General Quantum Fidelity Susceptibilities for the J1-J2 Chain
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t General Quantum Fidelity Susceptibilities for the J − J Chain
Mischa Thesberg ∗ and Erik S. Sørensen † Department of Physics & Astronomy, McMaster University1280 Main St. W., Hamilton ON L8S 4M1, Canada. (Dated: November 29, 2018)We study slightly generalized quantum fidelity susceptibilities where the differential change in thefidelity is measured with respect to a different term than the one used for driving the system towardsa quantum phase transition. As a model system we use the spin-1 / J − J antiferromagneticHeisenberg chain. For this model, we study three fidelity susceptibilities, χ ρ , χ D and χ AF , whichare related to the spin stiffness, the dimer order and antiferromagnetic order, respectively. Allthese ground-state fidelity susceptibilities are sensitive to the phase diagram of the J − J model.We show that they all can accurately identify a quantum critical point in this model occurring at J c ∼ . J between a gapless Heisenberg phase for J < J c and a dimerized phase for J > J c .This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by amarginal operator and is therefore particularly difficult to observe. I. INTRODUCTION
The study of quantum phase transitions, especially inone and two dimensions, is a topic of considerable andongoing interest. Recently the utility of a concept withits origin in quantum information, the quantum fidelityand the related fidelity susceptibility, was demonstratedfor the study of quantum phase transitions (QPT).
Ithas since then been successfully applied to a great num-ber of systems.
In particular, it has been applied tothe J − J model that we consider here. . For a recentreview of the fidelity approach to quantum phase tran-sitions, see Ref. 13. Most of these studies consider thecase where the system undergoes a quantum phase tran-sition as a coupling λ is varied. The quantum fidelityand fidelity susceptibility is then defined with respect tothe same parameter. Apart from a few studies, rel-atively little attention has been given to the case wherethe quantum fidelity and susceptibility are defined withrespect to a coupling different than λ . Here we considerthis case in detail for the J − J model and show that, ifappropriately defined, these general fidelity susceptibili-ties may yield considerable information about quantumphase transitions occurring in the system and can be veryuseful in probing for a non-zero order parameter.Without loss of generality, the Hamiltonian of anymany-body system can be written as H ( λ ) = H + λH λ , (1)where λ is a variable which typically parametrizes an in-teraction and exhibits a phase transition at some criticalvalue λ c . In this form H λ is then recognized as a termthat drives the phase transition. Using the eigenvectorsof this Hamiltonian the ground-state (differential) fidelitycan then be written as: F ( λ ) = |h Ψ ( λ ) | Ψ ( λ + δλ ) i| . (2)A series expansion of the GS fidelity in δλ yields F ( λ ) = 1 − ( δλ ) ∂ F∂λ + . . . (3) where ∂ λ F ≡ χ λ is called the fidelity susceptibility . Fora discussion of sign conventions and a more completederivation see the topical review by Gu, Ref. 13. If thehigher-order terms are taken to be negligibly small thenthe fidelity susceptibility is defined as: χ λ ( λ ) = 2(1 − F ( λ ))( δλ ) (4)The scaling of χ λ at a quantum critical point, λ c , is oftenof considerable interest and has been studied in detail andprevious studies have shown that χ λ ∼ L /ν , χ λ /N ∼ L /ν − d , (5)with N = L d the number of sites in the system. Aneasy way to re-derive this result is by envoking finite-sizescaling. Since 1 − F obviously is dimensionless it followsfrom Eq. (4) that the appropriate finite-size scaling formfor χ λ is χ λ ∼ ( δλ ) − f ( L/ξ ) . (6)If we now consider the case where the parameter λ drivesthe transition we may at the critical point λ c identify δλ with λ − λ c . It follows that ξ ∼ ( δλ ) − ν . As usual,we can then replace f ( L/ξ ) by an equivalent function˜ f ( L /ν δλ ). The requirement that χ λ remains finite for afinite system when δλ → f ( x ) ∼ x ∼ L /ν ( δλ ) , from which Eq. (5) follows.Here we shall consider a slightly more general casewhere the term driving the quantum phase transition isnot the same as the one with respect to which the fi-delity and fidelity susceptibility are defined. That is, oneconsiders: H ( λ, δ ) = H + δH I , H = H + λH λ . (7)The fidelity and the related susceptibility is then definedas F ( λ, δ ) = |h Ψ ( λ, | Ψ ( λ, δ ) i| , (8) χ δ ( λ ) = 2(1 − F ( λ, δ )) δ (9)The scaling of χ δ at λ c for this more general case wasderived by Venuti et al. where it was shown that: χ δ ∼ L d +2 z − v , χ δ /N ∼ L d +2 z − v . (10)Here, z is the dynamical exponent, d the dimensionalityand ∆ v the scaling dimension of the perturbation H I .In all cases that we consider here z = d = 1. We notethat Eq. (10) assumes [ H , H I ] = 0, if H I commuteswith H then F = 1 and χ δ = 0. The case where H λ and H I coincide is a special case of Eq. (10) for which∆ V = d + z − /ν and one recovers Eq. (5).A particular appealing feature of Eq. (5) is that when2 /ν > d , χ λ /N will diverge at λ c and the fidelity sus-ceptibility can then be used to locate the λ c without anyneed for knowing the order parameter. Secondly, it canbe shown that the fidelity susceptibility can be ex-pressed as the zero-frequency derivative of the dynami-cal correlation function of H I , making it a very sensitiveprobe of the quantum phase transition. On the otherhand, if a phase transition is expected one might thenuse the fidelity susceptibility as a very sensitive probeof the order parameter through a suitably defined H δ inEq. (7). This is the approach we shall take here usingthe J − J spin chain as our model system.The spin-1 / J − J chain is a very wellstudied model. The Hamiltonian is: H = X i S i · S i +1 + J X i S i · S i +2 (11)where J is understood to be the ratio of the next-nearestneighbor exchange parameter over the nearest neighborexchange parameter ( J = J ′ /J ′ ). This model is knownto have a quantum phase transition of the Berezinskii-Kosterlitz-Thouless (BKT) universality class occurringat J c between a gapless ’Heisenberg’ (Luttinger liquid)phase for J < J c and a dimerized phase with a two-folddegenerate ground-state for J > J c . Field theory ,exact diagonalization and DMRG , have yieldedvery accurate estimates of the Luttinger Liquid-Dimerphase transition, the most accurate of these being due toEggert which yielded a value of J c = 0 . et al. of this model using thefidelity approach used the same term for the driving andperturbing part of the Hamiltonian as in Eq. (1) with thecorrespondence H = P i S i · S i +1 , H λ = P i S i · S i +2 , λ = J . . Chen et al. demonstrated that, thoughno useful information about the Luttinger Liquid-Dimerphase transition could be obtained directly from the ground-state fidelity (and similarly the fidelity suscep-tibility), a clear signature of the phase transition waspresent in the fidelity of the first excited state. Some-times this is taken as an indication that ground-statefidelity susceptibilities are not useful for locating a quan-tum phase transition in the BKT universality class. Herewe show that more general ground-state fidelity suscep-tibilities indeed can locate this transition.Specifically, we will study three fidelity susceptibilities, χ ρ , χ D and χ AF , which are coupled to the spin stiffness, a staggered interaction term and a staggered field term,respectively. In section II we present our results for χ ρ while section III is focused on χ D and section IV on χ AF . II. THE SPIN STIFFNESS FIDELITYSUSCEPTIBILITY, χ ρ We begin by considering the J − J model with J = 0but with an anisotropy term ∆, what is usually called the XXZ model: H XXZ = X i [∆ S zi S zi +1 + 12( S + i S − i +1 + S − i S + i +1 )] . (12)The Heisenberg phase of this model, occurring for ∆ ∈ [ − , defined as: ρ ( L ) = ∂ e ( φ ) ∂δ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 . (13)Here, e ( φ ) is the ground-state energy per spin of themodel where a twist of φ is applied at every bond: H XXZ (∆ , φ ) = X i [∆ S zi S zi +1 + 12( S + i S − i +1 e iφ + S − i S + i +1 e − iφ )] . (14)The spin stiffness can be calculated exactly for the XXZmodel for finite L using the Bethe ansatz, and exact ex-pressions in the thermodynamic limit are available. Interestingly the usual fidelity susceptibility with respectto ∆ can also be calculated exactly.
Since the non-zero spin stiffness defines the gaplessHeisenberg phase it is therefore of interest to define afidelity susceptibility associated with the stiffness. Thiscan be done through the overlap of the ground-state with φ = 0 and a non-zero φ . With Ψ (∆ , φ ) the ground-stateof H XXZ (∆ , φ ) we can define the fidelity and fidelity sus-ceptibility with respect to the twist in the limit where φ → F (∆ , φ ) = |h Ψ (∆ , | Ψ (∆ , φ ) i| , (15) χ ρ (∆) = 2(1 − F (∆ , φ )) φ . (16)To calculate χ ρ the ground-state of the unperturbedHamiltonian was calculated through numerical exact di-agonalization. The system was then perturbed by addinga twist of e iφ at each bond and recalculating the ground-state. From the corresponding fidelity, χ ρ was calculatedusing Eq. (16). Our results for χ ρ /L versus ∆ are shownin Fig. 1. For all data φ was taken to be 10 − and pe-riodic boundary conditions were assumed. In all cases itwas verified that the finite value of φ used had no effecton the final results. The numerical diagonalizations weredone using the Lanczos method as outlined by Lin et al. Total S z symmetry and parallel programming techniques -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 ∆ χ ρ / L L=10L=12L=14L=16L=18L=20L=22L=24L=26L=28
FIG. 1. (Color online.) χ ρ L vs. ∆: The spin stiffness fidelitysusceptibility ( χ ρ (∆) /L ) as a function of the z-anistropy ∆.At the ∆ = 0 point the spin-current operator J and kineticenergy T commute with the XXZ Hamiltonian and thus sucha perturbation does not change the ground-state, and the fi-delity is one. Thus, χ ρ is zero at this point. were employed to make computations feasible. Numeri-cal errors are small and conservatively estimated to be onthe order of 10 − in the computed ground-state energies.In order to understand the results in Fig. 1 in moredetail we expand Eq. (14) for small φ : H XXZ (∆ , φ ) ∼ H XXZ (∆) + φ J − φ T + . . . , (17) J = i X i ( S + i S − i +1 − S − i S + i +1 ) , (18) T = 12 X i ( S + i S − i +1 + S − i S + i +1 ) . (19)Here, J is the spin current and T a kinetic energy term.The first thing we note is that, when ∆ = 0 both J and T commute with H XXZ (∆ = 0). The ground-state wave-function is therefore independent of φ (for small φ ) and χ ρ ≡
0. This can clearly be seen in Fig. 1.In the continuum limit the spin current J can be ex-pressed in an effective low energy field theory with scal-ing dimension ∆ J = 1. However, we expect sublead-ing corrections to arise from the presence of the opera-tors ( ∂ x Φ) with scaling dimension 2 and cos( √ πK Φ)with scaling dimension 4 K . Here, K is given by K = π/ (2( π − arcos(∆))). For ∆ = 0 both of these terms willbe generated by the term T in Eq. (17). With thesescaling dimensions and with the use of Eq. (10) we thenfind: χ ρ /L = A L + A + A L − + A L − K (20)In Fig. 2 a fit to this form is shown for 3 different val-ues of ∆ = 0 . , . .
75 in all cases do we observean excellent agreement with the expected form with cor-rections arising from the last term L − K being almost
10 15 20 25 30 L χ ρ / L ∆ = 0.25∆ = 0.50∆ = 0.75 FIG. 2. (Color online.) χ ρ vs. L (the XXZ model at dif-ferent values of ∆): This graph shows the scaling of χ ρ withsystem size for different values of the z-anisotropy ∆. Thepoints represent numerical data and the lines represent fits tothe scaling form predicted for the spin stiffness susceptibility χ ρ /L = A L + A + A L − + A L − K . It can be seen thatthere is good agreement. un-noticeable until ∆ approaches 1. We would expectthe sub-leading corrections L − and L − K to be absentif the perturbative term is just φ J .We now turn to a discussion of a definition of χ ρ in thepresence of a non-zero J but restricting the discussionto the isotropic case ∆ = 1. In this case we define: H ( φ ) = X i [ S zi S zi +1 + 12( S + i S − i +1 e iφ + S − i S + i +1 e − iφ )] + J X i [ S zi S zi +2 + 12( S + i S − i +2 e iφ + S − i S + i +2 e − iφ )] . (21)That is, we simply apply the twist φ at every bond of theHamiltonian. As before we can expand: H ( φ ) ∼ H (0) + φ ( J + J ) − φ T + T ) + . . . , (22) J = i X i ( S + i S − i +1 − S − i S + i +1 ) , (23) J = i X i ( S + i S − i +2 − S − i S + i +2 ) , (24) T = 12 X i ( S + i S − i +1 + S − i S + i +1 ) , (25) T = 12 X i ( S + i S − i +2 + S − i S + i +2 ) . (26)Our results for χ ρ /L versus J using this definition areshown in Fig. 3 for a range of L from 10 to 32. In theregion of the critical point at J = 0 . χ ρ /L vanishes yielding near scale invariance.How well this works close to J c is shown in the inset of χ ρ /L L=10L=12L=14L=16L=18L=20L=22L=24L=28L=320.235 0.240 0.245J χ ρ / L x FIG. 3. (Color online.) χ ρ L vs. J and Inset: The generalizedspin stiffness susceptibility, χ ρ as a function of the secondnearest-neighbor exchange coupling J . The system acquiresa clearly size invariant form in the vicinity of the critical point J ∼ .
24 (as well as tending to a global minima). Insetshows the the minima for system sizes L=16,20,24,28,32 with J c indicated as the vertical dashed line. A clear dependenceof the J value of χ ρ /L minima on the system size can beseen. Fig. 3. This alone can be taken to be a strong indicationof χ ρ /L s sensitivity to the phase transition. In fact, thisscale invariance works so well that one can locate the crit-ical point to a high precision simply by verifying the scaleinvariance. This is illustrated in Fig. 4B where χ ρ /L isplotted as a function of L for J = 0 . J = J c and J = 0 .
25. From the results in Fig. 4B the critical point J c where χ ρ /L becomes independent of L is immediatelyvisible.As can be seen in the inset of Fig. 3 χ ρ /L reaches aminimum slightly prior to J c . The J value at whichthis minimum occurs has a clear system size dependencewhich can be fitted to a power-law and extrapolated to L = ∞ yielding a value of J c = 0 . J c in the thermodynamic limit.This is shown in Fig. 4A. Comparison of this value withthe accepted J c = 0 . χ ρ /L is non-zero at the critical point, J c . Thisvalue is very small but we have verified in detail thatnumerically it is non-zero.The scale invariance of χ ρ /L is clearly induced by thedisappearance of the marginal operator cos( √ πK Φ)at J c . We expect that in the continuum limit the absenceof this operator implies that the spin current commuteswith the Hamiltonian resulting in χ ρ being effectivelyzero at J c . The observed non-zero value of χ ρ /L wouldthen arise from short-distance physics.Note that, as mentioned previously, we take the spinstiffness to be represented by a twist on every bond, bothfirst and second nearest neighbor and not merely on theboundary as is sometimes done. This choice is not just
10 12 14 16 18 20 22 24 L × -5 × -5 × -5 χ ρ /L J AB FIG. 4. (Color online.) A: The J value of χ ρ minima as afunction of system size, as well as a (power law) line of best fit.As the system size tends towards infinity the power law bestfit predicts a minima at J = 0 . χ ρ at J = 0 . J = 0 .
25 (the second highest,linear curve) and at the critical point J = 0 . χ ρ L at the critical pointas well as non-constant scaling on either side of the criticalpoint can clearly be seen. a matter of taste. Imposing a twist only on the bound-ary (usually) breaks the translational invariance of theground-state and, through extension, effects the valueand behavior of the fidelity itself. Another point of noteis the use of a twist of only φ between next-nearest neigh-bors. Geometric intuition would suggest that a twistof 2 φ should be applied between next-nearest neighborbonds. However, for the small system sizes available forexact diagonlization it is found that a simple twist of φ on both bonds yields significantly better scaling. III. THE DIMER FIDELITY SUSCEPTIBILITY, χ D We now turn to a discussion of a fidelity susceptibilityassociated with the dimer order present in the J − J model for J > J c . This susceptibility, which we call χ D , is coupled to the order parameter of the dimerizedphase by design. Usually in the fidelity approach to quan-tum phase transitions one considers the case where theground-state is unique in the absence of the perturba-tion. This is not the case here, leading to a diverging χ D /L in the dimerized phase even in the presence of agap. Specifically, we consider a Hamiltonian of the form: H = X i [ S i · S i +1 + J S i · S i +2 + δh ( − i S i · S i +1 ](27)Thus, in correspondence with Eq. (7) we have H I =( − i S i · S i +1 and we choose the driving coupling to be J χ D /L L=12L=16L=20L=24 χ D / L L=16L=24
FIG. 5. (Color online.) χ D L vs. J . The generalizeddimer fidelity susceptibility χ D /L as a function of the secondnearest-neighbor exchange parameter J . A clear intersectionof all curves can be seen in the vicinity of the proposed crit-ical point at J ∼ . − .
25. The inset explicitly shows thecrossing of L = 12 and L = 24. The dashed vertical linesindicate J c . J . This perturbing Hamiltonian represents a conjugatefield for the dimer phase. The scaling dimension of H I isknown , ∆ D = , and from Eq. (10) we therefore find: χ D ∼ L − D = L (at J c ) (28)Due to the presence of the marginal coupling we can-not expect this relation to hold for J < J c . However,the marginal coupling changes sign at J c and is there-fore absent at J c where Eq. (28) should be exact. For J < . that logarithmic correctionsarising from the marginal coupling for the small systemsizes considered here lead to an effective scaling dimen-sion ∆ D > . At J = 0 Affleck and Bonner estimated∆ D = 0 .
71. Hence, using this results at J = 0, wewould expect that χ D ∼ L . which we find is in goodagreement with our results at J = 0.We now need to consider the case J > . J = 1 / and the twodimerized ground-states are exactly degenerate even forfinite L . For J c < J < / χ D is formally infinite at J = 0 andas L → ∞ for J c < J < / χ D to divergeexponentially with L . At J c we expect χ D to exactlyscale as L and for J < J c we expect χ D ∼ L α eff with α eff <
3. Hence, if χ D /L is plotted for different L wewould expect the curves to cross at J c . However, thecrossing might be difficult to observe since it effectivelyarise from logarithmic corrections.Our results for χ D /L are shown in Fig. 5, where acrossing of the curves are visible around J ∼ . − . L χ D
12 16 20 24 28 L J A B
FIG. 6. (Color online.) A: Scaling of χ D vs. L at thecritical point J = 0 . J value of the intersection of χ D L betweensystems of size L and L + 2 plotted as a function of L . Thecurve can be fitted with a power-law line of best fit. The lineof best fit is found to converge to J = 0 . As an illustration, the inset of Fig. 5 shows the crossingof L = 12 and L = 24. In order to obtain a more preciseestimate of J c the intersection of each curve and the curvecorresponding to the next largest system were tabulated( L and L + 2). These intersection points as a functionof system size were then plotted Fig. 6A and found toobey a power-law of the form a − bL − α with α ∼ . a = 0 . J c = 0 . To further verify the scaling of χ D at J c we show inFig. 6B χ D at J c as a function of the cubed system size, L . The strong linear scaling is in contrast to the scal-ing a small distance away from the critical point (notshown) where the scaling was found to be distorted bylogarithmic corrections. IV. THE AF FIDELITY SUSCEPTIBILITY, χ AF Finally, we briefly discuss another fidelity susceptibil-ity very analogous to χ D . We consider a perturbing termin the form of a staggered field of the form P i ( − i S zi with an associated fidelity susceptibility, χ AF . The scal-ing dimension of such a staggered field is ∆ AF = andas for χ D we therefore expect that χ AF ∼ L at J c . How-ever, in this case it is known that the effective scalingdimension for J < J c is smaller than resulting in χ AF ∼ L α eff with α eff > J < J c . On the otherhad, in the dimerized phase χ AF must clearly go to zeroexponentially with L . Hence, if χ AF is plotted for differ-ent L as a function of J a crossing of the curves shouldoccur.Our results are shown in Fig. 7 where χ AF /L is plot-ted versus J for a number of system sizes. It is clear χ A F / L χ AF L=16 χ AF L=18 χ AF L=20 χ AF L=24 χ AF L=28
FIG. 7. (Color online.) χ AF /L versus J . χ AF is expected toapproach zero exponentially with the system size for J > J c ,to scale as L at J c and to scale as L α eff with α eff > J < J c . A crossing close to the critical point J c (dashedvertical line) is then visible. from these results that χ AF indeed goes to zero rapidlyin the dimerized phase as one would expect. Close to J c the scaling is close to L where as for J < J c it isfaster than L . Hence, as can be seen in Fig. 7, a crossingoccurs close to J c . V. CONCLUSION AND SUMMARY
In this paper we have demonstrated the potential ben-efits of extending the concept of a fidelity susceptibil- ity beyond a simple perturbation of the same term thatdrives the quantum phase transition. By using the spin-1 / χ ρ can be used to suc-cessfully estimate the transition point at J ∼ . χ D ,this time coupled to the order parameter susceptibilityof the dimer phase. Again, we were able to estimate thecritical point at a value of 0.241. Finally, we discussedan anti ferromagnetic fidelity susceptibility that rapidlyapproaches zero in the dimerized phase but diverges inthe Heisenberg phase. Although susceptibilities linked tothese quantities appeared the most useful for the J − J model we considered here, it is possible to define manyother fidelity susceptibilities that could provide valuableinsights into the ordering occurring in the system beingstudied. ACKNOWLEDGMENTS
MT acknowledge many fruitful discussions with SedighGhamari. ESS would like to thank Fabien Alet for sev-eral discussions about the fidelity susceptibility and IAffleck for helpful discussions concerning scaling dimen-sions. This work was supported by NSERC and by theShared Hierarchical Academic Research Computing Net-work. ∗ [email protected] † [email protected]; http://comp-phys.mcmaster.ca S. Sachdev,
Quantum Phase Transitions (Cambridge Uni-versity Press, 1999). H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P.Sun, Phys. Rev. Lett. , 140604 (2006). P. Zanardi and N. Paunkovi´c,Phys. Rev. E , 031123 (2006). H.-Q. Zhou and J. P. Barjaktarevi,Journal of Physics A: Mathematical and Theoretical , 412001 (2008). W.-L. You, Y.-W. Li, and S.-J. Gu,Phys. Rev. E , 022101 (2007). P. Zanardi, M. Cozzini, and P. Giorda,Journal of Statistical Mechanics: Theory and Experiment , L02002 (2007). M. Cozzini, P. Giorda, and P. Zanardi,Phys. Rev. B , 014439 (2007). M. Cozzini, R. Ionicioiu, and P. Zanardi,Phys. Rev. B , 104420 (2007). P. Buonsante and A. Vezzani,Phys. Rev. Lett. , 110601 (2007). D. Schwandt, F. Alet, and S. Capponi, Physical ReviewLetters , 170501 (2009). A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi,Physical Review B , 064418 (2010). S. Chen, L. Wang, S.-J. Gu, and Y. Wang,Phys. Rev. E , 061108 (2007). S. Gu, International Journal of Modern Physics B , 4371 (2010). P. Zanardi, P. Giorda, and M. Cozzini, Physical ReviewLetters , 100603 (2007). L. Campos Venuti and P. Zanardi,Phys. Rev. Lett. , 095701 (2007). C. De Grandi, A. Polkovnikov, and A. W. Sandvik,arXiv.org cond-mat.other (2011). S. Gu and H. Lin, EPL (Europhysics Letters) , 10003(2009). S. Chen, L. Wang, Y. Hao, and Y. Wang,Phys. Rev. A , 032111 (2008). F. D. M. Haldane, Phys. Rev. B , 4925 (1982). S. Eggert and I. Affleck, Physical Review B , 10866(1992). T. Tonegawa and I. Harada, J. Phys. Soc. Jpn. , 2153(1987). S. Eggert, Phys. Rev. B , R9612 (1996). R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, andS. Ramasesha, Phys. Rev. B , 6581 (1995). S. R. White and I. Affleck, Phys. Rev. B , 9862 (1996). B. S. Shastry and B. Sutherland,Phys. Rev. Lett. , 243 (1990). B. Sutherland and B. S. Shastry,Phys. Rev. Lett. , 1833 (1990). N. Laflorencie, S. Capponi, and E. S. Sørensen, Eur. Phys.J. B , 77 (2001). M. Yang, Physical Review B , 180403 (2007). J. Fjærestad, Journal of Statistical Mechanics: Theory andExperiment , P07011 (2008). H. Q. Lin, Phys. Rev. B , 6561 (1990). J. Sirker, R. Pereira, and I. Affleck,Physical Review B , 035115 (2011). I. Affleck and J. C. Bonner, Phys. Rev. B , 954 (1990). C. K. Majumdar, Journal of Physics C: Solid State Physics3