Ground-state phase diagram and universality of sequential topological valence-bond-solid quantum transitions in a mixed tetramer chain.

The ground-state ordering of a quantum mixed-spin Heisenberg tetramer chain composed of an alternate sequence ofs = 1 andS=3/2dimers is studied in detail as a function of two considered exchange interactions ascribed to similar and dissimilar spin pairs. At zero magnetic field, the ferrimagnetic mixed spin-(1, 1, 3/2, 3/2) Heisenberg tetramer chain displays, depending on a mutual interplay between two considered exchange interactions, three distinct gapped valence-bond-solid phases separated by gap-closing quantum critical points. Using density-matrix renormalization group calculations we construct the full ground-state phase diagram as a function of the interaction ratio and magnetic field, which exhibits besides three gapped valence-bond-solid phases special Kosterlitz-Thouless and topological quantum critical points. A tangential finite-size scaling analysis is employed to obtain precise estimates of the zero-field valence-bond-solid transitions and unveil their common logarithmic correction to a power-law scaling of the correlation length.

Magnetization plateaus and enhanced magnetocaloric effect of a spin-1/2 Ising-Heisenberg and Heisenberg double sawtooth ladder with four-spin interaction.

The ground state, entropy, and magnetic Grüneisen parameter of the antiferromagnetic spin-1/2 Ising-Heisenberg model on a double sawtooth ladder are rigorously investigated using the classical transfer-matrix technique. The model includes the XXZ interaction between the interstitial Heisenberg dimers, the Ising coupling between nearest-neighbor spins of the legs and rungs, and additional cyclic four-spin Ising term in each square plaquette. For a particular value of the cyclic four-spin exchange, we found in the ground-state phase diagram of the Ising-Heisenberg ladder a quadruple point, at which four different ground states coexist together. During an adiabatic demagnetization process, a fast cooling accompanied with an enhanced magnetocaloric effect can be detected near this quadruple point. The ground-state phase diagram of the Ising-Heisenberg ladder is confronted with the zero-temperature magnetization process of the purely quantum Heisenberg ladder, which is calculated by using exact diagonalization based on the Lanczos algorithm for a finite-size ladder of 24 spins and the density-matrix renormalization group simulations for a finite-size ladder with up to 96 spins. Some indications of the existence of intermediate magnetization plateaus in the magnetization process of the full Heisenberg model for a small but nonzero four-spin Ising coupling were found. The DMRG results reveal that the quantum Heisenberg double sawtooth ladder exhibits some quantum Luttinger spin-liquid phase regions that are absent in the Ising-Heisenberg counterpart model. Except this difference, the magnetic behavior of the full Heisenberg model is quite analogous to its simplified Ising-Heisenberg counterpart and, hence, may bring insight into the fully quantum Heisenberg model from rigorous results for the Ising-Heisenberg model.

Distribution of Bipartite and Tripartite Entanglement within a Spin-1/2 Heisenberg Star in a Magnetic Field.

The spatial distribution of entanglement within a spin-1/2 Heisenberg star composed from a single central spin and three peripheral spins is examined in the presence of an external magnetic field using the Kambe projection method, which allows an exact calculation of the bipartite and tripartite negativity serving as a measure of the bipartite and tripartite entanglement. Apart from a fully separable polarized ground state emergent at high-enough magnetic fields, the spin-1/2 Heisenberg star exhibits at lower magnetic fields three outstanding nonseparable ground states. The first quantum ground state exhibits the bipartite and tripartite entanglement over all possible decompositions of the spin star into any pair or triad of spins, whereby the bipartite and tripartite entanglement between the central and peripheral spins dominates over that between the peripheral spins. The second quantum ground state has a remarkably strong tripartite entanglement between any triad of spins in spite of the lack of bipartite entanglement. The central spin of the spin star is separable from the remaining three peripheral spins within the third quantum ground state, where the peripheral spins are subject to the strongest tripartite entanglement arising from a two-fold degenerate W-state.

Towards lattice-gas description of low-temperature properties above the Haldane and cluster-based Haldane ground states of a mixed spin-(1,1/2) Heisenberg octahedral chain.

The rich ground-state phase diagram of the mixed spin-(1,1/2) Heisenberg octahedral chain was previously elaborated from effective mixed-spin Heisenberg chains, which were derived by employing a local conservation of a total spin on square plaquettes of an octahedral chain. Here we present a comprehensive analysis of the thermodynamic properties of this model. In the highly frustrated parameter region the lowest-energy eigenstates of the mixed-spin Heisenberg octahedral chain belong to flat bands, which allow a precise description of low-temperature magnetic properties within the localized-magnon approach exploiting a classical lattice-gas model of hard-core monomers. The present article provides a more comprehensive version of the localized-magnon approach, which extends the range of its validity down to a less frustrated parameter region involving the Haldane and cluster-based Haldane ground states. A comparison between results of the developed localized-magnon theory and accurate numerical methods such as full exact diagonalization and finite-temperature Lanczos technique convincingly evidence that the low-temperature magnetic properties above the Haldane and the cluster-based Haldane ground states can be extracted from a classical lattice-gas model of hard-core monomers and dimers, which is additionally supplemented by a hard-core particle spanned over the whole lattice representing the gapped Haldane phase.

Unconventional spin frustration due to two competing ferromagnetic interactions of a spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices.

The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.

Unconventional Thermal and Magnetic-Field-Driven Changes of a Bipartite Entanglement of a Mixed Spin-(1/2,S) Heisenberg Dimer with an Uniaxial Single-Ion Anisotropy.

The concept of negativity is adapted in order to explore the quantum and thermal entanglement of the mixed spin-(1/2,S) Heisenberg dimers in presence of an external magnetic field. The mutual interplay between the spin size S, XXZ exchange and uniaxial single-ion anisotropy is thoroughly examined with a goal to tune the degree and thermal stability of the pairwise entanglement. It turns out that the antiferromagnetic spin-(1/2,S) Heisenberg dimers exhibit higher degree of entanglement and higher threshold temperature in comparison with their ferromagnetic counterparts when assuming the same set of model parameters. The increasing spin magnitude S accompanied with an easy-plane uniaxial single-ion anisotropy can enhance not only the thermal stability but simultaneously the degree of entanglement. It is additionally shown that the further enhancement of a bipartite entanglement can be achieved in the mixed spin-(1/2,S) Heisenberg dimers, involving half-odd-integer spins S. Under this condition the thermal negativity saturates at low-enough temperatures in its maximal value regardless of the magnitude of half-odd-integer spin S. The magnetic field induces consecutive discontinuous phase transitions in the mixed spin-(1/2,S) Heisenberg dimers with S>1, which are manifested in a surprising oscillating magnetic-field dependence of the negativity observed at low enough temperature.

Weak Singularities of the Isothermal Entropy Change as the Smoking Gun Evidence of Phase Transitions of Mixed-Spin Ising Model on a Decorated Square Lattice in Transverse Field.

The magnetocaloric response of the mixed spin-1/2 and spin-S (S>1/2) Ising model on a decorated square lattice is thoroughly examined in presence of the transverse magnetic field within the generalized decoration-iteration transformation, which provides an exact mapping relation with an effective spin-1/2 Ising model on a square lattice in a zero magnetic field. Temperature dependencies of the entropy and isothermal entropy change exhibit an outstanding singular behavior in a close neighborhood of temperature-driven continuous phase transitions, which can be additionally tuned by the applied transverse magnetic field. While temperature variations of the entropy display in proximity of the critical temperature Tc a striking energy-type singularity (T-Tc)log|T-Tc|, two analogous weak singularities can be encountered in the temperature dependence of the isothermal entropy change. The basic magnetocaloric measurement of the isothermal entropy change may accordingly afford the smoking gun evidence of continuous phase transitions. It is shown that the investigated model predominantly displays the conventional magnetocaloric effect with exception of a small range of moderate temperatures, which contrarily promotes the inverse magnetocaloric effect. It turns out that the temperature range inherent to the inverse magnetocaloric effect is gradually suppressed upon increasing of the spin magnitude S.

Magnetic-Field-Orientation Dependent Thermal Entanglement of a Spin-1 Heisenberg Dimer: The Case Study of Dinuclear Nickel Complex with an Uniaxial Single-Ion Anisotropy.

The bipartite entanglement in pure and mixed states of a quantum spin-1 Heisenberg dimer with exchange and uniaxial single-ion anisotropies is quantified through the negativity in a presence of the external magnetic field. At zero temperature the negativity shows a marked stepwise dependence on a magnetic field with two abrupt jumps and plateaus, which can be attributed to the quantum antiferromagnetic and quantum ferrimagnetic ground states. The magnetic-field-driven phase transition between the quantum antiferromagnetic and quantum ferrimagnetic ground states manifests itself at nonzero temperatures by a local minimum of the negativity, which is followed by a peculiar field-induced rise of the negativity observable in a range of moderately strong magnetic fields. The rising temperature generally smears out abrupt jumps and plateaus of the negativity, which cannot be distinguished in the relevant dependencies above a certain temperature. It is shown that the thermal entanglement is most persistent against rising temperature at the magnetic field, for which an energy gap between a ground state and a first excited state is highest. Besides, temperature variations of the negativity of the spin-1 Heisenberg dimer with an easy-axis single-ion anisotropy may exhibit a singular point-kink, at which the negativity has discontinuity in its first derivative. The homodinuclear nickel complex [Ni2(Medpt)2(µ-ox)(H2O)2](ClO4)2·2H2O provides a suitable experimental platform of the antiferromagnetic spin-1 Heisenberg dimer, which allowed us to estimate a strength of the bipartite entanglement between two exchange-coupled Ni2+ magnetic ions on the grounds of the interaction constants reported previously from the fitting procedure of the magnetization data. It is verified that the negativity of this dinuclear compound is highly magnetic-field-orientation dependent due to presence of a relatively strong uniaxial single-ion anisotropy.

Nature of intermediate magnetization plateaus of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice resembling a triangulated kagome lattice.

The spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice is exactly solved in a magnetic field within the framework of the generalized star-triangle transformation and the method of exact recursion relations. The generalized star-triangle transformation establishes an exact mapping correspondence with the effective spin-1/2 Ising model on a triangular Husimi lattice with a temperature-dependent field, pair and triplet interactions, which is subsequently rigorously treated by making use of exact recursion relations. The ground-state phase diagram of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice, which bears a close resemblance with a triangulated kagomé lattice, involves, in total, two classical and three quantum ground states manifested in respective low-temperature magnetization curves as intermediate plateaus at 1/9, 1/3, and 5/9 of the saturation magnetization. It is verified that the fractional magnetization plateaus of quantum nature have character of either dimerized or trimerized ground states. A low-temperature magnetization curve of the spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice resembling a triangulated kagome lattice may exhibit either no intermediate plateau, a single 1/3 plateau, a single 5/9 plateau, or a sequence of 1/9, 1/3, and 5/9 plateaus depending on a character and relative size of two considered coupling constants.

Magnetic behavior of a ferroferrimagnetic ternary alloy AB_{ρ}C_{1-ρ} with a selective site disorder: Case study of a mixed-spin Ising model on a honeycomb lattice.

Phase transitions, compensation phenomenon, and magnetization of a ferroferrimagnetic ternary alloy AB_{ρ}C_{1-ρ} composed of three different kinds of magnetic ions A, B, and C with the spin magnitudes 1/2, 1, and 3/2 are examined within the framework of a mixed-spin Ising model on a honeycomb lattice with a selective annealed site disorder on one of its two sublattices. It is supposed that the first sublattice of a bipartite honeycomb lattice is formed by the spin-1/2 magnetic ions, while the sites of the second sublattice are randomly occupied either by the spin-1 magnetic ions with a probability ρ or the spin-3/2 magnetic ions with a probability 1-ρ, both being subject to a uniaxial single-ion anisotropy. The model under investigation can be exactly mapped into an effective spin-1/2 Ising model on a triangular lattice through the generalized star-triangle transformation. For a specific concentration of the spin-1 (spin-3/2) magnetic ions, it is shown that the ferroferrimagnetic version of the studied model may display a compensation temperature at which the total magnetization vanishes below a critical temperature. The critical temperature strikingly may also become independent of the concentration of the randomly mixed spin-1 and spin-3/2 magnetic ions for a specific value of a uniaxial single-ion anisotropy. The spontaneous magnetic order may be notably restored at finite temperatures through the order-by-disorder mechanism above a disordered ground state, which results in an anomalous temperature dependence of the total magnetization with double reentrant phase transitions.

Enhanced magnetoelectric effect near a field-driven zero-temperature quantum phase transition of the spin-1/2 Heisenberg-Ising ladder.

The magnetoelectric effect of a spin-1/2 Heisenberg-Ising ladder in the presence of external electric and magnetic fields is rigorously examined by taking into account the Katsura-Nagaosa-Balatsky mechanism. It is shown that an applied electric field may control the quantum phase transition between a Néel (stripy) ordered phase and a disordered paramagnetic phase. The staggered magnetization vanishes according to a power law with an Ising-type critical exponent 1/8, the electric polarization exhibits a weak singularity, and the dielectric susceptibility shows a logarithmic divergence at this particular quantum phase transition. The external electric field may alternatively invoke a discontinuous phase transition accompanied with abrupt jumps of the dielectric polarization and susceptibility on the assumption that the external magnetic field becomes nonzero.

Peculiarities in pseudo-transitions of a mixed spin-(1/2, 1) Ising-Heisenberg double-tetrahedral chain in an external magnetic field.

Recently, it has been rigorously verified that several one-dimensional (1D) spin models may exhibit a peculiar pseudo-transition accompanied with anomalous response of thermodynamic quantities in a close vicinity of pseudo-critical temperature. In the present work we will introduce and exactly solve a mixed spin-(1/2,1) Ising-Heisenberg double-tetrahedral chain in an external magnetic field as another particular example of 1D lattice-statistical model with short-range interactions that displays a pseudo-transition of this type. The investigated model exhibits at zero temperature three ferrimagnetic phases, three frustrated phases, and one saturated paramagnetic phase. The ground-state phase diagram involves five unusual interfaces (phase boundaries), at which the residual entropy per site equals to a larger entropy of one of two coexisting phases. Four such interfaces are between a non-degenerate ferrimagnetic phase and a macroscopically degenerate frustrated phase, while one interface is between two non-degenerate ferrimagnetic phases. Though thermal excitations typically destroy all fingerprints of zero-temperature phase transitions of 1D lattice-statistical models with short-range forces, the mixed spin-(1/2,1) Ising-Heisenberg double-tetrahedral chain is quite robust with respect to thermal excitations and it displays peculiar pseudo-transitions close to all five aforementioned interfaces.

Breakdown of intermediate one-half magnetization plateau of spin-1/2 Ising-Heisenberg and Heisenberg branched chains at triple and Kosterlitz-Thouless critical points.

The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is rigorously solved in a magnetic field by the transfer-matrix approach. The ground-state phase diagram, the magnetization process and the concurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are examined in detail. Three different ground states were found depending on a mutual interplay between the magnetic field and two different coupling constants: the modulated quantum antiferromagnetic phase, the quantum ferrimagnetic phase, and the classical ferromagnetic phase. Two former quantum ground states are manifested in zero-temperature magnetization curves as intermediate plateaus at zero and one-half of the saturation magnetization, whereas the one-half plateau disappears at a triple point induced by a strong-enough ferromagnetic Ising coupling. The ground-state phase diagram and zero-temperature magnetization curves of the analogous spin-1/2 Heisenberg branched chain were investigated using density-matrix renormalization group calculations. The latter fully quantum Heisenberg model involves, besides two gapful phases manifested as zero and one-half magnetization plateaus, gapless quantum spin-liquid phase. The intermediate one-half plateau of the spin-1/2 Heisenberg branched chain vanishes at Kosterlitz-Thouless quantum critical point between gapful and gapless quantum ground states unlike the triple point of the spin-1/2 Ising-Heisenberg branched chain.

Universality and quasicritical exponents of one-dimensional models displaying a quasitransition at finite temperatures.

Quasicritical exponents of one-dimensional models displaying a quasitransition at finite temperatures are examined in detail. The quasitransition is characterized by intense sharp peaks in physical quantities such as specific heat and magnetic susceptibility, which are reminiscent of divergences accompanying a continuous (second-order) phase transition. The question whether these robust finite peaks follow some power law around the quasicritical temperature is addressed. Although there is no actual divergence of these quantities at a quasicritical temperature, a power-law behavior fits precisely both ascending as well as descending parts of the peaks in the vicinity but not too close to a quasicritical temperature. The specific values of the quasicritical exponents are rigorously calculated for a class of one-dimensional models (e.g., Ising-XYZ diamond chain, coupled spin-electron double-tetrahedral chain, Ising-XXZ two-leg ladder, and Ising-XXZ three-leg tube), whereas the same set of quasicritical exponents implies a certain "universality" of quasitransitions of one-dimensional models. Specifically, the values of the quasicritical exponents for one-dimensional models are: α=α^{'}=3 for the specific heat, γ=γ^{'}=3 for the susceptibility and ν=ν^{'}=1 for the correlation length.

Strong- and Weak-Universal Critical Behaviour of a Mixed-Spin Ising Model with Triplet Interactions on the Union Jack (Centered Square) Lattice.

The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the uniaxial single-ion anisotropy on phase transitions and critical phenomena is examined in particular. It is shown that the considered model exhibits a strong-universal critical behaviour with constant critical exponents when considering the isotropic model with four equal triplet interactions or the anisotropic model with one triplet interaction differing from the other three. The anisotropic models with two different triplet interactions, which are pairwise equal to each other, contrarily exhibit a weak-universal critical behaviour with critical exponents continuously varying with a relative strength of the triplet interactions as well as the uniaxial single-ion anisotropy. It is evidenced that the variations of critical exponents of the mixed-spin Ising models with the integer-valued spins S differ basically from their counterparts with the half-odd-integer spins S.

Enhanced magnetocaloric effect in the proximity of magnetization steps and jumps of spin-1/2 XXZ Heisenberg regular polyhedra.

The magnetization process and adiabatic demagnetization of antiferromagnetic spin-1/2 XXZ Heisenberg clusters in the shape of regular polyhedra (tetrahedron, octahedron, cube, icosahedron and dodecahedron) are examined using the exact diagonalization method. It is demonstrated that a quantum (xy) part of the XXZ exchange interaction is a primary cause for the presence of additional intermediate magnetization plateaux and steps, which are totally absent in the limiting Ising case. The only exception to this rule is the spin-1/2 XXZ Heisenberg tetrahedron, which shows just a quantitative shift of the level-crossing fields related to two magnetization steps. It is shown that spin-1/2 XXZ Heisenberg regular polyhedra exhibit an enhanced magnetocaloric effect in the proximity of magnetization steps and jumps, which are accompanied with a rapid drop (rise) of temperature just above (below) the level-crossing field when the magnetic field is removed adiabatically.

Presence or absence of order by disorder in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices.

The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three interconnected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperature, whereas the same model on triangulated Husimi lattices with a greater connectivity of triangles-in-triangles units exhibits at low enough temperatures an outstanding quantum order due to the order-by-disorder mechanism. The quantum reduction of both sublattice magnetizations in the peculiar quantum ordered state gradually diminishes upon increasing the coordination number of the underlying Husimi lattice.

Vigorous thermal excitations in a double-tetrahedral chain of localized Ising spins and mobile electrons mimic a temperature-driven first-order phase transition.

A hybrid spin-electron system defined on a one-dimensional double-tetrahedral chain, in which the localized Ising spin regularly alternates with two mobile electrons delocalized over a triangular plaquette, is exactly solved with the help of generalized decoration-iteration transformation. It is shown that a macroscopic degeneracy of ferromagnetic and ferrimagnetic ground states arising from chiral degrees of freedom of the mobile electrons cannot be lifted by a magnetic field in contrast to a macroscopic degeneracy of the frustrated ground state, which appears due to a kinetically driven frustration of the localized Ising spins. An anomalous behavior of all basic thermodynamic quantities can be observed on account of massive thermal excitations, which mimic a temperature-driven first-order phase transition from the nondegenerate frustrated state to the highly degenerate ferrimagnetic state at nonzero magnetic fields. A substantial difference in the respective degeneracies is responsible for an immense low-temperature peak of the specific heat and very abrupt (almost discontinuous) thermal variations of the entropy and sublattice magnetizations.

Magnetization process, bipartite entanglement, and enhanced magnetocaloric effect of the exactly solved spin-1/2 Ising-Heisenberg tetrahedral chain.

The frustrated spin-1/2 Ising-Heisenberg ladder with Heisenberg intra-rung and Ising inter-rung interactions is exactly solved in a longitudinal magnetic field by taking advantage of the local conservation of the total spin on each rung and the transfer-matrix method. We have rigorously calculated the ground-state phase diagram, magnetization process, magnetocaloric effect, and basic thermodynamic quantities for the model, which can be alternatively viewed as an Ising-Heisenberg tetrahedral chain. It is demonstrated that a stepwise magnetization curve with an intermediate plateau at half of the saturation magnetization is also reflected in respective stepwise changes of the concurrence serving as a measure of bipartite entanglement. The ground-state phase diagram and zero-temperature magnetization curves of the Ising-Heisenberg tetrahedral chain are contrasted with the analogous results of the purely quantum Heisenberg tetrahedral chain, which have been obtained through density-matrix renormalization group (DMRG) calculations. While both ground-state phase diagrams fully coincide in the regime of weak inter-rung interaction, the purely quantum Heisenberg tetrahedral chain develops Luttinger spin-liquid and Haldane phases for strongly coupled rungs, which are absent in the Ising-Heisenberg counterpart model.

Spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg interaction and quartic Ising interactions as the exactly soluble zero-field eight-vertex model.

The spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg interaction and quartic Ising interactions is exactly solved by establishing a precise mapping relationship with the corresponding zero-field (symmetric) eight-vertex model. It is shown that the Ising-Heisenberg model with the ferromagnetic Heisenberg interaction exhibits a striking critical behavior, which manifests itself through re-entrant phase transitions as well as continuously varying critical exponents. The changes in critical exponents are in accordance with the weak universality hypothesis in spite of a peculiar singular behavior that emerges at a quantum critical point of the infinite order, which occurs at the isotropic limit of the Heisenberg interaction. On the other hand, the Ising-Heisenberg model with the antiferromagnetic Heisenberg interaction surprisingly exhibits less significant changes in both critical temperatures and critical exponents upon varying the strength of the exchange anisotropy in the Heisenberg interaction.