Dynamical Stability of a Many-body Kapitza Pendulum

1.6 MB·English

by  Roberta Citro

DynamicalStabilityofaMany-bodyKapitzaPendulum
Roberta Citro,1 Emanuele G. Dalla Torre,2,3 Luca D’Alessio,4,5 Anatoli
Polkovnikov,5 Mehrtash Babadi,2,6 Takashi Oka,7 and Eugene Demler2
1Dipartimento di Fisica “E. R. Caianiello” and Spin-CNR,
Universita’ degli Studi di Salerno, Via Giovanni Paolo II, I-84084 Fisciano, Italy
2Department of Physics, Harvard University, Cambridge, MA 02138, USA
3Department of Physics, Bar Ilan University, Ramat Gan 5290002, Israel
4DepartmentofPhysics, ThePennsylvaniaStateUniversity, UniversityPark, PA16802, USA
5Department of Physics, Boston University, Boston, MA 02215, USA
6InstituteforQuantumInformationandMatter,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
7Department of Applied Physics, University of Tokyo, Tokyo, 113-8656 Japan
(Dated:January26,2015)
5 We consider a many-body generalization of the Kapitza pendulum: the periodically-driven sine-Gordon
1 model. Weshowthatthisinteractingsystemisdynamicallystabletoperiodicdriveswithfinitefrequencyand
0 amplitude. Thisfindingisincontrasttothecommonbeliefthatperiodically-drivenunboundedinteractingsys-
2 temsshouldalwaystendtoanabsorbinginfinite-temperaturestate.Thetransitiontoanunstableabsorbingstate
isdescribedbyachangeinthesignofthekinetictermintheeffectiveFloquetHamiltonianandcontrolledby
n
theshort-wavelengthdegreesoffreedom. Weinvestigatethestabilityphasediagramthroughananalytichigh-
a
J frequencyexpansion,aself-consistentvariationalapproach,andanumericsemiclassicalcalculations.Classical
andquantumexperimentsareproposedtoverifythevalidityofourresults.
2
2
PACSnumbers:67.85.-d,03.75.Kk,03.75.Lm

]
h
p I. INTRODUCTION MBL systems remain localized and no not thermalize. This
- isincontrasttoergodicsystems, whichareexpectedtother-
t
n Motivated by advances in ultra-cold atoms1–5, the stabil- malizetoaninfinitetemperatureforanyperiodicdrive6,9,10,15.
a Thesefindingsareinapparentcontradictiontoearliernumer-
ity of periodically-driven many-body systems is the subject
u ical studies7,19–21, who found indications of a finite stability
q ofseveralrecentstudies6–14. Accordingtothesecondlawof
thresholdinnon-integrablesystemsaswell.
[ thermodynamics, isolated equilibrium systems can only in-
To investigate this problem in a systematic way, we con-
crease their energy when undergoing a cyclic process. For
1 sider here a many-body analog of the Kapitza pendulum:
many-body interacting ergodic systems, it is often assumed
v the periodically-driven sine-Gordon model. This model
0 thattheywillheatmonotonously,asymptoticallyapproaching
is well suited for analytical treatments, including a high-
6 aninfinite-temperaturestate8–10,15. Incontrast,forsmallsys-
frequency expansion, quadratic variational approaches, and
6 temssuchasasingletwo-levelsystem(spin), thermalization
5 is not expected to occur and periodic alternations of heating renormalization-group methods. Unlike previously-studied
0 spin systems, the present model has an unbounded single-
andcooling(Rabioscillations)arepredicted. Aharmonicos-
. particle energy spectrum22. Thanks to this property, infinite-
1 cillatorcandisplayatransitionbetweenthesetwobehaviors,
temperatureensemblesarecharacterizedbyaninfiniteenergy
0 knownas“parametricresonance”16: dependingontheampli-
5 tude and frequency of the periodic drive, the oscillation am- densityandareeasilyidentified. Weshowtheemergenceofa
1 sharp“parametricresonance”,separatingtheabsorbing(infi-
plitudeeitherincreasesindefinitely,ordisplaysperiodicoscil-
: nitetemperature)fromthenon-absorbing(periodic)regimes.
v lations. Aninterestingquestionregardshowmuchofthisrich
Thistransitionsurvivesinthethermodynamiclimitandleads
i dynamicsremainswhenmany-degreesoffreedomareconsid-
X to a non-analytic behavior of the physical observables in the
ered.
r long time limit, as a function of the driving strength and/or
a This question was addressed for example by Russomanno
frequency. Weconjecturethatthistransitioncorrespondstoa
etal.6, whostudiedthetimeevolutionofthetransverse-field
mean-field critical point of the many-body Floquet Hamilto-
Ising (TI) model. This model is integrable and never flows
nian. Ourfindingenrichestheunderstandingofthecoherent
to an infinite-temperature state. In the translational-invariant
dynamicsofparametricallyforcedsystemandpavestheroad
case,thisresultcanberationalizedbynotingthattheTImodel
toward the search of unconventional dynamical behavior of
isintegrableandcanbemappedtoanensembleofdecoupled
closedmany-bodysystems.
two-level systems (spin-waves with a well defined wavevec-
tor), each of whom periodically oscillates in time and never
equilibrates. In this sense, the findings of Ref.s [9,11,12,14]
on periodically-driven disordered systems subject to a local II. REVIEWOFASINGLEKAPITZAPENDULUM
driving falls into the same category: many-body localized
(MBL)systemsareeffectivelyintegrablebecausetheycanbe Before entering the domain of many-body physics, we
describedasdecoupledlocaldegreesoffreedomaswell17,18. briefly review the (well understood) case of a classical sin-
Whenthedrivingfrequencyishigherthanagiventhreshold, gle degree of freedom. We consider the Hamiltonian of a
2
instabilityalternateforγ <γ .Inparticular,thesystemissta-
c
blefor g /γ2 > 0.25 andsmalldriving amplitudes. Inwhat
0
follows we will refer to these two stability regions as “large
drivingfrequency”and”largeg ”respectively. Asasidere-
0
mark, we note that this quadratic approximation is valid for
thequantumcaseaswell,providedthattheinitialstateisclose
tothe|φ = 0(cid:104)state. Inthiscase,theresultingstabilityphase
diagramisexpectedtobethesame.
SubsequentnumericalstudiesoftheclassicalHamiltonian
(1)leadtothestabilitydiagramreproducedinFig.128. Inthis
FIG.1:StabilitydiagramoftheclassicalKapitzapendulum,adapted plot,thewhiteareasrepresentregionsintheparameterspace
fromRef.[28].Inthecoloredareasatleastonefixedpointisstable, in which both extrema are unstable and the system flows to-
while in the white areas both minima are unstable and the system wardsaninfinite-temperaturestate,independentlyontheini-
is fvariational methodsully ergodic. The parameters g , g and γ
0 1 tial conditions. In contrast, in the colored regions at least
aredefinedinEq.(1)asg(t) = g +g cos(γt). Thegreenlines
0 1 one of the two extrema is stable, and the system is not er-
correspondtothestabilitythresholdofthefirstparametricresonance,
godic. In this case the pendulum can be confined to move
Eq.(2)
close to one of the extrema and will not in general reach
a steady state described by an effective infinite temperature.
Thestabilityofthelowerfixedpoint(greenlineinFig.1)is
periodically-drivensimple(non-linear)pendulum,alsoknown
asKapitzapendulum23,anddescribedbytheHamiltonian wellapproximatedbytheboundariesofEq.(2). Inparticular,
for g = 0 (where the upper and lower extrema are math-
0
1 ematically equivalent), the system is dynamically stable for
H(t)= p2−g(t)cos(φ), with g(t)=g +g cos(γt).
2 0 1 g1 < gc ≈ 0.45γ2, and dynamically unstable for g1 > gc.
(1) Asamainresultofthispaperwewillshowthatthistransition
Herepandφarecanonically-conjugatedcoordinatessatisfy- remainssharplydefinedeveninthemany-bodycase.
ing{p,φ}=−i,where{·,·}arePoissonbrackets.Forg =0
1
thesystemdisplaystwoclassicalfixedpoints: astableoneat
φ = 0andanunstableoneatφ = π. (Throughoutthispaper III. MANY-BODYKAPITZAPENDULUM
weassumewithoutlossofgeneralitythatg >0.).
0
In the presence of a periodic drive (g1 (cid:54)= 0), the unsta- Toexplorethefateofthedynamicalinstabilityinamany-
ble fixed point can become dynamically stable. This coun-
body condition we consider an infinite number of coupled
terintuitiveresultwasfirstobtainedbyKapitza23 inthehigh-
identicalKapitzapendula,depictedinFig.2(a). Thissystem
frequencylimitγ2 (cid:29)g0,g1. Byaveragingtheclassicalequa- is described by the periodically-driven Frenkel-Kontorova29
tionsofmotionoverthefastoscillationsofthedrive,Kapitza
model
found that the “upper” extremum φ = π becomes stable for
(cid:20) (cid:21)
lianrggeweonrokuignhitidartievdintgheamfipellidtuodfesvgib12ra>tiogn0aγl2m/2e.chTahniiscps2io4,neaenrd- H =Λ(cid:88) K2 Pi2− K1 cos(φi−φi+1)− gΛ(t2)cos(φi) ,
i
the Kapitza’smethod is usedfor description ofperiodic pro- (3)
cessesinatomicphysics25,plasmaphysics26,andcybernetical where P , φ are unitless variables satisfying {P , φ } =
i i j k
physics27. −iδ ,andg(t)isdefinedinEq.(1). TheenergyscaleΛde-
j,k
Forfinitedrivingfrequenciesγ2 ∼ g0, g1 thelowerfixed terminestherelativeimportanceofthecouplingbetweenthe
point(φ=0)canbecomedynamicallyunstableaswell. This pendula and the forces acting on each individual pendulum:
transitioncanbeanalyticallystudiedforexamplebyapplying in the limit Λ → 0 we expect to recover the case of an iso-
the quadratic approximation cos(φ) → 1−φ2/2 to Eq. (1) lated periodically-driven pendulum. In the continuum limit,
(valid for small g1). The resulting Hamiltonian corresponds themodel(3)canbemappedtotheperiodically-drivensine-
to a periodically-driven harmonic oscillator, with frequency Gordonmodel
√
ω = g , driving frequency γ, and driving amplitude g .
0 0 1 (cid:90) (cid:20)K 1 (cid:21)
Fdiosrplianyfisnipteasraimmaeltrdicrivreinsgonfarnecqeusenactieγs =(g12ω→/0n),=thi2s√sygst/enm, H = dx 2 P2+ 2K(∂xφ)2−g(t)cos(φ) , (4)
0 0
where n is an integer16. For finite driving amplitudes each
P(x) and φ(x) are canonically-conjugate fields,
resonanceextends toafiniteregion ofdrivingfrequencies16:
{P(x),φ(x(cid:48))} = −iδ(x−x(cid:48))andK istheLuttingerparam-
inparticularthefirstparametricresonance(n=1)extendsto
eter(weworkinunitsforwhichthesoundvelocityisu=1).
2g1 ≤(cid:12)(cid:12)γ2−4g0(cid:12)(cid:12) . (2) mThaexipmaarlamalelotewreΛd menotmeresnatusma:nφu(lxtr)av=iol(cid:82)etΛcudtqo/ff(,2sπe)tteiniqgxφthe.
−Λ q
Becausethesubsequentparametricresonancesoccuratlower The model (4) can also be realized using ultracold atoms
frequencies,onefindsthatthepointφ = 0isalwaysdynam- constrained to cigar-shaped traps. In this case, the field
ically stable for driving frequencies that are larger than the φ = φ −φ representthephasedifferencebetweenthetwo
√ 1 2
criticalvalueγ = 4g +2g ,whileregionsofstabilityand condensates,andthetime-dependentdrivecanbeintroduced
c 0 1
3
(a) (cid:82)T
(1/T) dtH(t),hasasimplephysicalinterpretation:when
0
thedrivingfrequencyisinfinite,thesystemperceivesonlythe
time-averagedvalueofH(t). Thisobservationhasbeensuc-
cessfully employed, for example, to engineer optical lattices
with negative tunneling amplitude3–5. In our case, the aver-
ageHamiltonianissimplydescribedbythetime-independent
sine-Gordon model (Eq. (4) with g = 0) and the drive has
1
no effect on the system. The third-order Magnus expansion
delivers:
(b)
(cid:90)
dx
H =H +H(cid:48)+H(cid:48)(cid:48)− [g cos(φ)+g˜cos(2φ)], (6)
eff LL 2π 0
𝜙1 𝑥′′ 𝜙2 𝑥′′
whereH istheLuttingerliquidHamiltonian(Eq.(4)with
LL
g =g =0),
𝜙1 𝑥′ 𝜙2 𝑥′ 1 0
(cid:90)
H(cid:48) =−g(cid:48) dxP2cos(φ), (7)
𝜙1 𝑥 𝜙2 𝑥 (cid:90)
H(cid:48)(cid:48) =−g(cid:48)(cid:48) dx(∂ φ)2cos(φ); (8)
x
FIG. 2: Proposed physical realizations. (a) A one-dimensional ar- g(cid:48) = g1K2, g(cid:48)(cid:48) = g1 and g˜ = Kg1 (cid:0)1g −g (cid:1). See Ap-
γ2 γ2 γ2 4 1 0
rayofKapitzapendula,whicharecoupledandsynchronously-driven pendixAforthedetailsofthisderivation.Eq.(6)isanalogous
realizesthetime-dependentFrenkel-Kontorovamodel(3). (b)Two totheeffectiveHamiltonianofthesingleKapitzapendulum7.
coupledone-dimensionalcondensates,whosetunnelingamplitudeis
The stability of the “upper” extremum φ = π is captured
periodicallydrivenintime,realizethetime-dependentsine-Gordon
by the interplay between g cos(φ) and g˜cos(2φ): this point
model(4). Moredetailsabouttheexperimentalrealizationsarepre- 0
becomes dynamically stable when g < 4g˜, or equivalently
sentedinSec.IX. 0
wheng < g12K,takingintoaccountthattheMagnusexpan-
0 γ2
sionisvalidinthelimit38ofg /γ2 (cid:28)1.
by periodically modulating the transversal confining poten- 0
The term H(cid:48) leads to the dynamical instability of the sys-
tial, as shown in Fig. 2(b). Because the fields P and φ are
tem: having a negative sign, it suppresses the kinetic energy
continuousvariable,theenergydensitiesoftheHamiltonians
∼ P2, eventually leading to an inversion of its sign. Using
(3) and (4) are unbounded from above and allows to easily
a quadratic variational approach, we can approximate H(cid:48) ≈
distinguish an absorbing behavior (in which the energy −4g(cid:48)(cid:82) dx(cid:104)cos(φ)(cid:105)P2. The stability of the φ = 0 minimum
densitygrowsindefinitelyintime)fromaperiodicone. This
canthenbestudiedthrough(cid:104)cos(φ)(cid:105)≈1,leadingtoarenor-
situation differs from previously-considered spin models,
malizedkineticenergyKP2/2−K2(g /γ2)P2 =K P2/2,
whoseenergydensityisgenericallyboundedfromabove. 1 eff
with
(cid:18) (cid:19)
g
K =K 1−2K 1 (9)
IV. INFINITEFREQUENCYEXPANSION eff γ2
Thisexpressionindicatesthatthesystemisdynamicallystable
Tounderstandtheeffectofperiodicdrivesondynamicalin-
forKg /γ2 < 0.5. Forlargerdrivingamplitudes(orsmaller
stabilitiesandlocalization,itisconvenienttodefinetheeffec- 1
drivingfrequencies)thekineticenergybecomesnegativeand
tive Hamiltonian (often termed “Floquet Hamiltonian”) H
eff thesystembecomesunstable.
as
Higher-order terms in the Magnus expansion can
U(T)=e−iHeffT , (5) be used to determine the qualitative dependence of
the critical driving amplitude as a function of Λ/γ
whereU(T)istheevolutionoperatoroveroneperiodoftime and Kg /γ2. For example, the fifth-order term of
1
T = 2π/γ. Inthecaseoftheharmonicoscillator,ithasbeen the Magnus expansion contains terms proportional
shownthattheparametricresonancecanbeeasilyunderstood to g (t)/γ4{(∂ φ)2,{(∂ φ)2,{P2,{P2,cosφ}}}} ∼
itnheteerimgesnomfotdheeseofffeHcteivffeaHreanmoirlmtoanliiazna3b5l:eianntdheansytaibnlietiarelgsitmatee P2∂x2c1os(φ),whixchrenormxalizesg(cid:48)byafactorof8γg122 (cid:16)Λγ(cid:17)2.
can be expanded in this basis, leading to a periodic dynam- This positive contribution leads to a decrease of the critical
ics.Incontrast,intheunstableregimetheeigenmodesofH drivingamplitudeasafunctionofΛ/γ. Asmentionedabove
eff
becomenotnormalizable,inanalogytoequilibriumHamilto- (seeEq.(3)),ΛsetsthecouplingbetweentheKapitzapendula
niansthatarenotboundedfrombelow(suchasH =x2−p2), and is indeed expected to shrink the stability region of the
andthedynamicsbecomesabsorbing. system.
The Magnus expansion36,37 is an analytical tool to de- Since the Magnus expansion generates an infinite number
rive the effective Hamiltonian in the limit of large drive fre- ofterms,onemaysuspectthatthefullseriescouldrenormal-
quency.Thefirst-ordertermoftheMagnusexpansion,H = ize the critical amplitude to zero (making the system always
eff
4
dynamically unstable). To address this point, we now resort malizedbyapproximately(g˜/Λ)2,whereΛisthetheorycut-
to the quantum version of the problem, where powerful ana- off. This is a strong indication that, for any finite Λ, the ex-
lyticaltechniquesareavailable. tremumatπcanstillbemadestablewithlargeenoughdriving
amplitudes39.
V. RENORMALIZATIONGROUPARGUMENTS
VI. QUADRATICEXPANSION
Let us now consider the quantum version of the Hamilto-
In analogy to the single Kapitza pendulum, the simplest
nian (4). In the absence of a drive (g = 0), this model cor-
1
respondstothecelebratedsine-Gordonmodel30,31. Itsground way to tackle the many-body Hamiltonian (4) is to expand
the cosine term to quadratic order. In particular, to analyze
state displays a quantum phase transition of the Kosterlitz-
the stability of the φ = 0 minimum we can use cos(φ) →
Thoulesstype: forK > K = 8π−o(g/Λ)thecosineterm
c
1−φ2/2. Inthisapproximationthesystembecomesequiva-
isirrelevantandthemodelsupportsgaplessexcitations,while
lenttoasetofdecoupledharmonicoscillatorswithHamilto-
for K < K the system has a finite excitation gap ∆. The
c (cid:80)
nianH = H ,where
presence of a gap is known to have significant effects on the q q
responseofthesystemtolow-frequencymodes,byexponen-
tiallysuppressingtheenergyabsorption. Incontrast,atlarge H = KP2+ 1 (cid:0)q2+Kg +Kg cos(γt)(cid:1)φ2 . (10)
drivingfrequenciestheexcitationgapisexpectedtohavelittle q 2 q 2K 0 1 q
effect. Eq. (10) corresponds to the Hamiltonian of a periodically
To analyze the limit of large driving frequencies, we pro- driven harmonic oscillator, whose stability diagram is well
pose to consider the Floquet Hamiltonian, as defined by the known. In the limit of g → 0, the system is dynamically
1
Magnus expansion64. Specifically, we consider the ground- stable as long as γ > 2max[ω ] = 2max[(cid:112)q2+Kg ] =
q 0
state properties of this Hamiltonian, which can be conve- (cid:112) (cid:112)
2 Λ2+Kg , or γ < 2min[ω ] = 2min[ q2+Kg ] =
niently studied through the renormalization group (RG) ap- √ 0 q 0
2 Kg . InanalogytoEq.(2),forfiniteg thestabilitycondi-
proach. The existence of a well-defined ground state for the 0 1
tionismodifiedto
FloquetHamiltonianimplicitlydemonstratestheergodicityof
thesystem: unstablesystemsaregenericallyexpectedtohave 2Kg <max(cid:2)γ2−4(cid:0)Kg +Λ2(cid:1),4(Kg )−γ2(cid:3) (11)
1 0 0
non-normalizableeigenstates35.
In our case, the first-order term of the Magnus expansion The resulting dynamical phase diagram is plotted in Fig. 3.
corresponds to the well-known sine-Gordon model. When In the limit of Λ → 0 we recover the stability diagram of
K >8πthisHamiltonianflowsunderRGtowardstheLut- the lower minimum of an isolated Kapitza pendulum: this is
eff
tinger liquid theory (Eq.(4) with g(t) = 0). Higher order demonstratedbythequantitativeagreementbetweenthegreen
termsaregivenbycommutatorsoftheHamiltonianatdiffer- curvesofFig.1andFig.3.
enttimes(seeAppendixA).Becausethetime-dependentpart
ofourHamiltonianisproportionaltocos(φ),eachcommuta-
tornecessarilyincludescos(φ),oritsderivatives.Withrespect
totheLuttinger-liquidfixedpoint,thesetermsareirrelevantin
anRGsense,andarenotexpectedtoaffecttheground-state-
properties of the Floquet Hamiltonian. If this is indeed the
case,the(asymptotic)expectationvaluesofphysicaloperators
suchas|φ |2and|P |2arefinite(andproportionaltoK /|q|
q q eff
and|q|/K respectively),indicatingthatthesystemdoesnot
eff
always flow to an infinite-temperature ensemble. When the
frequency is reduced, the amplitude of higher-order terms of
theMagnusexpansionincreases:althoughirrelevantinanRG
sense,ifsufficientlylarge,thesetermscanleadtoatransition
towardsanunstableregime.
Many-body quantum fluctuations have an important effect
on the stability of the inverted pendulum φ = π as well. As
mentioned above, this effect is related to the interplay be-
tween cos(φ) and cos(2φ) in Eq. 6. In the ground-state of
thisHamiltonian,quantumfluctuationschangethescalingdi-
mension of an operator cos(αφ) to 2−α2K/4. Thus for fi- FIG. 3: Stability diagram of the periodically-driven sine-Gordon
nite K , the term with g˜is less relevant than the term with model,asobtainedfromthequadraticexpansionofthelowermin-
eff
g fromarenormalizationgroup(RG)pointofview,making imum. Thesystemdisplaystwodistinctstabilityregionsatlargeγ
0
the upper extremum less stable than in the case of a simple andlargeg0,respectively(seetext).Thegreenlinereferstothelimit
pendulum. A simple scaling analysis reveals that the stabil- Λ/γ →0,wheretheproblemmapstothestabilityofthelowermin-
ity boundary between the two extrema at 0 and π is renor- imumoftheKapitzapendulum(greenlineofFig.1).
5
ForfiniteΛweobservetwodistinctstabilityregions,char- For a translation-invariant system, we find G ≡ G ≡
x,y x−y
acterized by a different dependence on the ultraviolet cut- 1/(2π)(cid:82)Λ dkG eik(x−y),whereΛistheUVcutoff,andthe
off Λ. The first stability region is adiabatically connected −Λ k (cid:82) (cid:82)
effectiveactionS ≡ dt dxL isgivenby
to the limit of infinite driving-frequency, γ → ∞ (the ori- eff eff
gin in Fig. 3)). Its boundaries are described by Kg1/2γ2 + (cid:90) Λ dk(cid:16)
Kg /γ2 +Λ2/γ2 = 0.25. For g → 0 and Λ → 0 this ex- S =Z(t)g(t)+ Σ G˙
0 0 cl 2π k k
pressionpreciselycoincideswiththeresultobtainedfromthe −Λ
large-frequencyMagnusexpansion,Kg /γ2 = 0.5. Thesta- 1 k2 (cid:17)
1 − KG−1−2KΣ G Σ − G ,
bility region is strongly suppressed by Λ and eventually dis- 8 k k k k 2K k
appears at Λ/γ ≈ 0.5. This indicates that it is related to the (cid:32) 1(cid:90) Λ dk (cid:33)
stabilityofthedegreesoffreedomattheshortestlengthscales where Z(t)=exp − G . (15)
2 2π k
Λ−1:itscharacteristhereforepredictedtobeanalogoustothe −Λ
stabilityofasingleKapitzapendulum.Incontrast,thesecond
The equations of motion are given by the saddle point of
stability region, at large g is roughly independent on Λ/γ.
0 theeffectiveaction. ByrequiringδS /δG = δS /δΣ = 0
Thisstabilityregionisdeterminedbytheresonantexcitation eff eff
weobtain
ofthelowestfrequencycollectiveexcitationofthesystemin
the static (γ → 0) regime, where the system is dynamically G˙ =4KG Σ , (16a)
k k k
stable due to the presence of a gap, ∆. In the quadratic ap-
√
proximation,thegapisgivenby∆≈ Kg0,andthestability Σ˙ = 1KG−2−2KΣ2 − k2 − 1Z(t)g(t) (16b)
conditionγ <2∆readsKg /γ2 <0.25. k 8 k k 2K 2
0
Thepresentquadraticexpansionbaresacloseresemblance
In the following calculations we assume the system to be
totheanalysisofPielawa32,whoconsideredperiodicmodula-
initiallyfoundinastationarystatesatisfying
tionsofthesoundvelocity(butkeepingg =0). Thevalidity
0
ofthesequadraticapproximationsishoweverunderminedby K 1
theroleofnon-lineartermsinnon-equilibriumsituations(and Gk = 2 (cid:112)k2+∆2, (17)
specifically noisy environments33 and quantum quenches34). 0
Even if irrelevant at equilibrium, non-linear terms enable where∆ isself-consistentlygivenas:
0
thetransferofenergybetweenmodeswithdifferentmomen-
(cid:32) (cid:33)
tum and are therefore necessary to describe thermalization. 1(cid:90) Λ dkK 1
Tcohuepqliunegsteifofnecwtshaircehswufeficwieilnltatdoddreesstsrohyerteheisdywnhaemthiecralminosdtea-- ∆20 =g0K exp −2 −Λ 2π 2 (cid:112)k2+∆20 . (18)
bilitydescribedabove.
Assuming ∆ (cid:28) Λ, the above equation gives ∆2 ≈
0 0
(g K/2)[∆ /(2Λ)]K/4π, which implies a critical point at
0 0
K = 8π (Kosterlitz-Thoulesstransition). Thecosineisrele-
VII. SELF-CONSISTENTVARIATIONALAPPROACH c
vant(irrelevant)forK <K (K >K ).
c c
In this initial gapped phase, the classical oscillation fre-
The above-mentioned quadratic approximation can be im- √
quency, Kg is renormalized by the factor Z due to quan-
proved by considering a generic time-dependent Gaussian 0
tum fluctuations (see Eqs. (15) and (16)). The introduction
wavefunction,alongthelinesofJackiwandKerman47:
of a modulation to the amplitude of the bare cosine poten-
(cid:18) (cid:90) (cid:20)1 (cid:21) (cid:19) tial leads to one of the two following scenarios (i) Unstable
Ψv[φ(x)]=Aexp − φ(x) 4G−x,1y−iΣx,y φ(y) (ergodic)regime: thedrivingfieldamplifiesquantumfluctua-
x,y tions (i.e. leads to “particle generation” via parametric reso-
(12)
nance)andclosesthegap,i.e.Z(t)→0.Oncethegapcloses,
Eq.(12)isaparticularcaseofthegenericwavefunctionpro-
posed by Cooper et al.48, valid when the expectation value it remains closed; we take this as an indication of the run-
away to the infinite-temperature limit (we can also study the
of the field and its conjugate momenta are zero: (cid:104)φ(x)(cid:105) =
(cid:104)P(x)(cid:105) = 0. The operator A ∼ (detG)1/4 is the normal- absorbedkineticenergyinthisformalismaswell);(ii)Stable
(non-ergodic)regime: quantumfluctuationsremainbounded,
ization constant to ensure the unitarity of the evolution at all
Z(t)staysfiniteatalltimes,andφremainslocalized. These
times:
two regimes are indicated in Fig. 4 as white and steel-blue
(cid:90)
(cid:104)Ψ |Ψ (cid:105)= D[φ]Ψ∗[φ]Ψ [φ]=1. (13) regions, andareinquantitativeagreementwiththeresultsof
v v v v
the quadratic approximation, Fig. 3. The apparent inconsis-
tencyforsmallΛ/γ (cid:28) 1andKg /γ2 (cid:28) 1issimplydueto
The functions G and Σ are variational parameter to 1
x,y x,y finite-timeeffects(seealsoAppendixB)
be determined self-consistently. To this end, we invoke the
TheeffectsofquantumfluctuationsisanalyzedinFig.4(b).
Dirac-Frenkel variational principle and define an effective
Thisplotdisplaysthestabilitydiagramforafixedandsmallg
classicalLagrangiandensityasfollows: 0
(g K/γ2 =10−4)asafunctionofg ,K andΛ. ForsmallK
0 1
(cid:90) we reproduce the large-frequency stability lobe of Fig. 4(a).
L = D[φ]Ψ∗[φ](i∂ −H[φ,∂/∂φ])Ψ [φ] (14)
eff v t v Indeed, the quadratic approximation is expected to become
6
exactinthelimitofK →0. FinitevaluesoftheLuttingerpa- VIII. SEMICLASSICALDYNAMICS
rameterK significantlyshrinkthevolumeofthenon-ergodic
(stable) regime. This result indicates that many-body quan-
Tofurtherdemonstratetheexistenceofaninstabilitytran-
tum fluctuations promote ergodicity. At the same time, our
sitionatafinitedrivingfrequency,wenownumericallysolve
analysis indicates that a finite region of stability can survive
the classical equations of motion associated with the Hamil-
inthethermodynamiclimit,eveninthepresenceofquantum
tonian (4). Specifically, we focus here on the stability re-
fluctuations.
gionatlargedrivingfrequenciesandalongg = 0(magenta
0
curve of Fig.s 3 and 4(a)). Following the truncated-Wigner
prescription40,46, we randomly select the initial conditions
fromaGaussianensemblecorrespondingtothegroundstate
of(4)withg(t) = 0andsolvetheclassicalequationsofmo-
tionassociatedwiththeHamiltonian(4).Althoughnotshown
here, wecheckedthatotherchoicesofinitialconditionslead
(a)K =0.1π
toqualitativelysimilarresults. Fig.5shows(a)thetimeevo-
lutionoftheaveragekineticenergyand(b)oftheexpectation
value (cid:104)cos(φ)(cid:105) for different driving amplitudes. The former
displays a sharp increase in correspondence of the expected
dynamicaltransition. Fig.6(a)showstheaveragekineticen-
ergyE ,andtheoscillationamplitudeδcos(φ)asafunction
∞
of driving frequency: Although E is smooth as a function
∞
ofthedrivingamplitude,itsfirstderivative(Fig.6(b))showsa
sharpkinkatacriticalvalueofKg /γ2. Non-discontinuities
1
in the second derivative of the energy are clear evidence of
continuous transitions. In contrast, δcos(φ) presents a kink
itselfatthecriticalvalueofKg /γ2.
1
From the position of the kink in either Fig. 6(b) or (c) we
compute the dynamical phase diagram shown in Fig. 7. We
find that larger K lead to a reduction of the stability region.
A similar result was obtained using the self-consistent varia-
(b)Kg /γ2 =10−4
0
(a)
n
σki100 Kg1/γ2=0.62
Kg /γ2=0.47
1
0 10 20 30 Kg /γ2=0.35
t (γ/2π) Kg1/γ2=0.2
(b) 1
Kg /γ2=0.11
0.4 1
Kg /γ2=0.046
φ〉os() 0.2 Kg11/γ2=0.019
c 0
〈
−0.2

0 10 20 30
t (γ/2π)
FIG. 4: The dynamical regimes (a) as a function of FIG. 5: (a) Time evolution of the normalized kinetic energy
(g0K/γ2,g1K/γ2,Λ/γ) for fixed K = 0.1, (b) as a function σkin(t) = Ekin(t)/Ekin(t = 0) − 1, with Ekin(t) =
of (K,g1K/γ2,Λ/γ) for fixed g0K/γ2 = 10−4. Blue (white) (1/K)(cid:104)(∂xφ)2(cid:105)forK = 0.4π, g0 = 0, Λ/γ = 0.04, L = 200,
regionscorrespondtopointsintheparameterspacewherethesys- N = 400. For small drives and large frequencies (lower curves)
temisstable(unstable). Solidlinesin(a)correspondtotheresults thesystemisstableandperiodicallyoscillateswithaperiodπ/Λ=
of the quadratic approximation (Fig. 3) and identify two distinct 12.5(2π/γ). Upon reaching a critical value of Kg1/γ2 the oscil-
stability regions, respectively at large γ and large g . In this plot lationsaresubstitutedbyanexponentialincreaseoftheenergy. The
0
thestabilitycriterionisarbitrarilysettoZ(T )/Z(0) > 0.95with dashedlineisaguidefortheeye.(b)Timeevolutionof(cid:104)cos(2φ)(cid:105)for
f
T = 100(2π/γ) (See also Fig. 9 for details about the finite-time thesameparametersasbefore. Theoscillationsbecomeverylarge
f
scaling.) aroundacriticalvalueofKg1/γ2.
7
(a) (b)
105 150 0.5
g1 K=0.1π
∂ 100
σkin100 ∂σ / kin 50 0.4 KK==01..46ππ
0 K=3.2π
0 0.2 0.4 0.6 0 0.2 0.4 0.6
Kg/γ2 Kg/γ2
1 1 0.3
(c) 2γ
0.15 Λ/γ=0.032 /1
g
Λ/γ=0.052 K
δφ cos()0.00.51 ΛΛΛ///γγγ===000...0128435 0.2
0 Λ/γ=0.37 0.1
0 0.2 0.4 0.6 0.8 1
Kg/γ2
1
0
FIG. 6: (a) Normalized kinetic energy σ at long times (t = 0 0.1 0.2 0.3 0.4 0.5
kin
100 × 2π/γ) as a function of the normalized driving amplitude Λ/γ
Kg /γ2 forK = 0.4π,g = 0,L = 200. Thisquantitydisplays
1 0
asharpkinkaroundacriticalvalueindicatedbythedashedline. (b) FIG.7: Stabilitydiagram: criticaldriveasafunctionoftheUVcut-
First derivative of the kinetic energy, showing a sharp discontinu- offΛ/γ forg = 0, asobtainedthrough thepresentsemiclassical
0
ityinitsderivative,confirmingthehypothesisofasecond-order-like approximation(+). Thedashedlineisaguidefortheeyes. Inthe
phasetransition.(b)Oscillationamplitudeof(cid:104)cos(2φ)(cid:105)atlongtimes limit Λ/γ → 0 all the curves tend towards the critical value of a
(T =30×2π/γ)asafunctionofthenormalizeddrivingamplitude singleKapitzapendulum,Kg /γ2 ≈ 0.45. Theerrorbarsreferto
1
Kg1/γ2.Thisquantitydisplaysasharppeakaroundacriticalvalue, thenumericaluncertaintyinthepositionofthepeakanddemonstrate
correspondingtothesharpincreaseinthekineticenergyidentifiedin thatthepeaksdonotbroadenandremainwelldefinedforfiniteΛ/γ.
(a).WithincreasingΛ/γthepeakmovestolowervaluesofthedrive Themagentacurvecorrespondstotheresultofthequadraticexpan-
amplitudeandbecomeslesspronounced. sion(Eq.(11)andmagentacurveinFig.3).
0.5
tional approach (Fig. 4). In particular, the stability diagram σ
K
ismainlyinsensitivetotheinitialconditions. Specifically,we 0.4 δcos
repeatedthesemiclassicaldynamicsforadifferentsetofini-
tialconditions(correspondingtoaninitiallygappedstatewith 2 0.3
γ
local correlations only) and observed a qualitatively similar g/1
dynamicalphasediagram. K 0.2
The above calculations refer to physical observables mea-
0.1
suredafterafinitenumberofdrivingperiods. Oneimportant
questionregardstheevolutionofthephasediagramofFig.3
0
asafunctionoftime. Inparticular,onemaywonderwhether
0 50 100 150 200
the stability region gradually shrinks and disappears in the T (γ/2π)
fin
infinite-time limit. In other words, can the dynamical tran-
sition be induced by applying an infinitesimal drive for very
FIG.8:Criticalvalueofthedrivingfieldasafunctionofthewaiting
long times? To address this question, in Fig. 8 we plot the timeT . Thecriticalfieldisdeterminedbytwoindependentmeth-
fin
criticaldrivingamplitudeasafunctionofthenumberofdriv- ods, namely by observing the peaks of σ as in Fig. 6(b), and of
K
ing periods. Although the critical driving amplitude initially δcos,asinFig.6(c). Bothmethodsshowaninitialdecreaseofthe
decreasesasafunctionoftime,weobservethatatlongtimes critical drive, followed by a saturation at a finite asymptotic value.
it tends to a finite asymptotic value: the stable regime occu- Numericalvalues:g0 =0,K =0.4π,Λ/γ =0.1
piesafiniteregionintheparameterspaceevenintheasymp-
toticlong-timelimit. Thenumberofdrivingperiodsplaysan
analogousroletothesizeofthesysteminequilibriumphase tions). Our findings are in contrast to an earlier conjecture
transitions: when appropriately rescaled, any physical quan- formulated in the context of MBL systems with randomness
tityisassociatedwithawell-definedasymptoticscalinglimit byOganesyanetal.49andsuggestingthatsuchatransitionhas
(seeAppendixC). apurequantumoriginanddoesnotoccurinclassicalsystems.
Thepresentcalculationsdemonstratetheexistenceofalo-
calized(non-ergodic)phaseinthethermodynamiclimit. Fol-
lowing D’Alessio et al.7, this result can be interpreted as a IX. EXPERIMENTALREALIZATIONANDOUTLOOK
(many-body) energy localization transition. Remarkably, the
present semiclassical approach involves the solution of clas- We now offer more details about the two physical realiza-
sical equations of motion (subject to quantum initial condi- tionsthatwereanticipatedinFig.2.Thetwoproposedexperi-
8
mentsbelongrespectivelytotheclassicalandquantumrealm. ordertostabilizethisphase. Asexplainedabove,inthepres-
The present calculations suggest that the two model should ence of a periodic drive it may be possible to stabilize it by
have a qualitatively similar stability diagram, but an experi- tuning the system in the region where only the “upper” ex-
mentalverificationisrequired.TherealizationoftheHamilto- tremum is stable. This situation is analogous to the recent
nian(3)usingclassicalelementsseemsparticularlyappealing proposal of Greschner et al.62. In relation to the dynamical
andrelativelyeasy: thismodeldescribesanarrayofpendula stability of this phase, one however needs to notice that the
attached to a common periodically oscillating support, and opticallatticeaffectstheLuttingerparameteraswell. Acor-
coupledthroughnearestneighborcouplings(seeFig.2(a)). rect description of these experiments therefore involves the
Aquantumversionoftheperiodically-drivensine-Gordon additional transformation K → K(t) = K0 +K1cos(γt),
model (4) can be realized using ultracold atoms50–53. By with possible significant consequences for the resulting sta-
trapping the atoms in cigar-shaped potentials it is possible bilitydiagram. Finally, itisworthmentioningotherpossible
to obtain systems in which the dynamics is effectively one realizationsofthemodel(4), includingforexampleRFcou-
dimensional. Specifically, the transversal confinement can pledspinorcondensates44andarraysofJosephsonjunctions45
be generated through laser standing waves (see for example inthepresenceoftime-dependentmagneticfields.
Ref. [54]), or through magnetic fields induced by currents Tosummarize,inthispaperwecombineddifferentanalyt-
running on a nearby chip (see for example Ref. [55]). In icalandnumericaltoolstostudytheperiodically-drivensine-
both cases, the amplitude of the transverse confinement can Gordon model. We applied a controlled high-frequency ex-
be easily modulated over time, allowing to realize the setup pansion, the Magnus expansion, to derive an effective (Flo-
showninFig.2(b).Hereatime-dependenttransverseconfine- quet) Hamiltonian. Employing ideas from the renormaliza-
ment induces a time-dependent tunneling coupling between tion group (RG) method, we propose the existence of a non-
two parallel tubes. In model (4) this coupling is modeled absorbing (non-ergodic) fixed point, in which the system is
(cid:82)
as dxcos(φ), where φ(x) = φ (x) − φ (x) is the local weakly affected by the periodic drive. At a critical value
1 2
phasedifferencebetweenthequasi-condensates41,56–60. Deal- of the driving frequency (or equivalently of the driving am-
ingwithasystemoftwocoupledtubes,theLuttingerparam- plitude), the system undergoes a dynamical phase transition
eter is d√oubled with respect to a single tube and equals to andflowstowardstheinfinitetemperatureabsorbing(ergodic)
K = 2 Γ, where Γ = mΛ/(cid:126)2ρ , Λ = µ is the chemical state. The transition occurs at a finite value of the drive am-
0
potential, m is the mass of the atoms, and ρ their average plitude and is therefore beyond the reach of perturbative ap-
0
one-dimensionaldensity65. Atomsonachiparecharacterized proaches. To provide a glimpse about the nature of the tran-
byrelativelysmallinteractionenergies,constrainingthemax- sitionweconsideredthelowest-orderMagnusexpansionsug-
imalvalueoftheachievableK. Typicalexperimentalvalues gestingthatthetransitioncouldcorrespondtothepointwhere
ofγ areintheorderofγ (cid:46) 10−2,orK (cid:46) 0.2. Theparame- thekinetictermintheFloquetHamiltonianbecomesnegative.
terg(t)issetbytheinstantaneous(single-particle)tunneling The existence of a transition at a finite value of the driv-
ratethroughJ⊥(t)=Kg(t)/µ. Theseexperimentsarethere- ingamplitudeisfurthersupportedbytwonumericalmethods:
foreconstrainedtog(t) > 0,org < g . Fig.s3and4show a self-consistent time-dependent variational approach, and a
1 0
thattheawideregionofstabilityisexpectedinthephysically semiclassicalapproach.Interestingly,theemergentphasedia-
relevant regime (small K and g < g ), demonstrating the gram(Fig.s4and7)isinqualitativeagreementwithasimple-
1 0
feasibilityoftheproposedexperiment. mindedquadraticexpansion(Fig.3).Thedynamicalphasedi-
Analternativeproceduretoperiodicallydrivethetwocou- agram displays two distinct stability islands, respectively for
pled tubes involves the modulation of the chemical poten- large driving frequencies γ and for large g = 1/T (cid:82)T g(t).
0 0
tial difference between them δµ(t) = µ (t)−µ (t) (while Theformerislandbecomesunstablewhenγ iscomparableto
1 2
keeping approximately fixed the tunneling element). In the the short wavelength cutoff Λ. This suggests that the transi-
bosonization language of Eq.(4), this corresponds to a time- tionisofmean-fieldnature,thusbearingthesamecharacteras
dependentfieldthatcouplestotheatomicdensity. Anappro- theparametricresonanceofasingleharmonicoscillator. The
priate gauge transformation allows one to map this problem latterislandisrelatedtotheexistenceoffiniteexcitationgap,
intoaphase-modulatedsine-Gordonmodelinwhichthedrive whichprotectsthesystemfromlow-frequencydrives. Inboth
entersthroughthephasedriveφ (t)asgcos(φ−φ (t)). We cases,weobservethatquantumfluctuationspromoteergodic-
0 0
postponethedetailedanalysisofthismodeltoafuturepubli- ityanddecreasethevalueofthecriticaldrivingamplitude.
cation: preliminarycalculationsindicatethatthestabilitydi- The observed transitions are analogous to the stability
agram is analogous to Fig. 3. However, we do not know yet threshold predicted in Ref.s [9,11,12,14] for many-body lo-
whetherthetwomodelsbelongtothesameuniversalityclass. calized(MBL)states,butdoesnotrequirelocalizationinreal
The sine-Gordon model is often discussed in the context space. Our findings are in contrast to the conclusions of
of isolated tubes under the effect of a longitudinal standing D’Alessio et al.15, who argued that generic many-body sys-
waves, or optical lattice, as well. At equilibrium this model tem should be dynamically unstable under a periodic drive.
describestheLuttingerliquidtoMottinsulatorquantumphase Ponte et al.9 showed that ergodic systems are always unsta-
transitionofonedimensionaldegenerategases(seeRef.61for bletotheperiodicmodulationofalocalperturbation: wefind
a review). Interestingly, the inverted pendulum φ = π cor- herethatglobalcoherentperturbationscandisplayaqualita-
responds to a distinct topological phase, the Haldane insula- tively different behavior (See also Russomanno et al.63 for a
tor42,43. At equilibrium non-local interactions are needed in specificexample).
9
Acknowledgments AppendixB:SinglePendulumLimit
WearegratefultoT.Giamarchi,B.Halperin,D.Pekker,A. As mentioned in the text, the limit Λ → 0 of Eq.(4) re-
Russomanno, K. Sengupta, A. Tokuno for many useful dis- coversthecaseofanisolatedclassicalKapitzapendulum. In
cussion. TheauthorsacknowledgetheorganizersoftheKITP Fig. 9 we show that this limit is correctly reproduced by the
workshop on “Quantum Dynamics in Far from Equilibrium self-consistent variational approach. With decreasing g , the
1
ThermallyIsolatedSystems”,duringwhichthisworkwasini- time required for the system to become unstable grows ap-
tiated, and the NSF grant No. PHY11-25915. EGDT and proximatelyas1/g .Thisexplainstheapparentinconsistency
1
EDthankthesupportoftheHarvard-MITCUA.RCacknowl- betweenthequadraticexpansionandtheself-consistentvari-
edgestheInternationalProgramofUniversityofSalernoand ationalapproach(Fig.4(a))forΛ/γ (cid:28)1andKg /γ2.
1
theHarvard-MITCUA.ThisresearchwassupportedbyTHE
ISRAELSCIENCEFOUNDATION(grantNo. 1542/14). AP 11
andLDacknowledgethesupportofNSFDMR-1206410and
AFOSRFA9550-13-1-0039. 0.5 10
9
0.4
AppendixA:Third-orderMagnusexpansion
2 8
γ
/1 0.3
Ifthedrivingfrequencyγisthelargestscaleintheproblem, g 7
K
Magnusexpansionappliesandgoinguptothirdorderonehas: 104
0.2 6
1 (cid:90) T 103
He(ff1) = T dtHˆ(t1)= τd102 5
0 0.1 101
=(cid:90) dx[KP2+ K1 (∇φ)2−g0cos(φ)] 0 10100−2 Kg110/−1γ2 100 4
1 (cid:90) T (cid:90) t1 0 0.1 0.2 0.3 0.4 0.5
He(ff2) = 2Ti dt1 dt2{Hˆ(t1),Hˆ(t2)}=0 Kg0/γ2
0 0
1 (cid:90) T (cid:90) t1 (cid:90) t2
H(3) = dt dt dt FIG. 9: Stability diagram obtained from the self-consistent varia-
eff 6Ti2 1 2 3 tional approach for Λ/γ = 0.01, K = 0.05π. The color code
0 0 0
(cid:16) (cid:17)shows log(τ ), where τ is defined as Z(τ )/Z(0) = 10−3 (dark
× {Hˆ(t1) , {Hˆ(t2),Hˆ(t3)}}+{Hˆ(t3),{[Hˆ(t2),Hˆ(t1)}} greenareascdorrespondtdoτ > T = 104(d2π/γ)). Orangedashed
d f
linesarefromthequadraticexpansionandsolidblacklinesarefrom
where the time-integral domain is ordered 0 < t < ... <
n Broer et al28. The inset plot shows τ as a function of Kg /γ2
t2 <t1 <T andT istheperiodofthedrivingT = 2γπ. along the resonance Kg0/γ2 = 1/4.d The dashed line refe1rs to
Inthepresentcasethesecond-ordertermvanishesH(2) = τd/∼(Kg1/γ2)−1.
eff
0: for time-reversal invariant perturbations, all even-order
terms are exactly zero. The third order Magnus expansion
leadstotheeffectiveHamiltonian(6),whereweusedthefol-
lowingidentities: AppendixC:Finitetimescaling
{P2,{P2,cos(φ)}}=−2i{P2,P sin(φ)}=4P2cos(φ);
In this Appendix we extend the analysis of Sec. VIII and
(A1)
determinethescalingofphysicalobservablesasafunctionof
thenumberofdrivingperiods. Figs.10(a)and11(a)showthe
{(∂ φ)2,{P2,cos(φ)}}=−2i{(∂ φ)2,P sin(φ)}
x x energy absorption rate and the average cosine as a function
=2(∂xφ)[∂xsin(φ)+sin(φ)∂x] ; ofthedrivingamplitude,atdifferenttimes. Theseplotsshow
(A2)
that all the curves tend to a well defined long-time limit. To
better highlight this asymptotic limit, we shift each curve to
1 (cid:90) 2π (cid:90) 2π (cid:90) 2π takeintoaccountthedependenceofthecriticalamplitudeon
dt dt dt
12π 0 1 0 2 0 3 Tfin (Fig.8). Specificallyweconsiderfinite-timecorrections
ofthetype: g → g (1+A/T ). Throughthistransforma-
[g(t )−g(t ) +g(t )−g(t )]=g ; c c fin
3 2 1 2 1 tionweobtainanexcellentdatacollapseonasingleuniversal
(A3)
asymptotic curve, as shown Fig.s 10(b) and 11(b). In both
1 (cid:90) 2π (cid:90) 2π (cid:90) 2π cases, the asymptotic curves become steeper and steeper as
dt dt dt
12π 1 2 3 the number of driving periods increases. This may indicate
0 0 0
1 thatphysicalobservablesareultimatelynotcontinuousinthe
g(t )[g(t )−g(t )]+g(t )[g(t )−g(t )]=g g − g2 .
1 3 2 3 1 2 0 1 4 1 Tfin →∞limit,withimportantconsequencesfortheuniver-
(A4) salpropertiesofthetransition.
10
(a) (a)
80 0.1
60
1/2σ/Tkin fin40 δcos0.05
20
0
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Kg(b1/)γ2 Kg(b1/)γ2 T fin=35
5 T fin=5 0.06 T fin= 65
1/2∂σ∂ / g / Tkin1 fin1234 TTTTT fffffiiiiinnnnn=====369115552555 δcos(1−A/T) fin00..0024 TTTT ffffiiiinnnn====91115258555
00 .1 0.15 0.2 0.25 0.3 0.35 0.4 0T.4 fi5n=1850.5 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Kg/γ2(1+B/T ) Kg/γ2(1+B/T )
1 fin 1 fin
FIG.10:Energyabsorptionrateasafunctionofthenormalizedperi- FIG.11:SameasFig.10fortheamplitudeofthecosineoscillations.
odicamplitude,foradifferentnumberofperiods(T ).(a)rawdata
fin
forasystemofsizeL=400,K =0.4πandΛ/γ =0.1;(b)same
dataonanormalizedx-axis,showingagooddatacollapsewhenthe
positionofthepeakisrescaledasg=g +A/T withA=20.
c
1 A.Eckardt,C.Weiss,M.Holthaus,Phys.Rev.Lett.95,260404 26 F.Bullo,SIAMJ.ControlOptim.41,542-562(2002).
(2005). 27 Y.Nakamura,T.Suzuki,M.Koimura,IEEETrans.Robot.Autom.
2 H.Lignier,etal.Phys.Rev.Lett.99,220403(2007). 13,853-862(1997).
3 J.Struck,etal.Science333,996-999(2011). 28 H.W. Broer, I. Hoveijn, M. van Noort, C. Simo and G. Vegter,
4 J.Strucketal.,Phys.Rev.Lett.108,225304(2012). JournalofDynamicsandDifferentialEquations16,897(2004).
5 C.V.Parker,L.-C.Ha,C.Chin,Nat.Phys.9,769-774(2013). 29 O.M.Braun,Y.S.Kivshar,Phys.Rep.306,1-108(1998).
6 A.Russomanno,A.Silva,andG.E.Santoro,Phys.Rev.Lett.109, 30 T. Giamarchi, Quantum Physics in One Dimension, Oxforf Sci-
257201(2013). encePublications,2003.
7 L.D’Alessio,A.Polkovnikov,AnnalsofPhysics333,19(2013). 31 A.O.Gogolin,A.A.Nersesyan,A.M.Tsvelik,Bosonizationand
8 A.Russomanno,A.Silva,andG.E.Santoro,J.Stat.Phys.P09012 stronglycorrelatedsystems,CambridgeUniversityPress,1998.
(2013). 32 S.Pielawa,Phys.Rev.A83,013628(2011).
9 P. Ponte, A. Chandran, Z. Papic, and D. A. Abanin, 33 E.G.DallaTorre,E.Demler,T.Giamarchi,andE.Altman,Phys.
arXiv:1403.6480(2014). Rev.B85,184302(2012).
10 A. Lazarides, A. Das, R. Moessner, Phys. Rev. E 90, 012110 34 A.Mitra,T.Giamarchi,Phys.Rev.B85,075117(2012).
(2014). 35 S.Weigert,arXiv:quant-ph/0106138,2001.
11 A.Lazarides,A.Das,R.Moessner,arXiv:1410.3455. 36 W.Magnus,Comm.onPureandAppl.Math.7,649(1954).
12 P.Ponte,Z.Papic,F.Huveneers,D.A.Abanin,arXiv:1410.8518. 37 S.Blanes,F.Casas,J.A.Oteo,J.Ros,Phys.Rep.470,151(2009).
13 S.ChoudhuryandE.J.Mueller,arXiv:1405.1398(2014). 38 Let us note that the Magnus expansion is valid in the limit of
14 D.Abanin,W.DeRoeck,F.Huveneers,arXiv:1412.4752. g /γ2 (cid:28) 1,leadingtotheadditionalrequirementthatg (cid:29) g ,
0 1 0
15 L.D’AlessioandM.Rigol,Phys.Rev.X4,041048(2014). thusonecanapproximateg˜withg˜= Kg12.
16 L.D.LandauandE.M.Lifshitz,Mechanics,3rdedition(1980). 4γ2
17 R.VoskandE.Altman,Phys.Rev.Lett.110,067204(2014). 39 Thedrivingamplitudemustbesmallinanycasetopreventchaotic
18 D. Huse and V. Oganesyan, arXiv:1305.4915 (2013). David A. behavior.
40 A.Polkovnikov,AnnalsofPhys.325,1790(2010).
Huse, Rahul Nandkishore, Vadim Oganesyan, Phys. Rev. B 90,
41 E. G. Dalla Torre, E. Demler, A. Polkovnikov, Phys. Rev. Lett.
174202(2014).
19 T.Prosen,Phys.Rev.Lett.,80,1808(1998). 110,090404(2013).
20 T.Prosen,J.Phys.A:Math.Gen.,31,L645,(1998). 42 E.G.DallaTorre,E.Berg,E.Altman,Phys.Rev.Lett.97,260401
21 T.Prosen,Phys.Rev.E,60,3949(1999). (2006).
22 Inthethermodynamiclimit,themany-bodyenergyisalwaysun- 43 E.Berg,E.G.DallaTorre,T.Giamarchi,E.Altman,Phys.Rev.
B77,245119(2008).
bound.
23 P.L.Kapitza,SovietPhys.JETP21,588-592(1951);P.L.Kapitza, 44 Y.KawaguchiandM.Ueda,Phys.Rep.520,253(2012).
45 SeeforexampleProceedingsoftheNATOARWCoherenceinSu-
Usp.Fiz.Nauk,44,7-15(1951).
24 IliyaI.Blekhman(transl.MinnaPerelman),VibrationalMechan- perconductingNetworks,J.E.MooijandG.Schoneds.,Physica
B152,(1988).
ics:NonlinearDynamicEffects,GeneralApproach,Applications,
46 J. Lancaster, E. Gull and A. Mitra, Phys. Rev. B 23, 235124
WorldScientific,Singapore(2000),509pp.
25 W.Paul,Rev.Mod.Phys.62,531-540(1990);I.Gilary,N.Moi- (2010).
47 R.JackiwandA.Kerman,Phys.Lett.A71,158(1979).
seyev, S. Rahav, S. Fishman, J. Phys. A, Math. Gen. 36, L409–
48 F.Cooper,P.Sodano,A.TrombettoniandA.Chodos,Phys.Rev.
L415(2003).