Entanglement in correlated random spin chains, RNA folding and kinetic roughening

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by  Javier Rodríguez-Laguna

Entanglement in correlated random spin chains, RNA folding and kinetic roughening
Javier Rodr´ıguez-Laguna,1 Silvia N. Santalla,2 Giovanni Ram´ırez,3 and Germ´an Sierra3
1Departamento de F´ısica Fundamental, UNED, Madrid, Spain.
2Departamento de F´ısica, Universidad Carlos III de Madrid, Legan´es, Spain.
3Instituto de F´ısica Te´orica UAM/CSIC, Madrid, Spain.
(Dated: January 13, 2016)
Average block entanglement in the 1D XX-model with uncorrelated random couplings is known
6
1 to grow as the logarithm of the block size, in similarity to conformal systems. In this work we
0 study random spin chains whose couplings present long range correlations, generated as gaussian
2 fieldswithapower-lawspectralfunction. Groundstatesarealwaysplanarvalencebondstates,and
their statistical ensembles are characterized in terms of their block entropy and their bond-length
n
distribution,whichfollowpower-laws. Weconjecturetheexistenceofacriticalvalueforthespectral
a
exponent,belowwhichthesystembehaviorisidenticaltothecaseofuncorrelatedcouplings. Above
J
that critical value,theentanglement entropy violates thearea law and grows as apower law of the
3
block size, with an exponent which increases from zero to one. Similar planar bond structures are
1
also found in statistical models of RNA folding and kinetic roughening, and we trace an analogy

between them and quantum valence bond states. Using an inverse renormalization procedure we
]
h determine the optimal spin-chain couplings which give rise to a given planar bond structure, and
c study the statistical properties of the couplings whose bond structures mimic those found in RNA
e folding.
m
-
t I. INTRODUCTION might have other fixed points of the SDRG. For exam-
a
ple,ifthe couplingsdecayexponentiallyfromthe center,
t
s they give rise to the rainbow phase, in which singlets ex-
. Entanglement in disordered spin chains has received
t tend concentrically [11–13]. In that case, all couplings
a much attention recently [1–4]. The main reason is
m that, as opposed to on-site disordered systems [5], long- arecorrelated. Butwemayalsodeviseensemblesofcou-
plingswhichpresentlong-rangecorrelations,butarestill
- distance correlations are not destroyed in this case, but
d random.
only modified in subtle ways. Thus, for the 1D Heisen-
n berg and XX models with uncorrelated random cou- A glimpse of some further fixed points can be found
o
plings, the von Neumann entropy of blocks of size ℓ is by observing the statistical mechanics of the secondary
c
[ known to violate the area law and grow as log(ℓ), simi- structure of RNA [14]. A simple yet relevant model is
larlytotheconformalcase[6,7]. Theprefactor,nonethe- constitutedbyaclosed1Dchainwithanevennumberof
1 less, is different: it is still proportionalto the the central RNAbases,whichwecallsites,whicharerandomlycou-
v
charge of the associated conformal field theory (CFT), pledinpairswithindicesofdifferentparity[15,16]. Each
8
0 but multiplied by an extra log(2) factor. Moreover, the pairconstitutesanRNAbond,andtheonlyconstraintis
4 R´enyientropiesdonotsatisfythepredictionsofCFT[8], that no bonds can cross. Therefore,the ensemble of sec-
3 because these models are not conformal invariant. ondary structures of RNA can be described in terms of
0 planar bond structures, just like ground states of disor-
A very relevant tool of analysis is the strong disorder
. dered spin-chains. Wiese and coworkers [16] studied the
1 renormalizationgroup(SDRG)devisedbyDasguptaand
0 Ma [9], whichshows that the groundstate of Heisenberg probability distribution for the bond lengths, and found
6 P (l) l−η, with η =(7 √17)/2 1.44.
or XX chains with strong disorder can be written as a B
∼ − ≈
1 product of random singlets, in which all spins are paired Furthermore, the studies of RNA folding included a
:
v upmaking SU(2)singletbonds. Furthermore,the renor- very interesting second observable. The planar bond
i malization procedure prevents the bonds from crossing, structure can be mapped to the height function of a dis-
X
i.e.,thebondstructurewillalwaysbeplanar. Thepaired cretizedinterface[16]. Wecandefinetheexpectedrough-
r
a spins are often neighbours, but not always. As it was ness of windows of size ℓ, W(ℓ), as the deviation of the
shown [1, 10], the probability distribution for the singlet heightfunction overblocksofsize ℓ,whichcanbe shown
bond lengths, P (ℓ) falls as a power-law, P (ℓ) ℓ−η, to scale in RNA folding structures like W(ℓ) ℓα, with
with η = 2. EnBtanglement of a block can bBe ob∼tained α=(√17 3)/2 0.56. Interestingly, η+α=≈2.
− ≈
just by counting the number of singlets which must be As we will show, the interface roughness is very simi-
cut in order to isolate the block, and multiplying by the lar to the entanglement entropy of blocks of size ℓ, and
entanglement entropy of one bond, which is log(2). theyarecharacterizedbysimilarexponents. IntheIRFP
Under the SDRG flow, the variance of the couplings phase for random singlets, notice that the entropy is
increases and its correlation length decreases, thus ap- characterized by a zero exponent, due to the logarith-
proaching the so-called infinite randomness fixed point mic growth, and η = 2. Therefore, it is also true that
(IRFP) [10]. Is this fixed point unique? Not necessarily. η+α=2. We may then ask, what is the validity of this
Ifthecouplingspresentadivergingcorrelationlength,we scaling relation? Does the RNA folding case correspond
2
to some choice of the ensemble of coupling constants for is that, for them, the Dasgupta-Ma renormalization rule
aspin-chain? Canweobtainotherfixedpointswhichin- becomes additive:
terpolate between the IRFP and the RNA folding cases?
We may keep in mind that the couplings in some spin
chain models (e.g., the XX model) can be mapped into t˜i =ti−1+ti+1 ti (3)
−
modulations of the space metric [22]. Thus, we are ob-
Once the SDRG procedure is finished we can read our
taining, in a certain regime, the relation between the
GS as a product state of singlet valence bonds.
statistical structure of the space metric and the statis-
tical properties of entanglement of the vacuum, i.e., the
ground state of the theory. 1
Ψ = + + (4)
This article is organized as follows. Section II intro- | GSi Y √2(cid:16)| −iij −|− iij(cid:17)
ducesourmodelandtherenormalizationprocedureused (i,j)∈P
throughout the text. Moreover, it discusses the conse-
where denotes a set of N/2 pairing bonds among the
quences of the planarity of the pairing structures which P
N spins. Many properties of these ground states have
characterize the states. In section III we establish our
been studied in the last thirty years [1–4, 10–13, 20].
strategy to sample highly correlated values of the cou-
One of the most salient of those properties is the fact
plings,andshownumericallythe behavioroftheentropy
thatthepairing whichresultsfromtherenormalization
and other observables. In section IV we focus on the re- P
procedure must be planar, i.e., it can be drawn without
lation between the RNA folding problem and our disor-
any two bonds crossing. States of the form (4) which
deredspinchains,anddetermine aninversealgorithmto
fulfill this requirementwill be called fromnow on planar
compute a parent Hamiltonian for any planar state, ex-
states.
emplifying it with the RNA folding states. How generic
are planar states is the question addressed in section V,
showing that they are non-generic through the study of A. Planar Pairings
their entanglement entropy. The article ends in section
VI discussingourconclusionsandideasforfurther work.
In more formal terms, let (even) N be the number
of nodes, and a bond is defined as an ordered pair
p = (p ,p ), where p ,p Z are the nodes joined.
II. DISORDERED SPIN CHAINS AND 1 2 1 2 ∈ N
In principle, (p ,p ) = (p ,p ). We define the covered
PLANAR STATES 1 2 6 2 1
nodes by the bond as C(p) p +1, ,p 1 . No-
1 2
≡ { ··· − }
tice that, if p and p are consecutive, C(p) = . Given
Letusconsiderforsimplicityaspin-1/2XXchainwith 1 2 ∅
two bonds, p and q, we say that p q if C(p) C(q).
N (even) sites and periodic boundary conditions, whose ⊂ ⊂
If neither C(p) C(q) or C(q) C(p), we say that
Hamiltonian is ⊂ ⊂
the bonds cross. A planar bond structure is defined as a
set of N/2 bonds which do not cross. Thus, the bonds
N form a nested graph. An important remark is that the
H = J SxSx +SySy (1) twonodesjoinedbybondmusthavedifferentparity. See
− i i i+1 i i+1
Xi=1 (cid:0) (cid:1) Fig. 1 for an illustration.
Letusassumethatthenodesareindexedcounterclock-
wheretheJ arethecouplingconstants,whichwewillas-
i wise. We can now define for eachnode i a value s to be
i
sume to be positive and strongly inhomogeneous. More
either+1or 1depending onwhetheritis the sourceor
precisely,we assume that neighboringcouplings are very −
the sink of a bond, as shown in Fig. 1. Of course, the
different. Notice that we do not impose them to be ran-
sum of all the s around the full system should be zero:
i
dom.
N s = 0. The s can be considered as slopes of a
In order to obtain the ground state (GS), we can em- k=1 k i
hPeight function,
ploy the strong disorder renormalization group (SDRG)
methodofDasguptaandMa[9]. Ateachrenormalization
step,wepickthemaximalcoupling,J ,decimatethetwo i
i
associatedspins, i andi+1,and establisha singlet bond h s . (5)
i k
≡
between them. The neighboring sites are then joined by kX=1
an effective coupling given by second order perturbation
Of course, this definition is not translation invariant,
theory:
sincewestartcountingatnode1. Inordertoavoidthat,
let h denote the absolute minimum of this height func-
0
J˜ =J J /J (2) tion. Then, we can define the absolute height function,
i i−1 i+1 i
H h h . Its meaning is the following: it denotes
i i 0
≡ −
amongthe nextneighboursofthe link, i 1andi+2. It the number of bonds passing above the link in the circle
−
is convenient to use a set of auxiliary variables, that we joining nodes i and i+1. By construction, this absolute
will call log-couplings: t = log(J ). The main reason height function has, at least, one zero.
i i
−
3
1 2 3 4 5 6 7 8 9 10 11 12
si + + + + + +
− − − − − −
hi 1 2 3 2 1 2 1 0 1 2 1 0
FIG.1. Illustratingtheplanarpairings. AsetofN =12sites,coupledbyaplanarpairing. Eachsitegetsaspinvalue,si =+1
or −1. The height function, hi is shown in thebottom row. A block is marked,containing sites 3 to 6.
B. Dyck Language and Catalan Numbers canprovethe followingtheoremwhichrelates the height
function and entanglement. Let [i..j] denote the block
There is a close analogy between planar pairings and i, ,j . Then,
{ ··· }
the Dyck language [17]. A Dyck word is a string of sym-
bols from the alphabet +, such that the number of S(B )=H +H 2 min H . (9)
+ counted from the left{is al−w}aysgreater or equal to the [i..j] i−1 j − k∈{i−1···j} k
number of . Equivalently, they are the set of prop- The meaning of that equation is the following. H
− i−1
erly balanced parenthesis. This means that their height represents the bonds that enter the block from its left
function hi, as defined in Eq. (5), is positive for all i. end, and Hj the bonds which exit from its right. For an
ThedifferencebetweenourplanarpairingsandtheDyck example, see Fig. 1. The block marked with the dashed
language resides entirely in the periodic boundary con- box is B . The number of bonds entering from the
[3..6]
ditions. If, in our circular planar structures, we break at leftisH =2,andthenumberofbondsleavingfromthe
2
theabsoluteminimumoftheheightfunction,theanalogy right is H = 2. But not all those bonds contribute to
6
with Dyck words becomes complete. the entropy. Some of them just fly over the block, and
Howmanydifferentplanarstatesarethereforasystem we can separate the block without touching them. Let
of fixed size N = 2M? Let us denote this value by PM. hF be the number of those flying bonds, in our example
Disregardingtheorderingofthesitesineachbond,which h =1,thebondfromsite1tosite8. Thelinksentering
F
merelycontributesageneral2M factor,wecanprovidea fromtheleft,H areeitheroverflying(h )ornot(h ):
i−1 F L
recursive relation. Site 1 must be linked to an even site, H = h +h . Similarly, on the right we have H =
i−1 F L j
2k. Then it creates two regions, one of size 2k 2 and h +h , and the block entropyis givenby h +h . We
− F R L R
the other 2M 2k. Thus, we get will proceed to prove that h is given by the minimum
− F
of the height function inside the block. Since the bonds
M which contribute to the entropy, hL and hR, do not fly
P = P P (6) overtheblock,theymusteitherendinsideit(h )orstart
M k−1 M−k L
kX=1 inside it hR. Since the bonds can not cut, the hL bonds
fromthe leftmusthaveendedbefore anyofthe h start.
along with P = 1, which is known in the literature as R
2 At that very moment, only the flying bonds will remain.
Segner’s recurrence [17], which gives rise to the Catalan
In Fig. 1, this moment takes place between sites 5 and
numbers:
6, h =1, h =1 andthe block entropy is S =2. Thus,
L R
theminimumvalueoftheheightfunctionis,exactly,h .
F
1 2M We have H +H 2h =h +h =S, as required.
P = (7) i−1 j F L R
M −
M +1(cid:18)M (cid:19) Notice that we can rewrite expression (9) as
S(B )=(H H )+(H H ), thus showing
[i..j] i−1 min j min
− −
a connection between the block entropy and the aver-
C. Entanglement of Planar States age variation of the height within the block, i.e. the
roughness of the interface. The main difference is that
Given a planar state of the form (4), we can easily the entanglement entropy gives special relevance to the
computetheentanglemententropyofanyblockB: using boundaries.
2 as the base for the logarithms, it coincides with the
number of bonds which must be cut in order to separate
it from the rest of the system [1, 2, 4]. III. CORRELATED RANDOM SPIN CHAINS
The statistical properties of the ground states of
S(B) [p B p B] (8)
≡Xp 1 ∈ ⊕ 2 ∈ Hamiltoniansoftheform(1)whenthecouplings{Ji}are
picked randomly and uncorrelated have been determined
where stands for the exclusive or (xor) symbol, which in a series of papers [1–4, 9, 10, 20]. The SDRG proce-
⊕
means that either p B or p B, but not both. We dure convergesto the so-called infinite randomness fixed
1 2
∈ ∈
4
point(IRFP).AlongtheRG,thevarianceoftheeffective 1.5
γ=0
couplingsgrow,andtheircorrelationlengthdecreases. It γ=1
1 γ=2
has been shown that the average entanglement entropy γ=3
of a block of size ℓ follows the expression [4, 20]:
ti 0.5
gs,
n
c N πℓ pli 0
S(ℓ) log(2) log Y +c′, (10) ou
≈ 3 (cid:18)π (cid:18)N(cid:19)(cid:19) g-c
o -0.5
L
where Y(x) is a scaling function and c is the central
-1
charge of the associated CFT, i.e., the one which cor-
responds to the homogeneous (conformal) case, with all
-1.5
the Ji equal. In our case, c=1. Surprisingly, expression 0 20 40 60 80 100 120 140
(10) is very similar to the conformal expression [6]: Nodeindex,i
1
c N πℓ 0)i 0.8
SCFT(ℓ)≈ 3 log(cid:18)π sin(cid:18)N(cid:19)(cid:19)+c′. (11) t(x)t(h 00..46
n,
The scaling function Y(x) is, in fact, rather similar to atio 0.2
sin(x) [4, 20]. el 0
Another relevant observable which helps characterize corr -0.2
g
the IRFP is the bond-length probability, i.e., given a sin- plin -0.4
glet bond (i,j), determine the probability distribution u
co -0.6
for its length l = |i−j|, PB(l). This value is directly Log- -0.8
related to the two-point correlationfunction [10]. In the
uncorrelatedcase,it is knownto behave,for l N, as a -1
power-law: P (l) l−η, where η =2 [10]. ≪ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B ≈ Distance,x
FIG.2. (A)samplesof{ti}forN =128andincreasingvalues
A. Correlated couplings of γ. (B) real-space correlation function for several values of
γ fromzero(black)toinfinity(red),obtainedanalyticallyfor
Our aimis to characterizethe groundstates of Hamil- N = 1000, from expression (14). The value γ = 1 appears
remarked in blue.
tonian(1)whenthecouplingsJ arerandom,butpresent
i
non-trivial correlations. If these correlations are short
ranged, they will be washed away by the renormaliza-
If γ =0, all momenta in expression (12) get the same
tion procedure, and return to the IRFP. Thus, we will
weight, and we obtain again an uncorrelated set of t .
consider the case of long-range correlations. i
As we increase γ, the larger momenta get less and less
Let us establish a procedure to obtain samples from
weight, and we are left with only the lowest momenta.
setsoflog-couplings t whichpresentlong-rangecorre-
{ i} This implies thatthe setof ti havestrongercorrelations.
lations, by employing a suitable Fourier expansion:
Fig. 2 (A) shows typical samples for increasing values of
γ.
The ensemble of log-couplings presents zero correla-
t = A sin(jk+φ ) (12)
j k k tionsinmomentumspace,butstrongcorrelationsinreal
Xk space for increasing γ. The correlationfunction is trans-
lation invariant by construction, and given by
where k are a set of allowed momenta, k = 2πn/N,
n
with n 1, ,2N . We do not include moment zero,
∈ { ··· }
since it would amount to a global constant which would
1
be irrelevant for the SDRG. The values A and φ are t(x)t(0) cos(kx) (14)
k k h i∝ k2γ
chosen as independent random variables. The phase φk k≥X2π/N
is taken to be uniformly distributed in [0,2π) and
For N = 1000, Fig. 2 (B) shows the correlation as a
function of the distance, normalized to have a maximal
A =k−γu , (13) valueofone. Forγ =0,thecorrelationisidenticallyzero
k k
forallx>0. Forγ itapproachesacosinefunction.
→∞
wheretheu areindependentgaussianvariateswithzero The value γ =1,which willhave specialrelevancein the
k
averageand varianceone, and γ is a fixed spectralexpo- restofthetext,appearsmarked. Weshouldremarkthat,
nent. althoughEq. (14)makesperfectsenseforallfinitevalues
5
B. Entanglement, Roughness and Bond-Lengths
Theaverageentanglemententropyasafunctionofthe
blocksizeℓ,forafixedvalueofN =1000,105realizations
andseveralvaluesofγ isshowninFig. 4(A).Theupper
partofthepanelisdevotedtoγ 1,whilethe lowerone
≥
shows more detail for γ 1. Notice that, for γ 1, the
≤ ≤
function S(ℓ) is nearly independent of γ. We propose a
finite-size fit of the form:
S(ℓ) A(NY(πℓ/N))χ, (15)
≈
where the scaling function Y(x) is determined via a
Fourier series expansion in the same line as [4, 20]:
∞
Y(x)=sin(x)+ α sin((2n+1)x). (16)
n
nX=1
Thebestfitvaluesofα aresmallandnearlyindepen-
n
dentofthespectralexponentγ. Wehavefoundα 0.05
1
≈
andα 0.005,bothslowlydecayingasγ increases. The
2
≈
inset of the top panel of Fig. 4 shows these scaling func-
tions for different values of γ.
The values of the exponent χ present more relevance.
Fig 4 (B) show these exponents, found by three different
strategies: (i) Finite-size, using a full fit to expression
(15)forN =1000,(ii)Localexponent,fittingtheentropy
for small blocks to a form S(ℓ) ℓχ also for N = 1000,
(iii) Global exponent, fitting S(≈N/2) to a form Nχ for
different values of N, up to N = 2000. The three ex-
pressionsdifferslightlyforlargerγ,althoughtheykeepa
generaltrend: forγ 1, χisveryclosetozero,while for
≤
γ we see χ 1. This signals a volumetric growth
→ ∞ →
ofthe entropyS ℓ. Thediscrepanciesbetweenthe val-
∼
ues of χ measured by the different strategies, as seen in
Fig. 4 (B), may be of numerical origin.
Interestingly,forγ 1,theS(ℓ)curvesarenearlyiden-
≤
tical, and the best finite-size fit to the whole function is
not given by the power-law expression (15), but expres-
sion(10),i.e. alogarithmicbehavior. Evenforγ 1,the
≈
best fit is logarithmic, but with a slightly larger prefac-
tor. Itisdifficulttodeterminewhetherthereisasmooth
crossoverbetweenγ =0andγ orasharptransition
→∞
FIG. 3. Bond diagrams for different samples with N = 64 atγ 1,belowwhichthe entropygrowslogarithmically,
≈
and γ running (downwards) from γ = 0 to γ = 3. To their i.e.: if the IRFP extends to the region γ 1.
≤
right, thecorresponding height profiles. Anotherinterestingobservableisprovidedbythestudy
oftheheightfunctionwhichcharacterizesthestate,given
by eq. (5). As we will show, the profiles are fractals, of
similarnaturetotheonesappearinginthestudyofrough
interfaces [18, 19]. Let us define the roughness,or width
ofN,theexpressiondivergesinthethermodynamiclimit
W, of the interface for a given length scale ℓ as the av-
for γ 1/2.
≤ erage deviation of the heights in windows of that size.
Fig. 3 shows some sample planar pairings for different Then, the Family-Vicsek Ansatz assumes that W ℓα.
∼
values of γ, in the range from γ =0 (no correlations) to Fig. 5 (A) showsthe roughnessas a function of the win-
γ =3(largecorrelations),alongwiththeircorresponding dow size ℓ, taking 105 realizations for each value of γ.
height diagrams. The top frame shows a log-log plot, while in the bottom
6
γ=1 100
160 γ=1.4 og) γγ==01
γ=1.6 (l γ=1.4
140 γ=2.2 W 10 γ=1.6
entropy,S(ℓ) 110200 0 0.2 0.4 0.6 0.8 1 γγ==2.63 Roughness 1.11 γγγγ====22..2630
nn 80 W γ=0.2
VonNeuma 4600 Roughness 00..79 γγγγγ=====0001....46821
20 0.5
S(ℓ) 0 1 10 ℓ 100 1000
y, 1
op 3 γ=0
nentr 2 γ=0 γ=0.8 (l) 0.1 γγγ===11..461
an γ=0.2 γ=1 PB γ=2.2
Neum 01 γγ==00..46 γ=1.2 bility, 0.01 γγ==2.63
Von 0 100 200 300 40B0lock50s0ize,6ℓ00 700 800 900 1000 proba 0.001
1.1 h
gt
1 en 0.0001
χ 0.9 nd-l
nent, 00..78 Bo 1e-05
o
xp 0.6
ye 0.5 1 10 100 1000
p
Entro 0000....1234 FiniGtLel-oosbcizaaelleeexxxpppooonnneeennnttt onents 1.82 LBoocRnadlo-uelLngenhetrngnotgephtsyh,sleeexxxpppooonnneeennnttt,,,αχη
p 1.6
0 1 2 3 4 5 x
e η
Spectralexponent,γ gth 1.4
n
e 1.2
l
FtiIoGn.o4f.th(eAb)lAocvkersaizgeefbolrodckiffveroenntNveaulmueasnonfeγn,twroitphyNas=a1fu0n00c-. and 1 α
Top panel: γ ≥ 1. Bottom panel: γ ≤ 1. Inset: scaling ness 0.8 χ
h
function, Y(x) as in Eq. (16). (B) Fitting exponent χ as a ug 0.6
o
function of γ for the threestrategies discussed in thetext. r
y, 0.4
p
ntro 0.2
E
0
one only the x-axis is logarithmic. The difference is no- 0 1 2 3 4 5
torious: for γ > 1, the roughness follows a clear power Correlationexponentγ
law,withexponentαwhichgrowsupto one(shownasa
straight line). For γ 1, instead, the behavior is better FIG.5. (A) Roughnessof blocks, W(ℓ),of differentsizes, for
≤
fit by a logarithmic function W(ℓ) log(ℓ). This pro- N =1000 and different values of γ. The top frame is in log-
∼
videsfurthersupporttotheconjecturethatthebehavior log,whilethebottomoneonlythex-axisislogarithmic. This
for γ 1 corresponds to the IRFP. waywecanshowthatforγ ≤1,thebestfitisforalogarithmic
≤ growth of the roughness. (B) Bond-length distribution, also
Panel (B) of Fig. 5 depicts the probability distribu-
for N = 1000, using the minimal length. Notice that the
tion for the bond-length. A power-law is established,
lines are parallel for γ = 0 and γ = 1, but shifted. They
i.e., P (l) l−η, and η is shown to depend on γ. For
B havethesame scaling exponent,but different prefactor. (C):
∼
γ 1, the curves appear to be parallel, i.e., show the fittingexponentsfortheentropy(χ),roughness(α)andbond-
≤
same exponent, andonly differing intheir prefactor[24]. length (η) as a function of γ. Below the γ =1 line, it can be
Fig. 5 (C) shows the values of the three exponents, arguedthattheroughnessandtheentropydonotbehavelike
entropy (χ), roughness (α) and bond-length distribution a power law, but instead they show logarithmic behavior.
(η) as a function of the correlation parameter γ. Notice
that χ is very similar to α, as suggested by relation (9)
which links the block entropy to the height fluctuations.
Both exponents grow with γ, starting near zero for un-
correlated spin chains and saturating at a value close to
7
1. The bond-length exponent η behaves in the opposite in the rainbow limit, we have χ+η 1, which suggests
→
way, starting at η = 2 for uncorrelated spin chains and a strong correlation between the bonds.
decreasingtowardszero. Theregionforγ 1ispeculiar:
≤
whilethebond-lengthη exponentisstill2,theothertwo
exponents are very close to zero, since the true behavior A. The inverse problem
is expected to be logarithmic.
The limit γ is also rather special. A look at HowstrongistheconnectionbetweentheRNAfolding
→ ∞
the lastpanelof3 showsthat rainbow-like structuresbe- and disordered spin chains? Can we obtain an ensemble
come more and more prominent. The limit in which of couplings J such that the ground states of Hamil-
i
only the lowest momentum modulation survives gives tonian (1) co{rre}spond to the planar states obtained in
rise to a perfect rainbow state, which presents volumet- RNA folding? This question leads us to the study of the
ric entanglement [11, 13], i.e., S ℓ. This explains more general inverse SDRG problem.
∼
the limit χ 1 for the entropy exponent for large γ. If we regard the SDRG as a mapping between sets of
→
Similarly, the height function becomes a nearly perfect couplings and planar pairings, we might be able to re-
wedge, which explains the α 1 behavior. In that ex- versethealgorithm,andobtainthesetofcouplingswhich
→
treme, the bond-length distribution is completely flat, give rise to a certain planar pairing. In other terms, a
since all bond-lengths show up once for each realization, parent 1D Hamiltonian for a given planar state. In this
thus η =0. section we will show that (1) every planar state has a
(non-unique) parent 1D Hamiltonian and (2) an explicit
algorithm to obtain the optimal set of couplings, in a
IV. RNA FOLDING AND SPIN CHAINS sense to be determined later.
The aim is to obtain the logarithmic couplings, t ,
i
{ }
As it was briefly discussed in the introduction, planar given the set of bonds, p . Our proposed algorithm
i
{ }
pairings also appear naturally in the study of the sec- works as follows (see Fig. 6 for an illustration):
ondarystructureoffolded RNA strands[14]. The model
developed by Wiese and coworkers [15, 16] works in the Sort the bonds in order of increasing length.
•
followingway: (1)apairofsiteswithdifferentparityare
Consider the bonds of lengthone, fix their internal
chosen randomly and paired; (2) further pairs are cho- •
log-couplings to 0. In the first row of Fig. 6, we
sen in the same way, always under the constraint that
put a zero under links 3 4 and 10 11.
no previous bonds can be crossed. In their seminal work − −
[16], the authors studied the roughness of the equivalent
Flank these zeroes with log-couplings of value 1 at
height function and the bond-length distribution, show- •
bothsides. SeethesecondrowofFig. 6,wherethe
ing that they both follow a power-law behavior, W ℓα
∼ arrows in the new values point to the zero which
andP (l) l−η. Thentheyprovedthatα+η=2. Ifwe
B ∼ they flank.
assume the scaling equivalence of the roughness and the
entropy, this result is also fulfilled in uncorrelated ran- Now consider the bonds of length three. Find the
•
dom spin chains, where we have α = 0 (because of the effective log-coupling which would appear as their
logarithmicbehavioroftheentropy)andη =2[1,4]. On renormalization value (which must be 2). Flank
the otherhand, this relationdoes notholdfor correlated them with log-couplings of value 2+1=3 at both
spin chains. sides, as in the third row of Fig. 6.
Wemayaskwhatistherangeofvalidityoftherelation
χ+η =2 (or α+η =2). Extending the results of [3, 4] Consider the rest of the bonds in order of increas-
•
we can provide a proof of that statement in the case of ing lengths. For each of them, find their renormal-
uncorrelated bonds. Indeed, let us consider a block of ization value and flank them with log-couplings of
size ℓ and let us number the sites from 1 to ℓ. The bond value one unit higher.
at site i will be cut by the block if it goes left and its
Log-couplingsmay never decreasealong the proce-
length is larger or equal than i, or if it goes rightand its •
dure. If two values collide, take the larger.
length is larger than ℓ i. So, we have an estimate for
−
the average entropy:
This procedure yields couplings which, by construc-
tion, always give rise to the desired bond structure.
ℓ ℓ Moreover, because the value of each bond is computed
1
S(ℓ) (P (l i)+P (l>ℓ i))= P (l i) using the SDRG itself, we ensure a certain optimality
B B B
≈Xi=1 2 ≥ − Xi=1 ≥ condition: among the sets of couplings yielding the de-
(17) sired state, our choice will always require the minimal
This equationimplies a double integration. IfP (l) span of coupling values. For example, this Hamiltonian
B
l−η,itleadstoS(ℓ) l−η+2, aswedesired. As itfollow∼s will yield the largest possible gap.
∼
from Fig. 5 (C), this is not true for the planar state Fig. 7(A)showsthecouplingswhichgiverisetoagive
ensembles generated with correlated couplings. In fact, instance of the RNA folding problem with N =100. We
8
1 2 3 4 5 6 7 8 9 10 11 12
0 0 0
− − − − − − − − −
1 0 1 1 0 1 1 0 1
− → ← → ← − → ← −
3 1 0 1 3 0 1 3 1 0 1 3
→ ← → ←
3 1 0 1 3 0 1 6 1 0 1 6
← →
ti 3 1 0 1 3 0 1 6 1 0 1 6
FIG. 6. Illustration of the inverse algorithm to obtain the optimal log-couplings which give rise, via the SDRG, to a given
planar bond structure. Each row corresponds to one of the steps of the algorithm. The log-couplings which have not been
assignedyetappearasa“−”. Thearrowswhichappearclosetoanewvaluepointtothebondwhichhascreated(ormodified)
that value.
have run 105 simulations of the RNA folding algorithm
and obtained the optimal couplings for different system
sizes up to N =1000. Fig. 7 (B) shows the (translation
invariant) correlation function for the log-couplings in
the N =50, 100 and 200 cases. The values present long
range correlations, but not a clear power-law behavior.
Moreover,the couplings field t is not gaussian. In the
i
{ }
insetofFig. 7(B)weshowthehistogram,inlogarithmic
scale,fort . The marginalprobabilitydistributionisnot
i
gaussian. Instead, it is a power-law, with an empirical
exponent close to 4/3.
−
V. GENERIC PLANAR STATES
Since wehavedeterminedthatallplanarstateshavea
1D parent Hamiltonian, we may still ask how dense are
planar states within the Hilbert space. In other terms,
how generic they are. We can define an ensemble of pla-
nar states for N sites under the condition that all possi- 1
L=50
ble planar pairings have the same probability. In order L=100
to sample that ensemble, we just apply a correction to 0)i 0.8 0.1 L=200
nwptahlerhgeoxioctcRrehbidtNodhuAnomrden,f,oiotsnlhtdocecihtnupogatasleilsarnanpomywlfapipsntilrhatienervesgqpi(osuaiut,iarrsjlai)nbptegworgosnhybdai.ac.rhbIeBniwlsuitatithlim,leesfcpoRoallenlmNodswotAwinitnguiftogthtlehdthttoinhhasgeeet correlation,t(x)t(h 00..46 Frequency011100.eee.0.0---00000010176511 115000000010 100
same probability. This can be corrected if the probabili- ed 0.2 Couplingvalue,t
z
ties for eachpair (i,j)arenotequal,but proportionalto mali
the number of planar pairings which are consistent with or 0
N
the presence of that bond.
-0.2
Let us consider a certain empty patch of length n in 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a planar pairing which is under construction, i.e., a set Distance,x
of contiguous spins which have not been paired yet. As
we know, there are Pn possible ways to create a planar FIG.7. (A)Theoptimal couplingswhich giverise toagiven
pairingonthatemptypatch. Thespinwithindex1must planarpairingobtainedfromtheRNAfoldingalgorithmwith
be pairedwith some spin inside the patch, let us refer to N = 100. (B) The correlation function for the log-couplings
its index as k. Then, after bond (1,k) is established, at different points for N = 50, 100 and 200, using 106 re-
alizations. Inset: histogram for the log-coupling values for
the number of different possible planar pairings will be
N = 100, 500 and 1000, also with 106 realizations. Notice
P P . Thus the probability with which bond (1,k)
k−2 n−k thepower-lawinertialrange, withanexponentclose to−4/3
should be taken is just P P /P , which is known
k−2 n−k n (straight line).
to be less than one by construction, as we see in Eq.
9
(6). Repeating this procedure,we cansample the planar described constitutes a family of local 1D Hamiltonians
pairing ensemble with equal probabilities. whose ground states violate the area law to any desired
We have found numerically the average block entropy degree.
asafunctionoftheblocksizeℓforthisensembleofstates, We have also considered the inverse renormalization
and found that it grows as S(ℓ) ℓχ, with χ 0.54. problem: given a (planar) valence bond state, to obtain
∼ ≈
The precisevalue is notveryrelevant,but itallowsus to its(1D)parentHamiltonian. Inthiswaywewereableto
conclude that planar states are highly non-genericquan- study the ensemble of random spin chains whose ground
tum states, because for generic states we should obtain states would correspond to the planar structures which
S(ℓ) ℓ, i.e., a volumetric growth of the entropy. show up in other physical situations, such as the RNA
∼
folding problem. These engineered random spin chains
present a behavior of the entanglement entropy and the
VI. CONCLUSIONS AND FURTHER WORK correlators which do not correspond to any value of γ.
This suggests that the phase diagram of random spin
In this article we have applied the SDRG to study chains with large correlations between the couplings is
the groundstate properties of a strongly disorderedran- richer than expected.
dom spin chain with long-range correlations between its Inhomogeneous spin chains can be mapped, in some
couplings. The states can be described as valence bond cases,to models whichrepresentthe motionoffermionic
states with planar bond structures, and they can have matter on a curved spacetime [22], where the metric is
arbitrarily large entanglement entropy. Concretely, we given by the coupling constants. Thus, our study shows
have chosen the couplings such that their logarithm is that the statistical properties of the metric show up as
expressed as a Fourier series with random coefficients, statisticalpropertiesofthe entanglementofthe vacuum,
falling as a power-law of the momentum k−γ. For γ 1 i.e., the ground state of the corresponding Hamiltonian.
the behavior is very similar to the infinite randomn≤ess Moreover,wecanalsofind,usingtheinverserenormaliza-
fixed point (IRFP) found for uncorrelated coupling con- tionalgorithm,the optimalspatialgeometrywhichgives
stants. Nonetheless, for γ > 1, the block entropy be- rise to a certain vacuum entanglement. These results
haves as a power-law of the block size, S ℓχ, with χ mayshedlightontherelationbetweenentanglementand
a function of the exponent γ which seems t∼o interpolate space-time [23].
smoothly between χ = 0 and χ = 1 as γ . The
→ ∞
bond length probability, which is related to the correla-
tor, is also characterized by a power-law, P (l) l−η, ACKNOWLEDGMENTS
B
∼
with η = 2 for γ 1 and falling to η 0 for γ .
≤ ∼ → ∞
This extreme, γ , corresponds to the case where We would like to thank J. Cuesta for insights into the
→ ∞
only the lowestmomentum k =2π/N contributes to the statistical mechanics of RNA folding. This work was
correlationbetweenthecouplings,andthestatebecomes funded by grants FIS-2012-33642 and FIS-2012-38866-
a rainbow state. As we have shown, the planar states C05-1, from the Spanish government, QUITEMAD+
can be mapped to a 1D interface, whose roughness be- S2013/ICE-2801 from the Madrid regional government
havesapproximately like the entanglement entropy,as it and SEV-2012-0249of the “Centro de Excelencia Severo
is suggested by expression (9). Remarkably, the system Ochoa” Programme.
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