asymptotic expansion

Prosen model ‘97

n* =Cω The bound is saturated

Implications

1) Broad prethermalization regime in driven many-body systems 2) Optimal order of the Magnus expansion 3) “Floquet topological insulators” can be very long-lived 4) Suggests ways of experimental preparation of “Floquet fractional Chern insulators” and other correlated states in driven systems OTHER APPLICATIONS: -Bounds of heating in bosonic lattice systems -Some rigorous results about quasi-adiabatic behavior in Floquet systems

n* ~ Cω

Part II: MBL in periodically driven systems

Kicked rotor: localization in a periodically driven system

U = e−iH0T0e−2iVFloquet operator H0 =12∂2

∂ϑ 2 V = K cosϑ

rotor kick



∂ϑ 2 V = K cosϑ

rotor kick

Mapping onto an effective Anderson model

On-site energy

U ψ = e−iϑ ψ

tan EnT0 −ϑ2

"

#$

%

&' Hopping operator tnm = n tan V̂( ) mtnm

λnλm



∂ϑ 2 V = K cosϑ

rotor kick

Mapping onto an effective Anderson model

On-site energy

U ψ = e−iϑ ψ

tan EnT0 −ϑ2

"

#$

%

&' Hopping operator tnm = n tan V̂( ) mtnm

λnλm

Quasi-random à dynamical localization! Hopping decays with | n−m |

Driven many-body systems: ergodic vs MBL

L spins +local driving V̂ U = e−iH0T0e−2iVMapping onto an effective hopping problem; sites=eigenstates of H0

Driven many-body systems: ergodic vs MBL

L spins +local driving V̂ U = e−iH0T0e−2iVMapping onto an effective hopping problem; sites=eigenstates of H0

ergodic

Hopping

Effective level spacing

Δ ~ 1D~ 12L

tnm ~1D~ 12L/2

MBL

Δ ~ 12L

tnm ~e−L/ξ

2L

H0 H0

(from ETH)

tnm >> ΔNonlocal hopping delocalization

tnm << Δ localization

Periodically driven many-body localized systems

1) Construct Floquet Hamiltonian iteratively (different from Magnus!) Show convergence and MBL at high driving frequency 2) Argue delocalization at low driving frequency

H (t) = HMBL + gV (t) many-body localized HMBL

ω driving frequency


1) Construct Floquet Hamiltonian iteratively (different from Magnus!) Show convergence and MBL at high driving frequency 2) Argue delocalization at low driving frequency 3) Establish phase diagram (fixed disorder, interactions, and g)



ω

Delocalized

MBL

0 ωc


1) Construct Floquet Hamiltonian iteratively (different from Magnus!) Show convergence and MBL at high driving frequency 2) Argue delocalization at low driving frequency 3) Establish phase diagram (fixed disorder, interactions, and g) 4) Driving: probe MBL (little/no heating), access MBL-ergodic transition



ω

Delocalized

MBL

0 ωc

Driven MBL systems: numerical results

1)  Spectral statistics: transition Between Poisson and Wigner-Dyson 2) Area-law for eigenstates in MBL phase 3) Log-growth of entanglement 4) Constructed a set of local integrals of motion

Kicked spin chain

Confirms analytical theory

Iterative procedure and its convergence

i ddtU(t) = H (t)U(t)Evolution operator:

Decompose U(t) = P(t)e−iHeff t P(t +T ) = P(t)

P+(t) H (t)− i ddt

"

#$

%

&'P(t) = Heff

Solve for iteratively, gradually eliminating time-dependent terms Perturbation theory in

P(t)

gν<<1 g2

νW<<1

Quasi-local Floquet Hamiltonian, which itself is MBL

Convergence criteria

Higher orders: combinatorics equivalent to time-independent MBL problem

Low frequency: delocalization via Landau-Zener transitions

Change

Consider instantaneous eigenstates of

H (λ) = HMBL + gV (λ)

λ =tT

Multi-level Landau-Zener problem

Diabatic crossing: not dangerous “Intermediate” crossings: the state get mixed, delocalization

At low frequency, many intermediate/adiabatic crossings à delocalization DA, De Roeck, Huveneers, arXiv:1412.4752 See also: Khemani, Nandkishore, Sondhi’14 (local ramp)

Documents

Slow heating, effective Hamiltonians, prethermalization€¦ · Dynamics is described by a quasi-conserved effective Hamiltonian . Driven many-body systems: idea of the approach 1)“Gauge