Table Of ContentNon-interacting central site model: localization and logarithmic entanglement growth
Daniel Hetterich1, Maksym Serbyn2, Fernando Dom´ınguez1, Frank Pollmann3, and Bj¨orn Trauzettel1
1Institut fu¨r Theoretische Physik, Universit¨at Wu¨rzburg, D-97074 Wu¨rzburg, Germany
2Department of Physics, University of California, Berkeley, California 94720, USA and
3Max-Plank Institute for the Physics of Complex Systems, D-0118 Dresden, Germany
(Dated: January 12, 2017)
WeinvestigatethestationaryanddynamicalbehaviorofanAndersonlocalizedchaincoupledtoa
singlecentralboundstate. ThecouplingtothecentralsitepartiallydilutestheAndersonlocalized
peaktowardsthenearlyresonantsites. Inparticular,thenumberofresonantlycoupledsitesremains
finiteinthethermodynamiclimit. Thisisfurthersupportedbyamultifractalanalysisofeigenstates
that shows the frozen spectrum of fractal dimension, which is characteristic for localized phases in
7
1 models with power-law hopping. Although the well-known Fano-resonance problem is seemingly
0 similar to our system, it fails to describe it because of the absence of level repulsion within the
2 energy spectrum. For weak coupling strengths to the central site, we identify a regime with a
logarithmic in time transport of particles and information.
n
a
J
In quantum mechanics the destructive interference of
0
wave functions in the presence of disorder may com-
1
pletely suppress diffusion, leading to the celebrated An-
] derson localization (AL) [1]. AL is driven by a compe-
n
tition between the disorder potential and the kinetic en-
n
ergy, and strongly depends on the spatial dimension [2].
-
s The Anderson transition (AT) between metallic and lo-
i
d calized phases in disordered systems presents an intrigu-
. ingexampleofergodicitybreakinginnon-interactingsys-
t
a
tem, and remains a subject of theoretical [3] and experi-
m
mental studies [4].
-
d The search for delocalization transitions in one-
n dimensional systems with uncorrelated disorder moti-
o vated the introduction of models with power-law hop-
c
pings. In particular, the power-law banded matrices [5] Figure 1. The wave function of the eigenstate that is local-
[
became a prime example allowing for a detailed study ized on the site i initially decays exponentially with the dis-
1 of AT criticality. Further, random-matrix-type mod- tance|k−i|withthesamelocalizationlengthasinthemodel
v withoutcentralsite(blueline). Non-zerohoppingtothecen-
els without any spatial structure were recently consid-
4 tral site, m > 0 leads to a saturation of the wave function
4 ered [6]. The power-law banded matrices can be viewed amplitude to a constant background that scales as c/L. We
7 as a one-dimensional system with a power-law hopping,
show data for disorder strength W = 10, coupling strength
2 1/rσ, controlled by the exponent σ. Such system is lo- m = 1.25 and L ∈ {29,210,211,212}. The inset displays the
0 calized when σ > 1, delocalized for σ < 1, and right at schematic√of the CSM, dashed lines represent the hopping
.
1 the AT for σ =1. terms m/ L.
0
In this work, we study a physical model, which real-
7
izes some of the phenomena observed in the power-law
1
: banded matrices. The key features of these toy models possiblerealizationofalocalizedphasewithinteractions
v
is the combination of long range hopping and disorder. — so called many-body localized (MBL) phase [12–14]
i
X Therefore, in order to mimic these Hamiltonians we con- — in such models is an intriguing open question. In this
r sider the central site model (CSM) which consists of an respect,theCSMmodelconsideredhereisanaturalsim-
a
Anderson localized fermionic chain, where all sites are plification of central spin models to the non-interacting
coupled to an additional site (see inset in Fig. 1). While case. Wewillseethat,despitebeingnon-interacting,the
the fermionic chain without central site is Anderson lo- CSM exemplifies exciting physical properties.
calized for any non-zero disorder, the central site allows TheCSMresemblesthephysicsoftheFanoresonance,
for hopping between arbitrary distant sites. where a single bound state is coupled to a continuum of
This model is motivated by central spin models [7–9] energy levels [15]. There, a particle placed on the bound
which adequately describe the hyperfine interaction be- state is always vanishing in the continuum, because no
tween an electron spin captured in a quantum dot with localized eigenstates exist. In the CSM the continuum is
the bath of nuclear spins in the host material [10, 11]. replaced by an Anderson insulator that initially shows a
Central spin models are intrinsically interacting, and a Poisson distribution of level spacings. Consequently, we
2
finddifferentregimesintheCSMdependingonthevalue
of coupling strength.
At weak couplings, the CSM retains a Poisson level
statistic in most parts of the spectrum. In what follows
we demonstrate that resonances in this part of the spec-
trumleadtologarithmicintimegrowthofentanglement
entropy – a feature that is usually attributed as a hall-
mark of the interacting many-body localized phase [16–
18]. As we increase the coupling constant to the central
site,thesystemcontinuouslyevolvesintoaregimewhere
itcanbedescribedwithintheFanoresonanceframework.
Herethelevelrepulsionpersistsinthefullspectrum,and
the dynamics of entanglement is faster than logarithmic.
Model.— The CSM is described by the quadratic
Hamiltonian H = H +H that consists of two parts.
ring c
The first describes a one-dimensional disordered ring of
(cid:16) (cid:17) Figure2. Thespectrumoffractaldimensionofparticipation
size L, Hring =(cid:80)Li=1hic†ici+J c†ici+1+c†i+1ci , where ratios,τq,behavesinaqualitativelysimilarwayforallvalues
of coupling m. The convergence to τ = 0 when q ≥ 1 indi-
c† and c are fermionic creation and annihilation opera- q
i i cates a “frozen” fractal spectrum. For m(cid:28)W, τ converges
q
tors. In H , the on-site ring energies h are assumed
ring i faster. Instead,form(cid:29)W,thefractalspectrumapproaches
to have random values uniformly distributed over the τ =(1−γq)(q−1) with γ =1, suggesting a behavior inter-
q
range h ∈ [−W,W], where W quantifies the disorder mediate between strong and weak multifractality [21]. Data
i
strength and is assumed to be dominant over the hop- is for eigenstates of the energy interval I =[2,4] for disorder
ping J, W >J. strength W =10 and systems with up to L=213 sites.
The second part is referring to the Hamiltonian of the
central site and its coupling to all ring sites,
The constant background in Fig. 1 results from the
averaging over individual eigenstates |E (cid:105) for different
m (cid:88)L i
H =(cid:15) c†c + √ c†c +c†c , (1) disorder configurations. Separately, each wave function
c 0 0 0 L i 0 0 i exhibitsstrongmultifractalfluctuations[22]. Wecharac-
i=1
terize these fluctuations by means of the moments of the
where in what follows we set the energy of the central participation ratios,
site, (cid:15) = 0, to be approximately in the middle of the
0
localized band. √Note, that we scale the coupling to the P =(cid:104)(cid:88)|(cid:104)k|E (cid:105)|2q(cid:105) ∼L−τq, (2)
q i i
centralsiteas1/ L,sothatoursystemhasawelldefined
k
thermodynamic limit. In the remainder of the paper we
set J = (cid:126) = 1, and measure energy and time in units of which we average over all eigenstates |E (cid:105) and over dis-
i
J and (cid:126)/J, respectively. order realizations. The scaling Pq ∼ L−τq defines τq =
Structure of eigenfunctions.— As already mentioned Dq(q−1)andthefractaldimensionDq [22]. Forthecase
above,thecouplingoftheALchaintothecentralsitein- of ideal metals or insulators, one expects the constant
duces a partial delocalization of the Anderson peak. We values Dq = d and Dq = 0, respectively, where d is the
can observe this in Fig. 1, where we show the average spatial dimension. Instead, multifractal wave functions
amplitude of the wave function |E (cid:105) localized at site i as exhibit a dependency on q.
i
a function of distance from this site, |k−i|. We observe Fig. 2 demonstrates that the coupling to the central
that the average probability of finding a particle along site partially destroys the insulating phase and gives rise
thesystemisstillexponentiallylocalizedaroundtheith- to a non-trivial spectrum τ . Nevertheless, for all values
q
site, with the AL length ξ [1, 2, 19] that is independent of m we find τ = 0 for q ≥ 1. The convergence to
q
of m and coincides with the localization length of the τ = 0 for q ≥ 1 is called “frozen” fractal spectrum.
q
HamiltonianH . Thus,themaineffectofthecoupling This observation is consistent with the presence of the
ring
to the central site is a small, on average homogenous, AL peak in Fig. 1, because for q ∼ 1, all participation
background,whichscalesinverselyproportionalwiththe ratios are dominated by the largest values of the wave
system size, i.e. like c/L [20]. Note that since the con- function.
stant background scales as c/L and ξ (cid:28) L, c can be For q (cid:28)1 instead, τ gives information about the dis-
q
naturally interpreted as the probability to find the par- tribution of the wave function outside of the AL peak.
ticle away from the localization center. Remarkably, we Themultifractalfluctuationsofthewavefunctionsreveal
observe this behavior for any value of m and not just in themselves in the deviation from the metallic spectrum
the limit of weak coupling to the central site. τmetal = d(q −1), which we would find for a constant
q
3
background. When m (cid:29) W, the fractal dimension ap-
proaches τ = (1−γq)(q −1) with γ = 1 (see dashed
q
line in Fig. 2). This behavior suggests a phase in be-
tween strong and weak multifractality [21]. For m(cid:28)W
in contrast, τ , and especially its slope at q =0, depend
q
on m.
Level statistics.— The existence of two different
regimes, depending on the value of the coupling m can
be understood by studying the influence of coupling to
the central site on the level statistics, which is Poisso-
nian in its absence. Perturbatively, an eigenstate with
energy (cid:15) will acquire a correction m2/((cid:15)L). Comparing
this shift to the mean level spacing δ = 2W/L, we de-
fine the crossover energy (cid:15) , where m2/((cid:15)L)∼δ. Hence,
∗
onlytheenergylevelswithin[−(cid:15) ,(cid:15) ]acquirelevelrepul-
∗ ∗
sion. At weak coupling strengths m (cid:28) W, this interval Figure 3. Density of the probability P(s) of level spacings
is negligible and Poisson statistics dominates. It is this s = (Ei −Ei−1)/δ between adjacent energy levels Ei and
feature that breaks the applicability of the conventional Ei−1 intheenergyrangeEi ∈[2,4]forW =10andL=210.
The level repulsion develops in the distribution of the level
Fano resonance picture, which relies on the presence of
spacingsuponincreasingvaluesofthecouplingtothecentral
level repulsion [15].
site. ThetailsofthedistributionsalwaysremainPoisson-like,
We verify the existence of the interval [−(cid:15)∗,(cid:15)∗] in as is revealed by their exponential decay shown in the inset.
Fig. 3, where we study eigenvalues within the energy
window I = [2,4]. At m (cid:28) W, eigenstates within I
are weakly perturbed and we observe the Poisson statis- of S (t) provides then a direct measure of the spread of
A
tics of level spacings. For larger values of m, the interval information throughout the system. In noninteracting
of perturbed eigenvalues overlaps with I, giving rise to systems, S (t)canbewrittenintermsofthecorrelation
A
a decrease of P(s) for s → 0 in Fig. 3. For m (cid:38) W, matrix CA =(cid:104)ψ(t)|c†c |ψ(t)(cid:105) [26, 27], with i,j ∈A, as
ij i j
the interval [−(cid:15) ,(cid:15) ] covers the whole band and thus,
∗ ∗
the entire spectrum shows level repulsion. Nevertheless, S (t)=−tr[CAlnCA+(1−CA)ln(1−CA)]. (3)
A
the level statistics is not described by a Wigner-Dyson
distribution, as the P(s) retains exponential tails. The In Fig. 4, we show the time evolution of S (t) in the
A
slowdecayofprobabilitytohavelargelevelspacing(sim- three regimes of the coupling strength. As expected for
pleexponential,ratherthanGaussian)suggeststhefinite m=0werecovertheALphase,characterizedbyaquick
level compressibility. This is natural: while a single cen- saturation of S (t) to a value independent of L. Also,
A
tral site can easily split degeneracies causing the level whenm(cid:38)W,theresultingS (t)exhibitsatypicallinear
A
repulsion at small s, it cannot lead to a strong mixing of time-dependence [23, 28]. However, in the case of non-
eigenstates with very different energies. Hence the fluc- zerobutsmallm(cid:28)W,wefindalogarithmicdependence
tuations in the level spacing between k-th nearest neigh- ofS (t)ontime,whichliesinbetweenbothphases. The
A
bor eigenstates would grow proportionally to k, corre- saturation value of S (t) scales linearly with the sys-
A
spondingtofinitecompressibility. Inthissense,thelevel tem size for any m > 0 [20]. As mentioned above, the
statistics is closer to the critical level statistics seen at logarithmic growth of entanglement has been associated
the AT [21]. so far to many-body localized phases where information
Dynamics and entanglement growth.— Next we spreads via interactions between the particles [17, 18].
demonstrate how the energy scale (cid:15) affects the dynam- Tothebestofourknowledge,thelogarithmicgrowthto-
∗
ical properties of the model. To this end, we study gether with a saturation value that scales linearly with
thespreadingofinformation, witnessedbyentanglement system size has never been observed in a non-MBL sys-
growth, which can in our noninteracting model only oc- tem before. Recently, a similar situation in the absence
cur due to particle transport. The entanglement growth of interactions but without a linear scaling with system
is thus strongly correlated with the motion of the par- size has been identified in a different model [29].
ticles. We consider the time evolution of the entangle- Inordertounderstandtheentanglementdynamics,we
ment entropy S (t) [23–25] which is frequently analyzed study how particles move through the central site that
A
to identify the localization transition (with and without connectsanypossiblebipartition. Thus,weconsiderhow
interactions). Using an equal size bipartition A, B of aparticleplacedataspecificsiteontheringevolveswith
the Hilbert space H in real space, we quantify the cor- the CSM Hamiltonian. Let us assume that this initial
relations among them by studying the von Neumann en- site has the strongest overlap with the eigenstate of en-
tropyS (t)ofthereduceddensitymatrix. Theevolution ergy(cid:15)oftheunperturbedchain. Theslowestdynamicsis
A
4
generated by the mixing with almost degenerate energy
levels. The coupling between these neighbored levels is
accordingtoperturbationtheoryJ ∼m2/((cid:15)L). Hence,
nn
theparticleleaksintosuchaneigenstateonthetimescale
t((cid:15)) ∼ (cid:15)L/m2. Eigenstates with (cid:15) > (cid:15) are hardly per-
∗
turbed by the central site. However, in rare cases with
probability J /δ, adjacent eigenstates are close to each ballistic,
nn
other and strongly mixed by the central site. Then, the
particleisequallylikelytobefoundinbotheigenstatesat
largetimes,contributingtoaprobabilitythattheparticle
leftitsinitialeigenstategivenbyn¯ ∼J /δ ∼m2/(W(cid:15)).
(cid:15) nn
The above intuition can be quantified by a three-site
model. Within this toy-model, we study the motion of
a particle starting on a site of potential (cid:15) through the
central site into the continuum. When the coupling to
Figure 4. Entanglement growth of the initial product state
the central site is weak, m (cid:28) W, we are in the pertur-
with L/2 fermions, where every other site is occupied,
bative regime where mixing occurs with a small number |0101...01(cid:105). Upon changing the value of m, we observe An-
of sites within [−(cid:15)∗,(cid:15)∗]. Hence in our case it suffices to dersoninsulatingbehavior(m=0,centralsiteisdecoupled),
consider the coupling to a single site of energy (cid:15)(cid:48) and a regime of logarithmic entanglement growth (W (cid:29)m>0),
average over all (cid:15)(cid:48) ∈[−W,W] after the equations of mo- and eventually a polynomial in time entanglement dynamics
for m ≥ W. The data is for L = 512 ring sites, disorder
tions are solved. This average neglects correlations be-
strength W =10, and coupling m=0 (green), m=1 (blue),
tween initially unoccupied ring sites, which is a reason-
and m = 10 (red). The inset illustrates that numerical re-
able assumption for m(cid:28)W. However, it accounts for a
sultsforthenumberofparticlesthatchangedtheirbipartition
smalllevelsplittingwhich,incontrasttotheconventional agree with analytical predictions for m=1.
Fanoresonance[15],ispresentinourproblemduetothe
Poisson distribution of level spacings in the unperturbed
model. Using the three-site model [20], the probability suchregimebecomessimplyaninverselevelspacing1/δ.
n (t) that the particle left its initial site becomes Inserting this into the first line of Eq. (4), we get n¯ ∝
(cid:15) (cid:15)
(cid:40)πm4 (cid:0) t (cid:1), √6 ≤t≤ (cid:15)L , 1th/e(cid:15)2l.inAeavrersapgrienagditnhgisoofvner(tt)h,eaninditbiaallleisnteicrgeyn(cid:15)tarnegsluelmtsenint
n (t)≈ WL (cid:15)2 (cid:15) 4m2 (4)
(cid:15) n¯ = πm2 (cid:0)1(cid:1), t≥ L(cid:15) , dynamics, as confirmed in Fig. 4.
(cid:15) 4W (cid:15) 4m2
Summary—WehaveconsideredthebehaviorofanAn-
whichdependsonthepotentialattheinitialposition. At derson localized fermionic chain perturbed by coupling
a given time t, sites with energy (cid:15) < (cid:15)t = 4m2t/L have allsitestotheadditionalcentralsite. Theeigenstatesaf-
saturated their dynamics. Summing the contributions terperturbationloosetheirlocalcharacter,butstillkeep
from all such sites, we get that the number of particles the localized statistics. Statistics of level spacings and
that are located at other sites is dynamical probes reveal the crossover between two dif-
1 (cid:90) (cid:15)t πm2 (cid:18)m2 (cid:19) ferent regimes. In the weak coupling regime, most of the
n(t)= 2W n(cid:15)(t)d(cid:15) = 4W2 ln L t +..., (5) eigenvalues of the system are Poisson-distributed, which
−(cid:15)t leadstoabreakdownoftheFanoresonancepicture. This
where ellipses denote possible constant terms. The log- regimeshowslogarithmicintimegrowthofentanglement
arithmic dependence here emerges from the summation entropycausedbythespreadingofparticles. Thesecond
over all possible energies of initial site, which translates regime, where level repulsion is present, can be mapped
into the integral of 1/(cid:15) over (cid:15) with limits set by time. In to the Fano resonance problem, and it is characterized
the inset of Fig. 4 we compare the result of the three- by ballistic motion of entanglement.
site model, Eq. (5), with the numerical data of the full Overallourfindingssupporttheintuitionthatasingle
modelandfindgoodagreementintheregimeoflogarith- degree of freedom, even if it is coupled non-locally to
mic growth of entanglement. As we outline in [20], the the Anderson insulator is not sufficient to delocalize the
entanglement entropy S (t) can be expressed by means system. In addition, our results suggest that logarithmic
A
ofn (t),whichexplainsthelogarithmictime-dependence entanglement growth can be realized in non-interacting
(cid:15)
of the entanglement entropy, plotted in Fig. 4. models of free fermions, and is related to the persistence
Finally, at m≥W, the energy scale (cid:15) becomes of the of the Poisson level statistics in parts of the spectrum.
∗
orderofthebandwidthW. Consequently,levelrepulsion Acknowledgments— We would like to thank Dmitry
ispresentforallenergystatesoftheHamiltonian. Then, Abanin, Christophe De Beule, Joel Moore, Romain
we expect much faster spreading of the particles, caused Vasseur, and Norman Yao for many stimulating dis-
by the strong level admixture: the saturation time in cussions. Financial support has been provided by the
5
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S1
SUPPLEMENTARY ONLINE MATERIAL A.2 Dynamics and breakdown of the Fano
description
A. Mapping to Fano resonance problem
Using self-consistent perturbation theory, we can cal-
A.1. Spectral weight and position of the resonance culate the expansion of eigenstates with the central site
over unperturbed eigenstates. This expansion reads [15]:
Physically, theCSMresemblesthephysicsoftheFano (cid:88) (cid:88)
c = ν α , c = W α . (S8)
resonance, when we have a single site coupled to a con- 0 i i i ij j
tinuum. Then,thereisaleadingorderself-energycorrec- i j
tion,whichdeterminesthenewpositionoftheresonance,
The coefficients W are expressed via Z =Z((cid:15) ) as
ij i i
m2 (cid:88) 1
Σ(iω)= , (S1) A2ν2 Z2
L iω−ε W2 = j +δ i . (S9)
α α ij ((cid:15) −(cid:15) )2 ijZ2+(2π/δ)2
i j i
whereε labelsenergiesoflocalizedeigenstates. Approx-
α Note, that the self-energy in Eq. (S5) defining Z is given
imatingthesumwithanintegral,wededucethefollowing
by:
self-consistent equation for the new position of the res-
onance ωr expressed via the dimensionless variable x as A2 ω
ω =Wx: Σ(ω)= arctanh for |ω|<W. (S10)
r δ W
m2 x−1
x= ln . (S2) Now, we are interested in the amplitude of the wave
2W2 x+1 function,thatisnotlocatedontheinitialeigenstatewith
On the one hand, when m(cid:29)W, we expect that x(cid:29)1. energy (cid:15) , which we denote as n (index i is omitted for
i (cid:15)
Then, we can expand the log and get: brevity). To calculate this amplitude, we sum:
m
x= √ (cid:29)1. (S3) (cid:88) (cid:88) 1 1
2W n = |W |2 =
(cid:15) ij ((cid:15) −(cid:15) )2Z2+(2π/δ)2
On the other hand, if the coupling is weak, we obtain j(cid:54)=i j(cid:54)=i i j j
x−1(cid:28)1, so we have a resonance which is very close to 1(cid:90) 1 m4/L2
= dω . (S11)
the band edge, δ (ω−(cid:15))2[ω−Σ(ω)]2+(πm2/W)2
x−1=2e−2W2/m2 (cid:28)1. (S4)
Todeducethedependenceontime,wecanlimitthesum
Depending on x, we can have a very different spectral in Eq. (S11) to states j with energy difference such that
weight on the resonance, |(cid:15)−(cid:15)j|≥1/t:
(cid:18) m2 2 (cid:19)−1 (cid:88)
Z =[1−∂ωReΣ(ωr)]−1 = 1+ LδW x2−1 . n(cid:15)(t)= |Wij|2
j(cid:54)=i
(S5)
2(cid:90) ∞ 1 m4/L2
For the strong coupling, when x(cid:29)1, we get: = dω .
δ ω2[ω−(cid:15)−Σ(ω−(cid:15))]2+(πm2/W)2
(cid:18) (cid:19) 1/t
1 1
Z ≈ 1− , (S6) (S12)
2 2x2
The integral yields the asymptotic expression
sothatthespectralweightisalmost1/2. Inthiscasewe
get very small spectral weight in the bath that scales as 1 m4
1/x2 ∝ W2/m2. On the other hand, when x (cid:28) 1, we n(cid:15)(t)= 3LW t3 (S13)
have
atsmalltimest(cid:28)1/(cid:15) . Forlongertimes,whenW/m2 >
Z ≈2e−2W2/m2 (cid:28)1, (S7) i
t>1/(cid:15), assuming that (cid:15)>m2/W =(cid:15) , we get:
∗
sothatthespectralweightthatremainsinthecontinuum
m4 t
is close to one. However, in both cases, we recover that
n (t)∝ , (S14)
the spectral weight remaining in the bath does not scale (cid:15) WL(cid:15)2
with the system size.
the linear growth of particle density that is located away
Sincetheonlywaytogetdelocalizedstatesarisesfrom
from initial eigenstate. Note, that for t = 1/δ this ex-
the hopping via the central site, we conclude that the
pressiongivesusthesaturationvaluethatcanbedirectly
constant weight, when distributed among L states in the
obtained from Eq. (S11):
continuum,willgivea∝1/Lconstantbackgroundofthe
wave function. The same result will be derived below m4 2 m4 1
n¯ =n (1/δ)= ∝ . (S15)
from the three-site model. (cid:15) (cid:15) 4W2(cid:15)2+(πm2/W)2 W2(cid:15)2
i i
S2
We see that n¯ is suppressed as 1/(cid:15)2 when the abso- turbatively treated) coupling term becomes
(cid:15)
lute value of energy is bigger than the crossover scale,
(cid:88)
|(cid:15)| > m2/W = (cid:15) defined in the main text. Such depen- V := (c†c +c†c ) (S18)
∗ i 0 0 i
dencewouldleadtoaballisticspreadingofparticles,and i≥2
consequentlyalinearspreadingofentanglemententropy. =n (cid:88)(f†f +f†f )+n (cid:88)(f†f +f†f ) (S19)
− i 1 1 i + i 0 0 i
In all above considerations, we however ignored the i≥2 i≥2
almost resonant pairs of sites. This is a legitimate as-
and the initial state can be written as
sumption when the initial bath of states has level repul-
sion, so that the probability to have degenerate energies 1
|ψ(t=0)(cid:105)=c†|∅(cid:105)= (n λ f†+n λ f†)|∅(cid:105) (S20)
is vanishing. It is this assumption that breaks down for (cid:113)1 A + + 0 − − 1
the present model, specifically outside the energy win- withn =1/ 1+λ2/A2and|∅(cid:105)isthe(empty)vacuum
± ±
dow [−(cid:15)∗,(cid:15)∗]. In order to rigorously consider the physics state. The total Hamiltonian is then given by H =H0+
emerging from such resonances, in the next section we AV andweareinterestedintheprobability|(cid:104)n|ψ(t)(cid:105)|2 to
present a toy model which allows for an analytical un- findtheFermiononanotherringsiten≥2withpotential
derstanding. h ∈[−W,W]. Tosolvethisproblem, weusetheAnsatz
n
(cid:88)
|ψ(t)(cid:105)= b (t)exp(−ih t)|φ (cid:105), (S21)
n n n
n
where |φ (cid:105) = c |∅(cid:105) for n ≥ 2 and |φ (cid:105) = f |∅(cid:105) for n ∈
B. Beyond Fano picture: three-site model n n n n
{0,1} are the eigenstates of H . Inserting |ψ(t = 0)(cid:105) in
0
the Schr¨odinger equation, this yields
B.1. Time dependent perturbation theory
d (cid:88)
idt bn(t)= ei(hn−hk)tVnkbk(t), (S22)
In order to derive the logarithmic growth of entan- k
glement entropy, we study the motion of single fermion
where h =λ for i ∈{0,1} and V =(cid:104)φ |V|φ (cid:105). This
i ± nk n k
below. In particular, we calculate n (t) for small times
(cid:15) exact solution is now approximated by the expansion (in
perturbatively and compare our analytical results with
powers of A)
numerical data for the central site model.
Wehavefoundoutthatitisimportanttotreatthecou- bn(t)=b(n0)(t)+Ab(n1)(t)+A2b(n2)(t)+..., (S23)
plingbetweeninitialsiteandcentralsiteexactly. Thisis
for which we find
becausethetwohybridizedstates,amixtureoftheinitial
sthiteeaexncdittahteiocnenmtroavlessitein,tcoontthaeinretmheaeinssdeenrtioaflpthheyssiycsstheomw. iddt b(nr)(t)=(cid:88)ei(hn−hk)tVnkb(kr−1) r ≥1 (S24)
Thus, we consider the following, unperturbed Hamilto- k
d
nian i b(0)(t)=0. (S25)
dt n
H0 =(cid:15)c†1c1+Ac†1c0+Ac†0c1+(cid:88)hic†ici , (S16) Using the initial conditions (see Eq. (S20)) b(00)(t =
i≥2 0) = n+λ+/A, b(10)(t = 0) = n−λ−/A, and b(n0)(t = 0) =
0 for all n≥2, this implies
where A is the coupling to t√he central site, and in the id b(1)(t)= 1 (cid:16)λ n2ei(hn−λ+)t+λ n2ei(hn−λ−)t(cid:17)
main text we used A = m/ L. If we diagonalize this dt n A + + − −
Hamiltonian, we obtain (S26)
and
H0 =λ+f0†f0+λ−f1†f1+(cid:88)hic†ici (S17) iddt b(n2)(t)= (cid:88) ei(hn−hk)tVnkb(k1)(t)=0. (S27)
i≥2 k∈{0,1} (cid:124) (cid:123)(cid:122) (cid:125)
=0
The time dependent probability that the Fermion is
(cid:113)
with λ = (cid:15) ± A2+ (cid:15)2. Within this basis, the (per- present at site n≥2 is then well aproximated by
± 2 4
S3
(cid:12) (cid:12)2
n (t,h ):=|(cid:104)φ |ψ(t)(cid:105)|2 ≈(cid:12)b(0)+Ab(1)(t)+A2b(2)(cid:12)
(cid:15) n n (cid:12) n n n (cid:12)
2n4λ2 [1−cos(h −λ t)] 2n4λ2 [1−cos(h −λ t)]
= + + n + + − − n − (S28)
(h −λ )2 (h −λ )2
n + n −
2n2n2λ λ {1−cos[(h −λ )t]−cos[(h −λ )t]+cos[(λ −λ )t]}
+ − + − + n + n m m p
(h −λ )(h −λ )
n + n −
Withthisresultathand, wesumthecontributionsofall 100
L sites with random potential h . In the numerical sim-
n
-1
ulation, this is automatically done by considering many 10
independent Fermions at a time and by averaging over
-2
disorder. Mathematically, we do this by 10
-3
L (cid:90) W 10
n (t)= dhn (t,h). (S29)
(cid:15) 2W −W (cid:15) 10-4
Using the principal value of the integral and extend- -5
10
ing the integration boundaries to (−∞,∞), which cor-
responds to a small mistake for A (cid:28) W, the integral -6
10
becomes 10-1 100 101 102 103 104 105
L 4A4π
n (t)≈ (t∆−sin(t∆)), (S30)
(cid:15) 2W ∆3 Figure S1. The probability n (t), that a Fermion, initially
(cid:15)
placed at a site of potential (cid:15), left its site and is located at
(cid:113)
where ∆ = 2 A2+ (cid:15)2 is the level splitting. For times anotherringsitehasdifferentbehaviordependingonthevalue
4 of (cid:15). Upon increasing (cid:15) ∈ {0.4,0.8,1.6,3.2,6.4} from top to
t(cid:29) 1, wehencehavederivedthelineargrowthofn (t).
∆ (cid:15) bottom,weseethatthetimeintervalwheren(cid:15)(t)hasalinear
Fortimest(cid:28) ∆1,wefindaninitialcubicgrowthofn(cid:15)(t), intimegrowthincreases,whilethesaturationvaluedecreases.
which coincides with Eq. (S13), but is not important
for the logarithmic growth of entanglement entropy. In
Fig.S1,wecomparetheperturbationtheorieswithexact c†0c1) and find the two hybridized states f0 and f1. Sub-
numerics and find good agreement. sequently, we couple these two states independently to a
single bath state of energy E.
Inasimpletwositeproblemswithenergygap∆ and
if
B.2. Saturation values couplingconstantA,theprobabilitytofindanexcitation
outside its initial position is (averaged over time) given
by
In this section, we specify the limit
2A2
1 (cid:90) T P = . (S33)
n¯ := lim dtn (t). (S31) if ∆2 +4A2
(cid:15) T→∞T 0 (cid:15) if
The probability we seek is thus well approximated by
Recall that n (t)=(cid:80)N (cid:104)ψ(t)|c†c |ψ(t)(cid:105) is the probabil-
(cid:15) i≥2 i i
ity that the single fermion left its initial site and is now n¯ (h )=|(cid:104)f |ψ (cid:105)|2P +|(cid:104)f |ψ (cid:105)|2P
present in the bath. In order to derive n¯ (t), we first (cid:15) i 0 0 f0i 1 0 f1i
(cid:15)
λ2n2 2n2A2
resolve it by energy, i.e. = + + +
A2 (λ −h )2+4A2n2
+ i +
(cid:88)N 1 (cid:90) T λ2n2 2n2A2
n¯ (E)dE = lim dt(cid:104)ψ(t)|c†c |ψ(t)(cid:105). (S32) + − − − . (S34)
(cid:15) i≥2T→∞T 0 i i A2 (λ−−hi)2+4A2n2−
E≤hi<E+dE
This probability is derived for one single bath state.
As in the previous chapter, we treat the initial site i=1 WeactuallyhaveL−1bathstatesuniformlydistributed
and the central site i = 0 exactly, i. e. we diagonalize in the energy window [−W,W], resulting in a level den-
theirtwo-siteHamiltonianH =0c†c +(cid:15)c†c +A(c†c + sity of L−1. The average effect of all L−1 bath sites is
0 0 0 1 1 1 0 2W
S4
B.3. Constant background level
Previously, we have described that all eigenstates |E (cid:105)
l
ofthefullHamiltonianarelocalizedatasitel. Onsitesk
farawayfroml,weseeonaverageaconstantprobability
|(cid:104)k|E (cid:105)|2 that the excitation |E (cid:105) is measured at site k.
l l
With the above presented toy model, it is possible to
deduce the correct scaling behavior of |(cid:104)k|E (cid:105)|2 ∼ m2 ,
l LW2
which will be outlined next.
LetusconsideragainasingleFermion, initiallyplaced
at site k with potential (cid:15). The time-averaged density
matrix
1 (cid:90) T (cid:88)
ω = lim dtρ(t)= (ρE) |E (cid:105)(cid:104)E | (S36)
T→∞T 0 l 0 ll l l
Figure S2. Saturation value of the entanglement entropy with n¯ (E )≡(ρE) :=|(cid:104)E |ψ(t=0)(cid:105)|2 =|(cid:104)E |k(cid:105)|2 con-
S = lim S(t) for different system sizes L and coupling (cid:15) l 0 ll l l
∞ t→∞ tains the probability n¯ (E ) to measure the initial ex-
values m ∈ {0,0.25,1,5,20} (bottom to top) and W = 10. (cid:15) l
citation at long times with an energy E , which is the
For all coupling constants m > 0 we find a linear scaling l
S∞ ∼ L, e.g. we compare the data set of m = 1 with overlap |(cid:104)k|El(cid:105)|2 we search for. Before, we have been in-
S = 0.007L (blue line). This feature, together with the terestedinthetotalprobabilitythattheinitialexcitation
∞
logarithmicintimegrowthofS(t)(seeFig.4)hasneverbeen is somewhere in the bath and found
observed before in a noninteracting system. (cid:90) ∞ L−1
n¯ = dh n¯ (h). (S37)
(cid:15) 2W (cid:15)
−∞
Now, wewanttheaverage valueof|(cid:104)E |k(cid:105)|2 =(cid:104)n¯ (E)(cid:105) ,
l (cid:15) E
thus given by
which is for L−1 bath states consequently given by
1 (cid:90) ∞ L−1 A2π
(cid:104)|(cid:104)E |k(cid:105)|2(cid:105) = dh n¯ (h)= (S38)
n¯ =(cid:90) ∞ dh L−1n¯ (h)= L−1 π (cid:2)λ2n3 +λ2n3(cid:3) l l L−1 −∞ 2W (cid:15) 2W(cid:15)
(cid:15) −∞ 2W (cid:15) 2W A + + − − Finally, we have to average over the potential (cid:15) of the
L−1A2π initial site k, which gives
A=(cid:28)(cid:15) +O(A3) (S35)
2W (cid:15) 1 (cid:90) W A2π A2π W
(cid:104)|(cid:104)E |k(cid:105)|2(cid:105) ≈2 d(cid:15) = ln .
l l,k 2W 2W(cid:15) 2W2 α
α>0
Here, we are weakly overestimating the probability n¯ (S39)
(cid:15)
due to two reasons. First, the integral should be taken In the main text of the Letter, we used A = √m and
from −W to W. However, as we explicitly checked, the L
found the constant background level ∼ m2 , which co-
error caused by extension of integration limits to infinity LW2
incides with this result, as the logarithmic contribution
is negligible for small A (cid:28) (cid:15) < W. Secondly, although
of the disorder W is weak compared to the quadratic
theassumptionthatallbathstatesareuncoupledisvalid
dependency and cannot be distinguished numerically.
for small A, it underestimates the physical outcome. In
thenumericalsimulation, theparticlehasmorethanone
possibility to enter the ring sites. The “fraction” of the
C. Relation between entanglement entropy and
particle that goes to an E ring site can no longer go
1 particle transport
to an E ring site and vice versa. Thus, the analytical
2
theory, which assumes only one bath state at a time, is
The entanglement entropy between two bipartitions A
overestimatingtheprobabilityn¯ . InFig.S1,wecompare
(cid:15) andB ofaHilbertspaceHisusuallymeasuredbymeans
the derived saturation value with the numerical gained
of the von Neumann entropy of the reduced density ma-
data and find perfect agreement in the regime A (cid:28) (cid:15)
trix ρ =tr [ρ(t)], i.e.
(lower curves). A B
S (t)=−tr[ρ lnρ ], (S40)
Eq. (S35) is not only completing the derivation of the A A A
logarithmicmotion,butitalsoshowsthatsaturationval- where ρ(t) describes the state of the system. For pure
ues should grow linear with system size. Indeed, we find statesρ=|ψ(t)(cid:105)(cid:104)ψ(t)|,S (t)=S (t)foralltandbipar-
A B
that also the entanglement entropy saturates at values titionsA,B,whichcaneasilybeshownusingtheSchmidt
√
(cid:80)
that grow linear with system size, see Fig. S2. decomposition |ψ(cid:105)= λ |i(cid:105) ⊗|i(cid:105) .
i i A B
S5
For lattice systems of L sites and free Fermions, it λ, which equals for the same reason λ = tr[C ]. The
A
is useful to describe a quantum state not with a vector entanglement entropy thus simplifies to
|ψ(cid:105)∈H=C2L, but by the correlation matrix
S (t)=−λlnλ−(1−λ)ln(1−λ). (S43)
A
C (t)=(cid:104)ψ(t)|c†c |ψ(t)(cid:105) (S41)
ij i j Using the relation λ= tr[C ], we find
A
where ci is a annihilation operator on site i, and thus, C λ=(cid:88)(cid:104)ψ(t)|c†c |ψ(t)(cid:105)=(cid:88)|(cid:104)i|ψ(cid:105)|2 =n (t) (S44)
is a correlation matrix of size L×L only. The matrix i i (cid:15)
i∈A i∈A
C contains all information about the full state |ψ(cid:105) or
ρ, because according to Wick’s theorem any correlation where n (t) is the probability, that the single fermion is
(cid:15)
functionsplitsforindependentFermionsintoproductsof present in the subspace A.
two-point correlators Cij. Hence, also the entanglement In the paper, n(cid:15)(t) is the probability, that a single
entropy SA(t) is expressible by means of C, which is fermion, initially placed at a site of potential (cid:15) changed
itsbipartitionattimet. Hence, wedohaveaccesstothe
SA(t)=−tr[CAlnCA+(1−CA)ln(1−CA)], (S42) entanglement entropy by
and C is the part of C with i,j ∈A. As the Fermions S (n (t))=−n (t)lnn (t)−(1−n (t))ln(1−n (t)).
A ij A (cid:15) (cid:15) (cid:15) (cid:15) (cid:15)
are independent from each other, it is sufficient to con- (S45)
sider a single fermion. The entanglement entropy is ad- By this equation, we have identified a direct connec-
ditive for independent degrees of freedom, hence, the to- tionbetweenn (t)analyticallyderivedinEq.(4)andthe
(cid:15)
tal entanglement entropy sums up to S (t) = (cid:80) Si (t), time evolution of the entanglement entropy S (t). The
A i A A
where Si (t) is the contribution of the ith Fermion. If (cid:15)-averaged expression for n (t) explains the logarithmic
A (cid:15)
only one fermion exists (in a pure state), the matrix C spread of a particle in the CSM, see Eq. (5). Thus, the
has only one nonzero eigenvalue, which is equal to unity. leading terms at intermediate time scales for the time
Hence,C isofrank1andallpossiblesubmatricesC are evolution of S (t) (which also shows a logarithmic time
A A
at maximum of rank 1 as well. For any bipartition A,B, dependence) is directly related to this peculiar particle
the matrix C has at maximum one nonzero eigenvalue motion.
A
