Melting of Ferromagnetic Order on a Trellis Ladder
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Melting of Ferromagnetic Order on a Trellis Ladder
Debasmita Maiti, Manoranjan Kumar ∗ S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata - 700106, India
Abstract
The ground state properties of a frustrated spin-1/2 system is studied on a trellis ladder which is composed of two zigzagladders interacting through rung interactions. The presence of rung interaction between the zigzag ladders induces anon-magnetic ground state, although, each of zigzag ladders has ferromagnetic order in weak anti-ferromagnetic leginteraction limit. The rung interaction also generates rung dimers and opens spin gap which increases rapidly with runginteraction strength. The correlation between spins decreases exponentially with the distance between them.
Keywords:
Frustrated magnetic systems, Dimer phase, Density Matrix Renormalization Group Method
1. Introduction
The interaction-driven quantum phase transition in low-dimensional frustrated systems like spin chain [1, 2], lad-der or any quasi-one dimensional system has been a fron-tier area of current research [3, 4]. Many realizations ofthese systems like (N H )CuCl [5], LiCuSbO [6], LiCuVO [7],Li CuZrO [8] are frustrated due to competing spin ex-change interactions. J − J Heisenberg spin-1/2 modelwith nearest neighbor (NN) ferromagnetic (FM) J andnext nearest neighbor (NNN) anti-ferromagnetic (AFM) J interactions is used extensively to understand the groundstate magnetic properties of many of these materials [9–12]. This model can explain the gapless spin fluid, gappeddimer and incommensurate gapped spiral phases [9, 12,13]. On the other hand, the ground state of quasi-1Dladder like structure with AFM leg and rung interactionsshows the presence of gapped short-range order in the sys-tem. This kind of phase is observed in SrCu O [14],(VO) P O [15, 16], CaV O , MgV O [17, 18] etc.The J − J chain can also be considered as a two-chain lattice with diagonal or zigzag like couplings. Henceit can be alternatively called as zigzag ladder. The iso-lated ladders like zigzag [9, 11] and normal ladder [3, 9]have been extensively studied. However, the theoreticalstudy of the effect of inter-ladder coupling on the laddersis still an open field. One of the extended networks of thecoupled ladders can form a trellis lattice like structure.The trellis lattice is composed of a number of normal lad-ders coupled through zigzag bonds; alternatively, it can beconsidered as coupled zigzag ladders through rung interac-tions as shown in Fig. 1. In this lattice, J and J are leg ∗ Corresponding author
Email address: [email protected] (ManoranjanKumar) J J J Figure 1: Two coupled zigzag ladders form trellis ladder. The brokenlines show the extension of trellis ladder to trellis lattice structure.The arrows show respective spin arrangements. The reference siteis labeled by ’0’ and the distances of all other sites are shown withrespect to it. The distance of the sites in the same zigzag ladder asthe reference site are written in bold numbers, and the sites on theother ladder are written in normal numbers. and rung couplings of a normal ladder, respectively and J is inter-ladder coupling through zigzag bonds.There are several theoretical studies for two coupledzigzag ladders, e.g. a two-leg honeycomb ladder is consid-ered in Ref. [19], where both J and J are AFM, but J can be either FM or AFM. This system shows two typesof Haldane phases for the FM J and, columnar dimerand rung singlet phases for the AFM J . Normand et al. considered the similar coupled ladders with all three AFM J , J and J interactions. They find dimerized chains forlarge J and small J limit, spiral long range order for bothlarge J and J limit, N´eel long range order in the small J < . J [20]. Ronald et al. have shownthe effect of inter-chain coupling on spiral ground state of J − J model [21]. The effect of inter-ladder couplingon spin gap and magnon dispersion has been discussed in[22] exploiting a theoretical model which has also been Preprint submitted to Journal of Magnetism and Magnetic Materials April 30, 2019 ompared with the experimental data of SrCu O andCaV O . The study of interladder coupling effect is impor-tant to explain the physical properties of some other ma-terials like LaCuO . [23], Sr Cu O [24], MgV O [17],NaV O [25, 26] etc. The weak interladder exchange inter-actions in these materials form an effective 2D trellis likestructure. Recently Yamaguchi et al. showed magneticfield induced spin nematic phase in the verdazyl radical β -2,3,5- Cl -V [27]. The J - J - J spin system promises azoo of exotic phases.In the current paper we consider a trellis ladder whichis composed of two zigzag ladders with FM J and AFM J and coupled by AFM J as shown in Fig. 1. The groundstate of an isolated zigzag ladder in small J limit is FM.Our main focus of this paper is to understand the effect ofrung interaction J on the FM ground state exhibited bya single zigzag ladder in the limit of J ≤ | J | , and studythe transition of ground state from the FM to the singletdimer state.This paper is divided into four sections. In section 2,the model Hamiltonian and the numerical methods areexplained. The numerical results are given in section 3.All the results are discussed and summarized in section 4.
2. Model Hamiltonian and Numerical Method
We consider four-leg ladder system where two zigzagladders are coupled through an AFM Heisenberg interac-tion as shown in Fig. 1. The diagonal interactions J inzigzag ladders are FM whereas J bonds along the legs areAFM. Two zigzag ladders interact with each other throughAFM rungs J . Thus we can write an isotropic Heisenbergspin-1/2 model Hamiltonian for the system as H = X a =1 , N/ X i =1 J ~S a,i · ~S a,i +1 + J ~S a,i · ~S a,i +2 + J ~S ,i · ~S ,i , (1)where a = 1 , ~S a,i is thespin operator at reference site i on zigzag ladder a . We setthe FM J = − J and J as variablequantities. We use periodic boundary condition along therungs, whereas it is open along the legs of the system.We use the exact diagonalization (ED) method for smallsystems and density matrix renormalization group (DMRG)method to handle the large degrees of freedom for largesystems. The DMRG is a state of art numerical tech-nique for 1D or quasi-1D system, and it is based on thesystematic truncation of irrelevant degrees of freedom atevery step of growth of the chain [28–30]. We use re-cently developed DMRG method where four new sites areadded at every DMRG steps [31]. For the renormaliza-tion of operators, we keep m eigenvectors correspondingto largest eigenvalues of the density matrix of the systemin the ground state of the Hamiltonian in Eq. (1). We have r C (r) J ξ J = 0.15J = 0.10.20.40.6 ξ = c J ν J = 0.15, ν = 1.712J = 0.1, ν = 1.576 Figure 2: The longitudinal spin-spin correlation C(r) shown for N =122, J = 0 .
15 and different J as indicated adjacent the respectivecurves. The solid lines represent respective exponential fits. Theinset shows the correlation length ξ vs. J plots for J = 0 . .
15 in log-log scale. kept m up to 300 to constraint the truncation error lessthan 10 − . We have used system sizes up to N = 400 tominimize the finite size effect.
3. Results
It is well known that the zigzag spin-1/2 ladder has aFM ordered ground state for FM J and AFM J ≤ | J | [12]. The AFM coupling, i.e., the rung interaction J be-tween two isolated zigzag spin-1/2 ladders retains the FMarrangement in each zigzag ladder; however, the spins ontwo zigzag ladders are in AFM arrangement to each other,as depicted in the schematic Fig. 1. In other words, theeffective ground state of the whole system is in S z = 0manifold, though the spins on each zigzag ladder are ar-ranged ferromagnetically. The ground state forms singletdimer along the rung in the large J limit, and groundstate wave function can be represented as the product ofsinglet dimers. However, to study the effect of J on theground state, we analyze longitudinal correlation function,longitudinal bond order and spin gap.In this paper we focus J / | J | < limit where eachzigzag ladder have the FM order in the ground state for J = 0. For a finite J , we calculate the longitudinal spin-spin correlations C ( r ) = < S zi S zi + r > , where S zi and S zi + r are the z-component of the spin operators at the referencesite i and the site at a distance r from i , respectively. In1, We have shown the distance r along the same zigzagladder with bold numerics, and otherwise in normal nu-merics, where the reference site is at the 0th position. Wenote that in J / | J | < limit all the spins are aligned inparallel on each zigzag ladder and have short range longitu-dinal correlation for finite J . As we increase the strengthof J , C ( r ) shows an exponential behavior as shown in themain Fig. 2 for J = 0 .
15. We notice that each zigzag2 J C ))0.084 exp (-0.418 J )0.087 exp (-1.499 J ) J = 0.15 |C R |C D C L D L CC R Figure 3: Correlation function C of the reference site with itsfirst neighbors along the rung ( C R , circle), diagonal direction ( C D ,square), and the leg ( C L , diamond), are shown as function of J .We have drawn | C R | to take care of the AFM J interaction. Thelines represent respective exponential fits. ladder shows FM arrangement as C ( r ) >
0, but it decaysexponentially with r , i.e. C ( r ) ∝ exp ( − r/ξ ). The correla-tion length ξ follows an algebraic decay with J for a given J , as shown in the inset of Fig. 2. ξ for J = 0 . J = 0 . ξ becomes less than 1 for J > . J = 0 .
15, and inthis parameter regime the system is completely dimerizedalong the rung.To study the effect of J on the bonds along the rung,diagonal and leg directions, we calculate correlations C R , C D and C L , respectively as shown in Fig. 3. The cal-culations of these three correlations are confined to thefirst neighbor along the respective directions. | C R | in-creases exponentially with J and follows | C R | = 0 . − . exp ( − . J )). C D and C L are represented by squareand diamond symbols, respectively, and both these bondorders exponentially decrease with J . The exponent ofthe C D and C L are 0.42 and 1.50 respectively. C L decaysfaster than C D , because J allows the magnon to decon-fine along the leg of the zigzag ladder; therefore weaker J reduces C L .The correlation function C ( r ) of the system shows theshort range spin order. Therefore, we explore the excita-tion energy or spin gap in the system. The rung inter-action dominates other interactions; thus we expect theopening of the spin gap ∆. We calculate ∆ for various J = 0 . , . , . ∞ in the thermodynamic limit. ∆ ∞ increasesalgebraically with J , as shown in the inset of Fig. 4 for J = 0 . γ for J = 0 . γ decreases with increasing J . This may be due to the de-localization of magnon along the leg of zigzag ladder. The ∆ J ∆ ∞ J = 0.1 J = 0.5 0.4 0.3 0.1 ∆ = ∆ ∞ + b/N J =0.10 , γ = 3.328 J = 0.15, γ = 3.129∆ ∞ = c J γ Figure 4: The extrapolation of spin-gap ∆ with respect to system size N for different J = 0 . , . , . J = 0 .
1. The solid linesare the fitted curves. In the inset, the spin gap in thermodynamiclimit, ∆ ∞ vs. J plots for J = 0 . large J / | J | enhances the interaction of spin along eachleg of zigzag ladder and the each leg of the system canhave quasi-long range correlation like a normal Heisenbergchain, in J / | J | >>
4. Discussion and Conclusions
In this paper the effect of J on FM order in each zigzagladder of a trellis ladder with FM J and AFM J < | J | is studied. We show that even a small J induces spingap in the system. The correlation between spins on azigzag ladder decays exponentially with distance. It maybe because of the confinement of magnon along the rungs.As shown in Fig. 4 the gap increases rapidly with J for J = 0 .
1. The correlation length of the system reduced toless than a unit lattice for J > . J ≈ .
15. Thisimplies the setting of the dimerized state.This model can also be mapped to a two interacting J − J Heisenberg spin-1/2 chains. This system is studiedrecently by Ronald et al. [21]. They have mostly studiedthe effect of inter-chain coupling on the spiral nature of theground state in large J and low J limit. There are manycompounds like CaV O [33], SrCu O [16, 22] etc, whichhave both strong J and J . However, our prediction areconfined to the J < | J | and large J limit.In summary, this model system goes from a FM orderedground state along a zigzag ladder in the J = 0 limit to arung dimer state in large J limit. The correlation length ξ of the system decreases algebraically and spin gap ∆ ∞ increases algebraically with exponent higher than γ > J for a given J . Acknowledgements
MK thanks DST for a Ramanu-jan Fellowship SR/S2/RJN-69/2012 and DST for funding3omputation facility through SNB/MK/14-15/137.
References [1] A. V. Chubukov, Chiral, nematic, and dimer states in quantum spin chains,Phys. Rev. B 44 (1991) 4693–4696. doi:10.1103/PhysRevB.44.4693 .URL https://link.aps.org/doi/10.1103/PhysRevB.44.4693 [2] S. Furukawa, M. Sato, S. Onoda, A. Furusaki,Ground-state phase diagram of a spin- frustrated ferromagnetic xxz chain: Haldane dimer phase and gapped/gapless chiral phases,Phys. Rev. B 86 (2012) 094417. doi:10.1103/PhysRevB.86.094417 .URL https://link.aps.org/doi/10.1103/PhysRevB.86.094417 [3] S. R. White, R. M. Noack, D. J. Scalapino, Resonating valencebond theory of coupled heisenberg chains, Phys. Rev. Lett. 73(1994) 886–889. doi:10.1103/PhysRevLett.73.886 .[4] T. Verkholyak, J. Streka, Quantum phase transitions in the ex-actly solved spin-1/2 heisenbergising ladder, Journal of PhysicsA: Mathematical and Theoretical 45 (30) (2012) 305001. doi:10.1088/1751-8121/45/30/305001 .[5] N. Maeshima, M. Hagiwara, Y. Narumi,K. Kindo, T. C. Kobayashi, K. Okunishi,Magnetic properties of a s = 1/2 zigzag spin chain compound (n 2 h 5 )cucl 3,Journal of Physics: Condensed Matter 15 (21) (2003) 3607.URL http://stacks.iop.org/0953-8984/15/i=21/a=309 [6] S. E. Dutton, M. Kumar, M. Mourigal, Z. G. Soos, J.-J.Wen, C. L. Broholm, N. H. Andersen, Q. Huang, M. Zbiri,R. Toft-Petersen, R. J. Cava, Quantum spin liquid in frustratedone-dimensional licusbo , Phys. Rev. Lett. 108 (2012) 187206. doi:10.1103/PhysRevLett.108.187206 .[7] M. Mourigal, M. Enderle, B. F˚ak, R. K. Kremer, J. M. Law,A. Schneidewind, A. Hiess, A. Prokofiev, Evidence of a bond-nematic phase in licuvo , Phys. Rev. Lett. 109 (2012) 027203. doi:10.1103/PhysRevLett.109.027203 .[8] S.-L. Drechsler, O. Volkova, A. N. Vasiliev, N. Tristan,J. Richter, M. Schmitt, H. Rosner, J. M´alek, R. Klingeler, A. A.Zvyagin, B. B¨uchner, Frustrated cuprate route from antiferro-magnetic to ferromagnetic spin- heisenberg chains: li zrcuo as a missing link near the quantum critical point, Phys. Rev.Lett. 98 (2007) 077202. doi:10.1103/PhysRevLett.98.077202 .[9] S. R. White, I. Affleck, Dimerization and incommensuratespiral spin correlations in the zigzag spin chain: Analogiesto the kondo lattice, Phys. Rev. B 54 (1996) 9862–9869. doi:10.1103/PhysRevB.54.9862 .[10] K. Okamoto, K. Nomura, Fluid-dimer critical point in s = 12 antiferromagnetic heisenberg chain with next nearest neighbor interactions,Physics Letters A 169 (6) (1992) 433 – 437. doi:https://doi.org/10.1016/0375-9601(92)90823-5 .URL [11] R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, S. Ramasesha,Density-matrix renormalization-group studies of the spin-1/2 heisenberg system with dimerization and frustration,Phys. Rev. B 52 (1995) 6581–6587. doi:10.1103/PhysRevB.52.6581 .URL https://link.aps.org/doi/10.1103/PhysRevB.52.6581 [12] M. Kumar, A. Parvej, Z. G. Soos, Level crossing, spin structurefactor and quantum phases of the frustrated spin-1/2 chain withfirst and second neighbor exchange, J. Phys.: Condens. Matter27 (31) (2015) 316001. doi:10.1088/0953-8984/27/31/316001 .[13] M. Kumar, S. Ramasesha, Z. G. Soos, Bond-order wave phase,spin solitons, and thermodynamics of a frustrated linear spin- heisenberg antiferromagnet, Phys. Rev. B 81 (2010) 054413. doi:10.1103/PhysRevB.81.054413 .[14] A. W. Sandvik, E. Dagotto, D. J. Scalapino, Spin dynamicsof srcu o and the heisenberg ladder, Phys. Rev. B 53 (1996)R2934–R2937. doi:10.1103/PhysRevB.53.R2934 .[15] D. C. Johnston, J. W. Johnson, D. P. Goshorn, A. J. Jacobson,Magnetic susceptibility of (vo) p o : A one-dimensional spin-1/2 heisenberg antiferromagnet with a ladder spin configurationand a singlet ground state, Phys. Rev. B 35 (1987) 219–222. doi:10.1103/PhysRevB.35.219 . [16] E. Dagotto, T. M. Rice, Surprises on the way fromone- to two-dimensional quantum magnets: The lad-der materials, Science 271 (5249) (1996) 618–623. doi:10.1126/science.271.5249.618 .[17] M. A. Korotin, I. S. Elfimov, V. I.Anisimov, M. Troyer, D. I. Khomskii,Exchange interactions and magnetic properties of the layered vanadates cav o , mgv o , cav o , and cav o ,Phys. Rev. Lett. 83 (1999) 1387–1390. doi:10.1103/PhysRevLett.83.1387 .URL https://link.aps.org/doi/10.1103/PhysRevLett.83.1387 [18] M. A. Korotin, V. I. Anisimov, T. Saha-Dasgupta, I. Dasgupta,Electronic structure and exchange interactions of the ladder vanadates cav 2 o 5 and mgv 2 o 5,Journal of Physics: Condensed Matter 12 (2) (2000) 113.URL http://stacks.iop.org/0953-8984/12/i=2/a=302 [19] Q. Luo, S. Hu, J. Zhao, A. Metavitsiadis, S. Eggert, X. Wang,Ground-state phase diagram of the frustrated spin- two-leg honeycomb ladder,Phys. Rev. B 97 (2018) 214433. doi:10.1103/PhysRevB.97.214433 .URL https://link.aps.org/doi/10.1103/PhysRevB.97.214433 [20] B. Normand, K. Penc, M. Albrecht, F. Mila,Phase diagram of the s = frustrated coupled ladder system,Phys. Rev. B 56 (1997) R5736–R5739. doi:10.1103/PhysRevB.56.R5736 .URL https://link.aps.org/doi/10.1103/PhysRevB.56.R5736 [21] R. Zinke, S.-L. Drechsler, J. Richter,Influence of interchain coupling on spiral ground-state correlations in frustrated spin- J − J heisenberg chains,Phys. Rev. B 79 (2009) 094425. doi:10.1103/PhysRevB.79.094425 .URL https://link.aps.org/doi/10.1103/PhysRevB.79.094425 [22] S. Miyahara, M. Troyer, D. Johnston, K. Ueda, Quan-tum monte carlo simulation of the trellis lattice heisenbergmodel for srcu o and cav o , Journal of the Physical So-ciety of Japan 67 (1998) 3918. arXiv:cond-mat/9807127 , doi:10.1143/JPSJ.67.3918 .[23] M. Troyer, M. E. Zhitomirsky, K. Ueda,Nearly critical ground state of lacuo . , Phys. Rev. B 55(1997) R6117–R6120. doi:10.1103/PhysRevB.55.R6117 .URL https://link.aps.org/doi/10.1103/PhysRevB.55.R6117 [24] M. Uehara, T. Nagata, J. Akimitsu,H. Takahashi, N. Mri, K. Kinoshita,Superconductivity in the ladder material sr . ca . cu o . ,Journal of the Physical Society of Japan 65 (9) (1996) 2764–2767. doi:10.1143/JPSJ.65.2764 .URL https://doi.org/10.1143/JPSJ.65.2764 [25] H. Smolinski, C. Gros, W. Weber,U. Peuchert, G. Roth, M. Weiden, C. Geibel,Nav o as a quarter-filled ladder compound, Phys. Rev.Lett. 80 (1998) 5164–5167. doi:10.1103/PhysRevLett.80.5164 .URL https://link.aps.org/doi/10.1103/PhysRevLett.80.5164 [26] Y. Tanokura, T. Morita, S. Ishima, S. Ikeda,H. Kuroe, T. Sekine, M. Isobe, Y. Ueda,Spin-gap mode in the charge-ordered phase of nav o studied by raman scattering under high pressures,Phys. Rev. B 81 (2010) 054407. doi:10.1103/PhysRevB.81.054407 .URL https://link.aps.org/doi/10.1103/PhysRevB.81.054407 [27] H. Yamaguchi, D. Yoshizawa, T. Kida, M. Hagi-wara, A. Matsuo, Y. Kono, T. Sakakibara,Y. Tamekuni, H. Miyagai, Y. Hosokoshi,Magnetic-field-induced quantum phase in s = 1/2 frustrated trellis lattice,Journal of the Physical Society of Japan 87 (4) (2018)043701. arXiv:https://doi.org/10.7566/JPSJ.87.043701 , doi:10.7566/JPSJ.87.043701 .URL https://doi.org/10.7566/JPSJ.87.043701 [28] S. R. White, Density matrix formulation for quantum renor-malization groups, Phys. Rev. Lett. 69 (1992) 2863–2866. doi:10.1103/PhysRevLett.69.2863 .[29] K. A. Hallberg, New trends in density matrix renormalization,Advances in Physics 55 (5-6) (2006) 477–526.[30] U. Schollw¨ock, The density-matrix renormalization group, Rev.Mod. Phys. 77 (2005) 259–315.[31] M. Kumar, Z. G. Soos, D. Sen, S. Ramasesha, Modified density atrix renormalization group algorithm for the zigzag spin- chain with frustrated antiferromagnetic exchange: Comparisonwith field theory at large J /J , Phys. Rev. B 81 (2010) 104406. doi:10.1103/PhysRevB.81.104406 .[32] Z. G. Soos, A. Parvej, M. Kumar, Numerical study of in-commensurate and decoupled phases of spin-1/2 chains withisotropic exchange j , j between first and second neigh-bors, J. Phys.: Condens. Matter 28 (17) (2016) 175603. doi:10.1088/0953-8984/28/17/175603 .[33] M. Onoda, N. Nishiguchi, Letter to the editor: Crystal structureand spin gap state of cav o , Journal of Solid State Chemistry127 (2) (1996) 359 – 362. doi:10.1006/jssc.1996.0395 ..