Contractor renormalization group (CORE) method is applied to the SU($N$)
chain and ladders in this paper. In our designed schemes, we show that these
two classes of systems can return to their original form of Hamiltonian after
CORE transformation. Successive iteration of the transformation leads to a
fixed point so that the ground state energy and the energy gap to the ground
state can be deduced. The result of SU($N$) chain is compared with the one by
Bethe ansatz method. The transformation on spin-1/2 ladders gives a finite gap
in the excited energy spectra to the ground state in an intuitive way. The
application to SU(3) ladders is also discussed.Comment: 4 pages, 4 figures, submitted to Phys. Rev. 

E. Schrödinger

Peng Li

Shun-Qing Shen

English

arXiv

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Contractor renormalization group theory of
                    
                      
                        SU

The contractor renormalization group (CORE) method is applied to the SU(N) chain and ladders in this paper. In our designed schemes, we show that these two classes of systems can return to their original form of Hamiltonian after CORE transformation. Successive iteration of the transformation leads to a fixed point so that the ground state energy and the energy gap to the ground state can be deduced. The result of SU(N) chain is compared with the one by Bethe ansatz method. The transformation on spin-1 2 ladders gives a finite gap in the excited energy spectra to the ground state in an intuitive way. The application to SU(3) ladders is also discussed. © 2005 The American Physical Society.published_or_final_versio

Li, P

Shen, SQ

HKU Scholars Hub

Title Contractor renormalization group theory of SU(N) chains andladdersAuthor(s) Li, P; Shen, SQCitation Physical Review B - Condensed Matter And Materials Physics,2005, v. 71 n. 21Issued Date 2005URL http://hdl.handle.net/10722/43474Rights Creative Commons: Attribution 3.0 Hong Kong LicenseContractor renormalization group theory of SUN chains and laddersPeng Li and Shun-Qing ShenDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, ChinasReceived 8 December 2004; revised manuscript received 7 March 2005; published 1 June 2005dThe contractor renormalization group sCOREd method is applied to the SUsNd chain and ladders in thispaper. In our designed schemes, we show that these two classes of systems can return to their original form ofHamiltonian after CORE transformation. Successive iteration of the transformation leads to a fixed point sothat the ground state energy and the energy gap to the ground state can be deduced. The result of SUsNd chainis compared with the one by Bethe ansatz method. The transformation on spin-1 /2 ladders gives a finite gap inthe excited energy spectra to the ground state in an intuitive way. The application to SUs3d ladders is alsodiscussed.DOI: 10.1103/PhysRevB.71.212401 PACS numberssd: 75.10.Pq, 64.60.Ak, 05.50.1qThe contractor renormalization sCOREd group methodcombines the contraction and cluster expansion techniqueswith the real space renormalization group approach to solvethe electron and spin lattice problems.1 It was first applied tospin-1 /2 Heisenberg chain and s1+1d-dimensional Isingmodel, and later to the frustrated antiferromagnets and theHaldane conjecture. The results are satisfactory andencouraging.1,2 Since then the method has been applied toinvestigate low energy physics in many strongly correlatedsystems.3–8 In this paper, we are concerned with a class ofmodels showing that CORE is at its critical point, whichmeans that the symmetry of the system is restored or thesame form of Hamiltonian is reconstructed after the COREtransformation, just like the spin-1 /2 Heisenberg chain.1 Un-doubtedly, this method is not limited to such a kind of sys-tems. Though in many systems the original Hamiltonian can-not be recovered, the lower energy physics are retained andstudied successively after the truncation and transformation.We consider the SUsNd chain and ladders in this work. Weshow that the same form of the Hamiltonian is recoveredafter dividing adequately the lattice into blocks and defininga truncation scheme, so that the CORE algorithm can bedone recursively. The range-2 result for SUsNd chain is com-pared with the Bethe ansatz solution by Sutherland.9 Thecomparison suggests that the CORE method can give goodresult especially for large N and relatively larger blocks. Wealso present the results of the real-space renormalizationgroup sRGd theory, which usually agrees qualitatively withthe one by range-2 CORE calculation.2 In many cases thelatter can be regarded as a refined method on the former. Thespin-1 /2 ladders have attracted a lot of attention since thediscovery of a finite spin gap in the two-leg ladders.10,11 An-other CORE scheme based on plaquette dividing of the lad-der had been applied to this system.7,12 Here we shall use adifferent scheme which shows a S=1 magnon gap in an in-tuitive way. Results up to range-3 are presented.It was shown by Morningstar and Weinstein that theCORE scheme of three-site block partition and two-statetruncation on SUs2d chain recover the original form ofHamiltonian. Then the resulting effective Hamiltonian can besolved iteratively and a quite satisfactory result can beobtained.1 The recovery of the form of the Hamiltonian oweshighly to the SUs2d symmetry and an adequate designedCORE scheme. As a generalization, we found that theirCORE scheme on SUs2d chain is a speciman picked out froma general CORE scheme on the SUsNd chain. Though theSUsNd chain had been exactly solved by Bethe ansatzmethod long time ago,9 it is still instructive to see howCORE works in the system.Let us start with a one-dimensional SUsNd chain in termsof the exchange operator, H=JS jPj,j+1. Here we limit ourdiscussion to the antiferromagnetic case by setting J=1.0.For a SUsNd system each site j has N quantum states uj ,alwith sa=1,2 , . . . ,Nd. The exchange operator Pj,j+1 swapstwo states on sites j and j+1, i.e., Pj,j+1uj ,a ; j+1,bl= uj ,b ; j+1,al. Usually Pj,j+1 can be expressed in terms ofthe SUsNd generators as Pj,j+1=SabJbasjdJabsj+1d, where theoperators Jbasjd satisfy the SUsNd algebra fJbasjd ,Jnmsj8dg=d j j8fdnaJbmsjd−dbmJnasjdg. Alternatively, Pj,j+1 can also be ex-pressed by spin operators.13,14 Many spin systems as well asspin-obital systems concerning SUsNd symmetry have beenstudied extensively.15–19In the CORE scheme, the first step is to divide the originalchain into a chain of blocks and retain adequate number ofenergy levels in each block. We found two obvious schemesto be readily applied to this system: one is sN−1d-site blockpatition with N-state truncation sscheme Ad, and the other issN+1d-site block patition with N-state truncation sschemeBd. The treatment on SUs2d case in Ref. 1 obviously fallsinto scheme B with Pj,j+1=2S j ·S j+1+1/2 when N=2. Wewill see the scheme B gives better results than scheme A.The existence of the two schemes can be understood fromthe single column Young tableaux with sN−1d or sN+1dboxes. In fact the SUsNd model on both sN−1d-site blockand sN+1d-site block have one unique N-dimensionalground state space. We denote the truncated space for asingle block by F j = huf j,1l , uf j,2l , . . . , uf j,Nlj. Then in therange-2 CORE calculation, we should retain appropriate N2low levels from the exact diagonalization of two blocks. Allthe retained low levels should have nonzero projection to theproduct space F j ^ F j+1, so the eligible levels are not alwaysthe lowest ones. Fortunately this job is easy to be done due tothe SUsNd symmetry. The range-2 CORE calculation leads tothe effective HamiltonianPHYSICAL REVIEW B 71, 212401 s2005d1098-0121/2005/71s21d/212401s4d/$23.00 ©2005 The American Physical Society212401-1Hs2d =1N 7 1oj s− C7 + K7P˜ j,j+1d , s1dwhere the sign 7 corresponds to the two schemes A s2d andB s1d, P˜ j,j+1 is a renormalized exchange operator connectingblocks j and j+1 after each block “contracts” to a single site.The coefficients C7 and K7 are listed in Table I. It can beconfirmed the range-3 Hamiltonian will include another op-erator P˜ j,j+2 and the range-4 Hamiltonian will include moreoperators like P˜ j,j+3, P˜ j,j+1P˜ j+2,j+3, P˜ j,j+2P˜ j+1,j+3,P˜ j,j+3P˜ j+1,j+2. Here we only give the range-2 results sincehigher range calculation will not change the physics. Forscheme A and B, we give the results for N=3,4 ,5 and 2,3,4,respectively.Successive application of CORE in Eq. s1d will lead therunning coupling approaching a gapless fixed point. And nophase transition is observed. The ground energy is read outasE0 = −C7sN 7 1d − K7, s2dwhere the sign 7 corresponds to the two schemes. Figure 1shows that the result of the range-2 CORE of scheme Bagrees quite well with the one by Bethe ansatz method. Thenumerical error can be reduced by higher range calculation.The range-4 result for N=2 by Weinstein shows the error isreduced to −0.0025.2In fact the traditional RG gives an effecive Hamiltonianhaving the same form of Eq. s1d. It can produce results con-sistent with CORE though not so good.2 The two schemesabove are still applicable and the corresponding coefficientscan be found in Table I. The advantages of CORE are obvi-ous. In many cases one can design more flexible schemes inCORE while selecting basic blocks and truncating at lowlevels.3–8 A more careful analysis shows that RG based onsN−1d-site block partition scheme sscheme Ad can give aneffective Hamiltonian for general N,HRG =1N − 1oj F− NsN − 2d2sN − 1d2+1sN − 1d2P˜ j,j+1G , s3dwhich exhibits a ground energy coinciding with the one byBethe ansatz method at large N, E0=−NsN−2d / sN2−N+1d ——→N→‘−1.The two-leg spin-1 /2 ladders aroused a lot of attentionwhen a finite spin gap was observed.11 A simple picture saysthat the ground state is a product state with the spins on eachrung forming a spin singlet. Then the lowest energy excita-tion is a S=1 magnon. Here we show that our scheme ofCORE produces exactly the same picture and refined resultscan be achieved following the CORE algorithm. We startfrom the HamiltonianH = ojfsS jA · S j+1A + S jB · S j+1B d + aS jA · S jBg , s4dwhere the indices A and B refer to the two rails of the lad-ders, a=Jrung/Jrail is the ratio between the rung and railscouplings, and we have set Jrail=1.Our first step is to divide the ladder into triads along therail direction fFig. 2sadg. The problem on the rail direction isjust the SUs2d chain that had been solved. Detailed calcula-tion shows that the effective interaction between the twoblocks along the rung can also recover the Heisenberg inter-action. Thus ladders with renormalized couplings can be ob-tained. The second step is to parse out the effective block-TABLE I. The coefficients C7 and K7 in Eq. s1d.Scheme ANRG Range-2 COREC−K−C−K−3 3414 1.0731 0.34114 16919 2.2485 0.17875 4516116 3.3762 0.1112Scheme BNRG Range-2 COREC+ K+ C+ K+2 131849 0.9956 0.49163 2.1693 0.2654 2.5982 0.30844 3.4111 0.1728 3.9432 0.2089 FIG. 1. sColor onlined The ground energy of SUsNd chain.Scheme B of CORE gives better results and the numerical errors areabout −0.0106, −0.0006, 0.0021 for N=2,3 ,4, respectively, com-pared to the results by Bethe ansatz ssee Ref. 9d.FIG. 2. sColor onlined sad Two-leg ladders. The basic blocks aretriads along the rail direction. The fixed point is a chain of decou-pled dimers. sbd The unsymmetric configuration of blocks involvedin range-3 CORE calculation.BRIEF REPORTS PHYSICAL REVIEW B 71, 212401 s2005d212401-2block interactions from all possible configurations ofconnected blocks. As defined by Morningstar and Weinstein,r connected blocks contain range-r8 interactions with r8=0,1 , . . . ,r fr8=0 corresponds to the constant term as in Eq.s5dg. To parse out all range interactions the exact diagonal-ization is employed on the connected blocks. We presentrange-2 and range-3 results here. It is notable that therange-3 blocks should include a configuration in Fig. 2sbd.This unsymmetric configuration may make the iteration pro-cedure more troublesome.The range-2 CORE result simply regains the original formof Hamiltonian except for a constant termHs2d =13oj f− Csad + dsS˜ jA· S˜ j+1A + S˜ jB · S˜ j+1B d + LsadS˜ jA · S˜ jBg ,s5dwhere d=0.491582, Csad and Lsad vary with a. The itera-tion on the range-2 effective Hamiltonian is always appli-cable because the retained four low levels are always onespin singlet and three spin triplets just like the SUs2d chaincase. After n steps of iteration on Eq. s5d we will get runningcoupling terms as hn=dnsS˜ jA ·S˜ j+1A +S˜ jB ·S˜ j+1B d+LnsadS˜ jA ·S˜ jB,where the coefficients are determined recursively, dn=dn,Lnsad=dn−1LhfLn−1sadg /dn−1j ,fl, L2sad=dLfLsad /dg,L1sad=Lsad, L0sad=a. So the rail coupling approacheszero dn→0 as n→‘, while the rung coupling goes to a fixedvalue Ln→‘sadÞ0 for a.0 swe observed that Ln→‘sad→0 only when a=0, which is in agreement with the conclu-sion drawn by DMRG10,20 and exact diagonalizationd.21 Sothe system flows to a fixed point exhibiting dimer coveringon each rung of the ladder. The spin gap is read out asDssad=Ln→‘sad. The ground energy E0 is obtained by cu-mulating the constant term. Table II gives an example ofiterations procedure for a=1.The range-3 CORE result at the first run of iteration con-tains the next-nearest-neighbor interactionsHs3d =13oj f− Csad + dsadsS˜ jA· S˜ j+1A + S˜ jB · S˜ j+1B d+ LsadS˜ jA · S˜ jB + VsadsS˜ jA · S˜ j+1B + S˜ jB · S˜ j+1A d+ gsS˜ jA · S˜ j+2A + S˜ jB · S˜ j+2B dg , s6dwhere Csad, dsad, and Lsad are different from the ones inEq. s5d, g=0.033975. g will vary with a in the successiveiterations, gn=gnsad , fl ,g1=g. After n-step iterations, wefound that the only nonvanishing coupling is still the inter-action along the rung Ln→‘sadÞ0, so the physical pictureobtained by the range-2 CORE does not change, i.e., theground energy and the spin gap are produced in the sameway.As we noted above, the unsymmetric configuration ofblocks in Fig. 2sbd brings some troubles to the range-3CORE iteration. Unlike the SUs2d chain, the desired lowlevels may not always stay at the lowest positions during theiterations. And sometimes it is hard to select out the eligibleset of levels from several possible candidates since each ofthem will lead to a recovered SUs2d symmetry. So differentiteration procedures with different results are inevitable.When these situations take place, we resort to the principle:retaining the iteration procedure that gives the lowestenergy,22 although in our observations the values of the re-sults only have small difference. The range-2 and range-3CORE results for the ground state energy and the spin gapare illustrated in Fig. 3. For a comparison, data by othermethods11,23–25 are presented together. The ground energyagrees well with those by other methods in the whole rangeof interchain coupling a. This means that CORE algrithmcan successively capture the low energy physics of the sys-tem. The gap has relatively larger deviation at intermediatevalues of a. Nevertheless the discrepancy can be remediedthrough higher range CORE calculation. The range-3 gap is alittle zigzag. This may be due to the unsymmetric configura-tion of range-3 blocks in Fig. 2sbd. It is noteworthy that RGgives a gap simply as DsRG=a, which captures the correctbehaviour of the gap at strong coupling limit a→‘.11We also applied CORE to the two-leg SUs3d ladders, H=S jfsPj,A;j+1,A+ Pj,B;j+1,Bd+aPj,A;j,Bg. The applicableschemes are presented in Figs. 4sad–4scd. Notice that allTABLE II. An example of the range-2 CORE iteration proce-dure at a=1.n E0 dn Lnsad0 0 1.0 1.01 −0.460796 0.491582 0.819192 −0.558041 0.241653 0.644995 −0.587214 0.028706 0.42014410 −0.587867 0.000824 0.38271515 −0.587869 0.000024 0.38160320 −0.587869 6.7907310−7 0.38157121 −0.587869 3.3382310−7 0.3815722 −0.587869 1.641310−7 0.38157FIG. 3. sColor onlined The ground energy and the gap for thespin-1 /2 two-leg ladder. The log-log plot shows that CORE and RGgive correct gap in strong coupling limit a→‘. Data by othermethods are adapted from Refs. 11 and 23–25.BRIEF REPORTS PHYSICAL REVIEW B 71, 212401 s2005d212401-3blocks are equivalent and a three-state truncation is made ineach scheme. Scheme sad should be valid when the runginteraction a is large enough. While for small a, the schemessad sa,1.0d and sbd sa,1.58d are appropriate and schemesbd is better than sad. All three schemes lead to the fixed pointwith zero gap. We see that scheme sad will be mapped to aSUs3d chain, which had been solved previously and gives azero gap. And after the first mapping we applied four-siteblock partition scheme on the chain in the successive itera-tion steps to produce the ground energy in Fig. 4. Whilescheme sbd and scd will return to a two-leg SUs3d ladder,hn=dnsPj,A;j+1,A+ Pj,B;j+1,Bd+LnsadPj,A;j,B, but we observedthat the running couplings of the rail direction dn and therung direction Ln will go to infinitesimals of the sameorder26 as we push the iteration steps to infinity, n→‘, so agapless phase is also obtained, which agrees with the resultof scheme sad. The SUs3d model on a four-leg ladder can beanalyzed in similar schemes and a gapless result is also ex-pected. The result is reminiscent of the SUs2d model on achain and on a three-leg ladder, which are also gapless. Butunfortunately the above schemes or their analogs are not ap-plicable for the two-leg SUs4d ladders, which exhibitsplaquette singlet-multiplet excitation.18,23,27 One may have toresort to other kind of schemes.In conclusion, we have studied the SUsNd chain and lad-ders by the CORE schemes. We have shown that the effec-tive Hamiltonian in the appropriate CORE schemes can re-gain its original form such that it approaches a fixed point byiteration of the CORE schemes. The ground state energy andthe lowest excitations can be deduced from the fixed point.The results show that the SUsNd chain and the two-leg SUs3dladders are gapless, while the two-leg spin-1 /2 ladder exhib-ites gapped phase originated from the rung dimmerization.This work was supported by the Research Grant Councilof Hong Kong under the Project No. HKU7038/04P.1 C. J. Morningstar and M. Weinstein, Phys. Rev. Lett. 73, 1873s1994d; Phys. Rev. D 54, 4131 s1996d.2 M. Weinstein, Phys. Rev. D 61, 034505 s2000d; Phys. Rev. B 63,174421 s2001d.3 E. Altman and A. Auerbach, Phys. Rev. B 65, 104508 s2002d.4 S. Capponi and D. Poilblanc, Phys. Rev. B 66, 180503sRd s2002d.5 E. Berg, E. Altman, and A. Auerbach, Phys. Rev. Lett. 90,147204 s2003d.6 D. Poilblanc, D. J. Scalapino, and S. Capponi, Phys. Rev. Lett.91, 137203 s2003d.7 S. Capponi, A. Lauchli, and M. Mambrini, Phys. Rev. B 70,104424 s2004d.8 R. Budnik and A. Auerbach, Phys. Rev. Lett. 93, 187205 s2004d.9 B. Sutherland, Phys. Rev. B 12, 3795 s1975d.10 E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, R5744s1992d.11 E. Dagotto and T. M. Rice, Science 271, 618 s1996d.12 J. Piekarewicz and J. R. Shepard, Phys. Rev. B 57, 10260 s1998d;60, 9456 s1999d.13 E. Schrödinger, Proc. R. Ir. Acad., Sect. A 47, 39 s1941d.14 P. Li and S. Q. Shen, New J. Phys. 6, 160 s2004d.15 D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 s1988d.16 N. Read and S. Sachdev, Phys. Rev. B 42, 4568 s1990d.17 Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81,3527 s1998d.18 S. Q. Shen, Phys. Rev. B 66, 214516 s2002d; 64, 132411 s2001d.19 K. Penc, M. Mambrini, P. Fazekas, and F. Mila, Phys. Rev. B 68,012408 s2003d.20 O. Legeza, G. Fath, and J. Solyom, Phys. Rev. B 55, 291 s1997d.21 H. Watanabe, Phys. Rev. B 50, 13442 s1994d.22 This selection rule should be an important key issue in manyiterative methods such as DMRG. See, e.g., S. R. White, Phys.Rev. Lett. 69, 2863 s1992d; Phys. Rev. B 48, 10345 s1993d; O.Legeza and J. Solyom, Phys. Rev. B 70, 205118 s2004d.23 T. Barnes, E. Dagotto, J. Riera, and E. S. Swanson, Phys. Rev. B47, 3196 s1993d.24 S. R. White, R. M. Noack, and D. J. Scalapino, Phys. Rev. Lett.73, 886 s1994d.25 N. Flocke, Phys. Rev. B 56, 13673 s1997d.26 In fact the ratio Ln /dn will reach at a fixed value 0.3597 forscheme sbd and 1.1176 for scheme scd as n→‘, although Ln→0 and dn→0.27 M. van den Bossche, P. Azaria, P. Lecheminant, and F. Mila,Phys. Rev. Lett. 86, 4124 s2001d.FIG. 4. sColor onlined The ground energy of the two-leg SUs3dladders. The insets sad, sbd, and scd show the schemes used in thecalculation.BRIEF REPORTS PHYSICAL REVIEW B 71, 212401 s2005d212401-4

Physical Review B

Contractor renormalization group theory of SU(N) chains and ladders