Table Of ContentSusceptibility and Low Temperature Thermodynamics of Spin-1/2 Heisenberg Ladders
Beat Frischmutha,b, Beat Ammona,b and Matthias Troyera,b,c
aTheoretische Physik, Eidgen¨ossische Technische Hochschule, CH-8093 Zu¨rich, Switzerland
bSwiss Center for Scientific Computing, Eidgen¨ossische Technische Hochschule, CH-8092 Zu¨rich, Switzerland
cInstitute for Solid State Physics, Univerisity of Tokyo, Roppongi 7-22-1, Minatoku, Tokyo 106, Japan
and J are positive, corresponding to antiferromagnetic
⊥
Thetemperaturedependenceoftheuniformsusceptibility
coupling.
and theground state energy of antiferromagnetic Heisenberg
Using the Quantum Monte Carlo loop algorithm with
ladderswithupto6legshasbeencalculated,usingtheMonte
improved estimators, we have calculated the tempera-
Carlo loop algorithm. Thesusceptibilitiesofeven-leg-ladders
ture dependence of the uniform susceptibility χ and the
showspingapswhiletheseofodd-leg-laddersremainfinitein
6
internal energy E. The QMC loop algorithm was first
9 thezerotemperaturelimit. Forsmallratiosofintra-tointer-
9 leg couplings, odd-leg-ladders can bemappedat low temper- developed by Evertz et al. [10] for the 6-vertex-model,
1 atures to single chains. For equal couplings, the logarithmic butcanalsobe appliedtoquantumspinsystems[11,12].
corrections at low temperatures increase markedly with the TheQMCloopalgorithmisanimprovedworldlinealgo-
n
numberof legs. rithm. The updates inthe loopalgorithmareglobaland
a
J no longer local as in the conventional Metropolis world
8 Recently, antiferromagnetic Heisenberg spin-1/2 lad- line algorithm. This has the great advantage that the
der systems have attracted much interest, following the autocorrelation times are reduced by several orders of
1
discoveryofafinitespingapinthe2-leg-ladder[1]. Later magnitude. It was thus possible to simulate very long
v
investigations showed that the crossover from the sin- ladders to very low temperatures. We considered sys-
5
2 gle Heisenberg chain to the two-dimensional (2D) an- tems up to 100x 6 sites and reachedtemperatures down
0 tiferromagnetic square lattice, obtained by assembling to T = J/50 without major problems. All results are
1 chains to form “ladders” of increasing width, is far from extrapolated to a Trotter time interval ∆τ → 0. The
0 smooth [2]. Heisenberg ladders with an even number of application of improved estimators (see e.g. Ref. [12]),
6
legs (chains), n , show a completely different behavior further reducesthe varianceofthe measuredobservables
9 l
/ thanodd-leg-ladders. Whileeven-leg-laddershaveaspin dramatically.
t
a gapandshortrangecorrelations,odd-leg-laddershaveno In Table I the ground state energies of the different
m gapandpower-lawcorrelations. Basedondensitymatrix ladders in the isotropic case (J = J ) are presented.
⊥
- renormalization group (DMRG) studies, White et al. [3] Considering ladders of lengths L, we first extrapolate to
d gaveanexplanationofthisfundamentaldifferenceinthe T →0. For the odd-leg-ladders we use the form
n
frameworkof the ResonantValence Bond (RVB) picture
o E (T)=E (0)+aT2, (2)
[4]. These theoretical predictions have been verified ex- L L
c
v: perimentally,inmaterialssuchas(VO)2P2O7 [5]andthe whereEL(T)istheinternalenergyoftheladderoflength
i homologousseriesofcupratesSrn−1Cun+1O2n[6],which L. This form is motivated by the infinite single chain,
X contain weakly coupled arrays of ladders. whichinalowtemperaturefieldtheorycanbedescribed
r Previous numerical studies, using exact diagonaliza- byamasslessboson. Ournumericaldataagreewellwith
a
tion [1,7], the Quantum Monte Carlo world line algo- Eq. (2) for the finite single chains, as well as for the fi-
rithm [8] or the Quantum Transfer Matrix method [9] nite 3- and 5-leg-ladders. The even-leg-ladders, on the
were restricted to small systems or could not be applied other hand, have spin gaps. The internal energy for T
at low temperatures. Using the new Quantum Monte well below the spin gap ∆ is determined by the thermal
Carlo(QMC)loopalgorithm[10],weovercometheselim- occupationofthelowestlyingS =1magnonbandwitha
itations and are able to investigate very large ladders to quadratic dispersion near the zone boundary minimum.
much lower temperatures. For the extrapolationto T →0 we therfore use the form
In this letter we consider ladders with nl legs (nl = EL(T) = EL(0)+b(T3/2 +2∆T1/2)exp(−∆/T). In a
1,2,...,6) of length L. The Hamiltonian of such secondstepthegroundstateenergiesE (0)forthefinite
L
spin−1/2 systems systems are extrapolatedto the bulk limit E (0), fit-
L=∞
tingE (0)toapolynomialin1/L. Thefinitesizecorrec-
H =J S~ S~ +J S~S~ (1) L
X i j ⊥X i j tions of the ground state energy, however, are negligibly
↔ l
small for ladders with L≥100.
is defined onL × n lattices. The sum markedby ↔ (l) In Fig. 1 we show the susceptibility per rung of n
l l
runs over nearest neighbors along the legs (rungs). We spins, χ(T), for the ladders with 1 ≤ n ≤ 6 in the
l
assumed periodic boundary conditions in the longitudi- isotropic case J = J . We always consider ladders long
⊥
nal direction of the ladder. The exchange constants J enoughsuchthatfinitesizecorrectionsarenegligible. For
1
T >J theresultsagreewellwithathirdorderhightem- χ(0)=(2πv)−1 =(Jπ2)−1 with infinite slope. The field
perature expansion. At low temperatures, we observe theoretical results can be compared to the exact Bethe
the predicted behavior. Ladders with even n show an ansatz data [16] and one finds that Eq. (4) holds to
l
exponential drop of the susceptibility indicating a gap within 1% for T <0.1J. These results are shown in Fig.
in the excitation spectrum. For larger n , the drop sets 2 together with our QMC data for low temperatures.
l
in at smaller T and is steeper. The gap ∆ decreases We turn now to the 3- and 5-leg-ladders. In the limit
nl
substantiallywithincreasingn . Foroddn onthe other J/J = 0 each eigenfunction is a direct product of one-
l l ⊥
hand, χ(T)remainsfinite alsoat T ≪J, as inthe single rungstateswhoselowestlyingmultipletisaspindoublet
chain. (s = 1/2). The ground state of the whole system is
Ladders with even n were already investigated in de- therefore2L-folddegenerate. AfinitevalueofJ lifts this
l
tail [3,9,15]. For temperatures T ≪ ∆ the susceptibility degeneracy. In this 2L-dimensional subspace M we can
forthe2-leg-ladderisdeterminedbythethermaloccupa- define an effective Hamiltonian H which includes all
eff
tion of S = 1 magnon band with a quadratic dispersion intra-leg interactions. To first order in J/J we obtain
⊥
near the zone boundary minimum [9]: [18]:
χ(T)∝T−1/2e−∆/T, T ≪∆. (3) ∞
H(1) =J S~ S~ , (5)
eff eff X j,tot j+1,tot
Provided a quadratic dispersion for the lowest lying j=0
magnonbranchinthe excitationspectrumnearits mini-
whereS~ isthetotalspinofthejth rungandJ =J
mum is assumed, Eq. (3) also holds for the 4- and 6-leg- j,tot eff
for the 3-leg-ladder, respectively J = 1.017J for the
ladder. We estimate ∆ by fitting the numerical QMC eff
data for low temperatunrels and find in the isotropic case 5-leg-ladder. H(1) has just the form of the Hamiltonian
eff
∆2 = 0.51(1)J for nl = 2, ∆4 = 0.17(1)J for nl = 4 of the single chain with an effective coupling Jeff and we
and ∆ = 0.05(1)J for n = 6. The value ∆ obtained can map the low lying energy states of the 3- and 5-leg-
6 l 2
for the 2-leg-ladder is in perfect agreement with former ladder to those of the single chain. In the following we
results [3,8,9]. Onthe other hand, the spin gapobtained will concentrate on the 3-leg-ladder.
by White et al. [3], using DMRG methods for the 4-leg- The susceptibility of the single chain χ1(T/J) scales
ladder ∆ =0.190J is slightly larger than our value. with 1/J. It follows, that for a 3-leg-ladder with small
4
The decrease of the spin gap with increasing nl can J/J⊥andatlowtemperature,whereonlytheabovemen-
be explained by delocalization of RVB singlets not only tioned low lying states in M are relevant, the suscepti-
along but more and more also across the ladder. The bility per rung χ3 scales with 1/Jeff and has the same
decrease of the spin gap, however, is much faster than functional dependence on T/Jeff as χ1(T/J), according
∆ ∝ 1/n , suggested in [3]. The spin gap for the 6- to Eq. (5). This can be seen in Fig. 3, where we show
nl l
leg-ladder ∆6 is already a factor 10 smaller than ∆2, Jχ3 for different ratios J/J⊥ as a function of T/J. For
suggesting rather an exponential decrease of ∆nl with smallJ/J⊥ thesusceptibilityperrungχ3multipliedbyJ
increasing nl. is veryclose to Jχ1 until a crossovertemperature,which
The susceptibility per rung of the odd-leg-ladders re- depends on J/J⊥. Above this temperature, the suscep-
mains finite in the low temperature limit and tends to a tibility of the 3-leg-ladder is larger, due to the presence
zero temperature value approximately independent of n of additional states in the 3-leg-ladder which are not in-
l
[see Fig. 1(b)]. For this we conclude that the odd-leg- cluded in the 2L-dimensional subspace M. These addi-
laddersbelongtothesameuniversalityclassasthesingle tional states have a finite gap ∆˜. The susceptibility of
chain. the 3-leg-ladder then reads
Thesinglechaincanbedescribedinalowtemperature
χ =χ (J )+χ˜, (6)
fieldtheoryby thek =1 Wess-Zumino-Wittennonlinear 3 1 eff
σ-modelorequivalentlybyafree,masslessboson,witha
whereχ˜isthecontributionoftheadditionalstates. From
(spin) velocity v =πJ/2. Based on this model χ(T =0)
our QMC data we find J ≈ J for all small J/J and
andtheleadingT dependencesofχhavebeencalculated eff ⊥
∆ (J/J )<∆˜ <∆ (J/J ).
[16,17] with the leading marginally irrelevant operator. 4 ⊥ ∼ ∼ 2 ⊥
In the isotropic case J = J , the gap ∆˜ in units of J
Two-looprenormalizationofthe marginalcouplingleads ⊥
is indeed smaller than for small J/J but remains finite
to ⊥
(∆4 <∼ ∆˜ <∼ ∆2). For T ≪ ∆˜ we can neglect the contri-
χ(T)= 2π1v + 4π1v hln(T10/T) − ln(l2nl(nT20(/TT0)/+T1)/2)i bfourtTion≪χ˜∆o,fwteheseaedtdhiatitotnhaelirstsalotepse.saCroemcopmarpinlegteχly3dtioffeχr1-
+O (ln T)−3 , (4) ent (see Fig. 3). Therefore we conclude that in this case
(cid:0) (cid:1)
the simple model, discussed above [Eq. (6)], no longer
where T is the cut-off-temperature. The susceptibil-
0 applies.
ity approaches its asymptotic zero temperature value
2
As J/J → 1 also next-nearest neighbor and longer
⊥
rangeinteractionsbetweenrung-spinsbecomeimportant
in the effective Hamiltonian H . Since these additional
eff
interactionsrespectthe SU(2)andtranslationalsymme-
try, Eq. (4) still applies for some values of v and T ,
0 [1] E. Dagotto, J. Riera, and D.J. Scalapino, Phys. Rev. B
according to [16]. Therefore, for very low temperatures 45, 5744 (1992)
the susceptibility of the 3-leg-laddercan be described by
[2] For a review, see E. Dagotto, T.M. Rice, Science (in
Eq. (4) also in the isotropic case (see Fig. 2). press)
Thespinvelocityintheisotropic3-leg-ladderv3 seems [3] S.R.White,R.M.Noack,andD.J.Scalapino,Phys.Rev.
tobeclosetothatofthesinglechainv . Thiscanbeseen Lett. 73, 886 (1994)
1
by two facts. First, both of the susceptibilities per rung [4] P.W. Anderson, Science 235, 1196 (1987)
seem to extrapolate to the same zero-temperaturevalue. [5] D.C. Johnston, J.W. Johnson, D.P. Goshorn, and A.J.
Secondly, White et al.[3]determined the spin gapofthe Jacobsen, Phys.Rev.B 35, 219 (1987)
[6] Z. Hiroi, M. Azuma, M. Takano, and Y. Bando, J. Solid
finite single chain and the finite 3-leg-ladder in function
StateChem95,230(1991);M.Azumaetal.,Phys.Rev.
of 1/L. The slopes of these curves as 1/L→0, πv and
1 Lett. 73, 3463 (1994)
πv , agree, at least within 5%. Assuming v = v , we
3 1 3 [7] D.Poilblanc, H.Tsunetsugu,andT.M.Rice,Phys.Rev.
getaroughestimateofthe cut-off-temperatureT0 inthe B 50, 6511 (1994)
isotropic 3-leg-ladder. The value is much smaller than
[8] T.Barnes,E.Dagotto,J.Riera,andE.S.Swanson,Phys.
in the single chain (see Fig. 2). We conclude therefore, Rev. B 47, 3196 (1993)
thatthe effective interactionsbetweenthe spinonsinthe [9] M. Troyer, H. Tsunetsugu, and D. Wu¨rtz, Phys. Rev. B
3-leg-ladder are much stronger than those in the single 50, 13515 (1994)
chain. [10] H.G. Evertz, G. Lana, and M. Marcu, Phys. Rev. Lett.
We conclude that the odd-leg-ladders belong to the 70, 875 (1993)
same universality class as the single chain and can be [11] H.G.Evertz,M.Marcu,Nucl.Phys.B(Proc.Suppl.)30,
277 (1993)
described in the zero temperature limit by a k=1 Wess-
[12] U.J. Wiese, and H.P. Ying,Z. Phys. B 93, 147 (1994)
Zumino-Witten non linear σ-model with a spin velocity
[13] H. Bethe, Z. Physik 71, 205 (1931)
vnl. These velocities seem to have more or less the same [14] E. Manousakis, Rev.Mod. Phys. 63, 1 (1991)
value for all nl. With increasing nl, however, we move [15] S.Gopalan,T.M.Rice,andM.Sigrist,Phys.Rev.B49,
further away from the conformal point and the logarith- 8901 (1994)
mic corrections,due to the leading marginally irrelevant [16] S.Eggert,I.Affleck,andM.Takahashi,Phys.Rev.Lett.
operator, increase markedly. 73, 332 (1994)
Finally, we want to point out, that the zero tempera- [17] K. Nomura Phys.Rev.B 48, 16814 (1993)
ture value of the susceptibility per site χ(site)(0) for the [18] N.HatanoandY.Nishiyama,J.Phys.A28,3911(1995)
nl
odd-leg-ladders decreases with increasing n . Since the
l
odd-leg-ladders seem to have more or less the same zero
temperature value of the susceptibility per rung, it fol-
(site)
lows that χ (0) ∝ 1/n. For n → ∞ the zero tem-
nl l l
peraturevalue χ(site)(0)thereforegoesto zerofor oddn
nl l
as well as for even n . The susceptibility per site of a
l
2D square lattice, however, is finite for T = 0. This is
therefore a further example that the crossover from the
single chain to the 2D−lattice is not a smooth one.
We wish to thank G. Sierra, H.G. Evertz, D. Wu¨rtz
and especially T.M. Rice for very instructive and stim-
ulating discussions. The calculations were performed on
the Intel Paragon of the ETH Zu¨rich. The work was
partially supported by the SchweizerischenNationalfond
(B.F.)andbyanETHinternalgrantNo. 9452/41-2511.5
(B.A.).
3
numberof legs obtained ground reference value
state energy
1 -0.4432(1) -0.44315... [13]
2 -0.5780(2) -0.578 [8]
3 -0.6006(3) -
4 -0.6187(3) -
5 -0.6278(4) -
6 -0.635(1) -
2D lattice -0.6693(1) [12]
TABLE I. Ground state energies per site for the differ-
ent ladders in the isotropic case. For the single chain we
have perfect agreement with the analytical result 1 −ln 2
4
from the Bethe Ansatz [13]. Furthermore, the result for the
2-leg-laddercoincideswiththegroundstateenergycalculated
by Barnes et al. [8], using Lanczos techniques. With increas-
ingwidththeresultsapproachtothegroundstateenergyper
site of the infinite 2D square lattice, which was calculated
by various methods. For an overview see [14]. The reference
valuegivenherewasrecentlyobtainedbyU.J.WieseandH.P.
Ying [12] using theQMC loop algorithm.
FIG.1. Susceptibility as a function of thetemperaturefor
thesinglechainandtheHeisenbergladderswithupto6legs.
Athightemperaturestheresult agreewellwithathirdorder
high temperature expansion. The low temperature region is
shownin(b)inalargerscale. Todistinguishthecurvessome
datapointsaremarkedbysymbols. Theerrorbarsaresmaller
than thesymbols.
FIG.2. Renormalizationgroupimprovedfieldtheory(solid
lines) [Eq. (4)] for different cut-off-temperatures T0 versus
Betheansatzdata[16]andQMCresultsforχ(T)atlowtem-
perature. Theerror bars are smaller than the symbols.
FIG. 3. Susceptibility per rung of the 3-leg-ladder χ3
(dashedlines)fordifferentratiosJ/J⊥ andofthesinglechain
χ1 in function of the temperature. The inset shows the low
temperatureregioninalargerscale. Todistinguishthecurves
some data pointsare marked by symbols. The error bars are
smaller than the symbols.
4
0.8
single chain 4-leg-ladder
(a)
2-leg-ladder 5-leg-ladder
3-leg-ladder 6-leg-ladder
high-T expansion
0.6
g
n
u
r
r 0.4
e
p
χ
0.2
0.0
0 1 2 3
0.4
(b)
0.3
g
n
u
r
0.2
r
e
p
χ
0.1
0.0
0.0 0.1 0.2 0.3 0.4
T / J
Bethe Ansatz (n =1)
l
QMC Data (n =1) T = 0.2 J
l 0
0.14 QMC Data (n =3)
T = 0.3 J
l
0
0.13
χ
J
0.12
T = 2.6 J
0
0.11
0.10
0.00 0.04 0.08 0.12
T / J
0.4
0.16
g
0.14
n
u
r
r
0.3 e
p
0.12
χ
J
g
n 0.10
u
0.0 0.4 0.8
r
r 0.2 T/J
e
p
χ
J
0.1
single chain J/J =0.25
⊥
J/J =0.1 J/J =0.4
⊥ ⊥
J/J =0.15 J/J =0.6
⊥ ⊥
J/J =0.2 J/J =1.0
⊥ ⊥
0.0
0.0 2.0 4.0
T / J
