The influence of further-neighbor spin-spin interaction on a ground state of 2D coupled spin-electron model in a magnetic field
Hana Čenčariková, Jozef Strečka, Andrej Gendiar, Natália Tomašovičová
aa r X i v : . [ c ond - m a t . s t r- e l ] J un The influence of further-neighbor spin-spin interaction on a ground state of 2Dcoupled spin-electron model in a magnetic field ✩ Hana ˇCenˇcarikov´a a, ∗ , Jozef Streˇcka b , Andrej Gendiar c , Nat´alia Tomaˇsoviˇcov´a a a Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Koˇsice, Slovakia b Institute of Physics, Faculty of Science, P. J. ˇSaf´arik University, Park Angelinum 9, 04001 Koˇsice, Slovakia c Institute of Physics, Slovak Academy of Sciences, D´ubravsk´a cesta 9, SK-845 11, Bratislava, Slovakia
Abstract
An exhaustive ground-state analysis of extended two-dimensional (2D) correlated spin-electron model consisting ofthe Ising spins localized on nodal lattice sites and mobile electrons delocalized over pairs of decorating sites is per-formed within the framework of rigorous analytical calculations. The investigated model, defined on an arbitrary 2Ddoubly decorated lattice, takes into account the kinetic energy of mobile electrons, the nearest-neighbor Ising couplingbetween the localized spins and mobile electrons, the further-neighbor Ising coupling between the localized spins andthe Zeeman energy. The ground-state phase diagrams are examined for a wide range of model parameters for bothferromagnetic as well as antiferromagnetic interaction between the nodal Ising spins and non-zero value of externalmagnetic field. It is found that non-zero values of further-neighbor interaction leads to a formation of new quantumstates as a consequence of competition between all considered interaction terms. Moreover, the new quantum states areaccompanied with di ff erent magnetic features and thus, several kinds of discontinuous field-driven phase transitionsare observed. Keywords: strongly correlated systems, Ising spins, mobile electrons, phase transitions, magnetic ordering
PACS: + q, 05.70.Fh, 71.27. + a, 75.30.Kz
1. Introduction
During several last decades a considerable amount ofe ff ort has been devoted to the investigation of coupledspin-electron systems due to the fact, that such materialsexhibit a wide range of unconventional properties [1, 2]with a direct application in the real life. Their appli-cation potential makes such materials very attractive forphysicists as well as engineers, but in spite of their enor-mous e ff ort, the exhaustive understanding of driven mech- ✩ This work was supported by the Slovak Research and DevelopmentAgency (APVV) under Grants No. APVV-16-0186, APVV-0097-12 andAPVV-14-0878. The financial support provided by the VEGA underGrants No. 1 / /
16 and 2 / /
15 is also gratefully acknowledged. ∗ Corresponding author
Email address: [email protected] (Hana ˇCenˇcarikov´a) anisms in such complex systems has not been achieved sofar. In general, it is assumed that the origin of mentionedcollective phenomena arises from a competition betweenelectron motion and magnetic behavior [3, 4], however,the importance of selected contributions is still highly de-bated. From the theoretical point of view, the special in-terest has been devoted to the relevance of additional in-teraction terms, which are often neglected in the first ap-proach analysis but could be responsible for the new in-teresting behavior as exemplified in Refs. [5–7].In the present paper we investigate the role of directspin-spin interaction on the formation of magnetic order,where we suppose that its presence fundamentally con-tributes to a magnetic diversity of real materials [8]. Asknown, the diversity of magnetic states is highly desiredin various field sensing devices and / or should be the base Preprint submitted to Physica B September 17, 2018 or presence of huge magnetocaloric e ff ect significant forthe refrigeration purposes. Consequently, their detailedexamination is, therefore, very valuable. For the theo-retical analysis we propose a relatively simple extendedspin-electron model on an arbitrary doubly decorated lat-tices with the localized Ising spins and delocalized mo-bile electrons, the simplified versions of which have beenpreviously studied in 1D [9, 10] as well as 2D cases [11–13]. In spite of the model simplicity, the previous resultspoint to the model convenience and present a good agree-ment with experimental observations. Our further analy-sis is primarily focused on the examination of the mag-netic ground-state phase diagrams under the influence ofexternal magnetic field, where an exhaustive descriptionof stable magnetic states is precisely done. Besides, weaccurately examine the stability area of each phase anddefine the exact boundary conditions among them. Fi-nally, we detect the presence of field-driven discontinuousphase transition and specify the conditions of their exis-tence.The paper is organized as follows. In Section 2 webriefly describe the investigated model and derive theeigenvalues of bond Hamiltonian as a necessary step indetermination of a ground-state energy. The most inter-esting results with the corresponding discussion are pre-sented in Section 3 and last, a few conclusions togetherwith future outlooks are collected in Section 4.
2. Model and Method
The proposed coupled spin-electron model on doublydecorated planar lattice is formed by immobile Ising spinslocalized at each nodal lattice site and by mobile elec-trons delocalized over the pairs of sites decorating eachbond. The energy terms occurring in the model Hamil-tonian correspond to the kinetic energy of mobile elec-trons, the Ising interaction between the mobile electronsand their nearest-neighbor Ising spins, as well as, the Isinginteraction between the nearest-neighbor Ising spins. Ofcourse, the Zeeman energy term must be included to studythe e ff ect of external magnetic field. The mutual commu-tativity between di ff erent bond Hamiltonians ˆ H k enableus to rewrite the total Hamiltonian ˆ H to the more conve-nient form: ˆ H = P Nq / k = ˆ H k , where N is a total number ofall nodal sites and q is the coordination number. Then the bond Hamiltonian can be defined as:ˆ H k = − t (ˆ c † k , ↑ ˆ c k , ↑ + ˆ c † k , ↓ ˆ c k , ↓ + ˆ c † k , ↑ ˆ c k , ↑ + ˆ c † k , ↓ ˆ c k , ↓ ) − J ˆ σ zk (ˆ n k , ↑ − ˆ n k , ↓ ) − J ˆ σ zk (ˆ n k , ↑ − ˆ n k , ↓ ) − h (ˆ n k , ↑ − ˆ n k , ↓ ) − h (ˆ n k , ↑ − ˆ n k , ↓ ) (1) − hq ( ˆ σ zk + ˆ σ zk ) − J ′ ˆ σ zk ˆ σ zk − µ (ˆ n k + ˆ n k ) , where the symbols ˆ c † k α ,γ / ˆ c k α ,γ ( α = γ = ↑ , ↓ ) denote thecreation / annihilation fermionic operators of the mobileelectron and ˆ n k α ,γ = ˆ c † k α ,γ ˆ c k α ,γ as well as ˆ n k α = P { γ } ˆ n k α ,γ are the corresponding number operators. ˆ σ zk α denotes the z -component of the Pauli operator with the eigenvalues σ = ±
1. The first term in Eq. (1) corresponds to the ki-netic energy of mobile electrons delocalized over a coupleof decorating sites k and k from the k -th dimer with thehopping amplitude t . The second and the third terms rep-resent the Ising interaction between the mobile electronsand their nearest-neighbors Ising spins described by theparameter J . The next three terms in the Eq. (1) corre-spond to the energy contribution induced by the externalmagnetic field acting on the localized as well as delocal-ized particles and the term J ′ denotes the Ising interactionbetween the nearest-neighbor Ising spins. Finally, µ is achemical potential of the mobile electrons.To perform an exhaustive analysis of the ground state itis necessary to obtain the eigenvalues of the bond Hamil-tonian. The bond Hamiltonian ˆ H k can be divided intoseveral disjoint blocks H k ( n k ) due to the commutativity ofˆ H k with the number operator of mobile electrons per bondˆ n k and the calculation procedure is significantly simpli-fied. Subsequently, the sixteen di ff erent eigenvalues E k ,corresponding to the di ff erent electron fillings have beenobtained: n k = E k = R , n k = E k , k = ± JP + Q / ± h + R − µ, E k , k = ± JP − Q / ± h + R − µ, n k = E k , k = ± JP ± h + R − µ, E k = E k = R − µ, E k , k = ± Q + R − µ, n k = E k , k = ± JP + Q / ± h + R − µ, E k , k = ± JP − Q / ± h + R − µ, n k = E k = R − µ, (2)where P = ( σ k + σ k ) / Q = p J ( σ k − σ k ) + t and R = − J σ k σ k − hL / q .2 . Results and discussion In this section we present the most interesting resultsobtained from the ground-state analysis of the model (1)in the presence of external magnetic field focusing on thediversity of stable magnetic structures. First of all, itshould be mentioned that the absolute value of the cou-pling constant J between the localized spins and mobileelectrons is set to unity and all others parameters willbe normalized with respect to this coupling. In addition,the applied magnetic field is always chosen positive, i.e. h > q = ff erent phasesmay become ground state. These phases together withtheir energies are collected in Tab. 1. As one can see,the model can stabilize both the ferromagnetic (F) as wellas antiferromagnetic (AF) type of long-range ordering inboth subsystems for an arbitrary integer electron concen-tration. In comparison with the previous studies of identi-cal model [13, 14], the mutual influence of all present in-teractions results into existence of novel magnetic phases,which are absent in the model without the magnetic fieldor the further-neighbor interaction J ′ . In this context,there arises a question whether all 15 ground states couldbe achieved by a simple modulation of just one externalparameter, for instance, the magnetic field h . If there ex-isted a conformable answer, then there would exist rel-atively simple way how to alter various magnetic stateswith a direct utilization in the real life.Let us analyze the obtained results in detail, dividingthem according to the type of the spin-electron interac-tion J (the F type if J > J ′ . > J ′ = J > t and external magnetic field h results in threedi ff erent types of magnetic phase diagrams depending ona relative strength of the hopping term. An arbitrary non-zero field favors the F spin-electron state in an uncom- pensated and empty / fully electron occupancy, while in thehalf-filled band case the magnetic field enforces discon-tinuous phase transitions. Of course, only the spin subsys-tem is F in the case n k = n k = J ′ > / existence of the field-driven phase tran-sitions detected in the half filling. This fact is clearly vis-ible in Fig. 1, where the respective phase boundaries aregiven by the following expressions:0 − I / III − IV : µ = ( − u ( − J − h − t ) , I − II / II − III : µ = ( − u ( − J − h + t ) , I − II / II − III : µ = ( − u ( J + h − t ) , I − II / II − III : µ = ( − u ( J + h + t + J ′ + h / q − √ J + t ) (cid:17) . (3)It should be mentioned that u = n k of both adjacentphases are less or equal to two ( n k ≤
2) or u = J ′ -invariantand hence only the phase boundaries I / III -II depend onthe spin-spin interaction J ′ . It can be found from Fig. 1that the F coupling J ′ > -II , which is completely in-dependent of J ′ :II − II : h = − J + t , II − II : h / q = (cid:16) − J − J ′ + √ J + t (cid:17) / ( q + , II − II : h / q = − t − J ′ + √ J + t . (4)The reduction of transition fields for the phase boundariesII -II and II -II relates with the fact that the F coupling J ′ > or state II becomes dominant. To conclude, the F inter-action J ′ > / reduces the ones existing in the formerphase diagrams.Contrary to this, the AF interaction J ′ < lectron filling Eigenvalue ( E ) Eigenvector ρ = E (0 ) = − h / q − J ′ E (0 ) = J ′ | i = Q Nq / k = | i σ k ⊗ | , i k ⊗ | i σ k | i = Q Nq / k = | i σ k ⊗ | , i k ⊗ | − i σ k ρ = E (I ) = − J − J ′ − h − h / q − t − µ E (I ) = J − J ′ − h + h / q − t − µ E (I ) = J ′ − h − √ J + t − µ | I i = Q Nq / k = | i σ k ⊗ √ ( |↑ , i k + | , ↑i k ) ⊗ | i σ k | I i = Q Nq / k = | − i σ k ⊗ √ ( |↑ , i k + | , ↑i k ) ⊗ | − i σ k | I i = Q Nq / k = | i σ k ⊗ ( α |↑ , i k + β | , ↑i k ) ⊗ | − i σ k ρ = E (II ) = − J − J ′ − h − h / q − µ E (II ) = − J ′ − h / q − t − µ E (II ) = J − J ′ − h + h / q − µ E (II ) = J ′ − h − µ E (II ) = J ′ − √ J + t − µ | II i = Q Nq / k = | i σ k ⊗ |↑ , ↑i k ⊗ | i σ k | II i = Q Nq / k = | i σ k ⊗ [ |↑ , ↓i k − |↓ , ↑i k + |↑↓ , i k + | , ↑↓i k ] ⊗ | i σ k | II i = Q Nq / k = | − i σ k ⊗ |↑ , ↑i k ⊗ | − i σ k | II i = Q Nq / k = | i σ k ⊗ |↑ , ↑i k ⊗ | − i σ k | II i = Q Nq / k = | i σ k ⊗ [ a |↑ , ↓i k + b |↓ , ↑i k + c ( |↑↓ , i k + | , ↑↓i k )] ⊗ | − i σ k ρ = E (III ) = − J − J ′ − h − h / q − t − µ E (III ) = J − J ′ − h + h / q − t − µ E (III ) = J ′ − h − √ J + t − µ | III i = Q Nq / k = | i σ k ⊗ √ ( |↑↓ , ↑i k − |↑ , ↑↓i k ) ⊗ | i σ k | III i = Q Nq / k = | − i σ k ⊗ √ ( |↑↓ , ↑i k − |↑ , ↑↓i k ) ⊗ | − i σ k | III i = Q Nq / k = | i σ k ⊗ ( β |↑↓ , ↑i k − α |↑ , ↑↓i k ) ⊗ | − i σ k ρ = E (IV ) = − h / q − J ′ − µ E (IV ) = J ′ − µ | IV i = Q Nq / k = | i σ k ⊗ |↑↓ , ↑↓i k ⊗ | i σ k | IV i = Q Nq / k = | i σ k ⊗ |↑↓ , ↑↓i k ⊗ | − i σ k Table 1: The list of eigenvalues and eigenvectors forming individual ground states. The probability amplitudes α and β used in the notation of theeigenvectors | I i / | III i have the explicit forms: α = (cid:16) √ J + t + J (cid:17)q (cid:16) J + t + J √ J + t (cid:17) and β = t q (cid:16) J + t + J √ J + t (cid:17) , while probability amplitudes a , b , and c used inthe notation of the eigenvector | II i have the explicit forms: a = J + √ J + t √ J + t , b = − ( √ J + t − J )2 √ J + t , and c = t √ J + t . −6−4−20246 µ IV I II II III J ′ =0.0t=1.0 −6−4−20246 µ IV I II II III J ′ =0.1t=1.0 −6−4−20246 µ IV I II III J ′ =0.5t=1.0 h µ IV I II II II III J ′ =0.0t=4.0 h µ IV I II II II III J ′ =0.1t=4.0 h µ IV I II II III J ′ =0.5t=4.0 Figure 1:
Ground-state phase diagrams in the µ - h plane for J = J ′ ≥ t . fields. Since the driving force of their existence origi- nates from the AF interaction J ′ <
0, naturally, the sta-4 µ IV II J ′ =0.0t=0.15 I III −6−4−20246 µ IV IV II II J ′ =−0.1t=0.15 I III I III −6−4−20246 µ IV IV II II J ′ =−0.5t=0.15 I III I III −6−4−20246 µ IV I II II III J ′ =0.0t=1.0 −6−4−20246 µ IV IV I II II III J ′ =−0.1t=1.0 −6−4−20246 µ IV IV I II II III J ′ =−0.5t=1.0 I III h µ IV I II II II III J ′ =0.0t=4.0 h µ IV IV I II II II III J ′ =−0.1t=4.0 h µ IV IV I II II II III I III J ′ =−0.5t=4.0 Figure 2:
Ground-state phase diagrams in the µ - h plane for J = J ′ ≤ t . bility of novel phases arises as a response to strength-ening of the AF interaction | J ′ | (see Fig. 2). All newphases are indeed characterized by the AF arrangementof magnetic moments in the spin subsystem (see Tab. 1),in accordance with the AF character of the interaction J ′ <
0. In the parameter space, where the e ff ect of cou-pling constant J is negligible ( n k = n k = h and the coupling constant J ′ <
0, whilethe value of hopping term t becomes unimportant. Thesame conclusion can be reached for the phase boundaries0 -0 or IV -IV emerging at h / q = − J ′ . By contrast,the increasing hopping term t has a significant e ff ect onthe electron subsystem within the novel phase I / III and, thus, it dramatically changes the stability of these phases.However, it can be understood from Fig. 3 that the hop-ping process of the mobile electrons e ff ectively decouplesthe localized spins within the novel phases I or III inthe limit of su ffi ciently strong hopping term t what is insharp contrast to the phases I / III with the F alignmentof the localized spins. Similarly to the case J ′ >
0, theAF further-neighbor interaction J ′ < ff ectspresence / existence of field-driven phase transitions. Con-trary to the former case, the increasing | J ′ | shifts the phaseboundary to the higher magnetic fields with exception ofthe phase boundary between II -II phases. It is notewor-thy that su ffi ciently strong value of J ′ can fully suppresspresence of the phase II for a strong electron correlation5 P α + =P β − P β + =P α − t P r ob a b ilit y σ k σ k k k J JtJ ′ σ k σ k k k J JtJ ′ σ k σ k k k J JtJ ′ σ k σ k k k J JtJ ′ α − stateβ − state phase I phase III Figure 3:
The occurrence probabilities of microstates within theground states I and III (see Tab. 1), where P ± α determinesthe probability of the microstate | ↑ , i k with the correspondingprobability amplitude α and P ± β stands for the probability of themicrostate | , ↑i k with the corresponding probability amplitude β . The upper index determines sign of the coupling constant J ,i.e., ’ + ’ for J > J <
0. Inset: the microstates en-tering a quantum superposition within the phase I and III withprobability amplitudes α and β , respectively. ( t =
4) and thus, it can reduce the number of field-drivenphase transitions. For completeness, let us quote analyti-cal expressions for remaining phase boundaries occurringin the phase diagrams for J > J ′ <
0, as derivedfrom a comparison of the energies given in Tab. 1.0 − I / III − IV : µ = ( − u ( − J − h − t − J ′ − h / q ) , − I / III − IV : µ = ( − u ( J − t − h − J ′ + h / q ) , I − II / II − III : µ = ( − u ( − J + J ′ + h / q ) − h + √ J + t ) , I − II / II − III : µ = ( − u ( − J ′ + t + h / q ) + h + √ J + t ) , I − II / II − III : µ = ( − u ( h − √ J + t ) , I − I / III − III : 2 h / q = − J − J ′ − t + √ J + t . (5) < J < ff ect ofmagnetic field dominates over all other forces. In an un-compensated electron limit ( n k = n k = J < and III with a di ff erent magnetism in both sub-systems. Since the e ff ect of applied field is smaller incomparison with the AF coupling J <
0, the localizedspins are aligned in opposite to the magnetic field in or-der to preserve the AF character of the spin-electron cou-pling J <
0. Naturally, the increment of external fieldleads to a suppression of the AF coupling J < andIII to the final F phases I and III , as evidenced by twofield-driven phase transitions. The first transition betweenthe phases I -I (III -III ) depends on all model parame-ters, 2 h / q = − J + J ′ + t − √ J + t , and it is shifted to thehigher fields as the hopping term t increases. The secondtransition also depends on all model parameters, but theincreasing hopping term t shifts its occurrence inversely.Consequently, both transitions can merge together intothe t -invariant phase boundary, h / q = − J /
2, betweenthe phases I -I or III -III for a su ffi ciently large hop-ping term t . It should be mentioned that the occurrenceprobabilities of two electron microstates in the phase I or III evolve inversely with respect to the J > J < located at a half filling.This phase is also characterized by a di ff erent magnetismof both subsystems due to the AF character of the spin-electron coupling J <
0. Surprisingly, the phase II oc-curs at relatively high magnetic fields in contrast to thephases with the AF ordering in one (II ) or both (II ) sub-systems emergent in a low-field region. It can be seenfrom Fig. 4 that the system exhibits field-driven phasetransitions also at half filling in the limit with absence ofspin-spin interaction J ′ =
0, where their number can betuned by the hopping term t . The rigorous expressions forthree of them are given by Eq. (4), while the remainingtwo field-driven phase transitions occur at:II − II : h / q = − J , II − II : h / q = (cid:16) J − J ′ + √ J + t (cid:17) / ( q − . (6)Let us turn our attention to the e ff ect of the further-neighbor interaction J ′ on the ground-state properties. As6ould be expected, the F interaction J ′ > J < J > J ′ > J < h and J for su ffi ciently large J ′ as well as t . By contrast,the AF interaction J ′ < and IV at low magnetic fields and it also generates thenovel phase II at relatively high magnetic fields at thehalf-filled band case on assumption that the hopping termis su ffi ciently strong. In this phase the AF coupling J ′ < ffi cient to preserve theirantiparallel orientation, but the magnetic field is strongenough to align the electron subsystem into the directionof external magnetic field. The most interesting result ofour investigations is the fact that the competing e ff ect ofthe AF spin-spin coupling J ′ <
0, the AF spin-electroncoupling J <
0, the hopping term t and the magnetic field h can produce various magnetic structures, which can bealtered only by the changes of external magnetic field. Ithas been found that the AF spin-electron coupling J < J >
0) and thus, it generatesnumerous field-driven phase transitions. Furthermore, theadditional spin-spin interaction J ′ may stabilize / produceselected magnetic structures, depending on the characterof the applied interaction, but the number of field-drivenphase transitions is in general reduced. To complete ouranalysis, the remaining phase boundaries between the rel- evant phases have the following form:0 − I / III − IV : µ = ( − u ( J − h + h / q − t ) , − I / III − IV : µ = ( − u (2( J ′ + h / q ) − h − √ J + t ) , − II / II − IV : µ = ( − u ( J − h + h / q ) , − I / III − IV : µ = ( − u ( − h − √ J + t ) , I − II / II − III : µ = ( − u (3 J − h + h / q + t ) , I − II / II − III : µ = ( − u ( J + J ′ − h + h / q + t ) , I − II / II − III : µ = ( − u ( − J + h − h / q − t ) , I − II / II − III : µ = ( − u ( J − h + t ) , I − II / II − III : µ = ( − u ( − J + h + t + J ′ − h / q − √ J + t )) , I − II / II − III : µ = ( − u ( − J ′ + t + h / q ) + h + √ J + t ) , I − II / II − III : µ = ( − u (2( J − J ′ + h / q ) − h + √ J + t ) , I − II / II − III : µ = ( − u ( − h + √ J + t ) , I − II / II − III : µ = ( − u ( h − √ J + t ) . (7)Finally, the conditions for the last phase transitions com-plete our study:II − II : h / q = − J − J ′ , II − II : h / q = ( J ′ + t ) / ( q − , II − II : h / q = − J + J ′ , II − II : h = √ J + t . (8)
4. Conclusion
In conclusion, we have examined the influence offurther-neighbor interaction on a diversity of magneticstructures in ground-state phase diagrams as well as thenumber of field-driven phase transitions. It was foundthat the mutual interplay between the kinetic term, theIsing interaction between the localized spins and mobileelectrons, the further-neighbor spin-spin interaction be-tween the localized spins and the non-zero magnetic fieldleads to very rich magnetic phase diagrams including theF, AF as well as combined F-AF magnetic structures. In-terestingly, it was found that the further-neighbor spin-spin interaction fundamentally influences the magneticground state and should be taken into account for a cor-rect description of the magnetization processes of cou-pled spin-electron systems. In addition, it was observedthat its inclusion strongly a ff ects presence / existence of7 µ IV I II II II III I J ′ =0.0t=0.5 I III III −6−4−20246 µ IV I I II II III II J ′ =0.1t=0.5 III −6−4−20246 µ IV I I II II III J ′ =0.7t=0.5 III −6−4−20246 µ IV I I II II II III III III I J ′ =0.0t=1.0 −6−4−20246 µ IV I I I II II II III III III J ′ =0.1t=1.0 −6−4−20246 µ IV I I II II III III J ′ =0.7t=1.0 −6−4−20246 µ IV I I II II II II III III III I J ′ =0.0t=2.0 −6−4−20246 µ IV I I I II II II II III III III J ′ =0.1t=2.0 −6−4−20246 µ IV I I II II II III III J ′ =0.7t=2.0 h µ IV I I II II II III III III I J ′ =0.0t=3.0 h µ IV I I II II II III III J ′ =0.1t=3.0 h µ IV I I II II III III J ′ =0.7t=3.0 Figure 4:
Ground-state phase diagrams in the µ - h plane for J = J ′ ≥ t . the field-induced phase transitions of metamagnetic na-ture at finite temperatures. However, it is necessary toperform an extended theoretical analysis to answer thisquestion satisfactorily. The work on this task is currentlyin progress [14]. References [1] P. Wachter, in K. A. Gschneider, L. R. Eyring (Eds.), Hand-book on the Physics and Chemistry of Rare Earth , vol. 19,North-Holland, Amsterdam (1994).[2] S.-W. Cheong and M. Mostovoy, Nature Mat. (2007) 13.[3] E. Velu, C. Dupas, D. Renard, J. P. Renard, and J. Seiden,Phys. Rev. B (1988) 668. µ IV I III II II I III J ′ =0.0t=0.09 −6−4−20246 µ IV IV I II II II II III I III J ′ =−0.1t=0.09 −6−4−20246 µ IV I III IV II II J ′ =−0.8t=0.09 −6−4−20246 µ IV IV I I II II II II III III I J ′ =−0.1t=1.0 III −6−4−20246 µ IV IV I I III II II II III J ′ =−0.3t=1.0 −6−4−20246 µ IV IV I II II III J ′ =−0.8t=1.0 −6−4−20246 µ IV IV I I II II II II II III III III I J ′ =−0.1t=2.0 −6−4−20246 µ IV IV I II II II III III I I III J ′ =−0.3t=2.0 −6−4−20246 µ IV IV I II II III J ′ =−0.8t=2.0 h µ IV IV I I II II II II III III III I J ′ =−0.1t=3.0 h µ IV IV I II II II III III I I III J ′ =−0.3t=3.0 h µ IV I II II III J ′ =−0.8t=3.0 Figure 5:
Ground-state phase diagrams in the µ - h plane for J = J ′ ≤ t .[4] M. Baibich, J. Broto, A. Fert, F. Dau, F. Petro ff , P. Etienne,G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. (1988) 2472.[5] O. I. Motrunich and P. A. Lee, Phys. Rev. B (2004)214516.[6] H. ˇCenˇcarikov´a and P. Farkaˇsovsk´y, Condens. MatterPhys. (2011) 42701.[7] S. Liu and A. L. Chernyshev, Phys. Rev. B (2013) 064415.[8] R. L. Carlin and A. J. Duyneveldt, Magnetic Properties ofTransition Metal Compounds , Springer Verlag New York(1977).[9] J. ˇCis´arov´a and J. Streˇcka, Phys. Lett. A (2014) 2801.[10] J. Streˇcka and J. ˇCis´arov´a, Mater. Res. Lett. (2016)106103.[11] F. F. Doria, M. S. S. Pereira, and M. L. Lyra, J. Magn. agn. Mater. (2014) 98.[12] J. Streˇcka, H. ˇCenˇcarikov´a, and M. L. Lyra, Phys. Lett. A (2015) 2915.[13] H. ˇCenˇcarikov´a, J. Streˇcka, and M. L. Lyra, J. Magn.Magn. Mater. (2016) 1106.[14] H. ˇCenˇcarikov´a, J. Streˇcka, and A. Gendiar, in progress.(2016) 1106.[14] H. ˇCenˇcarikov´a, J. Streˇcka, and A. Gendiar, in progress.