Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunnelling models

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by  Anton S. Buyskikh

Entanglement growth and correlation spreading with variable-range interactions in
spin and fermionic tunnelling models
Anton S. Buyskikh,1 Maurizio Fagotti,2 Johannes Schachenmayer,3 Fabian Essler,4 and Andrew J. Daley1
1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK
2D´epartementdePhysique,E´coleNormaleSup´erieure/PSLResearchUniversity,CNRS,24rueLhomond,75005Paris,France
3JILA, NIST, Department of Physics, University of Colorado, 440 UCB, Boulder, CO 80309, USA
4The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford, OX1 3NP, UK
(Dated: March 17, 2016)
Weinvestigatethedynamicsfollowingaglobalparameterquenchfortwo1Dmodelswithvariable-
rangepower-lawinteractions: along-rangetransverseIsingmodel,whichhasrecentlybeenrealised
6
1 in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tun-
0 nelling. For the transverse Ising model, the spreading of correlations and growth of entanglement
2 are computed using numerical matrix product state techniques, and are compared with exact solu-
tionsforthefermionictunnellingmodel. Weidentifytransitionsbetweenregimeswithandwithout
l
u an apparent linear light cone for correlations, which correspond closely between the two models.
J For long-range interactions (in terms of separation distance r, decaying slower than 1/r), we find
that despite the lack of a light-cone, correlations grow slowly as a power law at short times, and
1
that – depending on the structure of the initial state – the growth of entanglement can also be
] sublinear. These results are understood through analytical calculations, and should be measurable
s in experiments with trapped ions.
a
g
PACSnumbers: 67.85.-d,75.10.Pq,37.10.Ty
-
t
n
a I. INTRODUCTION cal solution is known. The second model is an exactly
u
solvable model that represents a generalisation of a Ki-
q
taevchain[31]withlong-rangetunnelling. Formoderate
. In recent years, advances in experiments with trapped
t
a ions [1–5], ultracold polar molecules [6, 7], and Rydberg system sizes and timescales, we compute the dynamics
m of the long-range transverse Ising model numerically by
atoms [8–10] have allowed for the experimental realisa-
using Matrix Product Operator (MPO) techniques [32–
- tion of highly-controllable spin models with interactions
d 36]. Forsystemswithlong-rangeinteractions,theseallow
that decay as a power law. Ions in Paul and Penning
n a particularly convenient implementation of propagation
traps, in particular, offer an opportunity to design spin
o
schemes for matrix product states (MPS), the state rep-
c models with variable range interactions by mediating in-
resentation that forms the basis for the time-dependent
[ teractions between internal spin states via the collective
density matrix renormalisation group methods [37–40].
motional modes of the ions [11, 12]. Parameters in these
2 We compare the qualitative behaviour of the two models
models can be controlled – and the spin dynamics mea-
v
intermsofspreadingofcorrelationsandgrowthofentan-
6 sured—time-dependently,openingnewopportunitiesto
glement after a global quantum quench, beginning with
0 study quench dynamics in closed quantum systems [13–
1 18]. a state that is initially uncorrelated. We also make use
2 ofaHolstein-Primakoffapproximationtogainfurtherin-
Thishasopenednewfundamentalquestionsrelatedto
0 sight into dynamics in the Ising model with long-range
the propagation of correlations in such systems – in par-
. interactions, for short times and initial states with all of
1 ticular, how to generalise the Lieb-Robinson bounds [19]
0 the spins aligned.
forthespreadingofcorrelationsinsystemswithnearest-
6 neighbour interactions to situations with long-range in- As was seen previously with local quenches in the
1
teractions. Whileaseriesofgeneralresultshavebeende- transverse Ising model [13, 16], we are able to classify
:
v rived,allowinggraduallytighterbounds[20–27],itispar- the behaviour of correlation spreading in both of these
Xi ticularly informative to identify exactly solvable models models into different regimes as a function of the de-
that reproduce and explain the qualitative behaviour of cay exponent α of power-law interactions (which decay
r
a the physical systems being studied. Up to now there are as 1/rα), where r is the separation distance. For cor-
relatively few models with long-range interactions (lim- relation spreading, the dynamics are divided into (i) a
ited mainly to longitudinal Ising models [28] and tun- regime of short-range interactions where α > 2, (ii) a
nelling bosons [29, 30]) for which the transitions in be- regime of intermediate and long-range interactions when
haviour are known. α < 2, with certain features also changing at α = 1.
While light-cone-like behaviour remains for intermediate
In this article, we study global quench dynamics in
an short-range interactions, in the case where α < 1,
two models with long range interactions. The first is
there is complete absence of a light cone for spreading of
the transverse Ising model with long-range interactions
correlations.
[11, 12], which was recently realised in a series of exper-
iments [2, 3, 13, 14], but for which no general analyti- Counterintuitively, though, in this regime the devel-
2
(a) A. Long-range transverse Ising (LRTI) model
The first model of interest is a 1D chain of M spin-
1/2 systems described by the long-range transverse Ising
model ((cid:126)≡1),
(b)
M M
(cid:88) (cid:88)
H = J σxσx+B σz, (1)
LRTI lj l j l
l>j l=1
where J is the spin-spin interaction matrix, B is the
lj
transverse field, and σx,z denote local Pauli matrices op-
l
eratingoneachspin. ItispossibletorealisethisHamilto-
FIG. 1: Illustration of the long-range models. (a) The long-
nian experimentally with 1D chains of trapped ions [11],
rangetransverseIsing(LRTI)model,whichhasbeenrealised
as an effective model for the dynamics of long-lived in-
experimentally in chains of trapped ions. The spins inter-
ternalstates(denoted|↑(cid:105) and|↓(cid:105) forspinl)oftheions.
act over a distance with amplitude Jlj, in the presence of a l l
transverse field term B. (b) The long-range fermionic hop- The interactions are generated by cooperative spin-flips
ping (LRFH) model. Spinless fermions tunnel between dis- of the internal states, induced by external laser fields,
tant sites in a 1D lattice with amplitude J , and pairing on and coupled via the motional modes of the ion chain. It
lj
neighbouring sites is induced with strength ∆. is possible in experiments to achieve a close approxima-
tion to algebraic decay of correlations J = J|l−j|−α,
lj
J > 0, with open boundary conditions, and 0 (cid:46) α (cid:46) 3.
By shifting the internal levels of individual ions an effec-
opment of both correlations and entanglement can be
tive magnetic field B can be realised, and here we focus
suppressed. For the fermionic model, we find that the l
on a transverse uniform field.
growth of correlations takes the form of a power law at
Our primary original motivation for studying quench
long distances, leading to slow growth at short times.
dynamics in this system is the easily realisable exper-
When the Hamiltonian and the initial state have simi-
imental sequence in which all of the spins are initially
larsymmetries, wealsoobserveforbothmodelsthatthe
prepared in a single state (say, |↓(cid:105) ), and then dynamics
growth of bipartite entanglement in spatial modes of the l
are allowed to proceed under the Hamiltonian (1). This
chain can be significantly suppressed at short times, in
corresponds to a global quench from B = ∞ with the
contrasttocaseswithshorterrangeinteractions. Wecan
ground state |ψ(t=0)(cid:105) = ⊗ |↓(cid:105) to some finite value of
understandthisbasedonachangeinthedispersionrela- l l
the field.
tionforlong-rangeinteractions,whichdivergesformodes
Thismodeldoesnothaveaknownanalyticalsolution,
with quasimomentum k → 0. The initial state then af-
but we are able to compute the dynamics using exact
fects the dynamics by determining to what extent new
diagonalisation for chain lengths up to M ∼ 20, or for
quasiparticles are produced with these momenta.
moderate times and longer chains up to M ∼ 100 by
Therestofthisarticleisarrangedasfollows: InSec.II, applying time-dependent variational principle (TDVP)
we discuss the long-range transverse Ising and fermionic techniques with MPO representation of the Hamiltonian
hopping models, giving details of the methods we use [32–36, 41]. For these calculations, the convergence in
to solve these. In Sec. III, we then present the time- theMPSbonddimensionD andtimestep∆twastested
dependent correlation dynamics for each of the models to ensure accuracy of the calculations. Furthermore, we
after a quantum quench, discussing in each case the dif- canqualitativelydescribetheshort-timedynamicsinthe
ferent regimes of short and long-range interactions. We Holstein-Primakoff approximation (see Sec. IIID).
thenmove onto discussgrowth ofentanglementin these
models in Sec. IV, before providing a summary and out-
look in Sec. V. B. Long-range fermionic hopping (LRFH) model
The second system is a 1D lattice with spinless
fermions that has the form of a generalised Kitaev chain
model [31] with long-range hopping [42]
II. MODELS WITH LONG-RANGE
INTERACTIONS
M
H = (cid:88) J¯ c†c +∆(cid:88)c†c† +h.c.. (2)
LRFH lj l j l l+1
In this section, we define the two systems with long-
l(cid:54)=j=1 l
rangeinteractionsthatwewillanalyseandcompare. The
first is the long-range transverse Ising model, and the Here, c is a fermionic annihilation operator on site l, J¯
l lj
second is an exactly solvable long-range hopping model is the hopping matrix with long-range couplings anal-
for spinless fermions. ogous to those in the spin model, and ∆ characterizes
3
interactions between fermions in the pairing term. For Fromthis, wecanobtainthetimeevolutionofthe single
analytical calculations in order to simplify expressions particle density matrix after the quench, beginning with
without the thermodynamic limit we will make use of the ground state α0|φ (cid:105)=0:
k 0
periodic boundary conditions, and choose
M−1
J¯lj =J(cid:12)(cid:12)(cid:12)(cid:12)Mπ sin(cid:20)π(lM−j)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)−α, (3) (cid:104)c†l(t)cj(t)(cid:105) = 2M1 k(cid:88)=0 e−i2πkM(l−j) [1−cosθkcosδθk
+sinθ sinδθ cos(2(cid:15)(k)t)], (5)
where J > 0. For numerical calculations we will follow k k
the case of the spin model, and choose open boundary
conditions with J¯ =J . We find that the behaviour in where
lj lj
each case agrees well in the limit of large system sizes.
δθ =θ −θ0 (6)
Also, in analogy with the spin model, we will consider k k k
quantum quenches, typically starting from the ground
isthedifferenceoftheBogoliubovanglesbeforeandafter
statefor largevalues of∆, andquenchingto smallerval-
the quench. In the following two sections we now look in
ues of ∆.
detail at the spreading of correlations and the growth of
The quadratic Hamiltonian of Eq. (2) can be diago-
entanglement, comparing the results of the LRTI model
nalized via Bogoliubov transformations, analogously to
to the LRFH model.
the recent results in Ref. [42]. In the case of peri-
odic boundary conditions, the Hamiltonian in momen-
tum space reads
III. SPREADING OF CORRELATIONS IN TIME
M/2−1
HLRFH = (cid:88) (cid:0)c†k cM−k (cid:1)× We begin by analysing the spreading of correlations
k=0 after a global quench. In order to do this, we look at
(cid:18)−2i∆J¯si(nk)(cid:0)2πk(cid:1) 2i∆−sJi¯n((cid:0)k2)Mπk(cid:1)(cid:19)(cid:18)c†ck (cid:19), ttwheo-sspitaeticaolrmreelaatnioonfftuhnectaibosnoslufoter vthaleuetwoof scyhsatreamctse.risFtoicr
M M−k
the LRTI system, we choose
where
ck = √1 (cid:88)M e−i2πMkrcr C˜d(t)=(cid:10)(cid:12)(cid:12)(cid:10)σl+(t)σl−+d(t)(cid:11)(cid:12)(cid:12)(cid:11)l (7)
M
r=1 and for the LRFH model,
and (cid:68)(cid:12)(cid:68) (cid:69)(cid:12)(cid:69)
C (t)= (cid:12) c†(t)c (t) (cid:12) . (8)
M−1 d (cid:12) l l+d (cid:12)
J¯(k)=2 (cid:88) J cos(2πkd/M). (4) l
l,l+d
Intheseexpressions, (cid:104)...(cid:105) indicatestheaverageinspace
d=1 l
oversitesl,whichdependsonwhethertheboundarycon-
Then via the Bogoliubov transformations
ditions are periodic or open.
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
αk u v ck Forquantumsystemswithfinite-rangeinteractions,in-
= ,
α† −v∗ u∗ c† formation is expected to spread with a finite group ve-
M−k M−k
locity limited by the Lieb-Robinson bound [19]. This
where u = cos(θ /2), v = isin(θ /2), and the angle θ
k k k forms a light cone for the spreading of information, and
is chosen to satisfy
the leakage of information outside of this light cone is
J¯(k)+2i∆sin(cid:0)2πk(cid:1) exponentially suppressed. Such a light-cone-like spread-
eiθk = (cid:113) M , ing of correlations was observed for quenches in a Bose-
J¯2(k)+4∆2sin2(cid:0)2πk(cid:1) Hubbard model in experiments with a quantum gas mi-
M
croscope [43]. We aim to identify to what extent a be-
theHamiltoniancanbediagonalised,andisfoundtohave
(cid:113) haviour with a linear light cone survives the generalisa-
eigenenergies E(k) = ± J¯2(k)+4∆2sin2(2πk/M) ≡ tion to algebraically decaying interactions.
±(cid:15)(k).
The pre-quenched Hamiltonian has another value ∆ ,
0
instead of ∆, so the pre-quenched Bogoliubov particles A. Comparison of the LRTI and LRFH models
and angle will be denoted α0 and θ0 respectively. The
k k
time evolution of the original fermionic operators will be In Fig. 2, we show examples of correlation spreading
(cid:18) (cid:19) (cid:18) (cid:19) afteraquantumquenchineachoftheLRTI(leftcolumn
ck(t) = cos(θk/2) −isin(θk/2) × of subfigures) and LRFH models (right column of sub-
c† (t) −isin(θ /2) cos(θ /2)
M−k k k figures), comparing the relative behaviour in a regime
(cid:18)e−i(cid:15)(k)t 0 (cid:19)(cid:18) αk (cid:19) of short-range interactions where α > 2 (Figs. 2(a,d)),
.
0 ei(cid:15)(k)t α† an intermediate regime for 1 < α < 2 (Figs. 2(b,e)),
M−k
4
(a) (d) the LRTI model, the edge is no longer completely linear
on the timescales of our calculations. The definition of
the edge changes significantly with the chosen threshold
value, as can be seen in Figs. 2(b,e). In the case of the
LRFH model we find that the maximum group velocity
divergeswhenα<2,whichwillbediscussedinSec.IIIC
below.
In the regime of long-range interactions (α < 1),
(b) (e)
the spread of correlations becomes even more extreme
and light cone effects disappear. Immediately after the
quench, correlations start growing over the whole sys-
tem, and no light cone can be properly defined (note the
different scales on the time axis in Fig. 2(c,f)). Simi-
lar behaviour at short to moderate distances is observed
for both models. However, for the LRFH model in this
regime (Fig. 2(f)), where we can perform calculations
(c) (f)
at much longer distances than for the LRTI model, we
notice that also in this case the correlations at longer
distancesaresuppressedatshorttimes,despitethelong-
range interactions. We will discuss this suppression in
more detail in Sec. IIIC below.
For the LRTI model, qualitatively similar behaviour
can be observed also in terms of the mutual information
between distant points [15], and has also been seen for
local quenches in this model, where an equilibrium state
FIG. 2: Correlation spreading after global quenches in
isperturbedbyflippingasinglespin,aswasdiscussedin
long-range spin and fermionic models, respectively. (a-c)
Ref. [16]. Although we see quite abrupt changes in the
log |C˜ (t)| for the LRTI model after the global quench
10 d behaviour of the LRTI model at α = 1 and α = 2 for
B/J = ∞ → 1 is applied. These results are obtained us-
system sizes of the order of M ∼ 100 spins, it is diffi-
ing TDVP approach with MPOs for chains of M = 100
cult to carry out these calculations for longer times and
spins (converged with MPS bond dimension D = 256). (d-
f) log |C (t)| for the LRFH model after the global quench larger systems, and to better delineate these boundaries.
10 d
∆/J = 10 → 1 is applied. These results are obtained us- In order to obtain exact results for a global parameter
ing exact numerical computations for M = 104 sites. (a,d) quench, we turn to the LRFH model, for which we will
α = 3, short-range interactions with a strongly suppressed discuss the behaviour for α > 1 and α < 1 respectively
leakage of correlations outside of the light cone d/t > vgmrax in the next two sections.
are observed for both models. Markers indicate contour In Fig. 3, we plot the correlation function log |C (t)|
lines at levels log |C˜ (t)| = [−4,−31,−3] for the LRTI 10 d
10 d 2 asafunctionofseparationdistanceforseveralfixedtimes
model and log |C (t)| = [−6,−5,−4,−3] for the LRFH
10 d tJ. In the case of short-range interactions, α > 2, one
model. (b,e) α = 3/2, intermediate-range interactions, the
can clearly see that the connection region between the
light cone is not sharply defined, but a light-cone effect is
fast decaying correlation wave and slowly decaying tail
observed. Markers indicate averaged contour lines at lev-
els log |C˜ (t)| = [−31,−3,−23] for the LRTI model and occurs over very short distances, as opposed to the case
10 d 4 4
log |C (t)| = [−6,−5,−4,−3] for LRFH. (c,f) α = 1/2, no of the intermediate-range correlations 1 < α < 2. This
10 d
light cone, instant spread of correlations through the entire leads to a very clearly defined light cone. We also note
system. The suppression of correlations at large distances in that for both these regimes (α > 1) the correlations de-
(f) is discussed in Sec. IIIC. cay algebraically outside the light cone, as opposed to
the exponential decay that would be expected for ini-
tially uncorrelated states in models with finite-range or
and a regime of long-range interactions when α < 1 exponentially decaying interactions [19].
(Figs. 2(c,f)).
Inthecasewhereα>2,weseeaclearlinearlightcone
inthedynamics. Becausethelightconeissharp,defining B. Dynamics with short-range interactions α>2
this with threshold values for the correlations leads to
a light cone that does not change substantially as the Aswealreadynotedabove,whenα>2,theboundary
threshold values are changed, as shown in Figs. 2(a,d). of the light cone is defined clearly (Fig. 3). We will see
One can see a strongly defined edge with algebraically below that this result is explained from the behaviour of
suppressed correlations outside. thedensityofstatesinvelocitynearthemaximumgroup
In an intermediate regime 1 < α < 2, the edge of velocity. In this case, we can define a light cone by iden-
the light cone broadens significantly, and in the case of tifying the position of rapid change in the correlations,
5
(a) saddle point equation
-2 tJ =5
)tj-4 tJ =20 2(cid:15)(cid:48)(cid:0)kj∗(cid:1)+u=0. (12)
( tJ =35
d
C0j-6 tJ =50 For u > vgmrax, the usual argument for the exponential
g1-8 decayoftheintegralduetothelackofastationarypoint
o
l fails due to a non-analyticity at k = 0. In this case,
-10 using 2π-periodicity of the integrand, its non-analytical
part was extracted. The contribution of this part decays
0 200 400 d 600 102 d 103
only algebraically in a distance d and prevails over the
(b)
exponentiallysmallcontributionoftheremaininganalyt-
-2 tJ =4
icalpart. Thus,weobtainthepowerlawexponentofthe
)j tJ =12
t-4 correlationdecayoutsideofthelightcone(seeEq.(11)).
( tJ =20
Cd On the other hand, for u < vmax, the existing saddle
gr
0j-6 point gives the major contribution to the integral. As
1
go a result, the correlations inside of the light cone decay
l-8
slowlyintime,O(t−1/2),whichcorrespondstoFick’slaw
-10 ofdiffusion. Wefindthatbothapproximationsagreewell
0 1000 2000 d 102 103 d withfullnumericalsolutionsuptofinitesizecorrections.
At the same time, in Fig. 2, and through the follow-
FIG.3: DeterminingthelightconeintheLRFHmodel. We ing analysis of the density of states in velocity, we see
plotthetwo-sitecorrelationfunctionlog |C (t)|afterglobal that the light cone effect is strong only when α > 2. To
10 d
quenches ∆/J = 10 → 1 in the LRFH model with M = 104 understand this we consider both the dispersion relation
sites at different times tJ (a) short-range interactions α = 3 and the density of states in velocity as a function of the
(b) mid-range interactions α = 3/2. Dashed lines indicate wave vector (Fig. 4),
different threshold levels δ=[−6,−5,−4,−3] that the corre-
loagtoiounsmfuanrckteiorsnfroeratchheesL,RseTeItmheodmealrakreersininFiFgi.g2.(2a(,db,)e.).InAenaaclh- D(k)= M (cid:12)(cid:12)(cid:12)dvgr(k)(cid:12)(cid:12)(cid:12)−1 = M (cid:12)(cid:12)(cid:12)d2(cid:15)(k)(cid:12)(cid:12)(cid:12)−1. (13)
casehere,ontherighthandside,wereproducethesameplots π (cid:12) dk (cid:12) π (cid:12) dk2 (cid:12)
on a double logarithmic scale, showing algebraically decay of
correlations outside the light cone as d→∞ in both cases. We see that for α > 2, the density of states in velocity
diverges exactly at the value of v =vmax, which means
gr gr
that depending on the type of quench (the final occu-
essentially equivalently to choosing a threshold value for pation number) it is possible to excite infinitely many
the correlations, as indicated in Fig. 3. quasi-particlespropagatingwiththemaximumgroupve-
Alternatively, we can approach the question of the locity. Those quasi-particles propagating through the
light-cone effect in this regime directly analytically for system will form the well-defined front of correlations.
the LRFH model. To see the general difference between And as one can see in Fig. 4(b) the maximum group ve-
the regime α > 2 and α < 2, we can analyse the dis- locity is finite in this regime, vmax <∞.
gr
persion relation (cid:15)(k), which is plotted in Fig. 4. We see
that for α>2 the dispersion as well as its derivative are
bounded, i.e. C. Dynamics with intermediate and long-range
interactions (α<2) in the LRFH model
vmax =max(cid:15)(cid:48)(k)<∞. (9)
gr
k
If we look more closely at the time-dependent part of The situation changes when a kink appears at k = 0
the Green’s function (5) in the thermodynamic limit in the dispersion relation for 1<α<2
ˆ
F(d,t)= 1 π dke−ikdsinθ sinδθ cos(2(cid:15)(k)t), (10) (cid:15)(k)∝(cid:15)0+ckα−1, (14)
2 2π k k
−π
inthisregimethedispersionisbounded,butthevelocity
we see that the corresponding integral has a typical be-
diverges. Forα<1eventhedispersiondivergesforsmall
havior in the “space-time scaling limit” [44] of d,t→∞,
momenta k
with u = d/t fixed. Using the stationary phase approxi-
mation we analytically obtain (cid:15)(k)∝kα−1, k →0. (15)
(cid:40)
O(1/dα+2), u>vmax,
F(d,t)≈ gr (11) Hence, for α < 2 there no longer is a finite maximum
(cid:80) A cos(B ), u<vmax,
j j j gr group velocity and for α < 1 the spectrum of quasi-
excitation become unbounded. This result can be seen
wπ/h4er+e tA(cid:0)kj∗u=+(cid:0)21(cid:15)6(cid:0)πk(cid:15)∗(cid:48)(cid:1)(cid:48)(cid:1)(cid:0),kj∗a(cid:1)ntd(cid:1)−k1∗/2asrientθhkej∗ssionluδtθikoj∗n,sBofjth=e athnealryetsiucaltlloyfitnhethpeolcyalsoegaorfitohpmenbebhoauvnidouarryincoEnqd.it(i2o2n),faosr
j j j
6
(a) dent part of the correlation function is small and scales
,=3 like a power-law in time.
,=3=2
) ,=1=2
k
0( D. Holstein-Primakoff approximation for the LRTI
model
(b) Inordertogetamoredetailedpictureofthedynamics
intheLRTImodelatshorttimes,weconsiderananalyt-
ical treatment in the Holstein-Primakoff approximation.
)
k This relies on the initial ordering of the spins along the
(
D z-axis (from the initial state we chose previously), and
should be a good approximation as long as this order
remains. We note that this is expected to be a better
0
approximation for non-zero values of the transverse field
-1 -0.5 0 0.5 1
k=: B, as this supports retention of the ordering.
Considering this initial state with |ψ (cid:105) = ⊗ |↓(cid:105) , and
0 l l
FIG. 4: (a) The dispersion relation (cid:15)(k) and (b) density of writing (cid:126)≡1, we can rewrite Eq. (1) using the transfor-
states in velocity D(k) for the LRFH model for ∆ = J and
mation
variousinteractionranges. Inthecaseofshort-rangeinterac-
tions (α>2) one can see the smooth behaviour of (cid:15)(k) lead- Sz =a˜†a˜ −S
icnogrretospaonfidniintge mdeanxsiimtyumofgsrtoautepsvienlovceitlyocvitgmyraxD=(k∗(cid:15)(cid:48))(kd∗i)v.erTgehse, Sll− ≡Sllxl−iSly =(cid:113)2S−a˜†la˜la˜l ,
whichjustifiesthestronglightconeeffect. Inthecaseα<2, S+ ≡Sx+iSy =a˜†(cid:113)2S−a˜†a˜
vmax ∝ kα−2 for k → 0, but density of states in velocity is l l l l l l
gr
suppressed,D(k)∝k3−α. Asaresultthereisnodomination
where S = σ/2 are spin operators for spin-1/2 and a˜
of infinite velocity excitations and the light cone broadens. l
In the case α < 1, the quasiexcitation spectrum become un- are bosonic annihilation operators. Starting from the
bounded as well. initially fully polarized state we consider dynamics only
ontimescaleswhentheinitialorderisstillpreserved, i.e.
(cid:10)a˜†a˜ (cid:11)/2S (cid:28)1, then we can take
α < 2. As a result the light cone boundary becomes l l
washed out for these regimes. Sz ≈a˜†a˜ −S
Even though the group velocity diverges for α < 2,  l √l l
S− ≈ 2Sa˜ .
the density of states in velocity around k = 0 is sup- l √ l
pressed, D(k) ∝ k3−α. The combination of both these Sl+ ≈ 2Sa˜†l
facts means that although the correlations can to some
Uptoaconstantshift,wecanthenwritetheLRTImodel
extent build instantly through the entire system, they
as
grow very slowly, v (k)D(k) ∝ k → 0 for k → 0.
gr
Because of this suppression the correlation spread for H = S(cid:88)J (cid:16)a˜ a˜ +a˜†a˜ +h.c.(cid:17)+2B(cid:88)a˜†a˜ .
fermions in Fig. 2(f) does not seem as immediate as in HP lj l j l j l l
l(cid:54)=j l
the case of spins in Fig. 2(c), which will be discussed in
Sec. IIID. Taking the Fourier transformation to momentum space
In terms of the saddle point approximation, the situ- operators,
ation is very different from the case of α > 2, but the
alepapsrtooxnimeasatidodnleispsotiinllt,vebreycaguosoedt.hNeogwrotuhperveelioscaitlywaisysuant- ak = √1 (cid:88)M e−i2πMkla˜l
M
bounded, vmax =∞. Therefore we have only the second l=1
gr
part of the solution of Eq. (11).
we can then rewrite the Hamiltonian as
In order to understand the behaviour in the case of
small α it is useful to consider the contribution of the M(cid:88)/2−1(cid:104) (cid:16) (cid:17)
smallest saddle point. In particular we focus on d(cid:29)ut, HHP = 2 SJ(k) aM−kak+a†ka†M−k
inwhichcasethissaddlepointk∗occursclosetozero. We k=0
(cid:16) (cid:17)(cid:105)
canworkouttheresultofthesaddlepointapproximation +(SJ(k)+B) a†a +a a† ,(17)
to be k k M−k M−k
t3/2 (cid:40)(t/d)43−−2αα , α<1 where J(k) is defined in Eq. (22). Then via the Bogoli-
F(d,t)∝ . (16) ubov transformations for the bosonic field
d2 (t/d)34α−−2α1 , 1<α<2
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
γk u v ak
= ,
Hence, although there is no light cone, the time depen- γ† v∗ u∗ a†
M−k M−k
7
where u=cosh(φ /2), v =sinh(φ /2). Eq. (17) can be (a) (d)
k k
diagonalized if
2SJ (k)
e2φk =1+ .
B
Then the dispersion relation reads
(cid:114)
2SJ (k) (b) (e)
(cid:15)(k)=2B 1+ . (18)
B
Notethatinorderfor(cid:15)(k)tobereal,werequiretheex-
pression under the root to be real. Then, using Eq. (22),
we obtain the following constraint
|B/J|≥−2sign(B/J)Li (−sign(B/J)),
α
(c) (f)
whichsetsthelimitsforB valuessuchthattheHolstein-
Primakoff transformation is stable for all modes k.
Using the limiting behaviour of the polylogarithm, we
then obtain the divergence of the dispersion relation for
α<1,
α−1
(cid:15)(k)∝k 2 , (α<1, k →0). (19)
Ingeneralthedispersionrelationanddensityofstatesin
FIG. 5: Correlation spreading in the LRTI model after the
velocity have the same featured as for the LRFH model
global quench B/J = ∞ → 2 is applied in the system of
in Fig. 4.
M =100 spins. (a-c) log |C˜ (t)| calculated numerically via
We observe a similar divergence in the Bogoliubov an- 10 d
MPS/MPOmethods(asinFig.2(a-c)),and(d-f)log |Cˆ (t)|
gle δφ =φ −φ0, where φ0 is the pre-quenched Bogoli- 10 d
k k k k calculated analytically via the Holstein-Primakoff approxi-
ubov angle for the Hamiltonian with the field B = ∞
0 mation. (a,d) α = 3, short-range interactions case. (b,e)
instead of B. When α < 1, modes near k = 0 will dom- α=3/2,intermediate-rangeinteractions. (c,f)α=1/2,long-
inate the spread of entanglement, and we will observe a range interactions.
transition of behaviour with respect to α > 1, when the
dispersion relation is regular for k = 0. This explains
the transition in behaviour that we observed at α = 1 IV. ENTANGLEMENT GROWTH
numerically, as outlined in Sec. IVA2.
Analogous to the other models we consider the corre- In this section we quantify the time-dependence of en-
lation matrix for the bosonic particles tanglementinspatialmodesofthesystem. Inordertodo
this, wecansplitthesystemintotwoparts. Thesecould
(cid:68)(cid:12)(cid:68) (cid:69)(cid:12)(cid:69)
Cˆ (t)= (cid:12) a†(t)a (t) (cid:12) . (20) be a part in the centre, and the rest of the system bulk,
d (cid:12) l l+d (cid:12)
l or a single bipartite splitting of the 1D system (e.g., the
half-chain splitting directly in the centre of the system).
after a global quench of the transverse field.
If we refer to these two parts of the system as A and B,
InFig.5wecanseethecomparisonofcorrelationfunc-
thenassumingthequantumstateofthewholesystemto
tions for the LRTI model calculated with MPS/MPO
be pure, if it cannot be written as a product of states of
methods and via the Holstein-Primakoff transformation
both parts, i.e. |ψ(cid:105) (cid:54)= |ψ (cid:105)⊗|ψ (cid:105), then A and B are
after the system is quenched from the fully polarized A B
entangled. As a measure of spatial entanglement we use
state. The initial order of the state is preserved only
the von Neumann entropy of the reduced density matrix
at times of a fraction of tJ for moderate fields B, which
for one part of the system, ρ =tr (|ψ(cid:105)(cid:104)ψ|), given by
doesnotallowustousethisapproximationforquantita- A B
tiveanalysisforlongertimes. Byincreasingtheabsolute
S =−tr(ρ log ρ ). (21)
value of the field one can extend the lifetime of the or- vN A 2 A
der, and the Holstein-Primakoff transformation will be In the case of local interactions (or interactions that ex-
valid for longer times. At the same time, the exchange ponentially decay with distance), the bipartite entan-
term becomes stronger for longer range interactions and glement entropy grows linearly in time after a quench
the initial spin order lasts for shorter times. It should [45, 46]. This is related to a finite speed of excitations
be noted, although the quantitative agreement lasts for propagating in the system, which leads to a linear rate
short times, key qualitative aspects of the dynamics are ofquantuminformationexchangebetweenpartitionsand
typically captured over longer timescales. lineargrowthofentanglement. Wecanthennaturallyask
8
to what extent the behaviour that is known for short- (a) (b)
:::
range interactions persists for the models we consider 8 j#### i
here. SNv4 SNv6 j""""::::::i
j"#"# i
Belowwefirstconsiderthegrowthofentanglementfor :::
the LRTI model, which we again compute using matrix 2 ,= 4 j## ""i
1
,=2 2
product operator techniques. We then analyse analyti-
,=1=2
cally the entanglement growth in the LRFH model. In 0 0
0 5 tJ 10 -1 0 log tJ 1
each case, we find that the behaviour mostly follows the 10
(c) (d)
same form as for short range interactions. The counter-
,=3=2 ,=1=2
intuitive exception to this is that when the initial state 4 4
N N
is reasonably symmetric, for long-range interactions the Sv Sv
3
ssinhutoberlritancteotairomn—esdaiuic.tmeu.,atliilnmyescueerpntptaariennsgslreeesmgitemhneetsggrrhooawwvtitnhhgocfalonennbgt-earnavgnelgreye- 2 j<<<A12xxx800ijjAAA00ii 2 j<<A1xx00ijAA0i
ment relative to short-range interactions. 0 22j 0i 1 20j 0i
0 2 4 6 tJ 8 -0.5 0 log tJ 1
10
FIG. 6: Bipartite entanglement growth after the global
A. Entanglement growth in the long-range
quench in LRTI model with open boundary conditions. (a)
transverse Ising model
Half-chain entanglement entropy as a function of time for
M = 20 spins beginning in the fully polarised state |ψ (cid:105) =
0
In Fig. 6, we plot representative calculations for the |↓↓↓...(cid:105) along the axis of the magnetic field B = J (from
growth of entanglement entropy in the LRTI model, be- exactdiagonalisation). (b)Thesameas(a),butforα=1/2,
ginning from a selection of different initial states. In andstartingfromaselectionofinitialstates(2fullypolarised
Fig. 6(a) we see clearly the change in characteristic be- states in the opposite directions, N´eel ordered state, and a
productstatewithspinsdowninthelefthalfofthechainand
haviour as we go from the nearest neighbour interaction
up in the right half). (c) Linear and (d) sublinear growth of
limit (α→∞) to long-range interactions, beginning in a
entanglement entropy (shown on a logarithmic scale) for the
fully polarised state with all spins down.
LRTI model and selection of initial states, now for M = 50
In Fig. 6(b), we focus on the case where α < 1, and
spins, computed with MPS methods (converged with MPS
identify clearly sublinear behaviour in the entanglement bond dimension D=256).
entropygrowthatshorttimes. Thisisapproximatelylin-
ear on a logarithmic scale, but depends strongly on the
structureoftheinitialstate,withmuchslowergrowthat 1. This can be seen from the equality of all curves at
intermediate times for the initially fully polarised state, short times in Fig. 6(c). In contrast, for α < 1, the
compared with other spin configurations or with short- disturbance in the initial state has immediate effect on
range interactions. The intuitive explanation for this re- entanglementindependentofitspositioninthechain, as
striction of entanglement growth at intermediate times seen in Fig. 6(d).
is that the structure of the initial state, when combined
withthesymmetryoftheHamiltonian,preventsthesys-
tem from accessing large sections of the Hilbert space
1. Behaviour in limiting cases
at short to intermediate times, as it becomes somewhat
stuck in its initial symmetry sector. The ultimate limit
Before treating the variation in behaviour with chang-
of this occurs when α → 0, where the Hamiltonian is
ing α in general, it is worthwhile to consider the limiting
fully symmetric. In that case, if we begin with a com-
cases of very short or very long-range interactions. In
pletelypolarizedspinstate,thesystemwillbeatalltimes
the case of the nearest-neighbour interactions (α → ∞)
restricted to completely symmetric spin states, substan-
the transverse Ising model can be studied analytically
tially limiting the maximum entanglement entropy that
and the dynamics after global quantum quenches has
can be reached [15].
been considered in numerous works, see e.g. [44, 47–
In Figs. 6(c,d) we consider initial states with a single
57]. In this limit the Ising chain can be mapped onto a
spin flipped compared with the fully polarised state. We
modeloffreefermions, whichhasthetwofolddegenerate
see that both the linear behaviour for α > 1, and the (cid:113)
sublinear behaviour for α < 1 are quite robust to small dispersionrelation(cid:15)(k)=2 (J −B)2+4JBsin2(kλ/2)
changes in the initial state like this at short times. At for k (cid:54)= 0, where λ is the distance between spins. The
the same time, a small difference between the two cases fastest quasi particles in this model move at the Lieb-
can be observed, because in calculating the half-chain Robinson velocity v = vmax = 2λJ for B ≥ J
LR gr
entanglement entropy, we notice that the changes in the and v = vmax = 2λB for B < J. By performing
LR gr
dynamicsinducedbythespinflipstaketimetopropagate the global quench of the system parameters we excite
acrossthedividebetweenthedifferentpartsofthesystem counter-propagating entangled pairs of quasi particles at
and affect the half-chain entanglement entropy for α > certain points of the system. The spatial entanglement
9
between two parts of the system grows as one quasi par- (a) (b)
ticle of a pair crosses the border. Since the pairs are 70.4 70.4
spreadingwithafinitemaximumspeedthespatialentan- = =
7 7
S0.3 S0.3
glement entropy is limited by the Lieb-Robinson bound,
which leads to a linear growth of the entanglement en-
0.2 M =80 0.2 7=20
tropySvN ≤C1vLRt+C2,whereC1andC2areconstants, M =160 7=40
as is discussed in more detail in Sec. IVB1. 0.1 M =320 0.1 7=80
In the opposite case of all-to-all interactions with α= bulkprediction bulkprediction
0 0
0, the model can also be analytically solved via map- 0 1 tJ=7 2 0 1 tJ=7 2
ping with the Lipkin-Meshkov-Glick model, for which
entanglement properties have been studied [58, 59]. By
FIG.7: TheentanglemententropyinLRFHmodelwithα=
starting from a fully polarised initial state with spins
0.8, computed using exact numerical techniques for various
aligned with the external field B, the dynamics is fully
systemsizesM andsubsystemsizesµaftertheglobalquench
restricted to the subspace of Dicke states with finite well of∆/J =4→1/5. (a)Entanglementdynamicsfortheequal
defined maximum spatial entanglement [60]. Hence, the partition case with µ=M/2. The circular markers show the
entanglement entropy will always be restricted S ≤ limit described by (23) for large subsystems in the bulk (i.e.,
vN
log (M/2+1), where M is the number of original spins, µ,M → ∞ and µ (cid:28) M). (b) Entanglement dynamics for a
2
intuitively explaining the slow and bounded growth of selection of system and subsystem sizes, which is shown to
entanglement with respect to the case of long-range in- converge to the prediction (23) with increasing µ for µ (cid:28)
M =640. The circular markers again indicate (23) for large
teractions.
subsystems in the bulk.
2. Transition at α=1
changeinbehaviourforparticularinitialstatesandlong-
range interactions. We can understand this analytically
In order to understand the change in behaviour that from the behaviour of and contribution to the entangle-
occurs at α=1, we need to go well beyond the intuitive ment growth from the quasiparticles with k →0.
discussion above for α→0. In Sec. IVB1, we will show
that the analogous transition in the LRFH model occurs
asaresultofchangesinthedispersionrelation,andshow 1. Quench from ∆=4J to ∆=J/5
how this directly relates to qualitative changes in the
growth of bipartite entanglement. Weconsidertheentanglemententropyasafunctionof
Although the LRTI model is not analytically solvable, time, both for a bipartite splitting in the centre of the
as we showed in Sec. IIID, we can treat this model for system, and the entanglement of blocks of size µ with
initialstateswithallofthespinsalignedalongthez-axis theremainderofthesystem(especiallyinthethermody-
in the Holstein-Primakoff approximation, which gives an namic limit where M → ∞). In Fig. 7 we show numer-
exactly solvable model that is relevant in the description ical results for the growth of the entanglement entropy
of short to intermediate time dynamics when the order- in a quench within the LRFH model from ∆ = 4J to
ing in the alignment of the spins is still present. This ∆ = J/5. Fig. 7(a) shows the half-chain entanglement
allows us to identify the change in qualitative behaviour entropy µ = M/2, and in Fig. 7(b), we consider a small
at α=1. Within that approximation, we observed (also chainµ(cid:28)M. Ineachcase,weseeafairlytypicalgeneric
as shown in Sec. IIID) that for α < 1, there is a diver- form for the entanglement growth, in which it saturates
genceintheexcitationspectrum,andhenceinthegroup at the value and time that is proportional to the size
velocity, as k → 0. For initial states where the quench of the subsystem, µ. It should be noted that the half-
excites significant excitations near k = 0, the dynamics chain entropy per unit length, is generally smaller than
of these quasiparticles then dominate the growth of en- for smaller subsystems µ.
tanglement, changing the entanglement entropy growth It is useful to co compute an analytical expression for
away from linear. This is completely analogous to the this growth in the thermodynamic limit, where µ (cid:28)
case of the LRFH model, which we will now discuss. M →∞. Herewecanevenconsideropenboundarycon-
ditions as the opposite to the derivations in Sec. IIB. In
thislimit,itisconvenienttodefinetheFouriertransform
B. Entanglement growth in the long-range of the long range term (analogous to Eq. (4))
fermionic hopping model
(cid:88)∞ e−idk (cid:104) (cid:105)
J(k) = J =J Li (eik)+Li (e−ik)
In this section we discuss analytical forms for the |d|α α α
growth of the entanglement in the LRFH model, which d=−∞
d(cid:54)=0
we also evaluate numerically. We will see the same qual- = 2JRe[Li (eik)] (22)
α
itative behaviour as we saw for the LRTI model: linear
growth for all α for generic initial states, but a marked where Li (z) is the polylogarithm of order n and argu-
n
10
ment z. In terms of this, we get the same dispersion re- (a) (b)
lation and the Bogoliubov angle as in Sec. IIB but with 0.3 0.4
7 7
J(k) instead of J¯(k). = =
S70.2 ,=0:4 S7
Ifweconsiderpre-quenchHamiltoniansthatarereflec-
,=0:8 0.2
tion symmetric and can be diagonalised by the Bogoli-
0.1 ,=1:2
ubov transformation, then after the quench the entan-
,=2
glement entropy per unit length of large subsystems far 0 0
away from the boundaries (i.e., in the thermodynamic 0 0.5 tJ=7 1 10-2 10-1 tJ=7100
(c) (d)
limit) is given by [46, 51]
ˆ 7 70.3
S π dk (cid:16) t(cid:17) =0.2 =
µµ ∼ π min 1,2|ε(cid:48)(k)|µ G(cosδθk), (23) S7 S70.2
0
0.1
where 0.1
1+x 1+x 1−x 1−x 0 0
G(x)=− log − log , (24) 10-2 10-1 tJ=7100 10-2 10-1 tJ=7100
2 2 2 2
and δθk is the difference of the Bogoliubov angles before FIG.8: TheentanglemententropyinLRFHmodelcomputed
and after the quench, as in Eq. (6). We plot this ex- using exact numerical techniques with a selection of interac-
pression in Fig. 7, and see that it fits very well for larger tion range exponents α, showing results for ∆ = J/5, and
subsystems in Fig. 7(b). twodifferentinitialstatestwocases: (a,b)Beginningfromthe
We see that Eq. (23) can be always bounded by the groundstatefor∆=4J,and(c,d)beginningfromtheground
velocity term state of a critical Ising model, as detailed in the text. (a,c)
ˆ The half-chain entanglement entropy µ = M/2 = 40, and
π dk (cid:18) t(cid:19) (b,d)thesubsystementanglemententropyforµ=M/8=40.
π min 1,2|ε(cid:48)(k)|µ G(cosδθk) Markers denote the bulk prediction from (23). (a) For α≥2
0 ˆ there is a time window of clean linearity. At short times the
≤ 2t π dk|ε(cid:48)(k)|G(cosδθ ). (25) leadingbehaviourseemstobelinear,butthesubleadingterms
µ π k (intJ/µ)arealsoaffectedbythedivergenceofthedispersion
0
relation. (b) The agreement with the bulk prediction is ex-
In the long-range case with α < 1, though, the velocity cellent, but at sufficiently large tJ/µ the discrepancy due to
(9) diverges as kα−2 when k →0 and the short time be- thefinitenessofM isclearlyvisible,especiallyforthelargest
haviour of the entropy could be strongly influenced by values of α. (c) There are evident differences with respect to
the modes close to zero momenta if these are affected by (a) due to the fact that quasiparticles with momentum close
the quench. However, we find for the quenches we plot to zero turn out to give a finite contribution to the entropy.
Whileforα≥2theentropyseemstogrowlinearly,forsmallα
inFig.7,thechangeintheBogoliubovangleδθ →0,so
k
theleadingcontributionatshorttimegrowthlogarithmically
thequenchdoesnotgeneratenewquasiparticlesatk =0,
intime. (d)Theagreementwithprediction(23)isstillgood,
and existing quasiparticles just pick up a phase. Conse-
althoughtheleadingcorrectionseemstobemorecomplicated
quently,theseplaynoroleinthechangeofentanglement.
than a simple constant.
Onaformallevel,thebehaviourofG(whichensuresthat
the modes at k = 0 are barely affected by the quench)
is sufficient to “cure” the divergence of the velocity and
of the critical transverse-field Ising chain (with Hamil-
the integrand in the right hand side of Eq.(25) becomes
tonian H = (cid:80) c†(c† +c −c )+h.c.), this has
finite. CTI l l l+1 l+1 l
In Figs. 8(a,b) a weak dependence on α for the qual- cosθk0 ∼ |k|/2 and hence G(cosδθk) = log2. The be-
itative behaviour of the entanglement entropy is shown haviour of quasiparticles with momentum close to zero
clearly for this quench. In Fig. 8(a) this is shown for then dominates the dynamics at short times.
µ=M/2, and in Fig. 8(b), this is shown for the subsys- Figs. 8(c,d) show the entropy per unit length for the
tem in the bulk, µ(cid:28)M. timeevolutionunderthelong-rangeHamiltonian(2)with
However,ifwechooseadifferentinitialstate,forwhich ∆=J/5,againshowingthehalf-chainentropyandasys-
the quasiparticles (as represented by the Bogoliubov an- tem in the bulk respectively. These calculations start
gle) change substantially in the quench for k → 0, then from the ground state of the CTI Hamiltonian (with
the behaviour can depend substantially on α. In order open boundary conditions). For α < 1 the short time
to show this dependence — analogous to what we saw in behaviour is not dominated by the linear term and a
theprevioussectionfortheLRTImodel,weneedtocon- completely different behaviour emerges. If we take the
sider ground states with a different structure than the leading order terms at short times, we find that this be-
∆ = 4J ground state. Specifically, we look for situa- haviourispolynomial,andseethatS ∝t1/(2−α)atshort
µ
tions where |ε(cid:48)(k)|G(cosδθ ) in (23) diverges as k → 0, times. The timescales over which this power law holds
k
e.g., by choosing the ground state of a critical model becomeshorter,andgotozeroasα→1. Thatis,inthis
as the starting point. If we choose the ground state casewehavegrowththatisfasteratveryshorttimes,but