O(N) symmetry-breaking quantum quench: Topological defects versus quasiparticles

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by  Michael Uhlmann

O(N) symmetry-breaking quantum quench: Topological defects versus quasiparticles
Michael Uhlmann1, Ralf Schützhold2, and Uwe R. Fischer3,4
1Department of Physics and Astronomy, University of British Columbia, Vancouver B.C., V6T 1Z1 Canada
2Fachbereich Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany
3Eberhard-Karls-Universität Tübingen, Institut für Theoretische Physik, D-72076 Tübingen, Germany
4Seoul National University, Department of Physics and Astronomy, 151-747 Seoul, Korea
Wepresentananalyticalderivationofthewindingnumbercountingtopologicaldefectscreatedby
0 anO(N)symmetry-breakingquantumquenchinN spatialdimensions. Ourapproachisuniversalin
1 thesensethat wedonot employany approximationsapart fromthelarge-N limit. Thefinalresult
0 is nonperturbativein N, i.e., it cannot beobtained byan expansion in 1/N, and we obtain far less
2
topological defects than quasiparticle excitations, in sharp distinction to previous, low-dimensional

n investigations.
a
J
7 I. INTRODUCTION further approximations will be needed, i.e., our results
2 will be quite universal. Moreover, similar to analogous
large-N approachesincondensedmatterandfieldtheory
In contrast to the vast amount of literature regarding
] (assuming that there is no critical value of N where the
h static properties (e.g., universal scaling laws) of phase
systemchanges drastically),we expect our results to ap-
c transitions – both thermal and at zero temperature –
e ply qualitatively also to finite N (e.g., N =3), which are
we are just starting to understand their dynamical fea-
m accessible to experimental tests. Bose-Einstein conden-
tures, especially the behaviour during a time-dependent
- sweep(quench)throughthecriticalpoint. Thistopichas sates,inparticular,permitthetime-resolvedobservation
t of the defect formation mechanism due to the compar-
a attractedincreasinginterestinrecentyears,see,e.g.,[1–
t atively long req-equilibration time scales of these dilute
s 9] and, again, some universal properties became evident
t. [10]. For example, during a symmetry-breaking(second- ultracold quantum gases [18, 19].
a
order)dynamicalphasetransition,thedivergingresponse
m
time inevitably entails nonequilibrium processes and so
- the initial quantum (and thermal) fluctuations are am- II. EFFECTIVE ACTION
d
n plified strongly, ultimately determining the final order
o parameter distribution. If the final phase permits topo- As a first step, we construct a general effective action
c logical defects (e.g., vortices in superfluids), they will for an O(N)-model in terms of the N-component field
[ generally be created in such a quench via (the quantum φ=(φ ,...,φ ),whichdeterminestheorderparameter.
1 N
2 version of) the Kibble-Zurek mechanism. The latter oc- To this end, we start from the equation of motion with
v curs in many diverse physical settings, for instance, in an arbitrary function f
7 nonequilibrium phase transitions during the early uni-
7 verse [11–13] or in condensed matter systems [14]. φ¨=f(φ,φ˙,∇2φ,∇2φ˙,∇4φ,∇4φ˙,...). (1)
8
Unfortunately,duetotheinherentcomputationalcom-
0 In order to avoid run-away solutions and to facilitate a
. plexityofsuchscenarios,explicitcalculationsaredifficult
5 proper quantum description, we have assumed the ab-
in general, and thus often rather uncontrolled assump-
0 sence of time derivatives of third or higher order. The
tions andapproximations(e.g.,Gaussianity [6, 15]) have
9 initialstate (before the transition)obeysthe O(N) sym-
beeninvoked. Forexample,thecorrelationfunctionafter
:0 the transition has been used to infer the number of cre- metry: hφˆai=0andhφˆa(x)φˆb(x′)i∝δab,etc. As stated,
v atedquasiparticleexcitations(see,e.g.,[16]). Thequasi- in all of our calculations, we employ the large-N limit
Xi particle number is, then, supposed to directly yield an assuming N 1. In this case, O(N) invariant combi-
ar ebsytimthaetequfeonrchth.eFtoorposlpoegciciaall cdaesfeesctsudcehnsaitsietshege(neexraactteldy ninadteiopnesndseuncthq≫ausaφnˆt2it=iesφˆo21n+an··e·q+uaφˆl2Nfooatriengsu[2m0s].oCfomnsaindy-
solvable) one-dimensional quantum Ising model (where ering commutators of such combinations, we obtain the
the only excitations are topological defects, i.e., kinks well-known fact that their leading contribution (in the
[1, 3, 17]), such an approach might give the correct an- large-N limit) behaves as a c-number whereas the (clas-
swer–butingeneral,thiswillnotbethe case,aswewill sical and quantum) fluctuations scale with √N (cf. the
argue below. law of large numbers). Therefore, we may approximate
In the following, we consider a rather general O(N)-
symmetry breaking quantum quench and study the cre- φˆ2 = φˆ2 + (√N), φˆ2 = (N), (2)
h i O h i O
ationoftopologicaldefects(hedgehogsinthecaseconsid-
ered) via calculating their winding number. In order to arriving at a semi-classical (mean-field) expansion valid
base our derivation on a well-defined expansion, we con- in the large-N limit. As a result, we may approximate
siderthelarge-N limit. Apartfromthelarge-N limit,no the nonlinear terms in the equation of motion (1), for
2
ˆ3 ˆ2 ˆ
example by φ φ φ, arrivingat a linearizeddescrip- speaking,weconsiderthesimultaneouslimitN and
tion. This leads≈uhs toi the most general linear and local D and assume that these limits commu→te.∞Since
O(N)invarianteffectiveactioncontaininguptofirsttime all→rele∞vant quantities such as φˆa(r,t)φˆb(r′,t) depend
derivatives of the fields φ=(φ ,...,φ ) on r r′ only (isotropy), thhe large-D limitibasically
1 N | − |
just affects the integration measure dDk, which strongly
= 1 φ˙ F( ∇2)φ˙ φ G( ∇2)φ , (3) supports this assumption. With a Fourier expansion of
L 2(cid:16) · − − · − (cid:17) Eq.(3),weobtainthetwo-pointfunctionafterthequench
with arbitrary Fourier space functions F(k2) and G(k2).
φˆ (r,t)φˆ (r′,t)
a b
h i
δ J (kL)
III. PHASE TRANSITION = (2πa)Nb/2 Z dkkN−1 (νkL)ν Ck±e±2iωkt+Dk , (5)
(cid:2) (cid:3)
From Eq. (3), we derive a Klein-Gordon type disper- with L = r r′ . The Bessel functions J with in-
ν
| − |
sion relation [to (k2)] for the linearized fluctuations, dex ν = N/2 1 arise from the integration over all k-
O − ±
directionsandthe factorsC andD depend onthe ini-
G(k2) k k
ω2(k)= =m2c4+c2k2+ (k4). (4) tialstate(forexamplethetemperature)aswellasquench
F(k2) O dynamics, and are roughly independent of N. As ex-
pected, we obtain an exponential growthof the unstable
Initially, all modes are stable, ω2(k) 0, since we lin-
≥ modes, which have ω2 < 0, after the phase transition –
earize around the initial [O(N)-symmetric] state. Af- k
which then seeds the creation of topological defects. Of
ter the O(N)-symmetry breaking transition, however,
ˆ course, due to the growing modes, the linearization in
the state φ = 0 is no longer stable and the system
h i ˆ Eq. (3) will fail eventually – but for N , the time
“wants” to roll down to a state with φ = 0. Typ- ↑ ∞
t until which the linearization and thus Eq. (5) applies
h i 6
ically (for second-order transitions, i.e., without meta-
does also grow. Therefore, we may distinguish basically
stability), this implies that some of the modes become
threephasesfollowingthequench: First,wegetaperiod
unstable, ω2(k)<0, cf. Fig.1. Since Eq.(3) is already a
of exponential growth of the modes where the linearized
resultofthe large-N limit, weassumethatω2(k)isinde-
description in Eqs. (3) till (5) applies. Then, nonlin-
pendentofN 1(otherwisethe groupandphaseveloc-
ear effects set in and lead to a saturation of this growth
≫
itieswouldeitherdivergeorvanishinthelimitN ).
and possibly an oscillation around the new energy min-
→∞
Furthermore, modes with sufficiently large k should be
imum. Finally, the topological defects created by the
stable ω2(k ) > 0, so that the unstable interval in
quench start to “feel” the attraction between hedgehogs
which ω2(k)↑<∞0 is assumed to be finite.
and anti-hedgehogs leading to their approaching each
So far, our results were independent of the number D
other and eventual annihilation. This general picture
ofspatialdimensionsduetoisotropy. Inthefollowing,we
has been qualitatively confirmed by numerical simula-
setN =Dinordertofacilitatethecreationoftopological
tions [29] for N = 2. The topological defects are seeded
defects in the form of hedgehogs (see below). So strictly
in the first phase (exponentialgrowth)and slowlydisap-
pear in the final phase. Therefore, we expect to obtain
ω2 ω2 a goodestimate for the maximum number of createdde-
fects at long length scales and intermediate times from
our linearized analysis.
In addition to the exponentially growing (in t) modes
at finite k, the integral (5) does also yield a huge contri-
k k butionfromlargek,becausethephasespacefactorkN−1
rapidlyriseswithkforlargeN. Thisgivesrisetoastrong
UV singularityofthetwo-pointfunction r r′ −O(N),
k k
∗ ∗ see the discussion below of regularizing∝th|is−UV| diver-
gence.
FIG.1. Twogenericexamplesfortheevolutionofthedisper-
sion relation (4) during a symmetry-breaking phase transi- IV. TOPOLOGICAL DEFECTS
tion, seealso [10]. Initially (dotted line), all k-valuesaresta-
ble,ω2(k)>0. Atthecriticalpoint(dashedline),thedisper-
After the symmetry-breaking transition, the ground
sionrelationtouchesthek-axis,andafterthetransition(solid
line),modesinafinitek-intervalbecomeunstable,ω2(k)<0. stateisdegenerateandcanbespecifiedbyanonvanishing
ˆ
The left panel corresponds to a case where ω2(k = 0) = m2 expectation value φ =0, which singles out a preferred
in Eq.(4) remains positive while c2 changes sign (see, e.g., direction given byhthie6unit vector n = φˆ / φˆ . Thus
[21]); whereas, in the right panel, ω2(k = 0) = m2 becomes theoriginalO(N)symmetryisbrokendohwnit|ohOi(|N 1),
negative. In both cases, however, there is a dominant wave i.e., rotations around the n-axis, and the ground-s−tate
vector k∗ (for large N), as indicated bythevertical arrows. manifold corresponds to the surface N−1 of a sphere in
S
3
N dimensions O(N)/O(N 1) N−1. Remembering order terms vanish for N so that (8) becomes in-
thehomotopygroupπN−1(−N−1≃)=SZ,weseethattopo- deedexactand,inviewoft→he∞asymptoticbehaviorofthe
S
logical point defects in the form of hedgehogs exist in Bessel functions [27], a Gaussian correlator
N spatial dimensions [22, 23]. These defects correspond
to nontrivial mappings from the ground-state manifold 1 k∗2L2
f(L)= exp (9)
N−1 onto the surface N−1 of a sphere in real space, N (cid:26)− 2N (cid:27)
Scharacterized by a windSing number N Z, which reads
[4, 24] ∈ follows. Thus, the typical linear domain size (correla-
tion length) is given by Lcorr = (√N/k∗). At ex-
N= εabc...εαβγ... dS na(∂ nb)(∂ nc)..., (6) tremely large distances L = (N/Ok∗) (where the first
α β γ O
Γ(N)kSN−1kI nontrivialzerooftheBesselfunctionJν islocated),there
are oscillatory deviations, but in this regime, the cor-
where N−1 =2πN/2/Γ(N/2)isthesurfaceareaofthe relator is already exponentially small. We emphasize
kS k
unit sphere in N dimensions. Starting with the O(N)-
that the Gaussian form of f(L) stems from the large
symmetric state as the initial state, we cannot simply
ˆ ˆ ˆ N limit of the exact expression in (8), and is not as-
insert n = φ / φ since φ vanishes. Therefore, we
sumed a priori. Furthermore, as may already be ob-
useaquantuhmiop|herai|tornˆ inhsteiad,whichmustbedefined
servedin Eq. (8), the emergence of a dominant wavevec-
appropriately,andallowsforaderivationoftheprobabil-
ity distribution of the quantum winding number Nˆ in a tor k∗ implies the cancellation of all time-dependence,
i.e., the time-dependence of the growing part of (5) ap-
given volume from the above generalexpression. In par-
proximately separates such that the time-averaged unit
ticular,theexpectationvalueofthewindingnumberisof
vector(7)becomes independentofg(t)andthus station-
course zero, Nˆ = 0, but its variance Nˆ2 is in general
ary (in the regime under consideration). So the emer-
nNoˆt2. SisetptilnagghudediirewctiltyhnˆUV∝ dφˆiv,ewrgeenseceeshthsaimtiitlahre tvoaroiatnhceer rgieonrciejuosftaifidesomthineainnttrsocdaulectiinonthoef ctohrereUlaVtorre,gku∗la,taorpogs(tte)-,
h i
quantities containing products of quantum fields at the whichonly affects the rapidly oscillatingmodes atlarger
samespace-timepoint. Thisisduetothefactthatquan- k but not the observables we are interested in.
ˆ
tumfluctuationsofφatarbitraryk-scales[cf.the strong
UV singularity mentioned after Eq.(5)] would in general
contribute to Nˆ2 . In fact, even the O(N)-symmetric V. SCALING LAWS
h i
initial ground state can be viewed as a “quantum soup”
of virtual hedgehog–anti-hedgehog pairs which are con-
Now we are in a position to derive the dependence of
stantly popping in and out of existence. Here, we are
Nˆ2 on N and the enclosed volume. Inserting Eq. (7)
not interested in those virtual short-lived defects, but in h i
into the winding number variance Nˆ2 from Eq. (6),
long-lived hedgehogs,which are created by the quantum h i
we obtain the expectation value of the product of 2N
quench. Therefore, we have to insert a time-averaged
fieldsφˆ ,whichfactorizesintoN two-pointfunctions(8).
unit vector defined via a
Since thesefunctions arecompletelyregular,wemayap-
nˆ(r)= 1 dtg(t)φˆ(t,r), (7) ply Gauss’ law to the two surface integrals occurring in
Z Z Nˆ2 , and get after some algebra [28]
h i
with a smooth smearing function g(t) and the normal-
ization Z = h[ dtg(t)φˆ(t,r)]2i1/2 +O(√N), where we Nˆ2 = NN! dNrdNr′ 1 ∂ ∂f N.
haveusedthelaRrge-N (mean-field)expansion. Thistime- D E N−1 2 Z LN−1∂L(cid:18)−∂L(cid:19)
average now suppresses all (rapidly) oscillating modes ||S || (10)
with ω2 > 0 and only leaves the growing modes ω2 < 0.
k k
After this UV-regularization, the integral in the two- For a sphere of radius R, V = r : r2 < R2 , we can
pointfunction(5)willbedominatedbyonlyafewmodes evaluate this expression and fina{lly obtain a si}ngle inte-
in the vicinity of a certain wavenumber k∗: In view of gral of the form
the phasespace factor kN−1 in (5), the dominant contri-
butionfor largeN [30]willarisefromthe largestk value π/2
for which ωk2 < 0, i.e., close to the zero of ωk2, cf. Fig.1. Nˆ2 = N!RN dθ cosθ∂f(2Rsinθ) N.(11)
Thus, wecanevaluate (5) insaddle pointapproximation D E π Z (cid:18)− ∂L (cid:19)
0
andobtainthe correlatorforthe time-averageddirection
vector in (7) Notethatthesegeneralexpressions(10)and(11)for Nˆ2
h i
hnˆa(r)nˆb(r′)i=2νΓ(νN+1)J(νk(∗kL∗)Lν)δab =f(L)δab,(8) alarregeneNit,hberutremstirgihcttebdeteomtphleoyBeedssfoelrfaunnyctcioornrsel(a8t)o,rnoofrthtoe
form nˆ (r)nˆ (r) = δ f(L) in any dimension N 2.
a b ab
where higher-order terms N−3/2 stemming from the Let ushdiscuss theiscaling of Nˆ2 with respect to R≥and
∝ h i
normalizationZ in(7)havebeenomitted. Thesehigher- N 1, using the Gaussian correlator (9). For radii far
≫
4
above the correlation length R Lcorr = √N/k∗, the a de-confined phase might occur if defect density and
≫
winding number variance (11) behaves as temperaturearesufficientlyhighandanyboundpairsare
brokenupagainbythermalquasi-particles. Inthatcase,
N−1
Nˆ2 = e−3/2k∗R 1+ (1/√N) . (12) defects and antidefects would be randomly distributed
D E (cid:18) √N h O i(cid:19) and a volume scaling of the winding number variance
Nˆ2 follows. Since we did not make any assumption in
We observe that Nˆ2 scales with the area RN−1 of the houriderivation apart from the large N limit (where the
h i
hyper-surfaceenclosingthedefects. Apartfromthepref- resultsbecomeexact),wecanclearlydistinguishbetween
actor e−3/2/√N, this area scaling is quite universial as the two phases (confined or de-confined).
it holds for spherical volumes in any dimension N 2
≥
– providedweassume short-rangecorrelations– andcan
already be inferred from Eq. (6): If we calculate Nˆ2 VI. STATISTICS
h i
using (6), we obtain two hyper-surface integrals. Due to
isotropy,the firstoneyields RN−1 while the secondinte- In a similar manner, we can calculate the higher mo-
gral averages over the distance r r′ between the two ments of the winding number. Again, by exploiting the
| − |
pointsonthesurface. Assumingshort-rangecorrelations fact that we have short-range correlations, the large-R
only, this second integralbecomes independent of R (for limit ofthe nextnontrivialmomentcanbe inferredfrom
large R) and gives h(N)k∗N−1 with some function h(N). pure combinatorics in the analysis of the four integrals
Note, however, that the assumption of short-range cor- occurring in
relations is crucial and nontrivial in this argument: For Nˆ4 =3 Nˆ2 2+ (RN−1)= (R2N−2). (13)
vorticesintwodimensions,forexample,weobtainedlog- h i h i O O
arithmic corrections to the “area” scaling, Nˆ2 RlnR Analogously, the leading terms of Nˆ2n are given by
h i∝ h i
[4], since the correlatorfell off quite slowly at large L. (2n 1)!! Nˆ2 n with(2n 1)!!=(2n 1)(2n 3)...5 3.
If the radius R shrinks and approachesthe correlation For−largehR,ithe windin−g number Nˆ− Z −can be a·p-
length R Lcorr = (√N/k∗), the winding number proximated by a continuous variable Nˆ∈ R and thus
∼ O
variance decreases rapidly (for N 1). A sphere with ∈
its full statistics is given by the inverse Mellin transform
≫
R = (√N/k∗) would then contain around one defect of(2n 1)!!=2nΓ(n+1/2)/√π, which yields the Gaus-
(or anOti-defect) on average, Nˆ2 = (1), which deter- sian pr−obability distribution p(N) exp γ2N2 , with
h i O ∝ {− }
mines the totaldefect density. Forevensmallerradii,far 1/γ2 = 2 Nˆ2 . We note that, like in Eq. (9), the Gaus-
below the correlation length R Lcorr = (√N/k∗), sianityishnotiassumedbutderivedfromfirstprinciplesin
≪ O
the above formulae would yield a scaling Nˆ2 R2N, a given limit – for small R (small N), for example, there
h i ∼
i.e., an exponential suppression (for large N). However, will be deviations from a Gaussian distribution.
the precise functional form (11) should not be trusted
upon in this regime since we have neglected (1/√N)-
corrections in our derivation, which is problemOatic if the VII. CONCLUSIONS
final result is exponentially small. From a more physical
point of view, the mean-field approximation (2) breaks Based on a very general O(N)-invariant effective ac-
down near the core of a defect (where n becomes ill- tion, we presented an analytical derivation of the wind-
defined), which renders Eq. (11) questionable for too ing number countingthe defects createdby a symmetry-
smallvolumina. ForsmallR,onewouldexpectavolume- breaking quantum quench in the large-N limit. Consis-
type scaling Nˆ2 RN, i.e., the typical behaviour for tent with previous calculations [25], our result (12) is
h i ∼
uncorrelated defects, which should be the case if there nonperturbative, i.e., it does not admit a Taylor expan-
is one hedgehog at most. The exponential suppression sion in 1/N. As another result, we find that the typi-
Nˆ2 exp (N) for large N and small R L cal distance between defects scales with the correlation
corr
h i ∼ {−O } ≪
shouldstillbecorrect,asthisjustreflectsthediminishing length (√N/k∗). By contrast, the typical distance be-
O
probablity of reversing field orientation in all directions tween quasiparticle excitations (e.g., Goldstone modes)
when increasing N. doesnotincreasewith N. Thiscanbe understoodbyre-
The area scaling (12) of the net defect number Nˆ2 calling that the total energy of the system (which scales
h i
can be interpreted as the occurrence of a confined phase withN)inagivenvolumehastobedistributedamongall
ofbounddefect-antidefectpairs. Onlypairswhereoneof the quasiparticle excitations, whose typical energy is de-
the partners is contained within while the other is out- terminedby the dispersionrelationω2(k) and thus inde-
sidetheintegrationvolumewouldyieldnetwindingnum- pendentofN. Therefore,weconcludethatthequasipar-
ber, whereasthose pairsentirely inside oroutside do not ticlespectrumalonedoesnotyieldanydirectinformation
contribute. Hence a scaling with surface area instead about the generation of topological defects in general.
of volume is natural for short-ranged correlations. On This situation is quite different in the one-dimensional
the other hand, there could, in principle, also exist a quantum Ising model, where topological defects (kinks)
de-confined phase of quasi-free hedgehogs similar to the are the only quasiparticle excitations [1, 3], which fre-
quark-gluon plasma of quantum chromodynamics. Such quently led to the assumption in the literature that this
5
is generic. We demonstratedhere that identifying quasi- the large-N limit without any further approximations.
particleexcitationanddefectnumberscreatedbyaquan- Moreover,asindicatedbelowEq.(12),weexpectthatthe
tum quench can be quite misleading. general picture does still apply qualitatively for smaller,
The crucial difference between quasiparticles (whose and thus experimentally accessible values of N, for ex-
number can be derived via a perturbative expansion in ample N = 3. In particular, this should be true for fast
1/N)andtopologicaldefects(whicharenonperturbative) quenches, where we have a well-defined period of expo-
canbeillustratedbythefollowingintuitivepicture: Con- nential growth of the unstable linear modes, while non-
sidering a discrete regular lattice with a unit direction linear effects (saturationof this growth,oscillations,and
vector n at each lattice site i, a quasiparticle excitation finally defect annihilation, see [29]) occur much later. In
i
occurs if n = n for two neighbours i,j. A topologi- thiscase,onemayfind(insteadof1/N)anothersmallpa-
i j
6
cal defect at the site i, one the other hand, means that rameter(e.g.,the diluteness ofthe gas)in orderto moti-
the unit vectors n of all neighbouring sites either point vatetheunderlyingeffectiveactioninanalogytoEq.(3).
j
awayortowardsthe site i. For largeN,this is obviously For N 1, universality will be partially lost and the
6≫
a much stronger condition. dependence on the dispersion relation, for example, will
It is also worth noting that the derived area scaling be stronger. For instance, it might then be necessary to
Nˆ2 RN−1 is inconsistent with the random defect introduce a time-dependent critical k∗ = k∗(t), which is
hgasmi o∝del(wheredefectsandanti-defectsaredistributed not close to the zero of ω2(k), but near the actual mini-
randomly in the sample volume, corresponding to a de- mum of ω2(k).
confined phase) since this model wouldpredict a volume
law, i.e., RN-scaling. We remark in this connection that
the area scaling [26] (correspondingto a confined phase)
we obtain can be interpreted by a random n-field model
ACKNOWLEDGMENTS
on the hyper-surface with the correlator Eq. (8), repre-
sentingageneralizationoftherandomphasewalkmodel
for N =2 (cf.the resultof [4] in which reasonableagree- M.U. acknowledges support by the Alexander von
mentwiththeexperimentreportedin[18]wasobtained). Humboldt Foundation and NSERC of Canada, R.S. by
Finally, we would like to stress that our result is quite the DFG (SCHU 1557/1-3,SFB-TR12), and U.R.F. by
universal, i.e., it is valid for very general dispersion re- theDFG(FI690/3-1)andtheResearchSettlementFund
lations of the O(N) model (cf. Fig.1) and just relies on of Seoul National University.
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6
found in the two-point (irreducible) correlator at next- [28] More details of the calculation will be presented else-
to-leading order in 1/N. where, M. Uhlmann, R. Schützhold, and U.R. Fischer
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tween the area scaling obtained here and the area scal- [29] N. Horiguchi, T. Oka, and H. Aoki, J. Phys.: Conf. Ser.
ingoftheground-stateentanglemententropyofamany- 150 (2009) 032007.
body system with short-range correlations. See, e.g., J. [30] For smaller N, however, the importance of the phase-
Eisert, M. Cramer, and M.B. Plenio, arXiv:0808.3773 spacefactorkN−1 subsidesandk∗willbeshiftedtowards
[quant-ph],to appear in Rev. Mod. Phys. (2009). themodeswiththefastestgrowthintime.Notethatthen
[27] Handbook of Mathematical Functions edited by M. adifferentexpansionparameterotherthan1/N mightbe
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