Magnetization processes in metallic Shastry−Sutherland systems: A comprehensive review

Abstract

Rare-earth tetraborides RB{}_4 are unique frustrated magnets in which strongly anisotropic localized 4f moments coexist with itinerant 5d electrons on the geometrically frustrated Shastry−Sutherland lattice. Their magnetization curves display remarkably rich sequences of fractional plateaus, ranging from the robust 1/2 state in ErB{}_4 and TmB{}_4 to intricate cascades in HoB{}_4 and TbB{}_4, whose microscopic origins cannot be captured by localized-spin models alone. This review provides a comprehensive and unified theoretical picture of the mechanisms that stabilize these commensurate magnetic textures. We begin with long-range oscillatory RKKY interactions that naturally emerge from the metallic character of the RB{}_4 compounds and show how accurate numerical treatment reveals a dense hierarchy of competing plateau phases. We then move to explicit spin-electron models, where electron itinerancy, Coulomb repulsion, transverse exchange terms, and realistic mixed-spin configurations jointly reconstruct the plateau stability diagram and reproduce the dominant experimental features. Further extensions, such as the coupled double-Ising (CDI) mapping and charge-stripe potentials, reveal how subtle electronic textures selectively suppress or enhance magnetization plateaus, enabling microscopic control over the competition between the 1/3 and 1/2 plateaus and linking magnetization processes to the magnetocaloric response. The theoretical frameworks reviewed here establish that the interplay of geometrical frustration, itinerant electrons, electronic correlations, and exchange anisotropy is essential for understanding the magnetic and thermodynamic behavior of metallic Shastry−Sutherland materials. The resulting picture not only explains the material-dependent plateau sequences observed in rare-earth tetraborides but also provides a versatile platform for designing new frustrated magnets with tunable fractional states and enhanced magnetocaloric performance.

Graphical abstract

Keywords

Shastry−Sutherland magnets / magnetization processes / rare-earth tetraborides / frustrated magnetism / spin−electron models / RKKY interaction

1 Introduction

Models defined on the Shastry–Sutherland lattice (SSL) constitute one of the most prominent realizations of geometrically frustrated spin systems in condensed-matter physics. It can be visualized as a square lattice with antiferromagnetic couplings between nearest-neighbour spins and additional antiferromagnetic couplings between next-nearest-neighbour spins in every second square. These diagonal interactions introduce strong geometrical frustration and lead to an extensive ground-state degeneracy that gives rise to a variety of exotic magnetic phenomena. Although originally conceived as a purely theoretical construction by Shastry and Sutherland [1], it gradually evolved into a prototypical platform for studying frustrated magnetism.

A major breakthrough occurred when a real material was discovered to host precisely the same lattice geometry. The first and most famous experimental realization of the SSL is the insulating quantum magnet SrCu{}_2(BO{}_3){}_2 [2–9]. In this compound, the Cu{}^{2+} ions form orthogonal dimer pairs arranged exactly as in the Shastry−Sutherland model [Fig. 1(a)], with a dominant intradimer coupling J_2 and a weaker interdimer coupling J_1 [Fig. 1(b)]. Because SrCu{}_2(BO{}_3){}_2 is an electrical insulator, its low-energy physics is governed entirely by localized spins, making it an ideal platform for testing theoretical approaches based on the Heisenberg Hamiltonian [10–14]. A broad spectrum of theoretical techniques ranging from the Monte Carlo methods, linked-cluster expansions, and continuous unitary transformations [15–17] to modern tensor-network methods [18–21], has been applied to this system. These studies successfully explain a number of hallmark features: the exact dimerized ground state [1], strongly localized triplet excitations [22], and a cascade of magnetization plateaus at m/m_s=1/8, 2/15, 1/6, 1/4, 1/3, and 1/2, observed experimentally in high magnetic fields up to 120 T [23–25]. The exceptional agreement between theory and experiment firmly established SrCu{}_2(BO{}_3){}_2 as the prototypical realization of the Shastry−Sutherland model and demonstrated that the SSL framework captures the essential microscopic physics behind the plateau formation in quantum magnets. This success naturally motivated the search for further materials with the same lattice geometry. In this context, two distinct directions have emerged. On one hand, several insulating SSL compounds have been identified, in which the magnetization process is governed predominantly by localized spin interactions [26–28]. A second, and in many respects even more remarkable, family of SSL-based compounds is found among the metallic rare-earth tetraborides RB{}_4 (R = La−Lu) [29–35]. Although their bonding arrangement is topologically equivalent to that in SrCu{}_2(BO{}_3){}_2, the tetraborides differ in several crucial aspects. Structurally, the SSL is slightly distorted: the diagonal angle is shifted, which leads to nearest- and next-nearest-neighbour exchange integrals that are nearly equal in magnitude, J_1\approx J_2 as shown in Fig. 1(c). This near-degeneracy enhances the geometrical frustration far beyond that found in SrCu{}_2(BO{}_3){}_2. Even more importantly, the tetraborides are metallic: 4f moments of the rare-earth ions coexist with itinerant 5d electrons that mediate additional interactions and couple strongly to the magnetic order [36]. The rare-earth ions occupy the positions of the SSL in the ab plane of the tetragonal P4/mbm crystal structure, again forming the characteristic network of squares and orthogonal diagonals. In this setting, the combination of localized and itinerant degrees of freedom on a strongly frustrated SSL creates an unusually rich magnetic behaviour.

The experimental discovery of series of fractional magnetization plateaus at low temperatures in the rare-earth tetraborides has attracted enormous attention during the last two decades [38–41]. Magnetization measurements have revealed a fascinating sequence of plateaus, the positions and widths of which depend strongly on the particular rare-earth element. For instance, in HoB{}_4 the plateaus occur at m/m_s=1/3,4/9, and 3/5 [41–43]; in TbB{}_4 at m/m_s=2/9,1/3,4/9,1/2, and 7/9 [44]; in ErB{}_4 a single broad plateau is observed at m/m_s=1/2 [41, 45]; and in TmB{}_4 the sequence consists of a dominant 1/2 plateau accompanied by a very narrow metastable ones [29, 36, 50, 51]. The existence of such fractional plateaus is considered the most prominent manifestation of geometrical frustration in these metallic systems. It reflects the stabilization of magnetically ordered states with a large unit cell and provides an important experimental benchmark for theoretical modelling.

In addition to the magnetization measurements, many other experimental techniques, such as specific-heat [52–54], magnetotransport [45, 46, 55], and magnetocaloric measurements [56–58], have been employed to explore the thermodynamic and dynamic properties of RB{}_4 compounds. These studies confirmed the strongly anisotropic magnetic behaviour and revealed a complex interplay between localized 4f moments of the rare-earth ions and itinerant 5d electrons that are responsible for the metallic conductivity. From this viewpoint, the rare-earth tetraborides represent a unique class of materials in which the localized and itinerant subsystems coexist and interact on the geometrically frustrated SSL. This dual character requires theoretical models capable of describing both local-moment interactions and itinerant-electron effects on an equal footing.

1.1 Historical development of theoretical approaches

The first theoretical efforts to explain the fractional magnetization plateaus in RB{}_4 naturally started from pure spin models. Because of the strong single-ion anisotropy imposed by the crystal field, the localized moments {\mathrm{R}}^{3+} can be, in the first approximation, treated as Ising spins. The simplest model that captures the geometry of the lattice is therefore the Ising model on the SSL, characterized by the couplings J_1 and J_2. Analytical and numerical studies of this model have shown that it stabilizes only a single fractional plateau at m/m_s=1/3 [59–61]. This 1/3 plateau corresponds to a configuration in which one third of the spins are aligned antiparallel to the external magnetic field, while the remaining two thirds point parallel to it, forming a stripe-like pattern. The success of this model lies in its ability to reproduce the basic mechanism of plateau formation in a frustrated geometry and to provide a clear physical picture based solely on localized spins with strong Ising anisotropy.

However, the limitations of this minimal model became apparent when compared with experimental data. In particular, the absence of the 1/2 plateau, the most pronounced feature in ErB{}_4 and TmB{}_4, and the inability to reproduce the more complex plateau sequences observed in other compounds indicate that the nearest- and next-nearest-neighbour Ising model is not sufficient to capture the magnetic behaviour of real RB{}_4. To improve the agreement with experiments, several generalizations of the classical Ising model were proposed. These extensions included longer-range interactions and additional multispin terms [62–68]. It was demonstrated that such generalized models can, for some combinations of parameters, stabilize additional plateaus, including the experimentally observed 1/2 plateau. Nevertheless, these modifications significantly increase the number of adjustable parameters, and it is often difficult to assign them a clear microscopic origin. In practice, the stability of individual plateaus in such models depends sensitively on the chosen values of the exchange constants, which limits their predictive power.

A further improvement was achieved by incorporating quantum fluctuations through Heisenberg or XXZ-type models on the SSL [69, 70]. In the spin-1/2 Heisenberg model, for example, the competition between J_1 and J_2 gives rise to a rich phase diagram including a dimerized singlet ground state, Néel-like order, and intermediate phases with complex spin textures. Quantum Monte Carlo and exact-diagonalization studies revealed that a moderate transverse component of the exchange can partially suppress the Ising-like 1/3 plateau and lead to the emergence of additional plateau at m/m_s=1/2. However, the parameter region in which these features appear is usually narrow, and the required anisotropy is often unrealistically large for real tetraborides. Therefore, although the inclusion of quantum effects enriches the theoretical phase diagram, it still does not provide a fully satisfactory quantitative description of the experimental magnetization curves.

The main conceptual problem of all purely spin models lies in their neglect of the metallic character of the rare-earth tetraborides. In these systems, the magnetic {\mathrm{R}}^{3+} ions interact not only via direct exchange but also indirectly through conduction electrons occupying the 5d bands. The evidence for this comes from measurements of electrical resistivity and magnetotransport, which reveal metallic conduction and strong coupling between magnetic order and the electronic subsystem [45, 46]. Consequently, the effective magnetic interactions [36] are expected to have a long-range, oscillating character of the Ruderman−Kittel−Kasuya−Yosida (RKKY) type [47–49]. Such interactions naturally extend beyond the nearest neighbours and can therefore provide a microscopic justification for the longer-range couplings that are otherwise added empirically to classical Ising models.

Another limitation of spin-only descriptions is their inability to capture the mutual influence between itinerant electrons and localized moments. In real materials, the conduction electrons can mediate effective interactions between localized spins, modify their exchange integrals, and, conversely, the magnetic ordering can affect the electronic density of states and transport properties. This mutual feedback is completely absent in models that consider spins as isolated from the itinerant background. It is thus reasonable to expect that a realistic theoretical description of RB{}_4 should explicitly include both subsystems and their coupling.

1.2 Motivation and scope of this review

The shortcomings of the pure spin approach have stimulated the development of several complementary theoretical frameworks that take the itinerant electrons into account either indirectly, through effective long-range RKKY interactions, or directly, by coupling localized spins with itinerant electrons in combined spin-electron Hamiltonians. The first line of research, based on the generalized Ising model with RKKY interactions, successfully reproduces the appearance of new magnetization plateaus and provides a microscopic mechanism rooted in the oscillatory exchange mediated by conduction electrons [71–75]. The second line, represented by spin-electron and extended Hubbard models on the SSL [76–83], offers an explicit treatment of both subsystems and reveals how their mutual coupling can lead to complex magnetic and thermodynamic behaviour.

In the present review, we aim to provide a comprehensive overview of these developments with a particular emphasis on the role of itinerant electrons in stabilizing various magnetic states on the SSL. Starting from the earliest RKKY-based studies, we will trace the gradual evolution towards more elaborate spin−electron and Hubbard-type models and discuss how these frameworks overcome the limitations of purely localized spin descriptions. While the detailed results of these approaches will be presented in the subsequent sections, the main goal of this introductory part is to outline the experimental motivation, summarize the key findings and shortcomings of the classical spin models, and justify the necessity of including the electronic degrees of freedom explicitly in the theoretical treatment of magnetization processes in the Shastry−Sutherland rare-earth tetraborides.

2 RKKY interaction and indirect electron coupling effects

In view of the metallic nature of rare-earth tetraborides and the evident limitations of purely localized-spin descriptions, the next logical step in the theoretical treatment of Shastry−Sutherland magnets is to consider the effect of conduction electrons on the effective magnetic interactions between localized moments. Since the direct inclusion of itinerant electrons in a microscopic model considerably increases its complexity, an attractive intermediate approach consists in taking the itinerant degrees of freedom into account indirectly, through the oscillatory exchange interaction mediated by the conduction electrons − the Ruderman−Kittel−Kasuya−Yosida (RKKY) mechanism. The RKKY interaction naturally introduces both ferromagnetic and antiferromagnetic couplings that alternate in sign and range depending on the Fermi momentum k_{\mathrm{F}}, thereby providing a microscopic origin for the long-range competing interactions that were earlier introduced empirically into the classical Ising models.

2.1 Early RKKY approach and the origin of magnetization plateaus

The first systematic attempt to describe the magnetization process of metallic rare-earth tetraborides using an RKKY-type interaction was carried out by Feng et al. [71]. Their work demonstrated that the long-range, oscillatory exchange mediated by conduction electrons can by itself generate a sequence of fractional magnetization plateaus on the SSL, without invoking higher-order or multi-spin interactions. In this approach the localized Ising spins S_i=\pm 1 interact via the three-dimensional RKKY exchange derived for an isotropic free-electron gas. The Hamiltonian is

(1)H={\displaystyle \sum _{ij}J(r_{ij})S_i{S}_j-h{\displaystyle \sum _i{S}_i,}}

where the indirect exchange coupling takes the analytic 3D RKKY form

(2)J(r_{ij})=-J_0{a}^3\left[{\displaystyle \frac{\cos (2k_{\mathrm{F}}{r}_{ij})}{r_{ij}^3}}-{\displaystyle \frac{\sin (2k_{\mathrm{F}}{r}_{ij})}{2k_{\mathrm{F}}{r}_{ij}^4}}\right],

with J_0>0 setting the overall scale, a the lattice constant, and k_{\mathrm{F}} the Fermi wave vector. Because J(r_{ij}) oscillates and changes sign with distance, the interaction naturally generates competing ferro- and antiferromagnetic bonds across multiple coordination shells.

Feng et al. [71] considered a 24\times 24 SSL cluster with periodic boundaries and evaluated all RKKY couplings up to a cut-off of r_{max}=6a. For the reference Fermi wave vector k_{\mathrm{F}}=2\pi /1.243, corresponding to the estimated conduction-electron density of TmB{}_4, the magnetization curve exhibits four distinct steps at low temperature: a q-Néel antiferromagnetic state L(0), a narrow 1/4 plateau corresponding to U(1/4) state, a wide and robust 1/2 plateau associated with a ferrimagnetic Z(1/2) configuration, and a fully saturated ferromagnetic Q(1) phase (see Fig. 2). The 1/4 plateau vanishes already at T\approx 0.1, whereas the 1/2 plateau remains stable up to T\approx 0.4. This strong thermal stability of the 1/2 plateau is a direct consequence of the RKKY-induced balance between further-neighbour couplings, which favours the Z(1/2) structure over all competing arrangements.

A key result is the strong sensitivity of the plateau sequence to the value of k_{\mathrm{F}}. This sensitivity originates from the fact that the sign and magnitude of the effective next-neighbour interactions (J_3, J_4, J_5,...) change rapidly with k_{\mathrm{F}}. For k_{\mathrm{F}} close to 2\pi /1.243 the Z(1/2) state is energetically preferred, stabilizing the 1/2 plateau. Decreasing k_{\mathrm{F}} towards 2\pi /1.100 reverses the sign of J_3 and weakens the ferromagnetic J_5 term, which favours the UUD configuration and generates a clear 1/3 plateau. In contrast, for smaller values such as k_{\mathrm{F}}=2\pi /1.300, the model supports new commensurate structures producing additional plateaus at m/m_s=1/6 and 5/12. These trends are in direct correspondence with the oscillatory character of the RKKY exchange and illustrate how the conduction electrons tune the competition between distinct magnetic superstructures.

Altogether, the work of Feng et al. [71] demonstrated that the RKKY interaction is not only capable of stabilizing the experimentally important 1/2 plateau, but can also generate a hierarchy of additional fractional steps whose stability is controlled by the electronic density. In this picture, the Fermi wave vector plays the role of a microscopic tuning parameter that selects between competing commensurate magnetic textures, providing a natural explanation for the strong sample-to-sample variation of plateau sequences observed in metallic rare-earth tetraborides.

2.2 Numerical refinement and phase-diagram complexity

In order to verify the conclusions of Feng et al. [71] and to test the robustness of the 1/2 plateau predicted by their model, Regeciová and Farkašovský revisited the problem in Ref. [72], introducing substantial improvements in both numerical precision and physical realism. Their study re-examined the same RKKY-based Ising model on the SSL [see Eq. (1) and Eq. (2) in Section 2.1 for the Hamiltonian and interaction definition], but used a careful finite-size scaling, refined statistical averaging, and an updated computational scheme combining the Metropolis algorithm [84] with parallel tempering [85]. The main goal was to determine whether the sequence of magnetization plateaus and corresponding phase boundaries found by Feng et al. [71] were intrinsic to the model or artefacts of finite-size limitations and insufficient sampling.

These improvements dramatically changed the resulting magnetization process. Instead of the simple four-phase structure reported earlier, the refined simulations revealed a much richer hierarchy of fractional plateaus. Besides the m/m_s=1/2 state, several additional commensurate phases were found, including plateaus at m/m_s = 1/3, 3/8, as well as multiple narrower intermediate steps. All of these plateaus persist across system sizes and appear systematically once metastable states are fully eliminated. The presence of these additional fractions demonstrates that the RKKY-Ising model has an intrinsically complex energy landscape whose structure cannot be resolved without careful annealing and large clusters.

The corresponding ground-state phase diagram obtained in the (k_{\mathrm{F}},h) plane is shown in Fig. 3. In contrast to earlier results, the improved diagram contains multiple ferrimagnetic and modulated phases separated by distinct first-order boundaries. The fractional plateaus arise from subtle competition between several coordination shells of the oscillatory RKKY exchange, and their stability depends sensitively on the value of the Fermi momentum. Small changes of k_{\mathrm{F}} around k_{\mathrm{F}}=2\pi /1.243, simulating electron or hole doping, reorganize the plateau sequence and shift the critical fields.

In summary, the refined numerical analysis showed that the appearance of multiple fractional plateaus is an intrinsic feature of the RKKY interaction on the SSL. The mixed 3D−2D formulation, which reflects the real electronic dimensionality of rare-earth tetraborides, naturally reproduces a broad set of commensurate spin textures and provides a microscopic origin for the material-dependent magnetization behaviour observed experimentally.

2.3 Two-dimensional RKKY interaction and the role of electronic dimensionality

While the mixed 3D−2D formulation provided a major improvement over the earlier model of Feng et al. [71], the use of a three-dimensional Fermi surface to describe conduction electrons remained a simplification that could not fully capture the layered electronic structure of the rare-earth tetraborides. Angle-dependent magnetotransport experiments on TmB{}_4 revealed a strongly anisotropic Fermi surface [55], implying that the effective exchange between localized moments should be derived for a two-dimensional electron gas. This observation motivated a further refinement of the theory, presented in Ref. [73], where the RKKY interaction was reformulated for two-dimensional conduction electrons.

In this version of the model, the localized spins on the SSL were again treated as classical Ising variables, but the coupling between them was obtained from the 2D analogue of the RKKY interaction [86]:

(3)\begin{array}{rl}J(r_{ij})=& \frac{k_{\mathrm{F}}^2}{2\pi }[B_0^{(1)}(k_{\mathrm{F}}{r}_{ij})B_0^{(2)}(k_{\mathrm{F}}{r}_{ij})\\ & +B_1^{(1)}(k_{\mathrm{F}}{r}_{ij})B_1^{(2)}(k_{\mathrm{F}}{r}_{ij})],\end{array}

where B_n^{(1)}(x) and B_n^{(2)}(x) are the Bessel functions of the first and second kind, respectively. This expression retains the characteristic oscillatory behaviour of the indirect exchange but exhibits a slower spatial decay than its three-dimensional counterpart, reflecting the longer-ranged character of the interaction in a two-dimensional electron gas. The modified form of J(r_{ij}) is thus more realistic for the metallic SSL layers in the tetraborides, where conduction electrons are largely confined to the ab plane but can still mediate effective couplings between distant sites.

Comprehensive Monte Carlo simulations were again performed using the combination of Metropolis updates and parallel tempering up to L=120\times 120 clusters. The resulting phase behaviour was mapped as a function of the Fermi momentum k_{\mathrm{F}} and external magnetic field h, producing the phase diagram shown in Fig. 4. The most notable feature of this diagram is the broad stability region of the m/m_s=1/2 plateau that dominates over a wide range of k_{\mathrm{F}} values in the central part of the phase diagram (near the point k_{\mathrm{F}}=2\pi /1.25), and the total absence of the m/m_s=1/3 plateau, in striking correspondence with the experimental magnetization of ErB{}_4. For slightly smaller k_{\mathrm{F}}, corresponding to lower carrier concentrations, a cascade of narrow plateaus appears below m/m_s=1/2, reminiscent of the magnetization process observed in TmB{}_4.

From a physical point of view, the transition from the three-dimensional to the two-dimensional RKKY form has several important consequences. First, the interaction oscillates with a different period and slower amplitude decay, which modifies the hierarchy of effective exchange integrals within the SSL. As a result, the competition among the first few coordination shells becomes less pronounced, and the magnetic frustration is partly relieved. This leads to a simplification of the magnetization curve: the multitude of fractional plateaus present in the 3D−2D model collapses into a single dominant plateau at 1/2, accompanied by a few narrow phases at lower magnetizations.

In this study, the authors systematically explored several values of the Fermi momentum, effectively simulating the influence of electron and hole doping on the two-dimensional RKKY interaction. Changes in k_{\mathrm{F}} modify the oscillation period of the indirect exchange and thus alter the relative stability of fractional plateaus. This provides a clear microscopic interpretation of how carrier doping can tune the magnetization

2.4 Three-dimensional RKKY coupling and the hierarchy of fractional plateaus

To complete the systematic analysis of dimensionality effects, Regeciová and Farkašovský extended their approach to the case where both the magnetic lattice and the electronic subsystem are treated as fully three-dimensional [74]. This “3D−3D” model was designed to explore the impact of interlayer magnetic coupling and to determine whether the inclusion of interplane interactions could reproduce the entire diversity of magnetization plateaus observed experimentally across the rare-earth tetraboride family.

The Hamiltonian and interaction form used here are the same as those introduced in Section 2.1 [see Eqs. (1) and (2)]. In this configuration, the exchange network couples each spin not only to its in-plane neighbours but also to sites in adjacent layers, thereby introducing an additional source of frustration that enriches the magnetic phase space.

Monte Carlo simulations were performed on cubic clusters ranging from L=12\times 12\times 12 up to L=20\times 20\times 20, using the standard Metropolis algorithm supplemented by parallel tempering to reduce metastability. For each pair (h,k_{\mathrm{F}}) the system was cooled down to the lowest temperature T=0.05, and the resulting spin configuration was used to evaluate the approximate ground-state energy. To minimize finite-size effects, this procedure was repeated for all cluster sizes, and the physical magnetization m was assigned by selecting the value m(L) corresponding to the lowest-energy configuration among all L. Applying this minimization procedure over the entire (h,k_F) parameter space yields the magnetic phase diagram shown in Fig. 5, which displays an exceptionally rich hierarchy of ferrimagnetic, striped, and modulated phases with numerous fractional magnetization plateaus. Stable plateaus were identified at m/m_s = 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 1/3, 3/8, 2/5, 5/12, 3/7, 1/2, 5/9, and 2/3, which collectively reproduce almost all fractional magnetization values reported in TmB{}_4, TbB{}_4, HoB{}_4, and ErB{}_4.

From a physical viewpoint, the inclusion of the third dimension leads to a much stronger competition among exchange paths. Whereas the two-dimensional model stabilizes essentially a single dominant 1/2 plateau, the three-dimensional one supports a cascade of fractional plateaus reflecting the coexistence of multiple commensurate superstructures. The simulation results clearly indicated that small variations of the Fermi momentum k_{\mathrm{F}}, which can correspond to changes in carrier concentration due to doping, strongly modify the sequence and stability of the plateaus. In particular, for k_{\mathrm{F}} values characteristic of rare-earth tetraborides, the model predicts the presence of the 1/8 plateau that had been previously observed experimentally but was absent in the 2D approximation.

In addition to the reference region corresponding to the undoped compounds, the study systematically examined slightly lower and higher k_{\mathrm{F}} values to simulate hole and electron doping. These variations in the Fermi momentum effectively model changes in carrier concentration and allow one to track how doping reorganizes the competition between exchange shells. The resulting shifts of the plateau boundaries and the emergence or suppression of specific fractional states demonstrate that doping provides a direct microscopic mechanism for tuning the magnetic phase diagram of metallic Shastry−Sutherland systems. The 3D−3D RKKY model therefore represents the most complete indirect-electron description to date, capable of reproducing the experimentally observed complexity of magnetization processes in rare-earth tetraborides.

2.5 Magnetocaloric effect within the 2D−2D RKKY model

In addition to reproducing the fractional magnetization plateaus, the RKKY framework also offers a microscopic explanation of the anomalous magnetocaloric properties observed in the metallic rare-earth tetraborides. The work [75] investigated this effect within the two-dimensional Ising model on the Shastry−Sutherland lattice coupled by the two-dimensional RKKY interaction. This formulation, in which both the magnetic and electronic subsystems are restricted to the plane, was motivated by the strongly anisotropic and quasi-two-dimensional Fermi surface experimentally confirmed in TmB{}_4.

The same Hamiltonian was adopted as in the preceding RKKY studies, with the coupling J(r_{ij}) given by the two-dimensional RKKY expression Eq. (3). Monte Carlo simulations on the 24\times 24 cluster combined with parallel tempering were employed to evaluate the temperature- and field-dependent entropy change \mathrm{\Delta }S(T,h)=S(T,h)-S(T,0) [75]. The calculated results, summarized in Fig. 6, show a pronounced non-monotonic field dependence of \mathrm{\Delta }S. A positive “heating” peak appears near the metamagnetic transition between fractional plateaus, followed by a negative “cooling” valley at higher fields. This alternating behaviour reproduces the main features of the experimental magnetocaloric response of TmB{}_4.

In this work, the magnetocaloric quantities were computed for several representative Fermi momenta, which effectively simulate changes in the carrier concentration due to electron or hole doping. Varying k_{\mathrm{F}} modifies the oscillatory character of the RKKY interaction and therefore shifts the position and amplitude of the heating and cooling peaks in \mathrm{\Delta }S. This analysis revealed that doping can strongly enhance or suppress the magnetocaloric efficiency, and that there exist two distinct intervals of k_{\mathrm{F}} corresponding to maximal relative cooling power. From an applied viewpoint, these results demonstrate that the magnetocaloric response of metallic Shastry−Sutherland magnets can be tuned by adjusting the electron density, e.g., through chemical substitution. Overall, these findings confirmed that the oscillatory, long-range 2D RKKY interaction governs not only the magnetic ordering but also the field-induced entropy variations underlying the magnetocaloric effect. Because the coupling depends explicitly on the Fermi momentum, tuning the carrier concentration by doping provides a direct microscopic route to control and optimize the magnetocaloric response of metallic Shastry−Sutherland magnets.

3 Explicit inclusion of electrons: Spin−electron models on the Shastry−Sutherland lattice

While the RKKY framework successfully captures the indirect influence of conduction electrons on the localized magnetic moments, a more complete microscopic description requires the electrons to be treated as explicit quantum degrees of freedom. Such an approach allows one to analyse not only the effective magnetic coupling mediated by the electron gas, but also the feedback of the magnetic background on the itinerant subsystem itself. In metallic Shastry−Sutherland compounds this feedback manifests through the interplay between electron mobility, spin ordering, and Coulomb correlations, which jointly determine both the magnetization process and the magnetocaloric response.

The spin−electron model provides a natural framework to study these effects. It consists of itinerant electrons hopping between nearest-neighbour sites of the SSL and interacting locally with the Ising-type spins via the exchange coupling J_z. By systematically extending this minimal model to include electron−electron repulsion U, external magnetic field h, and later its classical-Ising reduction, represented by the so-called coupled double-Ising (CDI) model, one obtains a unified picture connecting localized and itinerant limits. The following subsections summarise a series of theoretical studies that progressively develop this approach, from the simplest models without Coulomb interaction, through correlated variants with finite U, to the full spin−electron model and its CDI representation, highlighting their thermodynamic and magnetocaloric properties in relation to the experimental behaviour of the rare-earth tetraborides.

3.1 Spin-electron coupling without Coulomb interaction

A microscopic description of the magnetization processes in metallic rare-earth tetraborides requires explicit inclusion of itinerant electrons rather than treating them only as mediators of effective interactions. The first comprehensive analysis of this kind was carried out in Ref. [76], where the authors introduced a minimal spin−electron model on the SSL and demonstrated that itinerant degrees of freedom are essential for reproducing the rich hierarchy of magnetization plateaus characteristic of these materials. A complementary finite-temperature study based on the same model was later presented in Ref. [77].

In the spin−electron model the conduction electrons hop between nearest- and next-nearest neighbour sites of the SSL, while localized magnetic moments are represented by Ising spins S_i^z=\pm 1. The electrons couple locally to these spins through an anisotropic Ising-type exchange interaction J_z, and an external magnetic field h acts on both the electron and spin subsystems. The Hamiltonian reads

(4)\begin{array}{rl}{H}_0=& t_1\sum _{⟨ij⟩\sigma }{d}_{i\sigma }^{†}{d}_{j\sigma }+t_2\sum _{⟨⟨ij⟩⟩\sigma }{d}_{i\sigma }^{†}{d}_{j\sigma }\\ & +J_z\sum _i(n_{i\uparrow }-n_{i\downarrow })S_i^z-h\sum _i(n_{i\uparrow }-n_{i\downarrow })\\ & +J_1\sum _{⟨ij⟩}{S}_i^z{S}_j^z+J_2\sum _{⟨⟨ij⟩⟩}{S}_i^z{S}_j^z-h\sum _i{S}_i^z,\end{array}

where d_{i\sigma }^{†} (d_{i\sigma }) creates (annihilates) an electron of spin \sigma at site i, and n_{i\sigma }=d_{i\sigma }^{†}{d}_{i\sigma }. The parameters t_1 and t_2 denote the hopping amplitudes between nearest-neighbour ⟨ij⟩ and next-nearest-neighbour ⟨⟨ij⟩⟩ sites of the Shastry–Sutherland lattice, respectively, while J_1 and J_2 are the corresponding antiferromagnetic Ising exchange couplings between localized spins. The model described by (4) is a direct generalization of the model introduced by Lemanski [87] that has been used later for the study of various cooperative phenomena in interacting spin−electron systems [88–91].

For a fixed Ising background \{S_i^z\}, the electronic part of the Hamiltonian is quadratic and can be diagonalized exactly. The total energy for a given configuration is obtained by filling the lowest N one-electron levels according to the prescribed electron number. The ground state at each magnetic field h is found by iteratively modifying the spin configuration and accepting any update that lowers the total energy. This hybrid scheme, described in detail in Ref. [92], combining exact diagonalization of the electronic subsystem with classical energy minimization for the localized spins, was applied to very large clusters up to 120\times 120 sites [76]. As a result, finite-size effects in the magnetization curve were strongly suppressed and even narrow plateau regions became visible.

The principal finding of Ref. [76] is that the interplay between electron hopping and spin−electron coupling stabilizes a series of commensurate magnetization plateaus at m/m_s=1/2,1/3,1/5 on the L=120\times 120 cluster as well as 1/7,1/9,1/11 plateaus on the L=140\times 140, 90\times 90 and 110\times 110 clusters, where m_s is the saturated magnetization of the localized spins. This sequence is much richer than that of the pure Ising SSL, which supports only the 1/3 plateau. The plateaus at 1/2, 1/3, 1/5, and 1/7 appear as robust, finite intervals of the applied field, while those at 1/9 and 1/11 are narrower but clearly resolvable on sufficiently large clusters. Importantly, the calculated sequence closely mirrors the experimental hierarchy of plateaus in TmB{}_4, where plateaus at 1/7, 1/8, 1/9, and 1/11 precede a dominant 1/2 step.

The microscopic origin of these plateaus is a family of stripe-like ground states formed by the localized spins. The system organizes into parallel antiferromagnetic (AF) bands of even width w separated by fully polarized ferromagnetic (FM) stripes. The value of w uniquely determines the magnetization w=1\Rightarrow m/m_s=1/2,w=2\Rightarrow m/m_s=1/3,w=4\Rightarrow m/m_s=1/5,w=6\Rightarrow m/m_s=1/7 with larger even values (w=8,10) yielding the fractions 1/9 and 1/11. These stripe textures minimize the combined kinetic and magnetic energies: electrons gain kinetic energy in the FM regions (where their motion is less restricted), while AF bands lower the frustrated Ising exchange energy. The resulting commensurate AF/FM patterns represent the compromise that optimizes the total energy at each applied field. Representative stripe patterns for the major plateaus, together with the corresponding magnetization curve, are shown in Fig. 7.

The behaviour of the itinerant electrons reflects the spin textures. For m/m_s\le 1/2, the electron magnetization follows the localized spins closely, especially when J_z is large. In AF bands, electrons experience alternating local fields and reduced mobility, while in FM stripes their spectrum broadens. For m/m_s>1/2, the spins saturate rapidly, but the electrons polarize more gradually, consistent with the persistence of itinerant character even in high magnetic fields.

The finite-temperature stability of the stripe phases was analyzed in Ref. [77] using canonical Monte Carlo simulations on SSL clusters up to 18\times 18 sites. For representative fields corresponding to m/m_s=0, 1/3, and 1/2, pronounced peaks in the specific heat signal thermal transitions from the low-temperature ordered phases to a disordered paramagnetic state. The transition temperatures depend strongly on the magnetization fraction: the AF ground state (m=0) melts at a relatively high reduced temperature {\tau }_c\approx 1.8; the 1/3 stripe phase melts at roughly half this value; and the 1/2 plateau is stable only up to temperatures almost an order of magnitude smaller. These trends highlight the varying energetic cost of destroying AF regions of different widths and FM stripe patterns.

The nature of the thermal transitions is revealed by the energy histograms P(E) near {\tau }_c. For m/m_s=0 and m/m_s=1/2, P(E) remains single-peaked, which according to [93] indicates continuous second-order transitions. For the 1/3 plateau, however, P(E) becomes clearly double-peaked, consistent with a first-order transition and coexistence of ordered and disordered states. Representative specific-heat curves, and energy histograms are shown in Fig. 8.

Taken together, these results demonstrate that the U=0 spin−electron model on the SSL provides a minimal but remarkably accurate microscopic description of the magnetization processes in metallic rare-earth tetraborides. It reproduces the major plateaus at 1/3 and 1/2 as well as the smaller fractional ones at 1/5, 1/7, 1/9, and 1/11, and it captures their finite-temperature behaviour and transition orders. The explicit treatment of itinerant electrons is thus essential for understanding the experimentally observed hierarchy of fractional magnetization plateaus and the thermal properties of these highly frustrated magnetic metals.

3.2 Electron−electron correlation and anisotropic exchange effects

While the spin−electron coupling in the U=0 model already reproduces a hierarchy of fractional magnetization plateaus, the experimentally observed behaviour of metallic rare-earth tetraborides requires several additional microscopic ingredients. In particular, many RB{}_4 compounds display a dominant and very robust 1/2 plateau, whereas the 1/3 plateau is strongly reduced or even entirely absent, as in ErB{}_4 and TmB{}_4. The minimal Ising-type spin−electron model is not sufficient to account for this imbalance, and therefore two natural generalizations have been introduced and analysed in Refs. [78, 79]: on-site electron−electron repulsion of the Hubbard type, and anisotropic exchange interactions that include both local double-exchange processes and transverse Heisenberg terms. These two extensions modify the stability competition between the 1/3 and 1/2 plateau phases in qualitatively different but complementary ways.

The first extension incorporates the Hubbard interaction into the spin−electron Hamiltonian, adding a term that penalizes double occupancy on each site. The resulting model takes the form

(5)H_U=H_0+U{\displaystyle \sum _i{n}_{i\uparrow }{n}_{i\downarrow },}

where H_0 is given by Eq. (4) (with J_1=J_2 and t_1=t_2). The inclusion of U makes the electronic subsystem interacting, necessitating the use of exact diagonalization [94] for L=12 Betts cluster [95, 96] and the Projector Quantum Monte Carlo method [97–99] for clusters of size 12\times 12. These calculations reveal a clear and systematic trend [78]: electron−electron correlations strongly suppress the 1/3 plateau and simultaneously stabilize and widen the 1/2 plateau. Even moderate values such as U=2 increase the width of the 1/2 plateau by roughly a factor of three, while the 1/3 plateau shrinks steadily with increasing U, but it is still much larger than that of the 1/2 plateau. The agreement between ED and PQMC results indicates that these effects persist in the thermodynamic limit.

From a microscopic perspective, the Hubbard term disfavors the formation of narrow antiferromagnetic regions that underlie the 1/3 plateau, since these tend to localize electrons in environments prone to double occupancy. The 1/2 plateau, based on a wider commensurate AF−FM stripe pattern, is energetically more favourable when correlated electrons must avoid double occupancy. This mechanism naturally explains why many RB{}_4 compounds show a very broad and dominant 1/2 plateau, whereas the 1/3 plateau is suppressed or completely absent. Representative magnetization curves demonstrating the correlation-induced disappearance of the 1/3 plateau and simultaneous enhancement of the 1/2 plateau are shown in Fig. 9.

A second extension was developed in Ref. [79], motivated by the observation that real RB{}_4 magnets exhibit finite transverse components of both the local d−f exchange and the 4f−4f interaction. These transverse couplings introduce spin-flip terms into the Hamiltonian and modify the balance between the competing plateau phases even in the absence of Hubbard interaction. The extended spin−electron Hamiltonian becomes

(6)\begin{array}{rl}H=& H_U+2J_{xy}\sum _i(d_{i\uparrow }^{†}{d}_{i\downarrow }{f}_{i\downarrow }^{†}{f}_{i\uparrow }+d_{i\downarrow }^{†}{d}_{i\uparrow }{f}_{i\uparrow }^{†}{f}_{i\downarrow })\\ & +J_{xy}^{\prime }\sum _{⟨ij⟩}(f_{i\uparrow }^{†}{f}_{i\downarrow }{f}_{j\downarrow }^{†}{f}_{j\uparrow }+f_{i\downarrow }^{†}{f}_{i\uparrow }{f}_{j\uparrow }^{†}{f}_{j\downarrow })\\ & +J_{xy}^{\prime }\sum _{⟨⟨ij⟩⟩}(f_{i\uparrow }^{†}{f}_{i\downarrow }{f}_{j\downarrow }^{†}{f}_{j\uparrow }+f_{i\downarrow }^{†}{f}_{i\uparrow }{f}_{j\uparrow }^{†}{f}_{j\downarrow }),\end{array}

where H_U is given by Eq. (5) and in accordance with the geometry of the SSL in RB{}_4 compounds, the same value of the transverse Heisenberg exchange term J_{xy}^{\prime } is assumed for nearest- and next-nearest-neighbour spins. Exact Lanczos and truncated Lanczos calculations on L=12 and L=36 clusters show that the two transverse exchange parameters have completely opposite effects on the magnetization process. The on-site double-exchange term J_{xy} tends to destroy the 1/2 plateau and stabilizes the 1/3 plateau; once J_{xy} exceeds approximately J_z/2, the 1/2 plateau disappears altogether, while the 1/3 plateau remains present up to the isotropic limit J_{xy}=J_z. In contrast, the transverse Heisenberg term J_{xy}^{\prime } stabilizes the 1/2 plateau and suppresses the 1/3 plateau: for J_{xy}^{\prime }>0.4, the width of the 1/3 plateau is reduced by a factor of nearly 2.5, while the width of the 1/2 plateau increases by a factor of about 3.5 compared to the purely Ising case.

Consequently, the transverse exchange parameters provide two distinct microscopic routes for tuning the 1/3−1/2 plateau competition. The double-exchange term weakens the stripe patterns with wide antiferromagnetic regions characteristic of the 1/2 plateau, whereas the transverse Heisenberg interaction strengthens exactly those configurations. In materials with strong Ising anisotropy (J_{xy},J_{xy}^{\prime }\ll J_z) the double-exchange term is relatively unimportant, but even small J_{xy}^{\prime } can substantially modify the plateau widths. This observation highlights the need to include transverse 4f−4f exchange when modelling RB{}_4 magnets and simultaneously explains why the simplified Ising spin−electron model remains valid when these transverse terms are weak. Representative magnetization curves showing these opposite effects of J_{xy} and J_{xy}^{\prime } are presented in Fig. 10.

In summary, electron−electron correlations and anisotropic exchange processes represent two independent mechanisms that both drive the system toward the experimentally observed dominance of the 1/2 plateau and suppression of the 1/3 plateau. The Hubbard interaction disfavors the narrow antiferromagnetic bands that stabilize the 1/3 plateau, while the transverse Heisenberg exchange directly strengthens the stripe structure responsible for the 1/2 plateau. Together, these effects significantly improve the agreement between theoretical predictions and the magnetization curves of metallic rare-earth tetraborides.

3.3 Full correlated spin−electron model and CDI mapping

In the previous subsection we have discussed the influence of electron-electron correlations and anisotropic exchange couplings within the spin−electron framework. The next natural step towards a complete and realistic description of metallic Shastry−Sutherland magnets is to introduce the real spin values of both subsystems, corresponding to the 4f ions (S=1,3/2,2) and itinerant 5d electrons (s_d=1/2), and to examine how this mixed-spin structure modifies the ground-state and thermodynamic properties. This generalization, first analysed in Ref. [80], provided the foundation for a systematic mapping to the effective CDI model [81], which allowed one to study much larger clusters and to explore finite-temperature effects including the magnetocaloric response [82].

To model realistically the situation in rare-earth tetraborides, the localized spins S_i^z=\pm S are assigned the real spin values corresponding to the magnetic ions Ho{}^{3+} (S=2), Er{}^{3+} (S=3/2) and Tm{}^{3+} (S=1), while the itinerant d electrons possess the spin s_d=1/2. The spin subsystem is described by the Ising interaction between the first- (J_1) and second-nearest (J_2) neighbours, whereas the itinerant electrons are governed by the Hubbard Hamiltonian with on-site Coulomb repulsion U. Both subsystems are coupled through an anisotropic Ising-type interaction J_z, and the magnetic field h acts on the spins and electrons separately. Thus, the mixed-spin Hamiltonian is formally identical to the one discussed in the previous section [Eq. (5)], the only modification being the use of realistic spin values for the localized magnetic moments and itinerant electrons. In the next, the case J_1=J_2 and t_1=t_2 is considered, corresponding to identical coupling/hopping amplitudes along the square and diagonal bonds of the Shastry−Sutherland lattice.

Using exact diagonalization on the Betts cluster (L=12 sites) and the Projector Quantum Monte Carlo method on 6\times 6 lattices, the authors investigated how the Coulomb interaction U and the mixed-spin combinations affect magnetization processes. The resulting ground-state phase diagrams in the h−U plane (Fig. 11) show that for S=1 and S=3/2 the Coulomb interaction stabilizes the single macroscopic m/m_s=1/2 plateau and suppresses the 1/3 plateau, in excellent agreement with the experimental magnetization curves of TmB{}_4 and ErB{}_4. For S=2 (corresponding to HoB{}_4), the 1/2 plateau disappears and a different sequence with 1/3 and 7/15 plateaus is observed, again in accord with experiments.

Finite-temperature simulations (Fig. 12) further revealed that the specific heat capacity exhibits either a single-peaked or a double-peaked structure depending on the spin value S and the coupling J_z. The high-temperature peak originates from the spin subsystem, while the low-temperature peak arises from the itinerant electrons. For large U, the electron peak is shifted to very low temperatures (below 0.2 in reduced units), which explains why only the single high-temperature peak is detected experimentally, the lower electron contribution lies far below the 2 K range typically accessible in measurements.

The mixed-spin Hamiltonian is still fully quantum and numerically demanding. To overcome the size limitations, Ref. [81] introduced a mapping to the simpler CDI model valid under three physically realistic conditions: (i) the Coulomb interaction U between conduction electrons is large (U\gg t); (ii) the conduction band is half-filled (n=1); and (iii) the system exhibits strong easy-axis anisotropy. Under these assumptions, the itinerant-electron subsystem described by the Hubbard model can be replaced by an effective Ising branch with coupling J_{ij}^{\prime }=4t_{ij}^2/U [100], leading to the effective Hamiltonian

(7)\begin{array}{rl}{H}_{\mathrm{C}\mathrm{D}\mathrm{I}}=& \sum _{ij}{J}_{ij}{S}_i^z{S}_j^z+\sum _{ij}{J}_{ij}^{\prime }{s}_i^z{s}_j^z+J_z\sum _i{s}_i^z{S}_i^z\\ & -h\sum _i(S_i^z+s_i^z),\end{array}

which describes two interacting Ising lattices: the spin branch (S_i^z) and the electron branch (s_i^z). The exchange couplings J_{ij} (J_{ij}^{\prime }) correspond to the Ising interactions J_1 and J_2 (J_1^{\prime } and J_2^{\prime }) acting on the nearest- and next-nearest neighbors of the Shastry–Sutherland lattice, respectively. In the following, these couplings are taken equal, J_1=J_2=J (J_1^{\prime }=J_2^{\prime }=J^{\prime }). The inter-branch coupling J_z can be ferromagnetic () or antiferromagnetic (J_z>0).

Extensive Monte Carlo simulations on clusters up to 60\times 60 sites demonstrated that the CDI model reproduces the essential physics of the full spin−electron Hamiltonian even for moderate J_z. The comparison of magnetization curves for both models (Fig. 13) shows nearly perfect coincidence in the strong-coupling regime.

The CDI model allows simulations on much larger clusters and thus reveals a richer hierarchy of magnetization plateaus. Depending on the parameters (J^{\prime },J_z,S), several characteristic sequences appear, among which the dominant 1/2 plateau phase persists for S=1 and 3/2 (TmB{}_4, ErB{}_4), while the 1/3 phase is strongly suppressed. For S=2 (HoB{}_4), the 1/3 phase remains dominant, in full analogy with experiments.

Finite-temperature results for the CDI model reproduce the experimentally observed single high-temperature peak in the specific heat capacity and predict an additional low-temperature peak originating from the electron branch (Fig. 15). The latter is expected to occur below the temperature range accessible in experiments, which explains its absence in current data on TmB{}_4 and ErB{}_4.

The most recent extension of this framework [82] focused on the magnetocaloric properties of the CDI model. Using Monte Carlo simulations combined with the parallel-tempering technique, the field dependence of the isothermal entropy change \mathrm{\Delta }S(T,h)=S(T,h)-S(T,0) and the relative cooling power were computed for representative parameter sets corresponding to TmB{}_4 and ErB{}_4. The results (Fig. 16) reveal alternating positive and negative peaks of \mathrm{\Delta }S associated with successive metamagnetic transitions between fractional plateau states. The main cooling region is located near the transition between the low-field antiferromagnetic phase and the 1/2 plateau, where \mathrm{\Delta }S reaches its maximal magnitude. This alternating “heating-cooling” pattern of \mathrm{\Delta }S reproduces closely the experimental magnetocaloric behaviour of metallic Shastry−Sutherland compounds.

In summary, the full correlated spin−electron model with realistic mixed spins and its mapping onto the coupled double-Ising model provide a unified microscopic framework capable of describing both the magnetic and thermodynamic properties of rare-earth tetraborides. The inclusion of real spin values and Coulomb correlations accounts for the material-dependent stabilization of magnetization plateaus, while the CDI representation enables large-scale simulations that reproduce experimental features of specific heat and magnetocaloric response. These results demonstrate that despite its relative simplicity, the CDI model retains the essential physics of the full spin−electron Hamiltonian and offers an efficient route for further studies of correlated spin-electron systems on frustrated lattices.

3.4 Impact of charge stripes on magnetization processes

The most recent development in the modelling of magnetization processes in rare-earth tetraborides is the inclusion of charge inhomogeneity, motivated by experimental observations of charge stripes in structurally related borides [101–106]. In Ref. [83] a phenomenological spatially varying local potential V_i was introduced into the spin−electron Hamiltonian to account for possible charge ordering arising from static or dynamic Jahn-Teller distortions of boron clusters, similar to those observed in rare-earth hexaborides [105] and dodecaborides [106]. Although the charge modulation is introduced here in an ad-hoc manner, similarly to some previous works [107–109] addressing related problems, its inclusion provides a microscopic description of the magnetization processes fully consistent with the experimental data for TmB{}_4 and ErB{}_4.

The total Hamiltonian was written as

(8)H=H_0+{\displaystyle \sum _{i,\sigma }{V}_i{n}_{i\sigma },}

where H_0 denotes the conventional spin-electron model on the Shastry−Sutherland lattice given by Eq. (4). The term with V_i represents a periodic modulation of the local electronic potential and generates different types of charge stripes. The model was solved using an exact-diagonalization-based numerical scheme on L=6\times 6 and 12\times 12 clusters for all nonequivalent periodic potentials up to period p=4.

The calculations revealed that the form and intensity of the potential V_i have a dramatic impact on the stability of individual magnetization plateaus. Among all 1798 periodic potentials examined on the 6\times 6 cluster, the strongest effect was found for the configuration composed of full axial charge stripes alternating with chessboard stripes (Fig. 17). This pattern enhances the m/m_s=1/2 plateau and simultaneously suppresses the 1/3 plateau, leading to a very simple magnetization curve containing only these two main fractional plateaus and the zero-field phase. The enlargement of the 1/2 plateau and the reduction of the 1/3 plateau become more pronounced with increasing amplitude V and spin−electron coupling J_z.

Further tests on larger 12\times 12 clusters confirmed the robustness of this result. Two symmetry-inequivalent variants of the stripe potential, denoted V_i^{(1)} and V_i^{(2)}, were analysed (Fig. 18). Both produce nearly identical magnetization curves dominated by the 1/2 plateau and show complete suppression of the 1/3 plateau for large V. The ground-state spin configuration within the 1/2 phase is commensurate with the stripe pattern and coincides with the magnetic structure experimentally observed in TmB{}_4.

The calculated local electron occupations n_i=⟨n_{i\uparrow }+n_{i\downarrow }⟩ demonstrate that increasing V induces a strong modulation of the electronic density, producing alternating fully and partially occupied stripes analogous to the conducting channels observed in CeB{}_6 and LuB{}_{12}. This commensurability between charge and spin order provides a natural microscopic explanation for the robust 1/2 plateau and the disappearance of the 1/3 phase in TmB{}_4 and ErB{}_4.

To quantify the comparative efficiency of different theoretical models in stabilizing the 1/2 plateau, Ref. [83] evaluated the ratio {\mathrm{\Delta }}_{1/2}/{\mathrm{\Delta }}_{1/3} of the plateau widths for several representative frameworks (see Table 1). The spin-electron model with charge stripes (SEM+ChS) achieved {\mathrm{\Delta }}_{1/2}/{\mathrm{\Delta }}_{1/3}\to \mathrm{\infty }, outperforming all previous models including the Ising (0), anisotropic Heisenberg (0.46), Ising+J_3 (0.67), Ising+RKKY (1.86), and correlated spin−electron (0.47) models.

In conclusion, the introduction of charge stripes into the spin−electron framework yields a minimal and robust mechanism for the complete suppression of the 1/3 plateau and stabilization of the 1/2 plateau over a broad parameter range. The resulting magnetization curves reproduce the experimental behaviour of TmB{}_4 and ErB{}_4 without invoking long-range or higher-order interactions. Although the local potential V_i is phenomenological, its success suggests that intrinsic charge modulations – possibly driven by Jahn−Teller distortions of boron clusters – may indeed be the missing ingredient governing the magnetization processes in the rare-earth tetraborides.

4 Conclusion

The theoretical description of magnetization processes and thermodynamic behaviour in rare-earth tetraborides represents a uniquely challenging problem, combining strong geometrical frustration, itinerant‐electron effects, long-range interactions, and material-specific anisotropies. In this review we have traced the evolution of theoretical approaches from the earliest Ising-type descriptions on the Shastry−Sutherland lattice to progressively more realistic formulations that incorporate indirect RKKY exchange, explicit spin−electron coupling, on-site electron correlations, transverse exchange terms, mixed-spin generalizations, and the influence of charge inhomogeneity. Taken together, these studies now provide a coherent microscopic picture capable of explaining the experimentally observed hierarchy of fractional magnetization plateaus, the dominance of the half-magnetization state in several members of the RB{}_4 family, and the unusual magnetocaloric signatures characteristic of these metallic frustrated magnets.

The RKKY-based models represent the first major step beyond purely localized spin descriptions. Their significance lies in providing a direct microscopic origin for the longer-range competing interactions that were formerly introduced ad hoc into classical Ising Hamiltonians. By expressing the exchange integrals in terms of the Fermi momentum of conduction electrons, the RKKY framework establishes an explicit connection between electronic structure and magnetic ordering. The early work of Feng et al. demonstrated that even the simplest 3D RKKY form is capable of stabilizing the experimentally dominant 1/2 plateau in TmB{}_4. Subsequent refinements by Regeciová and Farkašovský revealed a far richer landscape: when realistic simulation sizes, parallel tempering, and mixed 3D−2D approximations are used, the RKKY interaction produces not only the 1/2 plateau, but also a cascade of fractional plateaus (1/3, 3/8, 7/18, ···) and a dense sequence of ferrimagnetic superstructures. The 2D RKKY formulation highlighted the role of electronic dimensionality by simplifying the magnetization curve into essentially a single robust 1/2 plateau with few subsidiary steps, closely reproducing the behaviour of ErB{}_4. In contrast, the fully 3D RKKY model gave rise to a hierarchical sequence of commensurate phases remarkably similar to those observed in TmB{}_4 and TbB{}_4, including the elusive 1/8 plateau. These findings collectively demonstrate that the oscillatory, long-range, Fermi-surface-controlled character of the RKKY interaction is a fundamental ingredient in understanding why different RB{}_4 compounds display different plateau sequences. They also highlight the microscopic sensitivity of the magnetic phase diagram to the carrier concentration, explaining why doping produces pronounced modifications of plateau stability and critical fields.

Despite the success of the indirect-electron approaches, their inability to describe the feedback of the magnetic background on the itinerant subsystem represents a conceptual limitation. Explicit spin-electron models overcome this difficulty by treating the conduction electrons as active quantum degrees of freedom coupled locally to the Ising-like rare-earth moments. The simplest U=0 model already produces a strikingly realistic hierarchy of magnetization plateaus, including 1/2, 1/3, 1/5, 1/7, 1/9 and 1/11, with the correct stripe-type magnetic textures. The fact that these structures emerge solely from the competition between electron hopping and local spin−electron coupling is of fundamental importance: it shows that commensurate AF−FM superstructures in RB{}_4 are not artifacts of long-range exchange, but intrinsic patterns favored by itinerant electrons on the frustrated SSL. The follow-up finite-temperature study clarified the thermodynamic stability of these plateaus and established that the melting of the 1/3 plateau is first-order while the 1/2 plateau melts continuously at much lower temperatures. This dichotomy naturally reflects the different widths of the AF bands underlying each plateau, reinforcing the view that the stripe geometry is the microscopic key to plateau formation.

The inclusion of electron−electron correlations represents another crucial milestone. The correlated spin-electron model with a finite Hubbard interaction revealed an especially important result: electron correlations strongly stabilize the 1/2 plateau and suppress the 1/3 plateau. Even moderate repulsion enhances the energetic cost of narrow AF regions and therefore penalizes the 1/3 stripes, while broadening and strengthening the 1/2 plateau. This provides a compelling and microscopically grounded explanation for the behaviour of ErB{}_4 and TmB{}_4, where the 1/3 plateau is absent or extremely narrow and the magnetization curve is dominated by the half-magnetization state. Notably, this conclusion emerges independently of the RKKY mechanism and demonstrates that electron correlations alone can drive the system toward experimentally observed plateau structures.

The model incorporating transverse exchange terms (J_{xy} and J_{xy}^{\prime }) revealed a complementary and equally enlightening perspective. The local double-exchange process J_{xy} destabilizes the 1/2 plateau while favouring the 1/3 plateau, whereas the transverse Heisenberg coupling J_{xy}^{\prime } has precisely the opposite effect: it stabilizes the 1/2 plateau and suppresses the 1/3 one. This competition elucidates the delicate balance between longitudinal and transverse components of the exchange in real RB{}_4 materials. The conclusion that even weak transverse interactions can significantly reorganize the plateau widths is important because tetraborides are known to exhibit substantial single-ion anisotropy, implying that the experimentally relevant region lies in the regime where J_{xy}^{\prime } produces an enhancement of the 1/2 plateau consistent with experimental observations.

The mixed-spin formulations and the subsequent mapping to the CDI model represent the final step toward a fully realistic microscopic theory. By incorporating the actual spin magnitudes (S=1, 3/2, 2) of Tm{}^{3+}, Er{}^{3+} and Ho{}^{3+} ions, the model reproduces the key material-dependent features of the magnetization curves: the stabilization of the 1/2 plateau for S=1 and 3/2 and the persistence of the 1/3 plateau for S=2. This provides a unified explanation for why TmB{}_4 and ErB{}_4 behave differently from HoB{}_4. The CDI mapping allows simulations on significantly larger clusters, confirming the stability of these phases in the thermodynamic limit and offering, for the first time, a realistic description of both magnetic and thermodynamic responses including the split specific-heat peaks and the magnetocaloric effect.

Finally, the most recent development, the incorporation of charge-stripe potentials, provides a simple yet remarkably effective mechanism for completely suppressing the 1/3 plateau and stabilizing the 1/2 plateau across a wide parameter range. This result is particularly relevant for TmB{}_4 and ErB{}_4, where hints of charge modulation have been reported in related borides, and where Jahn−Teller distortions of boron clusters are known to influence the electronic structure. The fact that a periodically modulated local potential can bring the spin−electron model into almost perfect agreement with experiments strongly suggests that charge inhomogeneity is not merely a secondary correction but may represent a missing microscopic ingredient in the full theory of metallic Shastry−Sutherland magnets.

Across all approaches considered in this review, RKKY, spin−electron, Hubbard-enhanced, anisotropic exchange, mixed-spin, CDI, and charge-stripe models, a consistent physical picture emerges. The magnetization processes in rare-earth tetraborides are governed by a robust interplay between itinerant electrons and localized moments, modulated by geometrical frustration and strong single-ion anisotropy. The half-magnetization plateau is an intrinsic and exceptionally stable feature of this interplay, reinforced by electron correlations, transverse spin interactions, and charge modulation. The one-third plateau, while natural in purely localized Ising models, is fragile in metallic systems and survives only when electron correlations, transverse exchange, or charge modulation do not suppress the narrow AF bands that generate it. The overall magnetization sequence and thermodynamic signatures are highly sensitive to the Fermi surface geometry and carrier concentration, explaining the strong material- and doping-dependence observed experimentally.

In conclusion, the combined body of work reviewed here demonstrates that the rare-earth tetraborides form a uniquely rich platform where frustrated magnetism, electronic correlations, anisotropy and itinerancy coexist and interact on equal footing. Theoretical progress over the past decade has transformed our understanding of these materials: from early spin-only models incapable of reproducing experimental curves, to sophisticated correlated spin−electron and CDI frameworks capable of explaining not only the plateau structure, but also specific heat, thermal transitions, magnetocaloric signatures, and charge-spin commensurability. The resulting picture is both conceptually coherent and quantitatively compelling. Yet, important open questions remain, most notably, the microscopic origin and dynamics of possible charge ordering, the role of quantum fluctuations beyond the Ising limit, effects of orbital component, and the degree to which conduction electrons participate in collective excitations in the plateau phases. Addressing these challenges will likely require a combination of advanced numerical techniques, improved experimental resolution, and further refinement of mixed spin−electron models. Nevertheless, the progress to date provides a solid foundation for such future work and firmly establishes the Shastry−Sutherland rare-earth tetraborides as a paradigmatic class of correlated frustrated electron−spin materials.

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