Magnetic phase diagram of the Ising model with the long-range RKKY interaction
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Magnetic phase diagram of the Ising model with thelong-range RKKY interaction
Lubom´ıra Regeciov´a and Pavol Farkaˇsovsk´yInstitute of Experimental Physics, Slovak Academy of SciencesWatsonova 47, 040 01 Koˇsice, Slovakia
Abstract
The standard Metropolis algorithm and the parallel tempering method areused to examine magnetization processes in the Ising model with the long-rangeRKKY interaction on the Shastry-Sutherland lattice. It is shown that the Isingmodel with RKKY interaction exhibits, depending on the value of the Fermiwave vector k F , the reach spectrum of magnetic solutions, which is manifestedin the appearance of new magnetization plateaus on the magnetization curve. Inparticular, we have found the following set of individual magnetization plateauswith fractional magnetization m/m s =1/18, 1/9, 1/8, 1/5, 1/4, 1/3, 3/8, 5/12,1/2, 3/5, 2/3, which for different values of k F form various sequences of plateaus,changing from very complex, appearing near the point k F = 2 π/ .
2, to verysimple appearing away this point. Since the change of k F can be induced bydoping (the substitution of rare-earth ion by other magnetic ion that introducesthe additional electrons to the conduction band) the model is able to predictthe complete sequences of magnetization plateaus, which could appear in thetetraboride solid solutions. Introduction
In the past decade, a considerable amount of effort has been devoted to understand-ing the underlying physics that leads to anomalous magnetic properties of metallicShastry-Sutherland magnets [1, 2, 3, 4]. However, in spite of an impressive researchactivity, the properties of these systems are far from being understood. In particular,this concerns the entire group of rare-earth metal tetraborides RB ( R = La − Lu )that exhibits the strong geometrical frustration. These compounds have the tetrag-onal crystal symmetry P /mbm with magnetic ions R located on an Archimedeanlattice (see Fig.1a) that is topologically equivalent to the so-called Shastry-Sutherlandlattice [5] (see Fig. 1b). It is supposed that the anomalous properties of these systems (a)J J J J J (b)J J J J J Figure 1:
The real structure realized in the (001) plane of rare-earth tetraborides (a), whichis topologically identical to the Shastry-Sutherland lattice (b). J , J , J , J and J denotethe first, second, third, fourth and fifth nearest neighbors on the real Archimedean lattice. are caused by the geometrical frustration that leads to an extensive degeneracy inthe ground state. The most famous manifestation of the geometrical frustration inthe above-mentioned tetraborides is the observation of the fascinating sequence ofmagnetization plateaus with the fractional magnetization. For example, for ErB themagnetization plateau has been found at m/m s = 1 / T bB at m/m s =2/9,1/3, 4/9, 1/2 and 7/9 [4], for HoB at m/m s =1/3, 4/9 and 3/5 [2] and for T mB at2 /m s =1/11, 1/9, 1/7 and 1/2 [1].Despite the metallic nature of rare-earth tetraborides, the first theoretical worksdevoted to magnetization processes in these materials ignored completely the exis-tence of the conduction electrons, and exclusively, only the spin models have beenconsidered as the generic models for a description of magnetization plateaus withfractional magnetization. Because of strong crystal field effects, which are presentin rare-earth tetraborides, the physically reasonable spin model seems to be spin-1/2Shastry-Sutherland model under strong Ising anisotropy [1]. Thus, the study of theIsing limit was the first natural step towards the complete understanding of magneti-zation processes in rare-earth tetraborides. The subsequent analytical [6, 7, 8, 9] andnumerical [10, 11, 12, 13] studies showed that the basic version of the Ising model onthe Shastry-Sutherland lattice, with nearest and next-nearest neighbor interactionsand its extensions, up to the 5th nearest neighbors, are able to describe some of theindividual plateaus observed in rare-earth materials, as well as the partial sequencesconsisting of two or even three right magnetization plateaus, but not the completesequences. These theoretical works point to the fact that for the correct description ofmagnetization processes in rare-earth tetraborides one has to consider the long-rangeinteractions. On the other hand, some theoreticians speculate that for a descrip-tion of complete sequences of magnetization plateaus in rare-earth tetraborides it isnecessary to take into account both, the spin and electron subsystems, as well asthe interaction between them. Indeed, our previous numerical studies showed [14]that the model based on the coexistence of both subsystems has a great potentialto describe, at least qualitatively, the complete sequence of magnetization plateausobserved experimentally in some rare-earth tetraborides, e.g., TmB . However, it isquestionable if it is the intrinsic property of a model, or only a consequence of thelarge number of variables (fitting parameters) that enter to the model as interactionparameters describing possible spin, electron and electron spin interactions.3n alternative model, which takes into account both, the long-range interactionsas well as the presence of conduction electrons, has been introduced recently by Fenget. al. [15]. Strictly said, it is a generalized Ising model, in which two spins on latticesites i and j interact via the RKKY interaction J ij mediated by conduction electrons.It is supposed that the RKKY coupling between the localized f and conduction s elec-trons is predominant in rare-earth compounds [1] and may play an important role inthe mechanism of formation of magnetization plateaus with fractional magnetizations.The model was studied numerically and various magnetization plateaus, dependingon the value of the Fermi wave vector k F of conduction electrons were confirmed.However, the importance of these results for a description of magnetization processes(magnetization plateaus) in rare-earth tetraborides is questionable, since they havebeen obtained under the assumption that these systems are electronically three di-mensional and the Fermi surface is isotropic, which contradicts the latest experimen-tal measurements of the angle-dependent magnetotransport in TmB revealing theanisotropic Fermi surface topology [16]. For this reason we have decided to examineeffects of the Fermi surface anisotropy on the magnetic phase diagram (magnetizationplateaus) of the two-dimensional Ising model with the long range RKKY interaction.For simplicity we consider here only the case of strong Fermi-surface anisotropy, rep-resented by the purely electronically two-dimensional system, for which the matrixelements of the RKKY interaction have been derived by B´eal-Monod [17] and havethe form: J ij = k F π [ B (1)0 ( k F r ij ) B (2)0 ( k F r ij ) + B (1)1 ( k F r ij ) B (2)1 ( k F r ij )] , (1)where r ij is the distance between the sites i and j on the real Archimedean lattice, k F is the Fermi wave vector, B (1) n ( x ), with n = 0 , B (2) n ( x ) are the Bessel functions of the second kind. In the current paperwe use this formula for the matrix elements of the RKKY interaction and constructthe comprehensive magnetic phase diagram of the two-dimensional Ising model with4KKY interaction on the Shastry-Sutherland lattice, in which both the magnetic andelectronic subsystems are considered strictly as two dimensional. To construct thisphase diagram we use the same method (the combination of the standard Metropolisalgorithm and the parallel tempering method) and the same conditions (the periodicboundary conditions and the cut-off radius of the RKKY interaction r ij = 6) as wereused by Feng et al [15]. The Hamiltonian of the Ising model with the long-range RKKY interaction on theShastry-Sutherland lattice can be written as H = X i,j J ij S zi S zj − h X i S zi , (2)where the variable S zi denotes the Ising spin with unit length on site i , h is themagnetic field and the matrix elements J ij are given by the formula (2).To verify the convergence of the Monte-Carlo results we have started our numer-ical studies of the model (3) on the finite cluster of L = 6 × × Monte Carlo steps(the initial 1 × Monte Carlo steps are discarded for equilibrium consideration).A comparison of exact and Monte Carlo results shows that the selected tempera-ture T=0.02 is sufficient to approximate reliably the ground-state properties of themodel and that 10 Monte Carlo steps (per site) is sufficient to reach well convergedresults. Moreover, these results reveal some interesting physical facts, concerningthe influence of the long-range RKKY interaction on the formation of magnetizationplateaus, which point to the importance of this interaction for a description of realrare-earth tetraborides. Indeed, our results show that the stability regions of different5 m / m s × F =2 π /1.1 mcex 0 2 4 600.20.40.60.81 h m / m s × F =2 π /1.2 mcex0 2 4 600.20.40.60.81 h m / m s × F =2 π /1.38 mcex 0 2 4 600.20.40.60.81 h m / m s × F =2 π /1.45 mcex2/3 2/3 Figure 2:
The magnetization curve of the Ising model with RKKY interaction calculatedfor four different values of the Fermi wave vector k F on the L = 6 × magnetization plateaus with fractional magnetization are very sensitive to the valueof the Fermi wave vector k F , which is directly connected with the concentration ofconduction electrons n e . However, the change of n e (and consequently k F ) can beinduced by doping (the substitution of rare-earth ion by other magnetic ion that in-troduces the additional electrons/holes to the conduction band) and thus, this simplemodel could yield the physics for a description of magnetization processes in dopedrare-earth tetraborides. For this reason, we examine in this paper the comprehensivemagnetic phase diagram of the model in the k F − h plane and predict the completesequences of magnetization plateaus at different k F . To reveal the basic structure ofthe magnetic phase diagram we have performed numerical calculations on the finite L = 24 ×
24 cluster (the cluster of the same size has been used also by Feng et al [15])6nd subsequently the exhaustive finite-size scaling analysis of the model is done. Theresults of our Monte-Carlo simulations obtained on the L = 24 ×
24 cluster are sum-marized in Fig. 3. One can see that the comprehensive magnetic phase diagram of the π /k F h L=24 ×
24 1/3 1/31/4 1/21/21/21/20 0 02/3 7/121/63/8 1 11.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.5012345678 00.10.20.30.40.50.60.70.80.911/9 1/18 1/81/21/4 1/3 7/122/35/83/417/24 1/4 1/12 5/243/85/1223/48
Figure 3:
The magnetic phase diagram of the model in the k F − h plane calculated for the L = 24 ×
24 cluster. two-dimensional Ising model with the two-dimensional RKKY interaction has muchcomplex structure than the one obtained by Feng et al [15] for the three-dimensionalone and larger is also the number of magnetization plateaus. In particular, we havefound the following set of individual magnetization plateaus with fractional magneti-zation (only the plateaus with the largest stability regions are listed): m/m s =1/18,1/12, 1/9, 1/8, 1/6, 1/4, 1/3, 3/8, 5/12, 1/2, 7/12, 2/3, which for different values of k F form various sequences of plateaus, changing from very complex, appearing nearthe points k F = 2 π/ . k F = 2 π/ .
38, to relatively simple appearing away thesepoints. Before discussing these sequences, let us present some interesting observationsconcerning the individual plateaus with the largest stability regions. From all phases7orresponding to magnetization plateaus with fractional magnetization the largeststability regions exhibit the 1/3 and 1/2 plateau phases. For the 1/3 plateau phasethis result is expected since it appears already in the simplest version of the Isingmodel on the Shastry-Sutherland lattice (when only the first ( J ) and second ( J )nearest neighbour interactions are considered), as well as in practically all extensionsof the model, which take into account the next nearest neighbor interactions. Whatis unexpected, however, is the fact that the 1/3 plateau phase is absent in the centralpart of the phase diagram (near the point k F = 2 π/ . k F . In the central part of the mag-netic phase diagram (near the k F = 2 π/ .
24) the model predicts only the 1/2 plateau,which perfectly accords with the real situation in the
ErB compound [2, 3], whilefor k F slightly smaller than k F = 2 π/ .
24, it predicts the main 1/2 plateau accompa-nied by a sequence of narrow magnetization plateaus with m/m s < /
2, similarly aswas observed in
T mB compound [1]. Unfortunately, the size of cluster used in ourcalculations consisting of L = 24 ×
24 sites is still small to verify the magic sequenceof magnetization plateaus observed in
T mB consisting of m/m s =1/2, 1/7, 1/9, and1/11 plateaus (to verify this, the cluster of size at least L = 1386 × A(0) B(0) C(0) D(0) E(0) F(0)G(0) H(0) A(1/3) B(1/3) C(1/3) D(1/3)E(1/3) A(1/2) B(1/2) C(1/2) D(1/2)
Figure 4:
The complete list of spin configurations corresponding to magnetization plateauswith the largest stability regions ( m/m s =0, 1/3, 1/2) from Fig. 3. cludes various types of axial and diagonal striped phases as well as homogeneous(quasi-homogeneous) distributions of single spins or n -spin clusters, confirming strong9nfluence of the long-range RKKY interaction on the ground state properties of themodel.Of course, one can object that these results can not be considered as definite sincethey were obtained on the relatively small cluster consisting of only L = 24 ×
24 sites.For this reason we have performed exhaustive numerical studies of the model on muchlarger cluster consisting of L = 120 × × (the case of L = 24 ×
24 cluster) to 2 × ,but it seems that this fact did not influence significantly the convergence of Monte-Carlo results (see Fig, 5). Indeed, a direct comparison of magnetic phase diagrams π /k F h L=120 ×
120 1/3 1/21/21/21/20 012/3 3/51/53/81.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.5012345678 00.10.20.30.40.50.60.70.80.911/9 1/18 1/81/21/4 1/4 5/12
Figure 5:
The magnetic phase diagram of the model in the k F − h plane calculated for the L = 120 ×
120 cluster. obtained on L = 24 ×
24 and L = 120 ×
120 cluster shows that they are practicallyidentical for k F < π/ . k F > π/ . k F = 2 π/ .
38. In this region the magnetic phase diagram10f the model obtained on L = 24 ×
24 cluster exhibits relatively complex structureformed by the main 1/6 plateau accompanied by smaller 1/4, 5/24 and 1/12 plateaus,while for the L = 120 ×
120 cluster there exists only one large 1/5 plateau. Thecorresponding plateau phase has the period 10 and therefore it can not appear inthe L = 24 ×
24 magnetic phase diagram, but was replaced by phases with nearestfractional magnetizations. For the same reason the 7/12 plateau phase is replacedby the 3/5 plateau phase. Since the L = 120 ×
120 cluster is indeed the robust oneand no significant finite-size effects have been observed comparing results obtainedfor L = 24 ×
24 and L = 120 × L = 120 × L = 140 ×
140 sites are compared. The cluster of L = 140 ×
140 has beenchosen for the reason that it is compatible with the 1/7 plateau, which absent on the L = 120 ×
120 cluster. One can see that for all values of the Fermi wave vector k F ∆ E π /k F h Figure 6:
The difference ∆ E = E − E between the ground-state energies (per site) ofthe model calculated on the L = 120 ×
120 and L = 140 ×
140 cluster. L = 120 ×
120 cluster is smaller than one corresponding to the L = 140 ×
140 clusterand thus no new plateaus compatible with the L = 140 ×
140 cluster, e.g. the 1/7plateau, are expected in the thermodynamic limit L → ∞ .One of possible explanations of absence of the 1/7 plateau in our results couldbe the fact that the 2D formula used in our work for the matrix elements J ij of theRKKY interaction is too crude approximation of the real situation in TmB , wherethe Fermi surface is anisotropic, but not 2D. Therefore, in a more realistic model oneshould consider, instead the 2D [17] or 3D [15] formula, an intermediate version ofthe RKKY interaction between 2D and 3D. It is also possible that the model basedon the indirect interaction between two spins mediated by conduction electrons isinsufficient to capture all experimentally observed features of magnetization processesin rare-earth tetraborides, which are metallic and for their correct description it willbe necessary to use a more complex model taking into account both, the electron andspin subsystems as well as direct interactions within subsystems and between them.Works in both directions, the generalization of the RKKY interaction for the case ofthe anisotropic Fermi surface and numerical studies within a more complex model,are currently in progress.In Table 1 we present several sequences of magnetization plateaus that followfrom our phase diagram for selected values of the Fermi wave vector k F representingtypical behaviours of the model from the regions k F < π/ .
24 and k F > π/ . T mB , ErB ), the set of k F with k F < π/ .
24 models quite realistically the experiment when the additionalholes are doped into the system, while the second one with k F > π/ .
24 models theexperiment, when the additional electrons are doped into the system. It is seen thatthe spectrum of magnetization sequences is very wide and thus the results obtained12 π/k F m/m s .
05 0 , / , / .
07 0 , / , / , / , .
095 0 , / , / , / , .
12 0 , / , / , .
17 0 , / , .
196 0 , / , / , / , .
25 0 , / , .
37 0 , / , / , / , . , / , / , / , / , .
45 0 , / , Representative sequences of magnetization plateaus identified at different valuesof k F . can serve as a motivation for experimental studies of the influence of doping on theformation of magnetization plateaus in the tetraboride solid solutions.In summary, we have presented a simple model for a description of magnetizationprocesses in metallic rare-earth tetraborides. It is based on the two-dimensional Isingmodel, in which two spins on the Shastry-Sutherland lattice interact via the long-rangeRKKY interaction J ij mediated by conduction electrons. The model is solved by acombination of the standard Metropolis algorithm and the parallel tempering methodand it yields the reach spectrum of magnetic solutions (magnetization plateaus), de-pending on the value of the Fermi wave vector k F and the external magnetic field h .In particular, we have found the following set of individual magnetization plateauswith fractional magnetization m/m s =1/18, 1/9, 1/8, 1/5, 1/4, 1/3, 3/8, 5/12, 1/2,3/5, 2/3, which for different values of k F form various sequences of plateaus, chang-ing from very complex, appearing near the points k F = 2 π/ . k F can be induced by doping (thesubstitution of rare-earth ion by other magnetic ion that introduces the additionalelectrons (holes) into the system) the model is able to predict the complete sequencesof magnetization plateaus, that could appear in the tetraboride solid solutions.13 his work was supported by projects ITMS 26220120047, VEGA 2-0112-18 and APVV-17-0020. Calculations were performed in the Computing Centre of the Slovak Academy ofSciences using the supercomputing infrastructure acquired in project ITMS 26230120002and 26210120002 (Slovak infrastructure for high-performance computing) supported by theResearch and Development Operational Programme funded by the ERDF. eferences [1] K. Siemensmeyer, E. Wulf, H. J. Mikeska, K. Flachbart, S. Gabani, S. Matas,P. Priputen, A. Efdokimova, and N. Shitsevalova, Phys. Rev. Lett. , 177201(2008).[2] S. Mataˇs, K. Siemensmeyer, E. Wheeler, E. Wulf, R. Beyer, Th. Hermannsd¨orfer,O. Ignatchik, M. Uhlarz, K. Flachbart, S. Gab´ani, P. Priputen, A. Efdokimova,and N. Shitsevalova, J. Phys. Conf. Ser. , 032041 (2010).[3] S. Michimura, A. Shigekawa, F. Iga, M. Sera, T. Takabatake, K. Ohoyama, andY. Okabe, Physica B , 596 (2006).[4] S. Yoshii, T. Yamamoto, M. Hagiwara, S. Michimura, A. Shigekawa, F. Iga, T.Takabatake, and K. Kindo, Phys. Rev. Lett. , 087202 (2008).[5] B. S. Shastry, B. Sutherland, Physica B and C , 1069 (1981).[6] Y. Dublenych, Phys. Rev. Lett. , 167202 (2012).[7] Y. Dublenych, Phys. Rev. E , 022111 (2013).[8] Y. Dublenych, Phys. Rev. E , 052123 (2014).[9] S.A. Deviren, JMMM , 508 (2015).[10] M. C. Chang and M. F. Yang, Phys. Rev. B , 104411 (2009).[11] W. C. Huang, L. Huo, J. J. Feng, Z. B. Yan, X. T. Jia, X. S. Gao, M. H. Qin,and J.-M. Liu, EPL , 37005 (2013).[12] H. ˇCenˇcarikov´a and P. Farkaˇsovsk´y, Phys. Status Solidi B , 333 (2015).[13] P. Farkaˇsovsk´y and L. Regeciov´a, Eur. Phys. J. B . 33 (2019).1514] P. Farkaˇsovsk´y, H. ˇCenˇcarikov´a, S. Mataˇs, Phys. Rev. B , 54410 (2010).[15] J. J. Feng, L. Huo, W. C. Huang, Y. Wang, M. H. Qin, J.-M. Liu, and Z. Ren,EPL , 17009 (2014).[16] S. Mitra, J. G. S. Kang, J. Shin, J. Q. Ng, S. S. Sunku, T. Kong, P. C. Canfield,B. S. Shastry, P. Sengupta, and Ch. Panagopoulos, Phys. Rev. B , 045119(2019).[17] M.T. B´eal-Monod, Phys. Rev. B , 88835 (1987).[18] B. H. Hou, F. Y. Liu, B. Jiao,and M. Yue, Acta Phys. Sin.61