Information propagation for interacting particle systems

0.25 MB·English

by  Norbert Schuch

Informationpropagationforinteractingparticlesystems
Norbert Schuch,1 Sarah K. Harrison,2 Tobias J. Osborne,3,4 and Jens Eisert4,5
1InstituteforQuantumInformation,CaliforniaInstituteofTechnology,MC305-16,PasadenaCA91125,USA
2DepartmentofMathematics, RoyalHollowayUniversityofLondon, Egham,Surrey, TW200EX,UK
3Institutfu¨rTheoretischePhysik,Leibniz-Universita¨tHannover,Appelstr.2,30167Hannover,Germany
4Institute for Advanced Study Berlin, 14193 Berlin, Germany
5Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
We show that excitations of interacting quantum particles in lattice models always propagate with a finite
speed of sound. Our argument is simple yet general and shows that by focusing on the physically relevant
observablesonecangenerallyexpectaboundedspeedofinformationpropagation.Theargumentappliesequally
toquantumspins,bosonssuchasintheBose-Hubbardmodel,fermions,anyons,andgeneralmixturesthereof,
onarbitrarylatticesofanydimension. Italsopertainstodissipativedynamicsonthelattice,andgeneralizesto
thecontinuumforquantumfields.OurresultcanbeseenasameaningfulanalogueoftheLieb-Robinsonbound
1 forstronglycorrelatedmodels.
1
0
2
How fast can information propagate through a system of
t
n interacting particles? The obvious answer seems: No faster
a than the speed of light. While certainly correct, this is not
J
theansweroneisusuallylookingfor. Forinstance,inaclas-
7
sical solid, liquid, or gas, perturbations rather propagate at
2
thespeed ofsound, whichis determinedby theway thepar-

] ticles in the system locally interact with each other, without
h
any reference to relativistic effects. We would like to under-
p
stand whether a similar “speed of sound” exists for interact-
-
t ingquantumsystems,limitingthepropagationspeedoflocal-
n
a ized excitations, i.e., (quasi-)particles. For interacting quan- FIG.1.Schematicrepresentationofthe“lightcone”ofparticlesini-
u tum spin systems, such a maximal velocity, known as the tially placed into a region R of a lattice (yellow circles) and then
q Lieb-Robinson bound [1–4], has indeed been shown. While propagatingintimetinawaygovernedbyaninteractingquantum
[ model,outsideofwhichtheinfluenceoftheseparticlesisexponen-
itseemsappealingthatthereshouldalwaysbesuchabound, tiallysuppressed.
2 systems of interacting bosons can show counterintuitive ef-
v fects, in particular since the interpretation of excitations in
6
termsofparticlesisnolongerfullyjustified;infact,anexam- TheoriginalLieb-Robinsonboundalreadyappliesinavery
7
5 pleofanon-relativisticsystemwherebosonscondenseintoa generalsetting,namely,toanylow-dimensionalquantumspin
4 dynamical state which steadily accelerates has recently been system, and to any fermionic system confined to a lattice. It
0. constructed [5]. This example suggests the disturbing possi- isthereforetemptingtoextendtheoriginalargumenttoother
1 bilitythatourintuitioniswrong,andonlyrelativisticquantum settings,inparticular,tosystemsofinteractingbosons;unfor-
0 theorycanprovideaproperspeedlimit. tunately,allattemptstodosohaverunintoinsuperablediffi-
1
Therearemanyimportantreasons,boththeoreticalandex- culties for systems with nonlinear interactions, including the
:
v perimental, to investigate information propagation bounds in Bose-Hubbardmodel. Thereasonforthefailureoftheorig-
i interactingparticlesystems. Itturnsoutthatsuchboundslead inal Lieb-Robinson argument is fundamentally connected to
X
directly to important, general results concerning the cluster- the unboundedness of the creation operator for bosons: The
r
a ing of correlations in equilibrium states [2]. Lieb-Robinson Lieb-Robinson velocity depends on the norm of the interac-
bounds facilitate the simulatability of strongly interacting tion,whichisunboundedfor,e.g.,bosonshoppingonalattice,
quantum systems—the mere existence of a Lieb-Robinson andexampleswithoutaspeedlimitcanbeconstructed[5].
boundforaquantumsystemcanbeusedtodevelopgeneral, In this Letter, we show how these difficulties can be over-
efficient, numerical procedures to simulate the dynamics of come by considering the right question concerning the prop-
latticemodels[6].Fromamorepracticalperspective,newex- agationofinformation. Ourapproachallowsustodetermine
perimentsallowonetoexplorethenon-equilibriumdynamics Lieb-Robinson type bounds for the maximal speed at which
of ultracold strongly correlated quantum particles—bosonic, informationcanpropagatethroughsystemsofinteractingpar-
fermionic, or mixtures thereof—in optical lattices with un- ticles in a very general scenario: In particular, it applies to
precedented control [7, 8]. In such experiments, it is impor- systems of interacting bosons, as well as to fermions, spins,
tanttounderstandhowtheparticlesmove:Forexample,when anyons, or mixtures thereof, both on lattices and in the con-
studyinginstancesofanomalousexpansion,itisfarfromclear tinuum. Moreover,itcanalsobeappliedbeyondHamiltonian
aprioriwhetheritispossibletoidentifyameaningfulspeed evolution, suchastosystemsevolvingundersomelocaldis-
ofsoundatall. sipativedynamics.
2
Thetypeofsystemwehaveinmindisexemplifiedbythe evolveunderHˆ . Tothisend,wederiveaboundontherate
BH
Bose-Hubbardmodel,amodelofbosonshoppingonanarbi- atwhichα (·)changes,whichinturnleadstoaboundonthe
j
trary lattice G of any finite dimension and interacting via an velocityatwhichparticlescanpropagatethroughthesystem.
on-siterepulsion, Itholdsthat
Hˆ =−τ (cid:88)(ˆb†ˆb +h.c.)+U (cid:88)nˆ (nˆ −1)−µ(cid:88)nˆ , α˙j(t)=−itr(cid:0)nˆj[HˆBH,ρ(t)](cid:1)=−itr(cid:0)[nˆj,HˆBH]ρ(t)(cid:1)
BH (cid:104)j,k(cid:105) j k 2 j j j j j =2τ (cid:88)Im(cid:104)tr(cid:0)ˆb†ˆb ρ(t)(cid:1)(cid:105), (3)
k j
(1)
(cid:104)j,k(cid:105)
where the first summation is over neighboring sites on the
lattice, ˆb is the boson annihilation operator for site j, and where the summation runs over all sites k neighboring j,
j
nˆ = ˆb†ˆb is the number operator. The natural distance in d(j,k) = 1. Sinceweareonlyinterestedinanupperbound
j j j
onthisrateofchange,wenowconsider|α˙ (t)|andapplythe
the lattice will be denoted by d(·,·), e.g., d(j,k) = |j −k| j
triangleinequalitytoobtain
foraone-dimensionalchain. Whilewewill,forclarity,focus
oreucrtldyisgceunsesrioalnizoenttohemBodoesels-Hofubthbearfdormmodel,ourargumentsdi- |α˙j(t)|≤2τ (cid:88)(cid:12)(cid:12)tr(ˆb†kˆbjρ(t))(cid:12)(cid:12). (4)
(cid:104)j,k(cid:105)
S
Hˆ =−τ(cid:88)(cid:88)(ˆb† ˆb +h.c.)+f({nˆ ,...,nˆ } ), To bound this term we use the operator Cauchy-Schwarz in-
s,j s,k 1,j S,j j∈G
s=1(cid:104)j,k(cid:105) equality,viewing
(2)
wheretheˆbs,j areannihilationoperatorsforbosons,fermions, tr(ˆb†kˆbjρ(t))=(cid:104)ˆbkρ1/2(t),ˆbjρ1/2(t)(cid:105)
orevenanyonsofspeciess = 1,...,S atsitej, andnˆ =
ˆb† ˆb ;thespeciescouldforinstancerefertoaninternasl,sjpin as a Hilbert-Schmidt scalar product of ˆbjρ1/2(t) and
s,j s,j ˆb ρ1/2(t), where ρ1/2(t) is the matrix square root of ρ(t).
degree of freedom. The interaction between the particles is k
Thisgivesriseto
characterized by f which can be an arbitrary function of the
local densities, and may involve higher moments of the par-
ticle number, or even non-local interactions. Moreover, our (cid:12)(cid:12)tr(ˆb†kˆbjρ(t))(cid:12)(cid:12)≤(cid:16)tr(ˆb†kˆbkρ(t))tr(ˆb†jˆbjρ(t))(cid:17)1/2.
argumentalsoappliestotime-dependentHamiltoniansofthis
form,aslongasthetunnelingamplitudeτ(·)isbounded. Combining this with Eq. (4), we obtain a set of coupled dif-
ThescenarioweconsiderisdescribedbytheBose-Hubbard ferentialinequalities
model on a lattice G, where in the initial state all sites are
empty(i.e.,(cid:104)nˆ (cid:105)=0)exceptforthesitesinaregionRwhich |α˙j(t)|≤2τ (cid:88)(αj(t)αk(t))1/2 , (5)
j
can be in an arbitrary initial state with finite average particle (cid:104)j,k(cid:105)
number. NotethattheregionRmayverywellencompassthe √
which,using xy ≤(x+y)/2,yieldsthelinearizedsystem
major part of the lattice. What we are interested in is how
fast these bosons will travel into the empty part G\R of the (cid:18) (cid:19)
(cid:88)
lattice, as a function of the distance d(·,·) on the underlying |α˙j(t)|≤τ Dαj(t)+ αk(t) ,
graph. Inparticular,wewouldliketofinda“speedofsound” (cid:104)j,k(cid:105)
forthebosons,thatis,avelocityv suchthatforanyregionS
whereDisthemaximalvertexdegreeoftheinteractiongraph.
inG\R withd(S,R) ≥ l [i.e.: d(s,r) ≥ l ∀s ∈ S, r ∈ R],
Weareinterestedintheworst-casegrowthofα (t)astpro-
and for all times t for which vt < l, the expectation value j
gresses. Thiswilloccurwhenwehaveequalityintheabove
of any observable Oˆ on S is equal to the expectation value
S expression(i.e.,thederivativeisaslargeaspossible),andthus
ofthevacuum,uptoacorrectionwhichdecaysexponentially
a bound γ (t) ≥ α (t) is given by the solution of the linear
awayfromthelightcone,eγ(vt−l). k k
systemofdifferentialequations
To start, we consider the Bose-Hubbard model Hˆ and
BH
focusonmeasurementsofthelocalparticlenumberoperators (cid:18) (cid:88) (cid:19)
γ˙ (t)=τ Dγ (t)+ γ (t)
nˆ . This corresponds to looking for bosons at the initially j j k
j
emptysites,andthuscapturesthemostnaturalnotionofpar- (cid:104)j,k(cid:105)
ticlespropagatingintoaregion. Letusdenotetheinitialstate
whichfulfillsγ (0)=α (0). Thissolutionhastheform
byρ(0),whichevolvesaccordingto j j
ρ˙(t)=−i[Hˆ ,ρ(t)] (cid:126)γ(t)=eDτteτMt(cid:126)γ(0),
BH
fort≥0. Asweareinterestedinthespeedatwhichparticles whereM istheadjacencymatrixofthelattice,i.e.,Mj,k =1
intheBose-Hubbardmodelpropagate,letustrytounderstand ifd(j,k)=1and0otherwise,and(cid:126)γ :=(γk)k∈L. Thisyields
howthelocalparticledensities anupperbound
α (t)=tr(nˆ ρ(t)), j ∈G, α(cid:126)(t)≤eDτteτMtα(cid:126)(0)
j j
3
for the expected particle number at time t for any site, for particlenumberoperator. ThisprovesaLieb-Robinsonbound
α(cid:126) :=(α ) . forthehighermomentsoftheparticlenumberoperator.
k k∈L
In order to understand how quickly particles propagate Let us now turn our attention towards arbitrary local ob-
from the initially occupied region R into a region S with servables Aˆ . Any such observable can be written as Aˆ =
j j
d(R,S) ≥ l, we need to consider the off-diagonal block of (cid:80) c (ˆb†)pˆbq,andwehavethusthat
p,q p,q j j
eDτteτMt correspondingtothosetworegions. Thus,inorder
toobtainalightconewithanexponentialdecayexp(vt−l) (cid:12)(cid:12)tr(Aˆjρ(t))(cid:12)(cid:12)≤(cid:88)|cp,q|(cid:12)(cid:12)tr[(ˆb†j)pˆbqjρ(t)](cid:12)(cid:12) (8)
outsideit,weneedtounderstandhowrapidlytheoff-diagonal p,q
eelτeMmte.nTtshiosfctahnebbeanddoendembyatarpixplMyinggrTowheournedmer6exfrpoomneRnetifa.t[i9o]n, ≤(cid:88)|cp,q| (cid:16)tr(cid:2)(ˆb†j)pˆbpjρ(t)(cid:3)tr(cid:2)(ˆb†j)qˆbqjρ(t)(cid:3)(cid:17)1/2 .
whichyieldsforthe(i,j)-thelementofexp(τMt)thebound p,q
Inturn,forp>0,
[exp(τMt)] ≤Cev0t−d(i,j)
i,j tr(cid:2)(ˆb†)pˆbpρ(t)(cid:3)=tr(cid:2)nˆ (nˆ −1)···(nˆ −p+1)ρ(t)(cid:3)
j j j j j
with velocity v = χ∆τ, where χ ≈ 3.59 is the solution of p
χlnχ=χ+1,0∆=(cid:107)M(cid:107)∞/2dependsonthelatticedimen- =(cid:88)dr,pαj(r)(t)≤C˜pevt−l (9)
sion,andC =2χ2/(χ−1)≈10.Togetherwiththeprefactor r=1
exp(Dτt),thisgivesaLieb-Robinsonvelocityv = v0+Dτ by virtue of Eq. (6), for some constant C˜ . If p = 0, we
p
[10]. For the scenario of an empty lattice with particles ini-
triviallyhavetr[ρ(t)]=1. Together,thisyieldsabound
tially placed in a region R, this implies that for any j with
d(j,R)≥l, (cid:12)(cid:12)tr(Aˆjρ(t))(cid:12)(cid:12)≤C(cid:48)evt−l
α (t)≤Cevt−l(cid:88)α (0)=CN evt−l , (6) ifc0,q =cp,0 =0forallpandq,and
j k 0
k∈R (cid:12)(cid:12)tr(Aˆjρ(t))(cid:12)(cid:12)≤C(cid:48)e(vt−l)/2
i.e., uptoanexponentiallysmalltail, theparticlespropagate (cid:80)
otherwise,wherewehaveassumedthat |c |isfinite,and
withaspeednofasterthanv,independentoftheirinitialstate. p,q
Here,N =(cid:80) α (0)=(cid:104)Nˆ(cid:105)isthetotalnumberofparti- used that w.l.o.g. c0,0 = 0. In both cases, this means that
0 k∈R k outsidethelightconegivenbyvt = l,tr(Aˆ ρ(t))decaysex-
clesinthesystem(i.e.,theexpectationvalueofthetotalpar- j
ticlenumberoperatorNˆ =(cid:80) nˆ ). Notethatwhilethis(un- ponentially;however,thedecayisondoublethelengthscale
j j inthelattercase.
surprisingly) means that the strength of the signal observed
Finally, observables acting on more than one site can be
may depend on the number of bosons initially put into the
boundedanalogouslytothelocalcase: Anytwo-siteoperator
system, the maximum propagation speed v does not depend
actingonsitesj,kcanbewrittenasthesumoftermsAˆ Aˆ ,
onN . Infact,forapurelyharmonicone-dimensionalmodel j k
0
and
forU = 0,theexactspeedofsoundisindeedlinearinτ,so
theHaabvoivnegbuonudnedrsitsotoidghhtouwpttooaosbmtaainllcaobnostuanndtporneftahcetopr.ropa- (cid:12)(cid:12)tr(AˆjAˆkρ(t))(cid:12)(cid:12)≤(cid:16)tr(Aˆ†jAˆjρ(t))tr(AˆkAˆ†kρ(t))(cid:17)1/2 .
gation speed of particles, we now turn to more general ob-
The terms on the r.h.s. are local observables which can be
servables. First, let us show how we can bound the higher
bounded as before by exp(vt−l), yielding the same expo-
momentsoftheparticlenumberoperator. Forp≥1,
nential bound for two-site—and recursively for many-site—
α(p)(t) = tr(cid:0)nˆpρ(t)(cid:1) observables. (Note that there exist cases where terms which
j j arebounded byexp[(vt−l)/2] onlyappear, and inaddition
= (cid:88)tr(cid:0)nˆ nˆp−1P ρ(t)P (cid:1) one of the Aˆ’s above could be the identity. Thus, bounds of
j j N N
N theformexp((vt−l)/κ)canoccur,whereκcangrowexpo-
≤ (cid:88)tr(cid:0)nˆ Np−1P ρ(t)P (cid:1) (7) nentiallyintheblocksize.This,however,stillimpliesthatthe
j N N
signalisexponentiallysmalloutsidethelightcone.)
N
While we have illustrated our arguments for the Bose-
(≤6)(cid:88)Np−1(cid:0)CNevt−l(cid:1)tr(ρ(t)) Hubbardmodel,theygeneralizestraightforwardlytothemore
N generalclassofmodelsdescribedbyEq.(2). First,itisclear
= C(cid:104)Nˆp(cid:105)evt−l , that we can replace the on-site replusion and chemical po-
tential in the Bose-Hubbard model by any type of interac-
where P projects onto the subspace with a total of N par- tion (even a non-local one) which only depends on the par-
N
ticles, and we have used that Eq. (6) applies to each sub- ticle numbers, since any such term vanishes in the commu-
spacewithfixedparticlenumberindependentlyastheHamil- tator [nˆ ,Hˆ] in Eq. (3). Second, for systems that contain
j
tonian commutes with P . Here, (cid:104)Nˆp(cid:105) denotes the (time- severaltypesofbosonsthesameargumentsapply: Suchsys-
N
independent)expectationvalueofthep-thmomentofthetotal tems can be modelled using multiple copies of the original
4
graph, each of which supports the hopping of one individ- oftheproblemandgaverisetoanexactlysolvableworst-case
ual boson species, and one obtains independent differential bound.
inequalities for the particle densities α (t) = tr[nˆ ρ(t)] Theideaofstudyinginformationpropagationbyrestricting
j,s j,s
foreachspecies. to a specific set of observables and investigating the result-
Beyondgeneralbosonicmodels,ourargumentsalsoapply ingworst-casedifferentialequationcanalsobeappliedtothe
tofermionsandmixturesofbosonsandfermions[11],andin study of continous systems. This can be done either by tak-
fact even to anyonic systems. Again, in a first step one can ing an appropriate continuum limit of a lattice model, or by
decouple the individual species of particles (which mutually directlyconsideringacorrespondingdifferentialequationfor
commute) to hop on independent graphs. Then, it is easy to theparticledensitywhichiscontinuousinspace.
checkthatourargumentsworkindependentlyofthestatistics Acknowledgements.—This work was supported by the EU
oftheparticles,since[nˆ ,Hˆ]inEq.(3)evaluatestothesame (COMPAS, MINOS, QESSENCE), the EURYI, the Gordon
j
expression in terms of the fermionic (anyonic) creation and and Betty Moore Foundation through Caltech’s Center for
annihilation operators. Even better, fermionic and anyonic the Physics of Information, the National Science Foundation
systems yield stronger bounds for the higher moments, and under Grant No. PHY-0803371, and the ARO under Grant
thusforthescenarioofgenerallocalobservables: InEq.(7), No. W911NF-09-1-0442. Part of this work was done at the
nˆp−1 can be bounded by 1 instead of Nˆp−1, which yields a Mittag-Leffler-Institute.
j
bound α(p)(t) ≤ CN evt−l on the higher moments. Corre-
j 0
spondingresultsalsofollowforspinsystems,asthesecanbe
describedashardcorebosons.
Ourargumentsworknotonlyforunitarytheories,butalso
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Toconclude,wehaveproventhatthereisaspeedlimitfor
[10] There are two ways to obtain better bounds on the velocity.
√
thepropagationofinformationinasystemofinteractingpar- First,wecanbound xy ≤ 1(λx+y/λ),whichgivesave-
2
ticles.Thisresultisparticularlyrelevantforthecaseofbosons locity bound λv +Dτ/λ for any λ > 0. Second, one can
0
onalattice, asbosonicsystemscannotbeassessedusingthe solve the non-linear differential inequality (5) by substituting
established techniques of Lieb-Robinson bounds due to the αj(t)=:βj(t)2,whichgiveslinearinequalities
unboundedness of the bosonic hopping operator. Our argu- β˙ (t)≤τ (cid:88)β (t)
j k
mentappliesequallytobosonic,fermionic,anyonic,andspin
(cid:104)j,k(cid:105)
systems,aswellasmixturesthereof,witharbitraryinteraction
(cid:112)
withinitialconditionsβ(0)= α(0).Theworst-casesolution
termsbetweentheparticles,andcanbegeneralizedtoalsoad-
of this system is β(cid:126)(t) = eτMtβ(cid:126)(0), which, using the previ-
dresssystemswithdissipation.
ous estimate of eτMt, yields a velocity v . In order to obtain
Thekeypointthatallowedustomakestatementsaboutthe 0
bounds on α (t), we need to square this bound. On the one
j
propagation of information in bosonic systems beyond Lieb-
hand, this implies that the correlations outside the light cone
Robinsonboundswasfirsttofocusonasubsetofobservables decayase2(vt−l);however,italsoyieldsanunfavorabledepen-
relevanttodetectingthepropagationofparticles, namelythe denceoftheprefactorontheinitialconditions,((cid:80) (cid:112)α (0))2,
j j
number of particles present at each site, and second to de- whichcandivergeforafixednumberofbosonsastheregionR
viseaclosedsystemofinequalitiesboundingtheevolutionof grows.
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