We investigate the application of a perturbative renormalization group (RG)
method to random antiferromagnetic Heisenberg chains with arbitrary spin size.
At zero temperature we observe that initial arbitrary probability distributions
develop a singularity at J=0, for all values of spin S. When the RG method is
extended to finite temperatures, without any additional assumptions, we find
anomalous results for S >= 1. These results lead us to conclude that the
perturbative scheme is not adequate to study random chains with S >= 1.
Therefore a random singlet phase in its more restrictive definition is only
assured for spin-1/2 chains.Comment: 5 pages, 3 figures. To appear in Physical Review 

Boechat, B.

Bonfim, O. F. de Alcantara

Continentino, M. A.

Saguia, A.

English

arXiv

We investigate the application of a perturbative renormalization group (RG) method to random antiferromagnetic Heisenberg chains with arbitrary spin size. At zero temperature we observe that initial arbitrary probability distributions develop a singularity at J = 0, for all values of spin S. When the RG method is extended to finite temperatures, without any additional assumptions, we find anomalous results for S \u3e= 1. These results lead us to conclude that the perturbative scheme is not adequate to study random chains with S \u3e= 1. Therefore a random singlet phase in its more restrictive definition is only assured for spin-1/2 chains

University of Portland

University of PortlandPilot ScholarsPhysics Faculty Publications and Presentations Physics2001Breakdown of the perturbative renormalizationgroup for S >= 1 random antiferromagnetic spinchainsA. SaguiaB. BoechatM. A. ContinentinoO. F. de Alcantara BonfimUniversity of Portland, bonfim@up.eduFollow this and additional works at: http://pilotscholars.up.edu/phy_facpubsPart of the Physics CommonsThis Journal Article is brought to you for free and open access by the Physics at Pilot Scholars. It has been accepted for inclusion in Physics FacultyPublications and Presentations by an authorized administrator of Pilot Scholars. For more information, please contact library@up.edu.Citation: Pilot Scholars Version (Modified MLA Style)Saguia, A.; Boechat, B.; Continentino, M. A.; and Bonfim, O. F. de Alcantara, "Breakdown of the perturbative renormalization groupfor S >= 1 random antiferromagnetic spin chains" (2001). Physics Faculty Publications and Presentations. 12.http://pilotscholars.up.edu/phy_facpubs/12arXiv:cond-mat/0011479v1  [cond-mat.dis-nn]  29 Nov 2000Breakdown of the perturbative renormalization group for S ≥ 1 randomantiferromagnetic spin chainsA. Saguia, B. Boechat and M. A. ContinentinoInstituto de F´ısica - Universidade Federal Fluminense24210-130, Nitero´i, RJ - Brazil;O. F. de Alcantara BonfimDepartment of Physics, Reed College, Portland Oregon 97202, USA(October 13, 2013)We investigate the application of a perturbative renormalization group (RG) method to randomantiferromagnetic Heisenberg chains with arbitrary spin size. At zero temperature we observe thatinitial arbitrary probability distributions develop a singularity at J = 0, for all values of spin S.When the RG method is extended to finite temperatures, without any additional assumptions, wefind anomalous results for S ≥ 1. These results lead us to conclude that the perturbative schemeis not adequate to study random chains with S ≥ 1. Therefore a random singlet phase in its morerestrictive definition is only assured for spin-1/2 chains.Low-dimensional random quantum magnetic systems have become an object of increasing interest in recent years.Despite the apparent simplicity, quantum spin chains show a wealth of physical properties which have attracted theattention of both theoretical and experimental physicists. It is generally accepted that quantum antiferromagneticspin-1/2 chains in the presence of disorder exhibit, at low temperatures, a random singlet (RS) phase characterized bypower law behavior of thermodynamic quantities [1,2]. This has, in fact, been observed in several experiments [3–7].Theoretically, the new phase was revealed by a real space renormalization group (MDH) approach [1]. These resultshave motivated many authors to study the physical properties of other random antiferromagnetic quantum (RAQ)chains with S ≥ 1. In this case there are additional new problems, as for example, what happens to the Haldanegap of integer spins chains [8] in the presence of disorder. In fact, the possibility of the energy gap be suppressed bydisorder, driving the system to a RS phase has become a very attractive matter for investigation. In particular, theS = 1 RAQ chains has been exhaustively studied by several methods [9–16]. Many authors have employed extendedversions of the MDH scheme to explore the ground state properties of the model. In the course of these investigations,some authors suggested the existence of random singlet phase in the spin-1 chain in the strong disorder regime [9–13].On the other hand, studies based on exact diagonalization [14], quantum Monte Carlo (MC) simulation [15] anddensity matrix renormalization group (DMRG) techniques [16] found that the Haldane phase is quite robust againstrandomness.In view of these controversial results and the fact that most calculations for spin-1 chains have been done at T = 0,we were motivated to extend the MDH analysis to finite temperature. Our goal was to generalize the MDH recursionrelations to study the thermodynamic behavior of chains with arbitrary spin size. We performed careful numericalprocedures to iterate our generalized MDH equations at finite temperature. The anomalous results we have obtainedfor the free energy and are presented in this communication indicate that the MDH approach breaks down for chainswith S ≥ 1. Therefore, we argue that it is impossible to conclude on the existence or not of the RS phase in the S ≥ 1RAQ chains, as proposed in previous zero temperature studies using the MDH method [9–13]. Actually, within theMDH approximation the existence of the RS phase is assured only for the spin-1/2 chains.The random magnetic systems are described by the Heisenberg Hamiltonian:H =L−1∑i=1Ji~Si.~Si+1, (1)where Ji > 0 is the nearest neighbor antiferromagnetic interaction and ~Si are quantum spin operators. The exchangecouplings are random variables distributed according to a given probability distribution PJ (Ji,Ω), with a cutoff Ω.The MDH method consists in eliminating the pair of spins with the strongest coupling (J2 = Ω) in the random chainby considering the interaction (J1 and J3) with the neighboring spins of this pair as a perturbation (see Fig. 1).The Hamiltonian for the strongly coupled pair of spins ~S2 and ~S3 is given byH0 = J2 ~S2.~S3. (2)1These spins are weakly coupled to the neighbors viaH1 = J1~S1.~S2 + J3~S3.~S4. (3)We calculate the free energy of the system formed by ~S1, ~S2, ~S3 and ~S4 through perturbation theory in second orderof H1 (see ref. [1]). After some simple but extensive calculations we come with the results:F ′ = F0 −13Ω[S(S + 1)]2(J21 + J23 )Vs(βΩ) (4)withF0 = −S(S + 1)Ω−1βln2S∑i=0(2i+ 1) exp[−12i(i+ 1)βΩ] (5)andJ ′ =23S(S + 1)J1J3ΩWs(βΩ), (6)where β = 1/kBT and the functions V and W are given byVs(y) =(2S + 1)2 −∑2Si=0(2i+ 1)e−12i(i+1)y [1− 12 i(i+ 1)y]4S(S + 1)∑2Si=0(2i+ 1)e−12i(i+1)y(7)andWs(y) =(2S + 1)2 −∑2Si=0(2i+ 1)e−12i(i+1)y [1 + 12 i(i+ 1)y]4S(S + 1)∑2Si=0(2i+ 1)e−12i(i+1)y. (8)Equations (4)-(8) are the generalization of the MDH equations for the thermodynamic properties of RAQ chains witharbitrary spin size. For S=1/2 we recover the original MDH recursion relations [1]. The S=1 case has already beenstudied in the limit of very low temperatures [10].When the decimation process is carried out for a spin-1/2 chain the distribution PJ (Ji,Ω), independently of theform of the original distribution of antiferromagnetic bonds, rather quickly approaches a fixed form singular at J = 0[2],PJ (Ji,Ω) ≈αΩ(JiΩ)−1+α Θ(Ω− Ji). (9)withα = −1lnΩ(10)The fixed form of PJ (Ji,Ω) determines the low-temperature behavior of the thermodynamic quantities. The expo-nent α gives rise to logarithmic corrections in the thermodynamic functions.The great merit and the reason for the success of the MDH theory [1], was to show that the characteristic RS phasebehavior of spin-1/2 chain is universal. More specifically, it is independent of the original distribution of exchangecouplings. The MDH approach represented a considerable progress with respect to previous methods [5,6,17] whichrequired an initial distribution of bonds with a particular form. Actually, in order to reproduce the power lawdependence of the low temperature thermodynamic quantities observed experimentally, the previous models requireda initial distribution already with a power law behavior [6,17].We turn now to the numerical simulation of chains with spins S=1 and S=3/2. Each chain is composed of Nspin-S objects with periodic boundary conditions (with N= 50,000). We carried out the averages over 10 differentconfigurations for each PJ (Ji,Ω) to obtain the free energy of the system. We start by choosing the exchange couplingfrom an uniform distribution PJ(Ji,Ω) given by:PJ(Ji,Ω) ={ 11−∆ for ∆ ≤ Ji ≤ 1,0 otherwise(11)2The parameter ∆ represents the strength of the initial disorder of the couplings. The case ∆ = 0 represents the strongdisorder limit because the distribution becomes extremely broad in a logarithmic scale; namely, infinitesimally weakbonds appear. The weak disorder regime is represented by finite values of ∆. The distribution, in this case, presentsa gap ∆.We have studied the flow of the initial coupling distribution, Eq. (11), for the spin-1 chain for the following cases:∆ = 0, ∆ = 0.05, ∆ = 0.1, and ∆ = 0.2. We found that for any ∆, successive elimination transformations give rise toweaker and weaker couplings as the cutoff Ω decreases, independently of the temperature. For sufficiently small Ω, i.e.,after a sufficient number of eliminations, the distribution PJ (Ji,Ω) becomes peaked at Ji = 0 and, as in the T=0 case[10–13], can be approximated by a power-law. We should point out that the convergence to a power-law depends onthe value of ∆. It is faster in the case of the gapless (∆ = 0) initial distribution. As in the case of spin-1/2 chains [1]the power-law exponent obtained for the S=1 chain is non-universal: it depends on temperature and cutoff, althoughweakly. In Fig. 2 we present, for completeness, the power-law behavior of PJ (Ji,Ω) with exponent α = 0.38 obtainedfrom an initial distribution with ∆ = 0, at a fixed temperature kBT = 0.10Ω.Now we focus on the free energy per spin and the specific heat of the spin-1 chain. In Fig. 3 we show the plot of thespecific heat versus temperature (given in units of Ω). Despite the power-law dependence of PJ(Ji,Ω) at low energiesdescribed above, we find a non physical behavior in the free energy of the system. For intermediate temperatures,the free energy gives rise to a negative specific heat (see Fig. 3). Nonetheless, at very low temperatures the freeenergy and thus the specific heat is well-behaved, being described by a power-law (see inset of Fig. 3). These very lowtemperature features are similar to the random singlet phase of the disordered spin-1/2 chain [1]. Moreover, at highertemperatures, the free energy curve is again well-behaved and gives rise to a physically meaningful specific heat.The presence of the factor 2S(S + 1)/3 in Eq. (6), which defines the renormalized coupling J ′, plays an importantrole in our discussion. In the case of the spin-1 chain it is 4/3. At low temperatures (kBT < 0.37Ω), the functionW1(βΩ) is larger than 3/4, consequently, when the couplings neighboring the cutoff Ω are not small enough, couplingslarger than those eliminated may be generated (see Eqs. (6) and (8)). In this low temperature regime, although thecutoff decreases rapidly, large couplings are generated with non negligible probability. When the couplings betweenthe strongly coupled pair of spins and its neighbors are not sufficiently weak, the perturbative scheme fails, givingrise to the spurious result observed in the free energy. At higher temperatures (kBT > 0.37Ω) the MDH method givescorrect results for the thermodynamic properties [10]. There the function W1(βΩ) < 3/4 cancel the coefficient 4/3 ofJ ′1 (see Eq. (8)), thus validating the present approach, independently of the starting distribution. In addition, whenone considers distributions with a finite gap (weakly disordered distributions), the breakdown of the MDH formalismis most pronounced. The probability of obtaining, at low temperatures, the product J1 J3 < 3/4Ω2 depends on thevalue of the gap. By increasing ∆, the initial distribution will not generate infinitesimally weak couplings to ensurethat the product is sufficiently small.In the spin-3/2 chain we performed the decimation process by starting with a uniform gapless probability distribu-tion. Despite the larger coefficient in Eq. (6), PJ (Ji,Ω) still develops a power-law behavior as the cutoff Ω becomessufficiently small. However, this convergence is much slower than that for a spin-1 chain. Now the couplings J ′ whichare larger than the decimated out coupling Ω are statistically predominant. The effect of the breakdown of the MDHmethod in the free energy of the S = 3/2 chain is more dramatic yielding a negative specific heat even in the verylow temperature limit.These results has led us to try to improve the MDH transformation in two directions: first we considered thedecimation of larger spin clusters. In order to gain an insight on whether or not this procedure would work, it wassufficient to carry out the decimation at zero temperature. The intent was to investigate whether the perturbationscheme, using larger clusters, would decrease the factor appearing in the expression for the new coupling J ′. Wehave carried out numerically the decimation process for RAQ chains with both spin S = 1/2 and S = 1, consideringclusters with 2, 4, 6 and 8 spins. In our numerical calculations we considered, for simplicity, Ji = Ω = 1, where theJ ′is stand for the bonds between neighboring spins in the cluster. The results for the case of a cluster with only twospins can be directly compared with the value 1/2 and 4/3 of the original MDH procedure (see Eq. (6)) in the S=1/2and S=1 RAQ chains, respectively. We list the results in Table I. In the case of a spin-1/2 chain the results indicatethat the MDH scheme is always improved. However, the resulting values for larger clusters in the spin-1 chain makesclear that the problem remains unsolved.We also tried to improve the MDH scheme by considering terms of higher order in the perturbation expansion, atzero temperature. In that case, our purpose was to investigate whether the higher order corrections would decreasethe factor 2S(S + 1)/3 in Eq. (6). We have thus calculated the T = 0 renormalized coupling, for the spin-1 chain, upto third order in Ω, which is given by3J ′ =43ΩJ1J3 +1Ω2(J21J3 + J1J23 ) (12)Once again, we come upon a result which reinforces that the MDH formalism is not suitable to describe the lowtemperature properties of the RAQ chains with S ≥ 1 (defined in Eq (1)). Moreover, in the fourth-order correction tothe renormalized bond biquadratic exchange couplings are generated. In this case we should consider these interactionsalready from the beginning in the Hamiltonian model. Finally we call attention that starting with a power lawdistribution, the MDH procedure works very well. In this case, only pairs of spins weakly coupled are present in thechain and the probability of the generated coupling being larger than the one eliminated is statistically negligible.The singular form of PJ (Ji,Ω) dominates the thermodynamic properties which are also described by power-laws. Ofcourse this is in strong contrast with the robust and universal character of the RS phase for the spin-1/2 chain, whichis indeed a new fixed point [1,2].In summary, at zero temperature there is not a safe criterion to discuss the possibility of the occurrence of the RSphase in the RAQ chains. In fact, it was necessary to extend the MDH scheme, for S ≥ 1, to finite temperature toestablish the failure of the perturbative approach. In view of the results we have obtained, we strongly claim that theMDH procedure is not adequate to study the RAQ chains with S ≥ 1 defined in Eq. (1). Moreover, the existence ofa RS phase at low temperature is a property assured only for spin-1/2 RAQ chains.The authors are grateful to the Brazilian agencies FAPERJ and CNPq for financial supports.[1] S. K. Ma, C. Dasgupta and C. K. Hu, Phys. Rev. Lett. 43, 1434 (1979); S. K. Ma, C. Dasgupta, Phys. Rev. B 22, 1305(1980).[2] D. S. Fisher, Phys. Rev. B 50, 3799 (1994); Phys. Rev. B 51, 6411 (1995).[3] J. C. Scott, A. F. Garito, A. J. Heeger, P. Nannelli and H. D. Gilman, Phys. Rev B 12, 356 (1975).[4] Y. Endoh, G. Shirane, R. J. Birgeneau and Y. Ajiro, Phys. Rev. B 19, 1476 (1979).[5] L. N. Bulaevskii, A. V. Zvarykina, Yu S. Karimov, R. B. Lyubovskii and I. F. Shchegolev, Zh. EKsp. Teor. Fiz. 62, 725(1972) [Sov. Phys. JETP 35, 384 (1972)].[6] G. Theodorou and M. H. Cohen, Phys. Rev. Lett. 37, 1014 (1976)[7] J. C. Fernandes, R. B. Guimara˜es, M. A. Continentino, H. A. Borges, J. V. Valarelli and A. Lacerda, Phys. Rev. B 50,16754 (1994)[8] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).[9] M. A. Continentino, J. C. Fernandes, R. B. Guimara˜es, B. Boechat, H. A. Borges, J. V. Valarelli, E. Hanapel, Alex Lacerdaand P. R. J. Silva, Philos. Mag. B 73, 601 (1996).[10] B. Boechat, A. Saguia e M. A. Continentino, Solid State Communications 98, 411 (1996)[11] R. A. Hyman and Kun Yang, Phys. Rev. Lett. 78, 1783 (1997)[12] C. Monthus, O. Golinelli and Th. Jolicoeur, Phys. Rev. Lett. 76, 839 (1997); C. Monthus, O. Golinelli and Th. Jolicoeur,preprint cond-mat/9712172.[13] A. Saguia, Beatriz Boechat and M. A. Continentino, Phys. Rev. B 58, 58 (1998); 62, 5541 (2000).[14] Y. Nishiyama, Physica A252 35 (1998); erratum ibid A258 499 (1998).[15] Y. Nishiyama, Eur. Phys. J. B6 335 (1998).[16] K. Hida, Phys. Rev. Lett. 83,3297 (1999).[17] G. Theodorou, Phys. Rev. B 16, 2264, 2273 (1977).4TABLE Icluster size spin-1/2 spin-12 0.500000 1.3333334 0.377992 1.8813736 0.312648 2.5977768 0.270240 -TABLE I. Renormalized coupling for chains with S = 1/2 and S = 1 obtained with clusters of size 2,4,6 and 8. We havetaken Ji = Ω = 1.FIGURES CAPTIONFIG. 1. Spins and coupling constants involved in the elimination transformation (a) to (b).FIG. 2. Power-law behavior of PJ(Ji,Ω) in the low-energy limit, at kBT = 0.10Ω. Inset: the Log-Log plot of PJ (Ji,Ω) yieldsthe exponent α = 0.38.FIG. 3. Low temperature behavior of the specific heat of the random spin-1 chain for ∆ = 0. Inset: the power-law behaviorwith exponent equal to 0.77.50 0.05 0.1 0.15Ji0100200300400500600700P(Ji)10−4 10−3 10−2 10−1 100100101102103FIGURE 2α=0.380.00 0.05 0.10 0.15 0.20 0.25T−0.200.000.200.400.600.80Cv(T)0.00 0.01 0.02 0.030.0000.0050.0100.0150.020FIGURE 3

Breakdown of the perturbative renormalization group for S \u3e= 1 random antiferromagnetic spin chains

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Breakdown of the perturbative renormalization group for S >= 1 random antiferromagnetic spin chains

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