Breakdown of the adiabatic limit in low dimensional gapless systems

0.34 MB·English

by  Anatoli Polkovnikov

Breakdown of the adiabatic limit in low dimensional gapless systems
Anatoli Polkovnikov1 and Vladimir Gritsev2
1Department of Physics, Boston University, Boston, MA 02215
2Department of Physics, Harvard University, Cambridge, MA 02138
It is generally believed that a generic system can be reversibly transformed from one state into
another by sufficiently slow change of parameters. A standard argument favoring this assertion is
based on a possibility to expand the energy or the entropy of the system into the Taylor series in
8 the ramp speed. Here we show that this argumentation is only valid in high enough dimensions
0 and can break down in low-dimensional gapless systems. We identify three generic regimes of a
0 system response to a slow ramp: (A) mean-field, (B) non-analytic, and (C) non-adiabatic. In the
2
last regime the limits of the ramp speed going to zero and the system size going to infinity do not

n commute and the adiabatic process does not exist in the thermodynamic limit. We support our
a resultsbynumericalsimulations. Ourfindingscanberelevanttocondensed-matter,atomicphysics,
J quantumcomputing, quantumoptics, cosmology and others.

7
1
Adiabatic or reversible (also known as quasi- groundstate andadiabaticallytunes the systeminto an-

] stationary)processesbyallmeansplayamajorroleboth other regime of parameters the system remains in the
h
in physics and technology. The adiabatic process is for- ground state and thus no entropy is generated. The
c
e mally definedas suchwhereno heatis transferredto the quantum mechanical adiabatic theorem implies thermo-
m system from the environment1. Typically such processes dynamicadiabatictheorem,however,therearesomesub-
occur at time scales which are fast enough compared to tleties involved2. If there is a gap in the system then
-
t the thermalization times with the environment but yet at zero temperature the ground state can be excited
a
t which are sufficiently slow so that the system always only through the so called Landau Zener mechanism3,
s
remains in thermal equilibrium. Another feature of an whichgivestheexponentiallysmallprobabilityofexcita-
.
t adiabatic process is the conservation of entropy. There tions and thus the exponentially small entropy increase.
a
m is a simple general argument showing that slow quasi- We note that the fact that this increase is exponential
stationary processes are adiabatic and thus reversible. rather than quadratic is a peculiarity of the zero tem-
-
d The argument goes as follows:1 Assume that in an iso- perature limit. However, systems which have a gap are
n lated system some external parameter κ is slowly driven rather exception than the rule. Indeed most of systems
o from some initial value κ to the final one κ . For sim- with broken continuous symmetries have gapless excita-
A B
c
plicityweassumethatκchangeslinearlyintime,though tions over the ground state (Goldstone modes). Thus
[
it is not necessary, and let δ be the rate of this change. solids have phonons (sound waves), ferromagnets and

2 We also refer to δ as to the ramp speed. We assume anti-ferromagnets have gapless magnon or spin-wave ex-
v that on the way the system does not undergo any dis- citations,superfluidshavegaplessBogoliubovexcitations
2
continuousphasetransitions(thoughsecondorderphase and so on. Even in gapped systems, for example super-
1
transitionsaregenerallyallowed). Thentheentropyden- conductors, one always has continuum spectrum above
2
0 sity(ortheentropyperunitvolume)ofthesysteminthe the gap, which is occupied by quasi-particles at finite
6. finalstate,SB,willbeafunctionofthisparameterδ and temperatures. If gapless excitations are present in the
we can expect that for small enough δ one can expand system then the Landau Zener mechanism does not help
0
S into the Taylor series: toprotectagainstcreatingexcitations. However,onecan
7 B
0 generally argue that the available phase space for the
: SB =SA+α′δ+β′δ2+... . (1) excitationsdecreasesas the adiabaticity parameterδ be-
v
comessmallerandonecanstillexpectthatEq.(2)holds.
i Ongeneralgroundswecanarguethatα′ 0becausethe
X
≡
entropycanonlyincreaseandthuscannotbesensitiveto
r Theadiabatictheoremcanbealsoformulatedforinte-
a the signof δ. Thus at small δ the excess entropy density
grablesystems (the simplestexample ofthese are nonin-
pumped into the system during this process is
teractingsystemsdescribedbyharmonictheories). Then
∆S β′δ2. (2) the entropy is no longer a good observable because inte-
AB
≈ grable systems do not thermalize. Instead one can use
In other words it vanishes quadratically as δ 0. We theexcitation(quasi-particle)densitynex. Thequantum
→
notetherearesomesubtletieswiththisexpressionatzero adiabatic theorem implies that nex does not change if
temperature, which we will mention later. the process is sufficiently slow. Then on generalgrounds
The adiabatic theorem in thermodynamics is inti- we can expect that the analogue of Eq. (2) will be
mately relatedto the adiabatic theoreminquantum me- ∆n (δ) β˜δ2. Whether the system is integrable or
ex
≈
chanics, which states that under slow enough external not, its energy is always a good observable, since it is
perturbations there are no transitions between energy defined for any state of the system in or out of equilib-
levels. Thus, for example, if one starts from a unique rium. In terms of energies the adiabatic theorem can be
2
formulated in the same spirit: at finite temperature and slowly increases the interac-
tion strength. We point that existence of the regime C
B(δ)= B(0)+βδ2, (3) isconsistentwithrecenttheoretical6,7 andexperimental8
E E
workssuggestingthatthereisnoadiabaticlimitinapar-
where B(0) is the energy of the state adiabatically con- ticular problem of a BCS-BEC crossoverin cold atoms.
E
nected to the initial state. Let us make another very important remark. A gen-
Thereisoneimportantcaveatintheconsiderationsde- eral argument predicting entropy conservation in a slow
scribedabove. Theyaresimilarinspirittothemean-field process assumes that the system is always in thermal
argumentsuggesting existence of the long range order in equilibrium. If, on the other hand, we are dealing with
systems with broken continuous symmetries. However, anintegrableoranearlyintegrablesystemthenthermal-
weknowthatatlowdimensionseitherquantum4 orther- izationtimescanbeeitherinfiniteorverylong. However,
mal5 fluctuationscanchangethepictureanddestroythe as we argued above the non-integrability of the system
long range ordercompletely. An ultimate reasonfor this and its thermalization are not necessary conditions for
is that at low dimensions the density of the low energy formulation of the adiabatic theorem in the generalized
states ρ(ǫ) is generically high. Indeed if ǫ(q) qz, where sense. The only genuine requirement is that the system
∝
qisthemomentumofanexcitationandǫ(q)isitsenergy, is initially in a stable equilibrium. In this work we rig-
thenwenecessarilyhaveρ(ǫ) ǫd/z−1,wheredisthedi- orouslystudy the limit where the systemeither does not
∝
mensionality of the system. The quantum or thermal thermalize or it rethermalizes only after the ramp and
fluctuations force excitations to occupy these low energy showthatallthreeregimesarepossible. We supportour
states and if their density is sufficiently high, they can results by performingnumericalsimulationsfor a partic-
qualitatively change the nature of the equilibrium state. ularinteractingnon-integrablesystemandshowthatthe
Thesameargumentscanbeappliedtotheadiabaticpro- effects of weak nonintegrability do not affect our conclu-
cesswe areinterestedin. Indeedno matter howslowthe sions.
rampis(inagaplesssystem)therewillbealwayscreated An alternative point of view on the existence of the
somelowenergyexcitations. Inlowdimensionstheseex- regimesBandCisthe breakdownofthe linearresponse
citations can significantly alter behavior of the system theory for slow perturbations9. In this work we concen-
andinparticularinvalidateEqs.(2,3). Inthe remainder trated on the spatially uniform case. However, one can
of the paper we will address this issue in detail. anticipate that similar breakdown of the linear response
Themainconclusionofourworkisthattherearethree will be relevant to a more general class of nonuniform
possible regimes of a system response to a slow process: low-frequencyperturbations. Weareleavingthedetailed
(A)mean-field,(B)non-analytic,and(C)non-adiabatic. analysis of such possibilities for a future work.
In the first regime (A) we find that indeed the energy of Apart from many applications in condensed matter
thesystemandotherthermodynamicquantitiesareana- and atomic physics our findings may be relevant to such
lyticfunctionsoftheadiabaticityparameterδandEq.(3) diverse fields as quantum optics, quantum computing,
isvalid. Thisregimeisalwaysrealizedatsufficientlyhigh in particular for adiabatic quantum computation10; in-
dimensions. In the second regime (B) one still has the flationary cosmology, which has the phase of adiabatic
adiabatic limit in the sense that there is a well defined
evolution of Gaussian scale-invariant quantum fluctua-
limit δ 0 but the correctionin δ to the energy density tionsintheslow-rollapproximation11;cosmicmicrowave
→
or other quantities behaves nonanalytically: background radiation12, and Hawking radiation of black
holes13. Inallthese fields the assumptionofadiabaticity
(δ) (0)+β δ ν, (4)
B B in slow processes is usually taken for granted, while our
E ≈E | |
results suggest that this might not be always the case.
whereν <2 is somepowerwhichdepends onthe dimen-
Analytical treatment. In this paper we consider a spe-
sionality and on the universal details of the spectrum.
cific low energy quadratic Hamiltonian:
And finally the third (C) non-adiabatic regime is most
unusual. There instead of Eq. (3) we have
ρ q2 1
= s φ 2+ κ Π2, (6)
B(δ) B(0)+β δ νLη, (5) H q 2 | q| 2 q q
E ≈E | | X
where L is the system size and η > 0 is another expo- where φ and Π are the coordinate and the conjugate
q q
nent. In this regime there is no adiabatic process in the momentum. We note that this Hamiltonian describes a
thermodynamic limit. In other words the limits δ 0 very wide class of gapless systems. Thus in solids φ
q
→
and L do not commute. This is a striking conclu- represents the phonon or the plasmonfield, in ferromag-
→ ∞
sion, which states that no matter how slowly we try to netsandanti-ferromagnetsφ describesmagnonsorspin-
q
drivethe system,ifthe latteris sufficientlylarge,wewill waves, in superfluids φ describes Bogoliubov’s excita-
q
neverreachtheadiabaticlimit. Asweshowbothanalyt- tions. This list can be easily extended further. Depend-
icallyandnumericallysuchregimeisrealizedin(butnot ingonthe system,thecouplingsρ andκ havedifferent
s q
limitedto)thesituationwhereonestartsfromanensem- meanings. Forexample,forsuperfluidsρ denotesthesu-
s
ble of noninteracting bosons in one and two dimensions perfluid density, and κ is related to the compressibility.
q
3
Dependence of κ on q also varies for different systems. ing and ∆ δ2 lnδ . If κ = 0 then the B regime is
q 0
E ∝ | |
Thus κ = const(q) in solids, anti-ferromagnets and su- realized in all three spatial dimensions and the energy
q
perfluidsandκ q2 isferromagnetsandnoninteracting density scales as
q
∝
Bose systems. The reasonwhy the Hamiltonian (6) is so
genericandapplicableto suchdifferentsituationsisthat δ(d+1)/4
. (8)
it describes Goldstone modes of systems with a broken E ∝ (ρ λ3)(d+1)/8
s
continuous symmetry. We further choose κ = κ+λq2.
q
Thischoiceallowsustocoverallthesituationsmentioned We note thatif one analyzesthe density ofexcitations
above. Ina superfluidκstands forthe compressibilityof nex thentheclassificationofdifferentregimesisdifferent.
thesystemandwewillusethisterminologyinthepaper. In particular for large κ0 one finds that the nonanalytic
Letusnowimaginethatoneslowlyincreasesκintime B regime is realized in one spatial dimension and the
mean-field A regime occurs above two dimensions. For
κ(t)=κ +δt, (7) κ = 0 the system is in the nonadiabatic C regime for
0 0
d=1 and it is in B regime in two and three dimensions.
where κ0 is the initial value of the compressibility. In The fact that the classification of the system’s response
principle, one can consider ramps of other parameters, according to the density of excitations is different from
butwedonotexpectanysignificantchangesinthe over- that based on the energy is not really surprising. Indeed
allpicture. Inthis paperwewillfocusonpositivesignof primarilylowenergymodesareoccupiedandtheydonot
δ, i. e. the situation where compressibility increases in contributemuchtothetotalenergy. Asimilareffectalso
time. It can be shown that the opposite process, where occurs in equilibrium.
κ decreases, gives similar results up to unimportant nu- If the original system is nonintegrable then it will
merical prefactors. We also comment that the choice rethermalizeatlongtimes. Thermalizationinclosednon-
κq(t) = κ(t)+λq2 allows us to analyze interesting pos- integrable systems is a separate issue, which we are go-
sibilities from the field theory point of view. Indeed for- ing to address in future work. We just point out that
mallyλq2 isanirrelevantcoupling,whichisunimportant the Liouville’s theorem stating that the entropy of an
at low energies (this simply follows from the fact that it isolated system is conserved in time is generally not an
scales to zero at small q). On the other hand, if one issue in non-integrable many-particle systems. Indeed
starts at zero or very small κ0 this term dominates the the amount of external noise needed to break this en-
behaviorof the systemat initialtimes and as we will see tropy conservation scales to zero exponentially with the
it qualitatively changes the system’s response. Thus λ number of particles so that non-integrable systems are
playstheroleofadangerouslyirrelevantvariable4 inour never completely isolated. Let us assume that after the
problem. On the other hand if κ0 is large then indeed ramp we can wait long enough so that the system does
one can safely set λ to zero. rethermalizeandletusseewhataretheconsequencesfor
Letusfirstassumethatthesystemisinitiallyprepared the entropy. Note that since the system is isolated from
in the ground state, i.e. consider the zero temperature theenvironment,thermalizationoccursatafixedenergy,
limit. SincetheHamiltonian(6)isnoninteractingitdoes whichwejustdeterminedabove. Thisenergythusshould
not lead to thermalization, therefore as we discussed the be related to the temperature of the final state T via
f
entropy is not a good concept. One can instead describe
the degree of non-adiabaticity of the system by comput- Td+1
=C f , (9)
ing either the density of excitations nex created during E d(κ ρ )d/2
f s
the ramporthe energydensity (energyper unit volume)
pumpedtothesystem . Sinceourmodelisnoninteract- whereC isanumericalconstant. EquatingEqs.(8)and
E d
ing,theevolutionofthewavefunction(ormoregenerally (9) we can find that in the case of κ =0
0
density matrix) can be explicitly obtained. We give de-
tails of such analysis in the section “Methods”. Here we ρ3/8 δd/4
T √κ δ1/4 s , =A . (10)
will outline the main results. f ∝ f λ3/8 ⇒Sf d(ρ λ3)d/8
s
Letusfirstassumethatthesystemisinitiallyprepared
at zero temperature. Then its wave function factorizes Weseethatthebehavioroftheentropyessentiallyfollows
totheproductofGaussianscorrespondingtotheground thatofthe energywitha slightlydifferentexponent. We
state of a harmonic oscillator for each momentum q (see notethatthefactthatexponentisdifferentisapeculiar-
Eq. (18)). We find there are two different regimes of the ity of the zero temperature case, where the entropy has
system behavior depending on whether the initial com- a singular limit. Apart from that the same classification
pressibility κ is finite or zero. The crossover between of different regimes A, B, and C applies to the entropy
0
the two regimes occurs at κ⋆ δ λ/ρ . Note that as ofthesystem,itstemperatureandotherthermodynamic
δ 0 we have κ⋆ 0. Fo0r ≈large initisal compressibil- quantities.
ity→, κ κ⋆, the0re→sponse of the spystem belongs to the Quite similarly one can analyze the response of the
A reg0im≫e in0all three spatial dimensions, i.e. the energy system to the ramp if it is initially prepared at some
density behaves as δ2. Actually in one dimension finite temperature T. Because atfinite temperatures the
E ∝
there is an additionallogarithmic correctionto this scal- excitationsarepresentinthesystemfromthebeginning,
4
we will be interested in excess quantities pumped to the very interesting and not very well understood problem,
system during the ramp like ∆n , ∆ , and ∆S. The which however requires a separate analysis and goes be-
ex
E
analysisoffinitetemperatureevolutionisstraightforward yond the scope of this paper. So we will focus on the
and we sketchit in Section “Methods” and Appendix C. analysis of the energy (density) pumped to the system,
The results are again strongly sensitive to the initial ∆ ,sincethisobservableisnotsensitivetothe detailsof
E
compressibility. Thusifκ islargetheninonedimension thermalization.
0
we get Toperformnumericalsimulationswechoosethe Bose-
Hubbard model on a square lattice described by the
δ T√κf Hamiltonian:
∆ =A| | (11)
E √ρsκ20 U(t)
= J (a†a +a†a )+ a†a (a†a 1),
In turn, if the system rethermalizes after the ramp, this Hbh − i j j i 2 j j j j −
Xhiji Xj
results in the nonanalytic correctionto the entropy: (16)
wherea anda† arethe annihilationandcreationopera-
dS δ j j
∆S ∆ =A | | . (12) torsofbosonsonthej-thsite,J representsthetunneling
≈ dE E √ρsκ30/2 matrix element and U is the interactions strength. The
sum in the first term is taken over the nearest neighbor
In this case obviously the nonanalytic B regime is real-
pairs. WetakeJ tobetimeindependentandtheinterac-
ized. In two and three dimensions ∆ δ2 (with loga-
E ∝ tion increasing in time according to U(t) = U0tanh(δt).
rithmic correctionsatd=2)andthus we havethe mean
SoU(t)firstincreaseslinearlyintimeandthensaturates
field A regime.
atsomesteadystatevalue. Forsmallenoughinteractions
If κ = 0 then in one and two dimensions the energy
0 U Jn , where n is the mean number of atoms per
0 0 0
density diverges with the system size: ≪
lattice site, in the quadraticapproximationthe Hamilto-
nian (16) maps to the Bogoliubov Hamiltonian (6) with
∆ =A T√κfδ1/3L7/3−d (13) ρ 2Jn , κ U, and λ ρ /4n2. In order to simu-
E d λρ1s/6 laste≈dynam0ics≈of the system≈wesemp0loy the semiclassical
approachdevelopedby one ofus14. In this approachone
and thus
expands the time evolution of the system in the small
d quantum parameter U/Jn0. We note for those more fa-
∆S =A′ T d+1 δ3(dd+1)L(37(−d3+d1))d. (14) miliar with the Keldysh technique15 that this approach
d λρ2s/3! treats all classicalvertexes exactly and expands the evo-
lution in quantum vertexes. In the leading order in this
In particular at d = 1 we have ∆S δ1/6L2/3 and at parameter one obtains the so called truncated Wigner
d=2: ∆S (δL)2/9. So in one and t∝wo dimensions the approximation (TWA)16,17, where the classical fields ψj⋆
behaviorof∝thesystemisnonadiabaticandtheCregime and ψ corresponding to operators a† and a satisfy the
j j j
is realized. In three dimensions there is no dependence timedependentGross-Pitaevskiiequationsofmotion. In
on the system size and we find the next order the classical fields are subject to a single
quantum jump during the evolution. We find that while
∆ =A T√κf √δ and ∆S =A′ √T √δ. (15) TWA approximation is adequate at finite temperatures,
E 3ρ1s/4λ5/4 3√ρsλ in the zero temperature limit one has to go beyond and
add the next correction. This finding agrees with a gen-
So that the system is in the non-analytic B regime. We eral statement that the semiclassical approximation can
noteagainthatthedensityofexcitationdivergesstronger break down at long times14,18. We present some details
than the energy. In particular at finite T and for κ =0 of the semiclassical approach in Appendix D and here
0
we find ∆n δ1/3L10/3−d, i.e. it diverges with the only will discuss the results.
ex
∝
system size in all three spatial dimensions. Firstwelookintoonedimensionalsystems,sincethere,
Numerical results: application to interacting bosons. according to the theory, we expect strongest effects of
While the analysis of the previous sections is formally nonadiabaticity. In order to avoid potential complica-
exact, it directly applies only to a special case of an in- tions related to strong quantum effects we choose the
tegrable harmonic system. Most of the real systems are parameters of the system deep in the superfluid regime
nonintegrable. However, at low energies the harmonic throughout the entire evolution: n = 20,J = 1,U =
0 0
approximation is usually very good. We already argued 0.25 so that the semiclassical parameter U /Jn 10−2
0 0
∼
that whether the system is allowed to re-thermalize at (see Ref. [18]) is verysmall. Note that despite U =0.25
0
longtimesornot,thequalitativepicturedoesnotchange. is relatively small the product U n = 5 is larger than
0 0
If the thermalization occurs then a good measure of the J =1 implying that at long times U(t) U the system
0
≈
heating is the entropy generated during the ramp, if not is in so called quantum rotor regime with healing length
then one should look into the generated density of exci- smaller than the lattice spacing. According to our the-
tations. Thermalization in a closed system is certainly a oretical expectations the system is mostly excited while
5
the healing length remains large, i. e. when U U . In peraturesT 0.01J thebehaviorof∆ isdominatedby
0
≪ ∼ E
this regime U(t) = U tanhδt U δt linearly increases the thermal effects. This trend becomes even more ap-
0 0
≈
with t thus we can directly compare numerical results parentifwe analyzedependence of∆ onL (seeFig.2).
E
with the analytical predictions of the previous section. Inagreementwiththeanalyticresults∆ stronglygrows
E

7 y L=320
sit5 L=256
T=0.08 n L=192
6 e
T=0.04 d L=128
y T=0.02 y 4 L=64
densit 45 TT==00.01 s energ3
gy 3 es2
r c
e x
n E
E 2 1
1 0
0.0 0.2 0.4 0.6 0.8 1.0
0
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 2: Dependence of ∆ on δ for different sizes at fixed
E
temperature T =0.02. As we argued in the previous section
(seeEq.(13)),thereisclearlynothermodynamiclimitandthe

heating becomes more severe with the growth of the system
10
size.
y withthesystemsize(seeEq.(13))andclearlythereisno
t
si thermodynamic adiabatic limit. One canalsocheck that
n
e the dependence of∆ onL agreeswiththe predictionof
d E
Eq. (13). Although we do not show the graphs but we
y
g 1 checkedthatatzerotemperature∆ isbestfittedby√δ
r
e E
n dependence and is almost insensitive to the system size,
E
again in agreement with the analytic results (8).
We also performed numerical simulations for the two-
dimensionalBose-Hubbardmodel. Weworkwithsimilar
parameters as in the one-dimensional case, except that
atmuchhighertemperatureT =0.2andslightlysmaller
0.01 0.1 1
linearsizes. ThereasonwehadtochoosehigherT isthat
atagiventemperaturethermaleffectsintwodimensions
dominatethesystembehavioratmuchlargerlinearsizes
FIG. 1: Dependence of the energy density pumped into the
thaninonedimension. Wefindthattheresultsareagain
system during the ramp ∆ on the parameter δ for different
E in a good agreement with predictions of Eq. (13). In
initial temperatures and a fixed system size L = 128. The
other parameters are J = 1, n0 = 20, and U0 = 0.25. The particular, we find that ∆E ∝ δ1/3 at small δ and that
interaction increases in time according to U(t) = U0tanhδt. ∆ slowly grows with the system size consistent with
The top graph is in the normal scale and the bottom graph ∆E L1/3.
E ∝
isinthelog-log scale. Atfinitetemperaturesthedependence Summary and outlook. In this paper we analyzed
of ∆ on δ agrees with the prediction of Eq. (13): ∆ √3δ possible breakdown of the standard adiabatic (or quasi-
E E ∝
(thepower1/3canbereadfrom theslopeofthedependence stationary) approximation in low-dimensional gapless
∆ (δ)inthelowergraph). Similarlyatzerotemperaturethe
E systems. We explicitly analyzed behavior of harmonic
behavior agrees with Eq. (8): ∆ √δ. Note also that at
systemswhicharedescribedbyquadraticphonon-likeex-
E ∝
small rampsthecurvesat differenttemperaturesareequidis-
citations. Despite this particular choice, we point again
tant indicating that δ T again in accord with Eq. (13).
E ∝ that the Hamiltonian (6) describes a wide class of phe-
nomenasuchasphononsinsolids,magnonsinferro-and
InFig.1weshowthedependenceoftheenergypersite antiferromagnets, Bogoliubov excitations in superfluids
pumpedintothesystemduringtherampasafunctionof andsoforce. Using both analyticalandnumericalmeth-
the parameter δ for different temperatures T at a fixed odswe showedthatgenericallyone canhavethree possi-
system size L = 128. Note that even at very low tem- bleheatingregimes. ThefirstAregimeismean-fieldlike.
6
shown that the scaling (17) is valid below some critical
dimension,wheretheexponentofδsaturatesat2andwe
are back to the A regime. We emphasize that our gen-
1.6
eral arguments presented here do not exclude a possibil-
ityofcrossingacontinuoussecondorderphasetransition
1.4
y during the ramp. For example in Refs. [20,21,22,23,24],
t
nsi1.2 which considered crossings of a critical point in various
e integrablespin chains,the scaling(17) wasattributed to
d
y 1.0 the Kibble-Zurek mechanism26. However, we stress that
g
r neitherthederivationofthisscalingnortheanalyzedsys-
e
n0.8 L=32x32 temsdonotinvolveanytopologicaldefects. Theseworks
E
L=64x64 rather showed that the considered problems belong to
0.6 the nonanalytic regime B of non-adiabaticity. The fact
L=128x128
thatEq.(17)is correctis the consequenceofthe validity
0.4
of the perturbation theory. As we showed in this paper
0 2 4 6 8 the perturbative expression can break down due to the
bunching effect if the excitations near the critical point
havebosonicnature. Thenthescaling(17)willnolonger
FIG.3: Dependenceof∆ onδintwo-dimensionalsystemfor be valid. From results of this paper on general grounds
E
two different sizes. The temperature is fixed at T =0.2, the one can also expect stronger dependence of ∆n on δ
ex
otherparametersarethesameasinFig.2. Theasymptotical andpossibleemergenceofthenon-adiabaticCregimeat
behavior of ∆ at small δ agrees with δ1/3 dependence pre-
E finite temperatures.
dictedbyEq.(13). Also∆ slowlyincreaseswiththesystem
size consistent with theanaElytic prediction ∆ L1/3. An obvious outcome of our analysis is that one has to
E ∝ be verycarefulwithmakingstatements aboutadiabatic-
ity in isolated or nearly isolated gapless low-dimensional
Thereonecanapplysimpleargumentsshowingthatvar- systems. This can be important in many various situa-
ious thermodynamic observables are analytic functions tionsrangingfromrealizingproposalsonadiabaticquan-
of the ramp speed δ. This regime is typically realized at tumcomputationandpreparationofinteractingsystems
highdimensions. InthesecondregimeB,whichwecalled in a given state via slow ramps to inflationary cosmol-
non-analytic,theenergy,entropyandotherquantitiesde- ogy and black hole radiation. Perhaps cold atoms are
pend on δ in a nonanalytic way: (δ) (0)+β δ ν, the systems where our results canbe immediately tested
B B
E ≈ E | |
where ν < 2. The exponent ν depends on the universal in experiments. There one has all the necessary experi-
critical exponents characterizing the gapless phase. And mentaltoolslikeisolationfromtheenvironmenttogether
finally in low dimensions one can have a very unusual withhightunability ofparametersofthe systemandthe
regime C, which we called nonadibatic. In this regime possibility to perform tuning in real time27. However,
the limits of δ 0 and the system size L do not as coherent dynamic becomes more and more important
→ → ∞
commute and for example the energy density behaves as in other fields like nano-physics, quantum optics, cavity
(δ) (0)+β δ νLη. So no matter how slowly one QED, etc., we expect that our results should be relevant
B B
E ≈ E | |
ramps the system, in the thermodynamic limit one can to more and more potential applications.
never reach the adiabatic regime.
We didnotattempt to classify allpossiblemodels and
modelsituations where differentregimescanbe realized.
METHODS.
This is probably a very difficult task. But as a matter
of principle we proved that all three regimes are possi-
ble. We point that there is an interesting connection Zero temperature. The Hamiltonian (6) is quadratic
between this work and some earlier works where slow and thus the time evolution can be found exactly. We
dynamics across a quantum critical point was studied. first look into the zero temperature case. For quadratic
See Refs. [19,20,21,22]. In particular, using perturbative Hamiltoniansitiswellknownthatiftheinitialwavefunc-
approach,whichwealsodiscussinAppendixA,oneofus tion is gaussian, it will remain gaussian for an arbitrary
showed19thatthedensityofexcitationsgeneratedduring time dependence of the parameters of the Hamiltonian.
this process scales as Sinceweassumethatthe systemwasoriginallyprepared
in the ground state the initial wave function is
∆n δdν/(zν+1), (17)
ex
∝
where z and ν are the dynamical and correlation length 1 φ2
q
Ψ( φ )= exp , (18)
esuxlptosnwenetrechlaatrearctceornizfiirnmgetdhebypheaxsaecttramnestithioodns.21T,2h2e.seBruet- { q} Yq (2πσ0,q)1/4 "−4σ0,q#
this is nothing but the B regime of heating, which we
proposed in this paper. And indeed in Ref. [19] it was where σ = 1/(2q) κ /ρ . If κ changes with time,
0,q 0,q s
p
7
σ acquires time dependence: one has to realize that finding wave function was possi-
q
bleonlyduetoexactintegrabilityoftheharmonictheory.
idσq =2ρ q2σ2 1κ (t). (19) So one should look into quantities, which are robust to
dt s q − 2 q effects of weak nonintegrability. One of such quantities,
which has a particular importance in thermodynamics is
ThisisaRiccatiequation,whichcanbe explicitlysolved
the energy density of the system:
through Airy functions. We give details of this solution
in the Appendix B. √ρsκf ddq σqeff
It is straightforwardto check that the number of exci- q 1 , (23)
tations per mode q is related to σq via: E ≈ 2 Z (2π)d σqeq − !
where κ is the final compressibility. The integral above
1 σeff f
n = q 1 , (20) converges in all dimensions leading to Eq. (8).
q 2"σqeq − # Finite temperature case. One cangeneralizethe previ-
ous results to the situation where the system is initially
where σeff = 1/ (σ−1) and σeq is the equilibrium preparedinthethermalstatecharacterizedbysometem-
q ℜ q q
(groundstate)valueofσinthefinalstate. Wepointthat peratureT. Inthiscasetheexcitationsarepresentinthe
at large values of κ the ratio σeff/σeq does not depend system from the beginning and we will be interested in
q q
on κ (see details in the Appendix B). The asymptotical analyzing their enhancement during the ramp. Instead
expressionsforn inthelimitoflargeandsmallq canbe ofthewavefunctionwehavetodealwiththedensityma-
q
easily found from Eq. (20) and the solution of Eq. (19) trix, but essentially the derivation is similar to the zero
presented in Appendix B. In particular, temperature case. We present the details of the analysis
in the Appendix C. The result of these calculations is
δ2 surprisingly simple:
n (21)
q ≈ 64q2ρ κ3
s 0,q
σeff σeff
q q
=r , (24)
at q ρ κ3 δ and σqeq(cid:12) q σqeq(cid:12)
s 0,q ≫ (cid:12)T (cid:12)T=0
(cid:12) (cid:12)
q (cid:12) (cid:12)
π δ1/3 where (cid:12) (cid:12)
n (22)
q ≈ 32/3Γ2(1/3)q1/3(ρsκ03,q)1/6 r =coth q√κ0,qρs . (25)
q
2T
(cid:20) (cid:21)
in the opposite limit, where Γ(x) stands for the gamma-
function. We note that the asymptotics (21) exactly co- In other words one needs to take zero temperature
incides with the result of the perturbation theory, which asymptotics for the width of the wave function and mul-
can be obtained from Eq. (A1). tiply them by rq. In the zero temperature limit rq 1
≡
Thetotaldensityofexcitationsn canbeobtainedby andweobviouslyreproducetheresultsatT =0. Athigh
ex
integratingnq overq. Letusconsidertwodifferentcases. temperatures T ≫q√ρsκ0,q we have
If κ δ λ/ρ then n is dominated by the integral
0 ≫ s ex 2T
of Eq. (21) with the cutoff q δ and we immediately r (26)
recover thpe perturbative result∼, i. e. n δ d for d < q ≈ q√ρsκ0,q
ex
2 and n δ2 for d > 2 (in two dime∝ns|io|ns n
ex ex
Bδ2|rlengδim|).esSoa∝rweeproescsoibvleer.tThahteasgitauinatoionnly, wtyhpeereAoannedsttayrpt∝es azenrdottheumspσeqerffatduirveercgaesse.atLeqt→us0ombsuecrhvefatshtaetrtthheannuinmbtheer
inthenoninteractingregimeismorecomplicated. Indeed of excitations of a q-mode is
Eq. (22) indicates that the number of excited modes at
small q diverges in this case as q−4/3. In two dimensions 1 σqeff 1
n = 1 =r n + (r 1). (27)
and above this is an integrable divergence and we again q 2"σqeq − # q q|T=0 2 q −
recover the perturbative result n δd/4. However in
ex
∝
one dimension this integral diverges at small q and thus The last term is nothing but the equilibrium occupation
should be formally cutoff at q 1/L. In this case one of the q-mode at the initial time. Therefore we see that
finds that n δ1/3L1/3. Thi∼s is precisely the regime multiplying the zero temperature result for n by r we
ex q q
C, which was m∝issing in the perturbative analysis. We immediately obtainthe number of additionalexcitations
alreadyhighlightedthatthereasonwhytheperturbation in the q-mode ∆n generated during the ramp.
q
theory fails is the Bose enhancement of the transition
Using Eqs. (21) and (22) at q ρ κ3 δ we get
probabilities due to their bunching tendency. s 0,q ≫
Once we know the complete wave function we can, in q
δ2T
principle, find arbitrary observables like various correla- ∆n (28)
tionfunctions,responsetoexternalprobesetc. However, q ≈ 32q3ρ3/2κ7/2
s 0,q
8
and in the opposite limit which is the case if one starts in from weak interactions
κ 0, then
0
2π δ1/3T →
∆nq ≈ 32/3Γ2(1/3)q4/3ρ2s/3κ0,q. (29) nex ≈Adρd/δ8dλ/34d/8, (A2)
s
Now evenat finite value of κ the total density of gener-
0
where A is a numerical constant. It is easy to check
ated excitations ∆n divergesin one dimension ∆n d
ex ex
δ1/3L1/3 and we are in the non-adiabatic C regime. W∝e that above 8 dimensions the exponent of δ saturates at
2 and does not further change with the dimensionality.
note that formally n also diverges at finite tempera-
ex
Expression (A2) suggests that in this particular situa-
ture at d = 1 even in equilibrium, but this divergence
tion the nonanalytic regime B is realized in all physical
is much weaker and scales as the logarithm of the sys-
dimensions. In one dimension it is particularly hard to
tem size. In two dimensions we have ∆n δ and
ex
thus the nonanalytic B regime is realized. A∝nd|fi|nally reach the adiabatic regime since nex scales only as δ1/4.
in three dimensions and above ∆n δ2 and thus the WenotethatthescalinginEq.(A2)isconsistentwith
ex
meanfieldAregimeworks(inthreedi∝mensionsthereare the one obtained in Ref. [19] for the crossing of the sec-
logarithmic corrections to δ2 scaling). If the initial state ond order phase transition: nex δdν/(zν+1), where ν
∝
is the critical exponent characterizing divergence of the
is noninteracting then the divergence of n at small q is
q
correlation length. In our case there is a diverging heal-
much more severe: n q−10/3 and in all three spatial
q
dimensions the non-adia∝batic C regime is realized: ing length ξ ∼ λ/κ instead of the correlation length
(see Ref. [25] for details) so that ν =1/2 and given that
p
δ1/3T z = 2 in the noninteracting regime one immediately re-
∆nex =Adρ2s/3λL1/3+3−d. (30) covInertshtehaotppνo/s(iztνe +lim1i)t=(ii1)/w4.here the initial value of κ is
large δ κ ρ /λ the situation becomes more diverse.
Thusinordertokeeptheexcitationdensityconstantone ≪ 0 s
Thus for dimensions d<2 Eq. (A2) yields
has to scale δ as L3d−10. This means, for example, that p
in one dimension if the system size increases by a factor
δd
of two, then in order to keep heating (∆n ) the same n A′ . (A3)
one has to decrease δ by a factor of 128! ex ex ≈ dρds/2κ30d/2
Acknowledgements We would like to acknowledge
On the other hand for d>2 the exponent saturates and
E. Altman, E. Demler, S. Girvin, V. Gurarie, M. Lukin,
we have
V. Pokrovsky, and N. Prokof’ev for useful discussions.
A.P.wassupportedbyAFOSRYIPandpartiallybyNSF λ1−d/2κd
under Grant PHY05-51164. V. G. is partially supported nex ≈A′d ρ 0 δ2. (A4)
s
by the Swiss National Science Foundation and AFOSR.
A.P. also acknowledges Kavli Institute for Theoretical Intwodimensions thereis anadditionallogarithmiccor-
Physics for hospitality. rectionto the scaling (A3). We see that in this situation
thecriticaldimensionabovewhichthe meanfieldregime
holds is d⋆ =2.
SUPPLEMENTARY MATERIAL The present analysis can be generalized to other situ-
ations. For example, in the case of ferromagnets κ 0
0
≡
APPENDIX A: FERMI GOLDEN RULE andthenonecantuneλ. Thenonefindsthatnex δd/2
ANALYSIS and the critical dimension is d⋆ = 4. We commen∝t that
one can also consider other scenarios of varying κ with
time. Forexample,ifκ (δt)r thenitiseasytoseethat
The easiest way to find the density of excitations n
ex ∝
n δdr/2(r+1). Asrincreasesthescalingofthedensity
produced during a slow increase of κ(t) in Hamilto- ex
of ex∝citations interpolates from δd/4 to δd/2 and changes
nian (6) is to use Fermi golden rule analysis. Indeed,
d⋆ from 8 to 4.
since we expect that n is small at small δ one can ex-
ex
This simple perturbative analysis shows the existence
pectthattheperturbationtheoryinδ shouldgiveagood
ofA(B)regimesfordimensionsabove(below)somecrit-
estimate ofn (δ) at smallδ. Using a generalexpression
ex
icalvalued⋆. Thisanalysis,however,missestheexistence
forthe density ofexcitationsderivedinRef. [19]wefind:
of the C regime. To justify the validity of the applica-
tion of the Fermi golden rule one has to require that
1 ddq ∞ dξ 4i 2 the probability of excitation of each momentum mode is
n = exp √ρ qξ3/2 , small. This requirement breaks down at low energies as
ex 32Z (2π)d (cid:12)(cid:12)Zκ0,q ξ (cid:18)3δ s (cid:19)(cid:12)(cid:12) can be readily seen from Eq. (A1). In the case when
(cid:12) (cid:12)(A1) the excitations have Fermionic character, which is e.g.
(cid:12) (cid:12)
where κ0,q = κ0 +λq(cid:12)2. This expression gives diff(cid:12)erent the case for crossing the critical point in the transverse
asymptotics in two opposite limits. (i) If δ κ2 ρ /λ, field Ising model or the XXZ chain4, the mistake of the
≫ 0 s
p
9
perturbative treatment is a simple factor of the order of The initial condition determining α is:
one (see Refs. [19,20,21,22]). The Goldstone modes de-
scribed by the Hamiltonian (6) on the other hand are Bi′( κ˜0,q)+αqAi′( κ˜0,q)
κ˜ =i − − . (B8)
harmonic oscillators and thus behave as bosons. Bosons 0,q Bi( κ˜ )+α Ai( κ˜ )
0q q 0,q
− −
unlike fermions have a bunching tendency, i. e. transi- p
tionprobabilitiescanbesignificantlyenhancedcompared This equation can be inverted to give
to the golden rule prediction.
κ˜ Bi( κ˜ ) iBi′( κ˜ )
0,q 0,q 0,q
α = − − − . (B9)
q − κ˜ Ai( κ˜q) iAi′( κ˜ )
p 0,q − 0 − − 0,q
APPENDIX B: EVOLUTION OF THE WAVE
FUNCTION UNDER THE RAMP AT ZERO In the limit κp˜ 1 these equation yields:
0,q
TEMPERATURE ≪
32/3Γ2(1/3)
α √3+i κ˜ . (B10)
HerewepresentsomedetailsofthesolutionofEq.(19). q ≈ π 0,q
For convenience we write it here again: p
Consequently
dσ 1
i dtq =2ρsq2σq2− 2κq(t). (B1) σqeff 2π 1 . (B11)
σqeq ≈ 32/3Γ2(1/3) κ˜0,q
This equationcanbe simplifiedby firstchanginginde-
pendent variablet to κq(t) and then by simple rescaling: In the opposite limit κ˜q 1 one fipnds α i and
0 ≫ q ≈
κ=κ˜ 3δρ2/s3q2, σq =σ˜q23δ1ρ/2s3q4, q =q˜ρ1s/δ81λ/43/8. (B2) σσqeqeffq ≈1+ 32κ˜103,q. (B12)
p p
Under these rescalings we also have κ˜ =κ˜+q˜8/3. Then
q
one can check that Eq. (19) is equivalent to APPENDIX C: EVOLUTION OF THE DENSITY
MATRIX AT FINITE TEMPERATURES.
dσ˜
i q =σ˜2 κ˜ . (B3)
dκ˜ q − q
q Wechoosetorepresentthedensitymatrixcorrespond-
ing to the initial thermal state in the Wigner form16,28.
ThisRiccatiequationcanbeexplicitlysolvedintermsof
For the harmonic system described by the Hamilto-
Airy functions Ai and Bi:
nian(6) onecanshowthatthis density matrixfactorizes
Bi′( κ˜ )+α Ai′( κ˜ ) into the product of Gaussians:
q q q
σ˜ = i − − , (B4)
q
− Bi( κ˜q)+αqAi( κ˜q) 1 φ 2 σ Π 2
− − W = exp | 0,q| 0,q| 0,q| , (C1)
0
4πr −4σ r − 4r
whereαq is anintegrationconstant,whichis determined Yq q (cid:20) 0,q q q (cid:21)
fromtheinitialconditions. Inthelimitκ˜ ignoring
q
unimportant fast oscillating terms we find→∞ where
1 2ℑαq . (B5) rq =coth q√κ20T,qρs] . (C2)
ℜ(cid:20)σ˜q(cid:21)→ κ˜q[1+|αq|2] (cid:20) (cid:21)
It is well known that in the noninteracting problem the
Note that the realpart ofp1/σ determines ψ 2 andthus
q | | timeevolutionofthefieldsφq andΠq isdescribedbythe
the probability distribution of the corresponding phase. classical equations of motion28. In particular,
Thefactthat1/σ 0asκ˜ shouldnotbesurpris-
q q
→ →∞
ing. Indeed the width of the ground state wave function d 1 dφ
in scaled variables is q +ρsq2φ=0, (C3)
dt κ dt
(cid:20) q (cid:21)
1 κ
σeq = κ˜ . (B6) subject to initial conditions
q q ≈ 2q ρ
r s
p φ (t=0)=φ , φ˙ (t=0)=κ Π , (C4)
q 0,q q 0q 0,q
Theprobabilityofexcitationsinthesystemisdetermined
bytheratioofσq andσeq,whichtakesawelldefinedlimit where φ0,q and Π0,q are randomly distributed according
at κ→∞. Introducing σqeff =1/ℜ(σq−1) we find to (C1). The other important feature of Gaussian en-
semblesis that inthe absenceofinteractionsthe Wigner
σqeff 1+ αq 2 distribution (C1) always preserves its Gaussian form.
= | | . (B7)
σeq 2 α Therefore finding φ2(t) and Π2(t) is sufficient to fix
q ℑ q h q i h q i
10
the whole distribution function at arbitrary times. Al- FIG. 4: Dependence of the energy density ∆ on the δ at
E
ternatively one can directly solve the Liouville equation zero temperature with and without the quantum correction
forthe densitymatrixinthe Wignerform28 andcome to (D4). For the details of the calculation and the parameters
the same conclusion. of the problem see Fig. 1. Obviously at small values of δ the
TWAbreaksdownandonehastoincludethecorrection(D4).
A general solution of Eq. (C3) is:
Intheclassicallimitthebosonicfieldsψ⋆andψ satisfy
φq(κ˜q)=C1Ai′( κ˜q)+C2Bi′( κ˜q), (C5) discrete Gross-Pitaevskiiequations: j j
− −
where as in Appendix B we changedthe variablesfromt
toκ˜ . TheintegrationconstantsC andC canbefound
q 1 2
from the initial conditions: i∂ψj = J ψ +U(t)ψ2 ψ . (D1)
∂t − i | j| j
C = πκ0,q dφ0,qBi′( κ˜ ) πφ Bi( κ˜ ), (C6) iX∈Oj
1 κ˜2 dκ − 0,q − 0,q − 0,q
0,q 0,q
πκ dφ
C =πφ Ai( κ˜ ) 0,q 0,qAi′( κ˜ ). (C7)
2 0,q − 0,q − κ˜2 dκ − 0,q Here the sum in the first term is taken over the nearest
0,q 0,q
neighbors of the site j. In the leading order in quan-
Fromtheseexpressionsitiseasytofindtheasymptotical tum fluctuations,so called truncated Wigner approxima-
behaviorof φ2 atlargeκ˜ andthusfindthewidthofthe tion (TWA), the fields ψ and ψ⋆ are subject to ran-
h qi j j
distribution σeff: dom initial conditions, which are distributed according
q
to the Wigner transform of the initial density matrix
σσqeqeffq = π2 rκ˜q0,q κ˜0,qBi2(−κ˜0,q)+κ˜0,qAi2(−κ˜0,q) aWbl(eψΩj⋆,(ψaj†j),.ajT)hiesegxivpeenctbaytiotnhevaalvueeraogfeanofatrhbeitrcaorryreosbpsoenrdv--
p +((cid:2)Bi′(−κ˜0,q))2+(Ai′(−κ˜0,q))2 .(C8) itnogr)WΩecyl(lψsj⋆y,mψjb)olon(futhlleyssoylmutmioentsriozefdthfeorGmroosfs-tPhietaoepvesrkai-i
It is straightforward to verify that in the zero te(cid:3)mper- equations:
ature limit r 1 this expression coincides with the
q
→
previous result (see Eqs. (B7) and (B9)).
APPENDIX D: QUANTUM EXPANSION OF THE
Ω(t) = Dψ⋆Dψ W(ψ⋆,ψ )Ω (ψ⋆(t)ψ (t)). (D2)
DYNAMICS OF A BOSE-HUBBARD SYSTEM. h i0 j j j j cl j j
Z
We find that this levelofapproximationgivesveryac-
1 curateresultsinmostofoursimulationsdescribedinthis
TWA paper. However,atzerotemperaturecaseitbreaksdown
y Quantum correction forveryslowrampsandwehadtoincludethenextquan-
sit tumcorrectiontotheTWA.Thelattermanifestsitselfin
n
e the form of a single infinitesimal quantum jump during
d
y the evolution:
g
r
e
n
E
ψ (t′) ψ (t′)+ǫ +iǫ . (D3)
i i 1 2
→
0.1
0.1 1
The quantum correction is the evaluated as a nonlinear
response of Ω to such a jump14:
cl