Correlations in Quantum Spin Ladders with Site and Bond Dilution

0.86 MB·English

by  Kien Trinh

Correlations in Quantum Spin Ladders with Site and Bond Dilution
Kien Trinh∗ and Stephan Haas
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA
Rong Yu
Department of Physics & Astronomy, Rice University, Houston, TX 77005, USA
Tommaso Roscilde
Laboratoire de Physique, CNRS UMR 5672, Ecole Normale Sup´erieure de Lyon,
Universit´e de Lyon, 46 All´ee d’Italie, Lyon, F-69364, France
(Dated: January 20, 2013)
We investigate the effects of quenched disorder, in the form of site and bond dilution, on the
2
physics of the S = 1/2 antiferromagnetic Heisenberg model on even-leg ladders. Site dilution
1
is found to prune rung singlets and thus create localized moments which interact via a random,
0
unfrustratednetworkofeffectivecouplings,realizingarandom-exchangeHeisenbergmodel(REHM)
2
in one spatial dimension. This system exhibits a power-law diverging correlation length as the

n temperature decreases. Contrary to previous claims, we observe that the scaling exponent is non-
a universal,i.e.,dopingdependent. Thisfindingcanbeexplainedbythediscretenatureofthevalues
J taken by the effective exchange couplings in the doped ladder. Bond dilution on even-leg ladders

7 leads to a more complex evolution with doping of correlations, which are weakly enhanced in 2-leg
ladders, and are even suppressed for low dilution in the case of 4-leg and 6-leg ladders. We clarify

] the different aspects of correlation enhancement and suppression due to bond dilution by isolating
l
e the contributions of rung-bond dilution and leg-bond dilution.
-
r
t
s I. INTRODUCTION bond disorder can be introduced in IPA-CuCl by a par-
. 3
t tial substitution of non-magnetic Br− for the likewise
a
m Low-dimensional quantum Heisenberg antiferromag- non-magnetic Cl−, affecting the bond angles in the Cu-
halogen-halogen-Cu superexchange pathways12–14. The
- nets are known to exhibit non-trivial properties due to
d reducedstrengthofmagneticinteractionsontheaffected
quantum fluctuations. A most striking example comes
n bonds leads to Bose-glass behavior when a strong mag-
fromspinladders: interpolatingbetweenthechaingeom-
o netic field is applied.
etryandthe squarelatticegeometry, theyhave provided
c
[ an interesting new playground to explore the physics From the theory side, a number of studies have ad-
oflow-dimensionalstronglycorrelatedelectronsystems1. dressed the effects of site and bond disorder on Heisen-
2
Specifically, it has been shown that AF Heisenberg spin- berg ladders. Ref. 15 examined the correlation length of
v
1/2 ladders with an even number of legs are quantum randomly site-diluted spin-1/2 Heisenberg 2-leg ladders
6
5 spin liquids with purely short-range spin correlations. at weak and intermediate interchain couplings, show-
0 Theirspincorrelationsdecayexponentiallyduetoafinite ing an apparent divergence of the spin correlations at
0 singlet-triplet gap in the energy spectrum. In contrast, low temperatures, due to the presence of the impurities.
5. Heisenberg ladders with an odd number of legs display A related divergence of the staggered susceptibility has
0 properties similar to those of single chains at low tem- been reported in Ref. 16. This behavior has been re-
1 perature,namelygaplessspinexcitations,andpower-law lated to that of the random-exchange Heisenberg model
1 spin correlations. This difference between even-leg and (REHM)17,18, describing the effective random interac-
v: odd-leg ladders was predicted by theory and confirmed tions between LMs2,19. In a more recent study, three
i by experiments in a variety of systems1. of us have investigated bond-diluted 2-leg ladders20,21
X
and observed enhancement of spin correlations due to
In this paper we focus on disordered spin ladders.
r bond dilution. However, a general understanding of the
a For the case of doping with static non-magnetic impu-
mechanisms for enhancement of correlations in site- and
rities, one distinguishes between site and bond disor-
bond-dilutedHeisenbergladdersisstilllacking. Itisalso
der. Site dilution occurs upon doping the magnetic ions
unclear whether such an enhancement of spin correla-
with non-magnetic ones (as for instance substitution of
Cu2+ with Zn2+). This leads to the formation of lo- tions generally exists for even-leg ladders with a number
cal moments (LMs) close to the dopant site2–4. Effec- of legs greater than two.
tive residual interactions between the LMs lead to long- In this paper, we present a study of site- and bond-
range antiferromagnetic ordering5–9, experimentally de- diluted spin-1/2 AF Heisenberg ladders with n=2, 4 and
tected in systems such as Sr(Cu Zn )O (Ref. 10) and 6legs. MakinguseofquantumMonteCarlosimulations,
1−z z 3
Bi(Cu Zn ) PO (Ref. 11). Bond disorder occurs in- wecanaddressthecorrelationlengthofthesystemdown
1−z z 2 6
stead when the dopant ions replace the ions which act toextremelysmalltemperaturesatwhichtheasymptotic
as bridges between the magnetic ions. For instance, T → 0 behavior sets in. Our main findings can be sum-
2
marized as follows: Here S(q) is the time-averaged structure factor
1. For site-diluted even-leg ladders, we observe a
1 (cid:88) (cid:90) β
strongenhancementofcorrelationsupondopingup S(q)= e−iq·(ri−rj) dτ (cid:104)Sz(τ)Sz(0)(cid:105), (3)
Nβ i j
to n = 6 legs. In the specific case of 2-leg ladders, ij 0
we find that the system realizes the physics of the
REHM with a discrete distribution of the effective with N = n×L corresponding to the total number of
couplings, leading to a non-universal behavior at sites. [...] denotes the disorder average, which is per-
av
low temperatures; hence the known predictions for formed over 300-600 disorder realizations.
the universal regime of the REHM with a contin-
uous distribution of couplings are expected not to
apply to realistic models of doped spin ladders.
III. CORRELATION LENGTH OF
SITE-DILUTED LADDERS
2. For bond-diluted ladders, we find that correlations
are suppressed by a low level of dilution, due to
dilution-induced dimerization. Hence the system A. Effective Interactions among Localized
Moments
is first driven to a gapless phase with short-range
correlations. The correlation length becomes log-
arithmically divergent for vanishing temperatures Antiferromagneticeven-legladderswithoutdopingdis-
only beyond a critical dilution. play a rung-singlet ground state1 with a finite gap ∆ to
triplet excitations. In this state, the n spins on the same
rung preferentially form a singlet state, and therefore ef-
II. MODEL AND QUANTITIES OF INTEREST fectively decouple from the rest of the ladder. This leads
to exponentially decaying correlations in the direction of
We study the Heisenberg model on n-leg ladders the legs, characterized by a finite correlation length ξ0.
At low enough concentration of dopants, the main ef-
H = J (cid:88)n (cid:88)L p(l) S ·S fect of site dilution on an even-leg ladder is that of turn-
l i,m i,m i+1,m ing the number of spins on a rung from even to odd: in
m=1i=1 this situation, the state on the rung turns from a singlet
n−1 L intoadoublet, whichcorrespondstoaneffectiveS =1/2
+ Jr (cid:88) (cid:88)p(i,rm)Si,m·Si,m+1 (1) localized moment (LM). This S = 1/2 moment remains
m=1i=1 exponentially localized (over a characteristic length ξ )
0
close to the impurity site3,26, but its finite overlap with
where S is the quantum spin operator at site i =
i,m
other LMs leads to an effective interaction which generi-
1,...,L along the m-th leg. The first term in the Hamil-
cally decreases exponentially with the distance. In the
toniandescribesinteractionsalongthelegswithcoupling
case of dominant rung interactions, J (cid:29) J , the in-
J , whereas the second term represents the rung interac- r l
l
tion with coupling J . The random variables p(l) , p(r) teraction between LMs is appropriately described within
r i,m i,m second-order perturbation theory (in J /J ) as resulting
express the site or bond dilution: in the case of site dilu- l r
fromtheexchangeofvirtualmassivetripletsbetweentwo
tion,p(i,lm) =(cid:15)i,m(cid:15)i+1,m andp(i,rm) =(cid:15)i,m(cid:15)i,m+1,where(cid:15)i,m LMs2. Inthecaseofa2-legladder,inwhichwecaniden-
takes value 1 if the site (i,m) is occupied (with prob- tify the location of a LM with a spin site next to a miss-
ability 1 − z) or 0 if it is empty (with probability z). ing rung partner (dangling spin), the effective coupling
In the case of bond dilution, p(r) = p(l) /2 take values between LMs has the form of a SU(2)-invariant Heisen-
i,m i,m
0 with probability z/3 if the bond is not occupied, and berg interaction, and at large separation between LMs,
take values 1 otherwise. The simulations are performed |r −r |(cid:29)1, the coupling strength takes the form2
i j
using the Stochastic Series Expansion (SSE) quantum
Monte Carlo (QMC) method based on the directed loop J2 exp(−|r −r |/ξ )
J(eff) ∼(−1)i+j+1 l i j 0 , (4)
algorithm23. Periodic boundary conditions (PBC) along ij ∆ (cid:112)|r −r |/ξ
i j 0
the leg direction are used. To access the regime of very
lowtemperatures,atwhichtheasymptoticlow-T behav- wherethestaggeringfactortakesvalue−1ifthetwoLMs
ior of the correlation length sets in, we have made use belong to the same sublattice and +1 otherwise.
a β-doubling scheme24, allowing us to efficiently access
Eq. (4) holds strictly speaking only in the case in
inverse temperatures up to β =4096.
which J (cid:28) ∆, but a similar exponential decay of the
l
The main objective ofthis paper is to study the corre-
effective LM coupling (without the square-root denomi-
lation length along the leg direction, which is calculated
nator)hasbeenobservednumericallyinthecaseofmuch
via the disorder-averaged second-moment estimator:25: stronger leg interaction26, as e.g. for J = J , for which
l r
(cid:115) ∆ ≈ 0.53Jl. Moreover the above expression is numeri-
ξ = L [S(π,π)]av −1. (2) cally found to account for the decay of the effective cou-
2π [S(π+2π/L,π)] plings already for moderate distances, |r − r | (cid:38) ξ .
av i j 0
3
Hence,towardsaneffectivemodelofthesite-dilutedlad-
ders, we will assume in the following that the exponen-
tial decay of LM couplings with decay rate ξ remains
0
valid even when perturbation theory is no longer appli-
cable – which is the case of interest. In this paper, we
deliberately choose J ≤ J to have a sizable correla-
r l
tion length ξ , for reasons which will become clear in
0
the following. Our assumption then implies that non-
perturbativeeffectsonlyaffecttheprefactortotheexpo-
nential in Eq. (4). In addition, we will assume that the
exponential decay present in Eq. (4) sets in for distances
between LMs of the order of ξ .
0
In the case of n-leg ladders with n > 2 a detailed
theory of effective LM couplings is not available to our
knowledge,butforwidelyspacedvacancies,theextended
structureofthelocalizeddoubletbecomesirrelevant,and
for J (cid:28) J a perturbation approach analogous to that
l r
of 2-leg ladders should be applicable, leading to effec-
tive staggering couplings between LMs with exponential
dependence on the distance.
B. Statistical properties of the couplings for 2-leg
ladders
Inthecaseoflowdoping,z (cid:28)1,theprobabilitydistri-
bution of having two nearest-neighboring LMs at a dis-
tance d along the leg direction in a 2-leg ladder is given
by
P(d)=2z exp(−2zd) (5)
withacorrespondingaverage(cid:104)d(cid:105)=1/(2z). Asfoundnu-
merically in Ref. 26, the effective coupling has the form
J(eff)(d) ≈ J exp(−d/ξ ) for d (cid:38) ξ . As an approxima-
0 0 0
tion,onemayassumethatthesameexponentialbehavior
survivesford(cid:46)ξ . Thentheprobabilitydistributionfor
0
the strength of the couplings between nearest neighbor-
FIG. 1: (color online). Probability distribution (P) and
ing LMs reads
cumulative distribution (E) of the effective couplings J(eff)
∞ between nearest-neighbor localized moments in site-diluted
(cid:16) (cid:17) (cid:88)
P Jn(enff) = P(d) δ[Jn(enff)−J0exp(−d/ξ0)].(6) 2-leg ladders, with parameters z =2%, ξ0 =7.5 (upper pan-
els), and z=2.5%, ξ =15.3 (lower panels). The red dashed
d=0 0
linesshowtheprobabilitydistributionsinthecontinuumap-
This result suggests the possibility of extracting analyt- proximation according to Eq. (8).
ically the probability distribution for J(eff), as done in
nn
Ref. 2. To this end, one may further take the continuum
distribution.2 Theaverageabsolutevalueofthecoupling
approximation, namely, approximate the summation in
between neighboring LMs takes then the form2:
Eq. (6) by an integral:
P (cid:16)Jn(enff)(cid:17) ≈ (cid:90) dl P(l) δ[Jn(enff)−J0exp(−l/ξ0)] .(7) [|Jn(enff)|]av = 21−−γγ J0 = 1+2z2ξz0ξ0 J0. (9)
However, a critical analysis of the above derivation
This allows then to write
showsthatthecontinuumapproximationisproblematic.
(cid:16) (cid:17) 1−γ (cid:18) J (cid:19)γ In fact, it requires that the distances d giving a signifi-
P J(eff) ≈ 0 , (8)
nn 2J0 Jn(enff) cant contribution to the sum in Eq. (6) be d(cid:29)1, which
is in contradiction with the fact that P(d) decreases
where γ = 1−2zξ . Hence, within the continuum ap- exponentially with d; the characteristic decay length is
0
proximation the n.n. couplings obey a simple power-law (cid:104)d(cid:105)=(2z)−1, which for z ∼2% takes values ∼20.
4
Fig. 1 shows the distribution P(J(eff)) determined nu-
nn
merically according to Eq. (6), i.e., by sampling the dis- (a)
100
crete distribution lengths P(d), for z = 2%, ξ = 7.5
0 2-leg ladder
(corresponding to J /J =1/2, and giving γ =0.7), and
r l 4-leg ladder
z = 2.5%, ξ0 = 15.3 (corresponding to Jr/Jl = 1/4, and 6-leg ladder
giving γ =0.235) – these parameter sets will be relevant
for our study of correlations in the following. We notice ξ
that the distribution shown in Fig. 1 is only quantita- 10
tively correct when d (cid:38) ξ , namely for J /J (cid:46) 1/e,
0 eff 0
while it is only an approximation otherwise.
It is clear that for d(cid:38)ξ , the distribution of n.n. cou-
0
plings obtained numerically deviates strongly from the
prediction of Eq. (8). Over most of its support, the dis- 1
0 0.1 0.2 0.3 0.4 0.5
tribution of n.n. couplings has a fundamentally discrete
15
structure,duetothefactthatthelargestvaluesofJ(eff),
nn
(b)
which also take the largest probabilities, are associated
with short separations d among n.n. LMs. These prob- β=1024
β=512
abilities grow as a power law of the absolute value of
10 β=256
the coupling, opposed to what predicted by Eq. (8) for β=128
γ >0. Obviously a coarse graining on the δ-peaks of the β=64
exactdistributionwouldgraduallyreconstructtheshape ξ β=32
β=16
of Eq. (8), but given the finite separation between the δ-
peakseveninthethermodynamiclimit,thiscoarsegrain- 5
ingisnotjustifiedoveralargerangeofJ(eff). Infactthe
nn
δ-peaks become dense only in the limit J(eff) →0, which
nn
corresponds to larger and larger separations among the
0
LMs, and hence to lesser and lesser probabilities. Only 0 0.1 0.2 0.3 0.4 0.5
z
in that limit the continuum approximation appears le-
gitimate (in fact numerically we recover the analytical
prediction for J(eff) (cid:46) 10−4J due to the finite width of FIG. 2: (color online). Correlation length ξ of site-diluted
nn 0 ladders with L = 128 and J = J = J. Upper panel: com-
our bins, Jn(enff) =10−5J0). parisonbetween2-leg,4-legalnd6r-legladdersatinversetem-
In conclusion, for z (cid:38)1% and ξ ∼10, P(J(eff)) has a perature βJ = 1024. Lower panel: correlation length of the
0 nn
2-leg ladder at different temperatures.
remarkablestructure: itassignsthelargestprobabilityto
the range in which J(eff) is a discrete variable, prevent-
nn
ing a straightforward approximation of the distribution length(cid:104)l(cid:105)≈z−n(correspondingtotheinverselinearden-
with a continuous function. This will have significant
sity of rungs which are fully removed by doping). When
consequences on the analysis of doped ladder systems in
considering doped ladders at finite temperatures no geo-
relation with models of randomly coupled spins.
metric bound on the correlation length (neither coming
from finite-size effects nor from ladder fragmentation) is
presentaslongasthecorrelationlengthsatisfiesthecon-
C. Correlations as a function of doping dition
ξ(T)(cid:28)L(cid:28)(cid:104)l(cid:105). (10)
Inbothcasesofn=2andn>2theunfrustratedeffec-
tivecouplingsbetweenLMscanleadtotheformationofa
Itisclearthatsuchaconditioncanonlybesatisfiedwhen
gapless ground state with algebraically decaying correla-
thedopingissufficientlyweak. Hence,foranyfixedtem-
tions between the sites, and hence an infinite correlation
perature, upon increasing the doping concentration the
length. WhenlookingatafixedfinitetemperatureT the
correlation length ξ(T;z) will cross over from a growing
correlationlengthisnecessarilyfiniteforone-dimensional
behavior at low doping z (cid:28) 1 to a decreasing behavior
SU(2) invariant systems with short-range interactions, for z (cid:38)0.1, hence going through a maximum for a given
but it is expected to increase because the average cou- optimal doping value z∗ = z∗(J /J ;n), which is quite
r l
plingbetweenLMsincreaseswithz,asinEq.(9),sothat
stable at sufficiently low temperatures. This is indeed
the effective temperature T/[|Jn(enff)|]av decreases. seeninFig.2(a),inwhichweobserve38 that,forJr =Jl,
Despite the increase of correlations due to disorder, z∗ ∼0.06 for n=2, 4, and 6.
the range of correlations is fatally bounded by the fact For any finite value of doping and at arbitrarily low
that any finite amount of site dilution will break an in- temperatures, the correlation length is upper bounded
finite n-leg ladders into finite segments, of characteristic by the average length of the ladder segments, (cid:104)l(cid:105). Yet,
5
dominated by a unique fixed point; the correlation
100
length diverges as T → 0 with a universal scaling
exponent, ξ ∼T−2α with α=0.22±0.01. This re-
sulthasbeenconfirmedbyquantumMonteCarlo28
foranon-singulardistribution(squarebox,γ =0).
• Non-universal regime: for a more strongly singular
ξ
distributionγ >γ therenormalizationgroupflows
10 c
zzzzz=====22236.....06094%%%%% (((((JJJJJrrrrr=====00000.....2255555,,, ααα,, αα=====000...00111..6751256890,,,94 LLL,, ===LL==2325853564681)))42)) ttcmohoauealpiiznlnaeidottiinoas-pnluicnnGoi-ruvPoepeurlisipenargallsnfidcdxhiseSatdSirniEbpsuoQutisniMiontn,Cg.wRiSnhetiatuchldhiSisepdrsaeecopgeniemnRddeeosnapoloesrdno-
confirmthattheexponentαdependsonthedoping
0.001 0.01 0.1 concentration29.
T
AsdiscussedinSec.IIIB,thedistributionofcouplings
FIG. 3: (color online). Temperature dependence of the cor-
relation length of site-diluted 2-leg ladders with J = 1, and between LMs in a doped 2-leg ladder is intrinsically dis-
l
power-law fits at low temperature. crete, approximating a continuous distribution only for
vanishing couplings. It is therefore interesting to ask
whether the doped 2-leg ladder reproduces the known
as shown in Fig. 2(b), at low enough doping the upper physics of the REHM model with a continuous distri-
bound is reached extremely slowly in temperature - in bution of couplings. If a continuous approximation to
fact,forthe2-legladderatT =J /1024andJ =J ,the the coupling distribution is in order, as in Eq. (8), the
l l r
correlationlengthisstillnotsaturatedtoitsfiniteT =0 system should exhibit a universal REHM behavior for
value for z (cid:46) 0.25. It is easy to understand that this 2zξ0 > 1−γc, and a non-universal behavior for 2zξ0 <
occurs because the average effective coupling [|J(eff)|] 1−γc. In other words, ladders with sufficiently strong
nn av
doping z or sufficiently large correlation length ξ in the
decreasesaszdecreases,leadingtoexceedinglyweakcor- 0
undopedlimitshouldexhibituniversalscalingproperties
relations even at extremely low temperatures.
at low temperature. To our knowledge the only numeri-
This means that for sufficiently low doping, the condi-
calstudyofcorrelationsinsite-dilutedladdersisRef.15,
tion ξ(T) (cid:28) (cid:104)l(cid:105) is actually satisfied over all numerically
which,makinguseofquantumMonteCarlo,investigates
accessiblelowtemperatures(andquitepossiblyalsoover
ladderswithJ /J =1/2anddopingsz =0.01, 0.04and
all experimentally accessible temperatures), so that the r l
0.1. Given that ξ ≈ 7.5 in this case15, for z = 0.04
segmented nature of the doped ladders become in fact 0
and0.1oneobtains1−2zξ =0.4and−0.5respectively,
irrelevant. In the following we will focus our discussion 0
which appear to be both safely on the universal side of
on this regime.
the REHM with continuous couplings. Indeed a fit to
the low-temperature behavior of the correlation length
to the scaling law ξ =A(z)+B(z) T−2α(z) gives an esti-
D. Doped 2-leg ladders and the random-exchange
mateofα=0.2±0.025(independentofz)whichappears
Heisenberg model
consistent with the predictions of Refs. 18,27.
AsdiscussedinsectionIIIB,fromthestatisticalpoint
of view the system of interacting LMs randomly dis-
E. Low-temperature scaling of the correlation
tributed on the bipartite 2-leg ladder can be approxi-
length
mated by a system of S = 1/2 spins interacting with
randomlydistributedcouplings,followinganintrinsically
discrete distribution. In general terms, a doped 2-leg We have performed QMC simulations to extensively
ladder represents a physical realization of the so-called study the low-temperature behavior of the correlations
random-exchange Heisenberg model (REHM)18,27 , but in site-diluted 2-leg ladders. Our QMC results, however,
with a special structure of the coupling distribution. do not support the conclusion in Ref. 15 about a uni-
Numerical real-space renormalization group versal REHM behavior of diluted 2-leg ladders. We in-
studies18,27 show that the REHM colorred for a vestigate doped ladders at similar concentrations with
continuous distribution of couplings P(J˜) has two respect to those of Ref. 15, down to significantly lower
regimes: temperatures-Ref.15stopsatT =Jl/500, whilewede-
scend down to T = J /2048. A collection of our results
l
• Universal regime: if the distribution is not singu- is shown in Fig. 3. In qualitative agreement with what
lar, or has a power-law singularity P(J˜) ∼ J−γ expected from the theory of the REHM model, we also
with γ < γ (where 0.65 (cid:46) γ (cid:46) 0.75), the observe the clear onset of a low-temperature power law
c c
low-temperature thermodynamics of the REHM is scaling of the type T−2α. Yet our central observation is
6
0.6
z=0% (Ref. 15)
z=1% (Ref. 15) 0.113 0.145 0.167
z=4% (Ref. 15)
z=10% (Ref. 15) 0.154
z=0%
z=1%, α=0.113 0.4 0.164
z=4%, α=0.178
z=10% 0.142 0.176
10 Jl
ξ J/r 0.199
0.175 0.196 0.204 0.186
0.2
0.185
1 0
0.001 0.01 0.1 0 0.01 0.02 0.03 0.04 0.05
T z
FIG. 4: (color online). Comparison of our data with those FIG. 5: (color online). Values of the α exponent extracted
by Ref. 15. Our data refer to ladders with L = 384, and from the low-temperature behavior of site-diluted 2-leg lad-
Jl =2Jr =1. ders.
thatthispower-lawscalingexhibitsanon-universal,dop- theuniversalREHMregimeisonlyobservedinanarrow
ing dependent exponent α=α(z). For all the parameter region around Jr/Jl ∼1/4 and z ∼2%.
sets we considered, a fit of the low-temperature correla-
tionlengthprovidesvaluesofα whicharesystematically
below the one exhibited by the universal regime of the F. Discussion
REHM with continuous couplings.
The data shown in Fig. 3 refer to 2-leg ladders with Theaboveresultsleadustoconcludethatdoped2-leg
Jr/Jl =1/4andz =2.0%and2.6%, withcorresponding ladders do not realize in general the universal physics of
γ ≈ 0.39 and 0.235 respectively (here we use ξ0 = 15.3, the REHM with a continuous coupling distribution. In
which we estimate independently with QMC); and 2-leg fact,thecouplingdistributionfortheLMsindoped2-leg
ladders with Jr/Jl = 1/2 (as in Ref. 15) and z = 2.0%, ladders has a fundamentally discrete structure, approx-
3.9%, and 6.4%, corresponding to γ ≈ 0.7, 0.415, and imating a continuous one only for vanishing couplings.
0.04 respectively. Although all the values of γ fall be- The weight assigned by the distribution to strong dis-
low the critical value γc, as shown in Fig. 3, in fact, all crete couplings is dominant over that of weak, quasi-
thecorrespondingdatasetsforthecorrelationlengthdis- continuouscouplingsinthecaseofsizabledopingz (sup-
playanon-universal power-lawscaling. Inparticular,we pressing the probability of large separation between the
observe that data sets with close values of the parame- LMs) and of large correlation length ξ (cid:29) 1 (increasing
0
ter γ (e.g. Jr/Jl = 1/4, z = 2.0% with γ ≈ 0.39, and the strength of the couplings between LMs, the distance
Jr/Jl = 1/2, z = 3.9%, with γ ≈ 0.415) show a clearly beingfixed). Nonetheless,thisistheregimeinwhichthe
different scaling. We also critically revisit the parame- parameter γ =1−2zξ is small, and in which one would
0
ter sets explored in Ref. 15. A comparison to our data, expect the universal REHM physics to be manifested in
as done in Fig. 4, shows that in the data of Ref. 15 the the continuum approximation. On the other hand, the
significant scattering due to the statistical uncertainty quasi-continuous part of the distribution (corresponding
is masking the correct asymptotic scaling regime at low toverysmallcouplings)acquiresabiggerweightforsmall
temperature. z and ξ , which determine a small average coupling ac-
0
A comprehensive summary of our results on the low- cording to Eq. (9) and hence increase the probability of
temperature scaling of the correlation length is provided thesmallvaluesofthecoupling. Inthisregimeγ iscloser
in Fig. 5, where we report the values of the α exponent to1,whichleadstotoostrongasingularityintheproba-
obtainedfordifferentdopingvaluesandJ /J ratios. The bility distribution for the universal regime of the REHM
r l
values of doping concentrations and ratios were chosen with continuous couplings to be manifested.
suchthattheaveragedistancebetweenLMs(cid:104)d(cid:105)isatleast As a consequence, we generally observe a low-
larger than ξ . The LMs, therefore, are not overlapping temperature power-law scaling of the correlation length
0
on average and can be described by an effective REHM with doping-dependent exponents. As far as the nu-
with couplings obeying Eq. (4). Furthermore, for all the merical results are concerned, this scaling appears to be
points shown in Fig. 5, γ < γ , we do not observe a the asymptotic one for T → 0, given that it sets in for
c
universalαvalue. Indeed,itisfoundthatαdependsnot temperatures well below the average LM coupling. On
only on the doping concentration but also on the ratio the other hand, according to a real-space renormaliza-
of couplings J /J . And an α exponent close to that of tion group (RSRG) approach18, the system at a lower
r l
7
temperature is increasingly sensitive to the part of the 100
distribution related to the weaker couplings; given that
2-leg ladder
thispartistheonewhichbestapproximatesacontinuous 4-leg ladder
6-leg ladder
distribution, one could naturally expect that at very low
temperatures the physics of the REHM with continuous
couplingsbereproducedbythedoped2-legladders. This
argument would then suggest that, when lowering the ξ 10
temperature, the system will sooner or later attain the
universalregimeoftheREHMwithcontinuouscouplings
ifγ <γ . Nonethelessonecanarguethatinadoped2-leg
c
ladder the RSRG flow has to be necessarily stopped at a
finite length, corresponding roughly to the average seg-
mentlength(cid:104)l(cid:105). Inpracticethisimposesanupperbound 10 0.1 0.2 0.3 0.4 0.5
z
to the correlation length and a lower bound to the tem-
perature through the condition Eq. (10); these bounds
FIG. 6: (color online). Correlation length of bond-diluted
might prevent one from attaining the low-temperature
2-, 4-, and 6-leg ladders with L = 128, J = J = J, and
l r
regime at which universal REHM physics would mani-
temperature T =J/1024
fest itself.
Finally,ourstudyleavesoneopenquestionconcerning
theroleofdiscretenessoftheinitialcouplingdistribution • ifarungbondisdiluted,twoLMsareliberatedbut
for the physics of the REHM. The numerical RG study stillinteractantiferromagnetically,sothattheycan
ofRef.27onlyaddressesinitialcontinuousdistributions; screen each other to form a rung singlet at a lower
a systematic RG study of the flow starting from realis- energy J(eff) (corresponding to the effective cou-
ticdistributionsstemmingfromthedoped-ladderphysics plingbetwruengenLMswhicharerungneighbors). This
would be highly desirable. screeningisveryeffectiveatlowdilutionbecausein
that case the J(eff) interaction is largely dominant
rung
overtheinteractionbetweentwoLMsbelongingto
IV. BOND-DILUTED LADDERS different rungs, due to the large spacing of two di-
luted rung bonds – this spacing is 3/z (given that
only one bond in three is a rung bond);
In the following, we consider the doping- and tem-
perature dependence of correlations in bond-diluted lad-
• ifalegbondisdiluted,therung-singletstateonthe
ders, which show a marked difference with respect to
twoadjacentrungsisfurtherreinforcedbecauseits
site dilution. This is due to a fundamental geometric
connectivity to the other rungs is lowered.
aspect which distinguishes site and bond dilution: dilut-
ing a site leaves a single unpaired spin, giving rise to a The dilution of rung bonds has therefore the effect of
LM,whileeliminatingabondleavesalwaystwo unpaired lowering locally the gap over the ground state, whereas
spins,locatedondifferentsublattices. Thecorresponding the dilution of leg bonds has the effect of increasing it30.
LMs are therefore interacting with an effective antiferro- The fact that there are twice as many leg bonds as rung
magnetic coupling, mediated by the the shortest path of bondssuggeststhatthesecondeffectdominatesoverthe
bonds connecting them. In the following, we will focus first. Therefore the spin gap of the undoped ladder is lo-
for simplicity on the case J =J =J. cally preserved or even enhanced, leading to short-range
l r
correlations even in presence of bond dilution. Nonethe-
less, standard consideration on rare-event physics lead
to conclude that even a very weak bond dilution leads
A. 2-leg ladders
the system to a gapless, Griffiths-like phase. In fact, the
gap can close locally if rare regions appear in which only
1. Evolution of correlations with doping
rung bonds are diluted – the limiting case being that
of the formation of local strands made of two uncou-
Fig.6showstheevolutionofthecorrelationlengthata pled S = 1/2 chains. These exponentially rare regions
fixed,lowtemperatureT =J/1024. Astrikingdifference leadtotheclosingofthespingapinthethermodynamic
withrespecttothecaseofsitedilution,showninFig.6,is limit; yet global correlations remain short ranged, due
thatthecorrelationlengthdoesnotincreaseimmediately to the localized nature of the rare, locally gapless re-
with doping: it remains essentially constant for n = 2, gions. This Griffiths phase is reminiscent of what has
while it even decreases for n>2. been observed by some of us in the Heisenberg model on
In the case of a 2-leg ladder, we can understand qual- theinhomogeneouslybond-dilutedsquarelattice20,21. In
itatively the behavior of correlations by considering that that case, the predominance of correlation-suppressing
bond dilution has two fundamentally different effects dilution was guaranteed by the inhomogeneity of doping
(sketched in Fig. 7(b)): probabilities (favoring the appearance of ladder-like or
8
4 4
(a) (b)
3 3
z=0%
ξ β=1024 ξ z=3.1%
β=512 z=6.25%
2 β=256 2 z=9.4%
β=128 z=12.5%
β=64 zz==1158..67%%
β=32 z=21.9%
β=16
1 1
0 0.1 0.2z0.3 0.4 0.5 10-3 10-2 10-1
T
FIG. 7: (color online). Sketch of the bond dilution effects 3.5 0.3
on 2-leg ladders. a) pure ladder; b) ladder at low dilution:
(c) (d)
enhancement of rung singlets by leg-bond dilution, and low-
energysingletsformedbetweenLMsafterrung-bonddilution; 3 0.2
c)ladderatstrongerdilution: couplingsbetweenLMsbelong- z) z)
ing to different diluted rungs. A( B(
2.5 0.1
0.227
2 0
0 0.1 0.2 0.3 0 0.1 0.2 0.3
z z
FIG.9: (coloronline). (a)Correlationlengthofbond-diluted
10 0.219
2-leg ladders with L = 128 and J = J = J at various tem-
l r
ξ peratures. (b) Temperature dependence of the correlation
lengthofabovesystem. Thecurveswithz>22%areomitted
duetoξislimitedbythegeometriclength. (c)and(d)Fitting
n=2: random dilution parameters of correlation length ξ ≈A(z)+B(z)|log(T/J)|,
n=4: random dilution
n=2: rung bond dilution as a function of bond dilution.
n=4: outer-rung bond dilution
1 n=2: geometric length
n=4: geometric length
we have estimated by generalizing an efficient algorithm
0 0.1 0.2 0.3 0.4 0.5
z recently developed for homogeneous percolation31,32. It
is found that at finite z, (cid:104)l(cid:105) ≈ a/z2. It follows the
FIG. 8: (color online). Correlation length for 2-leg and 4-
same asymptotic behavior as from the naive estimate
legladderswithdilutionof(center-)rungbonds. Theaverage
(cid:104)l(cid:105)∼1/z2, but with a renormalized factor a<1. Hence
lengthofladderssegments(cid:104)l(cid:105)isobtaineduponhomogeneous
the enhancement of correlations due to rung-bond dilu-
dilution.
tion competes with this geometrical restriction on corre-
lations for sufficiently strong doping. Quite remarkably,
we observe that the optimal doping for the enhancement
dimer-like structures).
of correlations in the homogeneously doped 2-leg ladder
For a sufficiently large bond dilution z > z∗ (0.07 (cid:46)
coincides with the value of dilution (z ≈ 22%) at which
z∗ (cid:46) 0.09), the correlation length starts increasing with
the correlation length of the rung-only diluted ladder
increasing z. This implies necessarily that the liberated
equals the average length of ladder segments (cid:104)l(cid:105). There-
LMs belonging to different rungs begin to correlate with
forewecanconcludethatthenon-monotonicbehaviorof
each other (Fig. 7(c), see also Refs. 21,22). Such corre-
the correlation length at intermediate dilution values is
lations appear because the spacing between diluted rung
completely determined by the competition between the
bonds decreases, and hence the couplings between LMs
correlationenhancementthroughrung-bonddilutionand
belonging to different rungs become of the same order
the correlation suppression due to ladder fragmentation.
of J(eff). We have isolated the effect of enhancement of
rung
correlations through rung-bond dilution by considering
uniquelythisformofdilution,namelybytakingp(r) =0
i,m 2. Evolution of correlations with temperature
with probability z, while p(l) = 1 with probability 1.
i,m
The resulting correlations ξ (z) (r = rung) as a function Asdiscussedintheprevioussubsection,forz =z∗ cor-
r
ofz for2-and4-legladdersareshowninFig.8,andthey relations start to increase as a function of bond dilution.
areseentoincreasequitefastwithdilution(followingthe Fig.9(a)showsthedopingdependenceofthecorrelation
approximate form ξ (z) ≈ ξ exp(az) for sufficiently low lengthof2-legladdersforhighertemperaturesthanthose
r 0
dilution). Yet in the original system the correlations in- consideredinFig.6: hereweobservethatthecorrelation
duced by rung-bond dilution are upper-bounded by the length does not appear to evolve significantly with tem-
characteristic length of the ladder segments (cid:104)l(cid:105), which perature for doping values z (cid:46) z∗, whereas it becomes
9
5 hasafinitecorrelationlength, namelyitisinaGriffiths-
like phase. Adding leg-bond dilution (with concentra-
tions 12% and 14.6% respectively), we recover a ran-
4 domly bond-diluted ladder (with concentrations 18.75%
and 21.9% respectively). For large enough dilution the
correlationisfoundtogrowlogarithmicallywithdecreas-
ξ 3 ingtemperature, asinEq.(11), whichseemsatypicalfor
a Griffiths-like phase. On the other hand, we observe
z=18.75%, random dilution that, for dilution zr =z/3, the correlation length for the
2 zz==261.2.95%%,, rraunndgo-mbo dnidlu dtiioluntion rung-bonddilutedladderwithconcentrationzr isanup-
r per bound to that of the randomly diluted ladder with
z=7.3%, rung-bond dilution
r
concentrationz,afactwhichissomewhatintuitive,given
that rung-bond dilution is the main mechanism leading
1
10-4 10-3 10-2 10-1 100 to the enhancement of correlations. Therefore it is a pri-
T
ori unclear whether the logarithmic growth of the ran-
domlydilutedladderpersiststoevenlowertemperatures
FIG. 10: (color online). (a) Temperature dependence of
than the ones explored here, because this would eventu-
correlation lengths for four cases of rung-bond and random
ally lead the correlation length of the randomly diluted
dilution.
ladder to exceed that of the rung-bond diluted one.
We can then conclude that the correlation length re-
significantlydependentontemperatureforz >z∗. More- sults are a priori consistent with two different scenarios
over, the low-temperature dependence of the correlation fortherandomlydiluted2-legladders. Inafirstscenario
length, see Fig. 9(b), indicates that in this regime the the system transitions from a Griffiths-like phase with a
correlation length grows logarithmically with decreasing finitecorrelationsforz <z∗toanewphasewithlogarith-
temperature as micallydivergingcorrelations,whichisreminiscentofthe
behaviorofasystemcontrolledbyaninfinite-randomness
ξ ≈A(z)+B(z)|log(T/J)| . (11)
fixed point (IRFP) – see the discussion below. In a sec-
ondscenariothecorrelationlengthexhibitsdifferenttem-
perature dependences at intermediate temperatures for
The dependence of the coefficients A(z) and B(z) on
increasing bond dilution. Yet the correlation length of
bond dilution is shown in Fig. 9(c) and (d). While A(z)
the randomly diluted ladders converges always to a fi-
decreases with increasing bond dilution, B(z) increases.
nite value, because it is expected to be upper-bounded
In particular B(z) appears to vanish for z (cid:46) z∗, and it
by that of rung-bond diluted ladders, which is seen to
seems to suggest that a phase transition occurs at finite
converge to a finite value. In this second scenario the
z =z∗ from a phase with correlations converging to a fi-
system is therefore in a Griffiths-like phase for all the
nitevalueatlowT toaphasewithcorrelationsdiverging
values of bond dilution we explored. In the following we
logarithmically with decreasing temperature. Obviously
find that the analysis of the temperature dependence of
thedivergentbehaviorofcorrelationlengthonlypersists
theuniformsusceptibilityhelpsclarifyingwhichscenario
up to lengths of the order (cid:104)l(cid:105), corresponding to the char-
is the most appropriate.
acteristic length of ladder segments, but the exceedingly
slow growth of ξ with decreasing T keeps it safely be-
low (cid:104)l(cid:105) for all the temperatures and doping values we
explored. 3. Temperature dependence of the uniform susceptibility
To gain a deeper understanding on the above results,
westudythetemperaturedependenceofrung-bonddilu- Fig. 11(a) shows the uniform susceptibility of a 2-leg
tion and random dilution separately. We consider rung- ladderforvariousvaluesofbonddilution. Intheundoped
bond dilution with concentrations z = z/3 such that system, the susceptibility of 2-leg ladders vanishes expo-
r
the fraction of diluted rung bonds is the same as for nentially at low temperatures, in agreement with well-
the cases of random dilution that we have investigated. known previous results, and showing the gapped nature
This allows us to quantitatively ascertain the separate of the spectrum. In contrast, the susceptibility of the
effectofleg-bondandrung-bonddilution. Inpresenceof dopedsystemdivergesfollowingaCurielawforlowtem-
rung-bond dilution only, the correlation length is seen to peratures. Furthermore, Fig. 11(b) shows that the Curie
converge to a finite value as T → 0, even at relatively coefficient increases as a function of bond dilution con-
high concentration. Two examples with concentration centration following a power law, C(z) ∼ z2.66. One
z = 6.25% and 7.3% are shown in Fig. 10 (data for might expect that Curie paramagnetism comes trivially
r
higher concentration, not shown here, exhibit a similar fromspinswhichhaveremainedisolatedafterbonddilu-
behavior). This implies that the system with rung-bond tion. Yet the concentration of such spins would scale as
dilutiononlyisgapless(becauseoftheunboundedsizeof z3 with dilution, given that one needs to dilute at least
rareregionswithe.g. dilutedadjacentrungbonds)andit 3 bonds to decouple one spin from the rest of the ladder.
10
10-1 10-1
(a) zz==03%.1% (b) zz==03%.1%
z=6.25% z=6.25%
10-2 zzzzz=====91112.25814....%5679%%5%% efficient) 10-2 zzzzz=====91112.25814....%5679%%5%%
χTu10-3 e co χu,c10-3
uri T
C
10-4 C(z) ( 10-4
C(z)~z2.66
10-5
(a) (b)
10-5
10-3 10-2 10-1 0.01 z 0.1 100 101 10-3 10-2 10-1 100
T |Log(T)| T
FIG. 11: (color online). (a) Low-temperature uniform FIG. 12: (color online). (a) Uniform susceptibility of even-
susceptibility of bond-diluted ladders with L = 256 and numbered spin clusters of randomly bond-diluted 2-leg lad-
Jl = Jr = 1. (b) The Curie coefficient increases as a power- ders. The solid lines are (a) logarithmic fits of the form
lawfunctionofthebonddilutionconcentrationC(z)∼z2.66. χ =C/(T|logT|β)and(b)power-lawfitsχ =C(cid:48)Tβ(cid:48)−1.
u,c u,c
Thereforetheparamagnetismobservedinthesystemhas
In both cases we can conclude that the contribution to
a collective nature. In fact, two contributions add to the
the uniform susceptibility coming from even-numbered
one of free spins:
clusters diverges more weakly than for a Curie law, and
• odd-numbered clusters, obtained after fragmenta- therefore that our system exhibits a highly non-trivial
tion of the ladder into two (or more) pieces, have a collective paramagnetism. The fitting coefficients C,β,
doublet ground state, behaving as a collective spin and C(cid:48),β(cid:48) for both cases are shown in Fig. 13. We ob-
1/2. Inoursimulationsonsystemswithanevento- serve that all these fitting coefficients evolve smoothly
tal number of sites, three contiguous diluted bonds withz,apparentlycontradictingthepictureofapossible
leadtoanisolatedsiteplusanodd-numberedclus- phase transition suggested by the behavior of the corre-
ter, so that the latter clusters have a probability lation length. In particular the non-universal power-law
∼ z3. Alternatively odd-numbered clusters can be dependence of Eq. (13) is typical of Griffiths behavior,
obtained by cutting the ladder at two distinct lo- as observed e.g. in anisotropically bond-diluted square
cations into two odd-numbered extended clusters lattices21. Therefore the uniform susceptibility seems to
(biggerthanonesinglesite). Yetonecaneasilysee favor the scenario for which the system is in a Griffiths
that this would require at least 5 diluted bonds at phaseforallvaluesofthebonddilutionweexplored. For
specific locations, giving rise to a probability ∼z5, completeness we should also mention a third, less likely
and consequently to a very small contribution to scenario,forwhichthecorrelationlengthisdiverginglog-
the uniform susceptibility. Therefore the contribu- arithmically as T →0 even for infinitesimal doping (but
tiontocollectiveparamagnetismcannotcomefrom theprefactorB(z)ofthelogarithmicdivergencebecomes
such clusters. lower than our resolution). Therefore the system would
be controlled by an IRFP for all doping values explored
• clusters with an even number of spins, but a non-
here; this scenario is somewhat consistent with the fact
equal number of spins on A and B sublattices, can
that a logarithmically corrected Curie law is consistent
also contribute to a diverging susceptibility. They
with the susceptibility at all doping values considered in
represent therefore a further candidate for collec-
this section, and that it could be another sign of IRFP
tive paramagnetism.
behavior.
To isolate the second contribution, we calculate the
uniform susceptibility coming from even-numbered spin
clusters only, χu,c. This quantity is shown in Fig. 12. 4. Comparison with random ladders
Wefindthateven-numbereduniformsusceptibilityatlow
temperaturescanbeequallywelldescribedbytwofitting
As sketched in Fig. 7, the rung LMs in a bond-diluted
laws: a logarithmically-corrected Curie behavior:
ladder realize a ladder with weaker random couplings
C whichareatoncerung,leganddiagonalonesbutwithout
χ = , (12)
u,c T|logT|β frustration. The rung couplings are randomized by the
varietyofdifferentlocalenvironmentswhichmediatethe
and a power-law corrected Curie behavior:
effectiveinteractionbetweentherungLMs; whiletheleg
C(cid:48) couplings and the diagonal couplings are obviously ran-
χu,c = T1−β(cid:48), (13) domized (in a correlated way) by the positional disorder