# Interplay of Spin and Spatial Anisotropy in Low-Dimensional Quantum Magnets with Spin 1/2

^{*}

## Abstract

**:**

## 1. Introduction

_{C}. Intensive theoretical studies of one-dimensional (1D) and two-dimensional (2D) spin systems were stimulated by the effort to understand the properties of three-dimensional (3D) phase transitions and critical phenomena [3].

_{BKT}.

_{3})

_{4}NMnCl

_{3}(TMMC) with spin S = 5/2 and even (C

_{6}H

_{11}NH

_{3})CuBr

_{3}(CHAB) with S = 1/2 [16]. Various theoretical approaches tried to find a quantum analog to the classical solitary excitations leading to the concept of quantum solitons which proved useful for the description of the ground-state properties of an S = 1 Heisenberg antiferromagnetic (HAF) chain in the famous Haldane’s conjecture [17,18]. Within the semi-classical approximation Haldane showed that the ground state of the HAF chain with integer spin is characterized by the presence of topological solitons. Consequently, the ground state is disordered with S = 0, separated from the excited S = 1 magnon state by the Haldane gap, arising from the presence of strong quantum fluctuations preventing the onset of the Néel order even at T = 0. Experimentally, the Haldane phase was most comprehensively studied in the S = 1 chain material Ni(C

_{2}H

_{8}N

_{2})

_{2}NO

_{2}ClO

_{4}(NENP), confirming the theoretical predictions [19,20].

_{C}superconductors triggered renewed intensive theoretical and experimental interest in the 2D quantum magnets [21,22]. In this context, the frustration effects became widely studied to understand the pairing processes. Sophisticated mathematical and computational methods enabled theorists to solve the variety of more complex low-dimensional quantum frustrated lattices including Shustry-Shuterland [23], Kagomé lattice [24], Kitaev honeycomb model [25] and others, having exotic properties and many of them still waiting for their discovery in the real world [26].

## 2. Spatial Anisotropy of the Exchange Coupling: From the Chain to the Square Lattice

#### 2.1. The S = 1/2 Heisenberg Antiferromagnetic Chain

_{3}with strong AF exchange coupling propagating along the c-direction, while much weaker ferromagnetic coupling along the a and b directions is responsible for the onset of the long-range order at T

_{N}= 39 K. The inelastic neutron scattering measurements in zero magnetic field probed the energy spectrum of KCuF

_{3}below and above T

_{N}. In comparison with low temperatures, the measurements made at 50 K indicated only little change in the scattering cross-section. The calculated spectrum of the S = 1/2 HAF chain is in good agreement with the experimental data [42] (Figure 2).

_{3}the inter-chain coupling, J′, is rather weak (J′ ≈ 0.01J), in Cs

_{2}CuCl

_{4}, the J′ achieves nearly 20% of the intra-chain value, J/k

_{B}≈ 4 K, and the onset of LRO occurs at T

_{N}= 0.62 K. The low J-value guarantees rather easy achievement of the critical magnetic field, B

_{c}, necessary for decoupling the spin chains as well as the saturation field, B

_{sat}, above which the ground state achieves a full ferromagnetic polarization. Theoretical studies of the S = 1/2 HAF chain in the magnetic field [14] showed that the field splits the triplet excitation continuum into the separate continua, the positions of which alter with increasing field. Above the saturation field, the excitations have a character of well-defined magnon dispersion. Correspondingly, in Cs

_{2}CuCl

_{4}, the 1D regime was expected to set at the fields lower than B

_{sat}≈ 6 T. The magnetic excitations were studied as a function of the field by measuring the inelastic scattering at low temperatures, T = 0.06 K [43]. It was found that the intensity of the magnetic excitation decreases with increasing field and the line shape changes above B

_{c}= 1.66 T, where the 1D regime occurs. Corresponding spectra were found to be in good agreement with the predictions for the S = 1/2 HAF chain in the magnetic field [43].

_{4}H

_{4}N

_{2})(NO

_{3})

_{2}, which proved to be an excellent realization of the S = 1/2 HAF chain with J = 0.9 meV and a negligible inter-chain coupling (J′/J < 10

^{−4}) [44].

#### 2.2. The S = 1/2 Heisenberg Antiferromagnet on the Spatially Anisotropic Square Lattice

_{0}= 1/2. The quantum fluctuations are strong enough to preserve spin-rotation symmetry such as the RVB state which may be relevant at high energies. While low-energy excitations are gapless magnons, recent experimental and theoretical studies showed that at higher energies, the existence of pairs of fractional S = 1/2 quasiparticles, 2D analogs of 1D spinons was established [50]. Using various theoretical approaches, excitation spectra and finite-temperature properties of the square lattice were investigated including specific heat, uniform and staggered susceptibility, correlation length etc. to provide useful tools for the identification of the model realization in the real world [51,52,53,54,55]. In these studies, the compound La

_{2}CuO

_{4}, with the exchange coupling J/k

_{B}≈ 1500 K proved to be a model system for the testing of the theories, especially those investigating ground-state properties. Naturally, the huge intra-layer coupling prevented any studies of the compound at moderate temperatures, T ≈ J/k

_{B}. Later, other 2D magnetic systems, mostly copper(II) coordination complexes, were identified as excellent realizations of the spin 1/2 HAF square lattice with much lower exchange coupling. The analysis of the specific heat of Cu(en)

_{2}Ni(CN)

_{4}(en = C

_{2}H

_{8}N

_{2}) and Cu(bmen)

_{2}Pd(CN)

_{4}(bmen = N, N′-dimethyl-1,2-diaminoethane), revealed excellent agreement with the theoretical prediction for the spin 1/2 HAF square lattice with J/k

_{B}= 0.36 K and 0.48 K, respectively [56,57]. Unlike La

_{2}CuO

_{4}, small coupling and corresponding saturation field B

_{sat}≈ 1 T, enable comfortable studies in the wide region of temperatures and magnetic fields. In these octahedral Cu(II) complexes comprising of weakly bound electroneutral covalent chains, the exchange coupling between Cu(II) ions is mediated predominantly through hydrogen bonds. On the other hand, in the tetragonal compound [Cu(pz)

_{2}(NO

_{3})][PF

_{6}] (pz = pyrazine) and monoclinic Cu(pz)

_{2}(ClO

_{4})

_{2}, the copper sites are connected within the square layers by bridging pyrazine molecules with the exchange coupling J/k

_{B}= 10.5 K and 17.5 K, respectively [58,59]. Even larger exchange coupling was indicated in Cu(HCOO)

_{2}∙4H

_{2}O with J/k

_{B}= 72 K [60]. Magneto-structural investigations [61,62,63] of monoclinic compounds (5MAP)

_{2}CuBr

_{4}and (5BAP)

_{2}CuBr

_{4}(5MAP = 5-methyl-2-aminopyridinium, 5BAP = 5-bromo-2-aminopyridinium) revealed that the magnetic interaction occurs between Cu(II) sites with four equivalent nearest neighbors through Br ∙∙∙ Br contacts forming 2D square layers with exchange coupling J/k

_{B}≈ 7 K. The layers of CuBr

_{4}tetrahedrons are separated by the bulk of organic cations which stabilize 3D structure. Systematic study of the compounds from the series A

_{2}CuX

_{4}[A = 5MAP, 5BAP, 5-chloro-2-aminopyridinium ≡ 5CAP, 5-cyano-2-aminopyridinium ≡ 5CNAP, etc., X = Br, Cl] found that the increasing size of the A cation improves the isolation of individual magnetic square layers but at the same time it reduces the strength of the intra-layer exchange coupling [62,63,64]. Apparently, the chemical modification of the structure can control magnetic properties demonstrating the flexibility of molecular magnetism.

_{c}value above which a long-range order is established in the ground state (m > 0). Conventional spin-wave theories as well as various numerical techniques predicted a final value, R

_{c}≈ 0.1–0.2, below which a 2D spin-liquid state with m = 0 can be stabilized [71,72]. For small R, a single-chain mean-field theory [73] predicted R

_{c}= 0 and a gradual increase of m, proportional to $\sqrt{R}$. Multi-chain mean-field calculations complemented by large-scale Monte Carlo simulations of the 2D Hamiltonian (Equation (4)), confirmed that R

_{c}= 0 and showed that for R → 0, m vanishes slower than $\sqrt{R}$ due to a logarithmic correction to this form [74]. Applying various techniques [72], the order-disorder ground-state transition was indicated for R ≈ 0.2 (Figure 4a). The sharp change of m for R < 0.2 was interpreted as a crossover in the magnetic behavior of the S = 1/2 HAF rectangular lattice, accompanied by a sharp change in the spatial dependence of spin correlations; with decreasing R, the rising quantum fluctuations gradually reduce the size of the order parameter and for small R, the system approaches 1D behavior with algebraic decrease of correlation functions [72]. This conclusion was further supported by quantum Monte Carlo studies [75] of finite-temperature properties of the rectangular lattice (Equation (4)). For small R, the temperature dependence of the uniform susceptibility, χ, follows that of a single chain, while deviations appear below temperatures k

_{B}T/J ≈ 5R. For larger R, the χ values lie between those of the chain and square lattice. The 1D - 2D dimensional crossover is evident also in the behavior of the correlation length, ξ, depicted in Figure 4b. For R > 0 and low temperatures, the quantity is well described by the relation [75]

_{s}, depends on the spatial anisotropy R. As can be seen in Figure 4b, the intra-chain correlation length gradually approaches 1D behavior when decreasing R. Alike susceptibility, for small R, the ξ values merge with those for a single chain at sufficiently high temperatures. Using these theoretical predictions, authors of reference [75] identified the quantum magnet Sr

_{2}CuO

_{3}as a realization of the S = 1/2 HAF on the rectangular lattice with the intra-chain coupling J/k

_{B}= 2200 K, R = 0.002 and the staggered magnetization, m ≈ 0.03. In comparison with the m ≈ 0.3 derived for the square lattice, the significant reduction results from the strong enhancement of quantum fluctuations.

_{2}(J/k

_{B}= 28 K, R = 0.3), Cu(pz)(N

_{3})

_{2}(J/k

_{B}= 15 K, R = 0.46) and Cu(2-apm)Cl

_{2}(2-apm = 2-amino-pyrimidine) with J/k

_{B}= 116.3 K and R = 0.084 [77].

_{2}where the first-principle calculations enabled to identify the material as the realization of the rectangular lattice with J/k

_{B}≈ 170 K and R ≈ 0.08 [78].

_{2}(PM = [C

_{6}H

_{2}(COO)

_{4}]

^{4−}, EA = [C

_{2}H

_{5}NH

_{3}]

^{+}) can be described by the model (Equation (4)) with various R in a wide range of temperatures and magnetic fields (Figure 5).

_{max}, as well as its position, T

_{max}, depend on the J and g-factor values. Correspondingly, the proper choice of g and J parameters can yield an excellent agreement with the data (Figure 5). Unlike the susceptibility, another bulk property, specific heat, can provide more valuable information since the maximum, C

_{max}, depends on the used model (Figure 6). The best agreement was found for R = 0.7 as predicted by the first-principle calculations [79].

_{2}O)

_{2}SO

_{4}. Previous analysis of powder thermodynamic data [65] identified the material as a potential realization of the partially frustrated S = 1/2 HAF on the spatially anisotropic triangular lattice (SATL). The lack of proper theoretical predictions enabled the analysis only in the frame of the limiting models of the SATL, i.e., the square lattice, chain and triangular lattice and could not provide a reliable information whether the spin system approaches the properties of the chain or the square lattice (Figure 6b). Considering d

_{x}

^{2}

_{-y}

^{2}ground state of the Cu(II) ion, it was assumed that potential exchange pathways form SATL with a dominant exchange coupling creating the square lattice, while weaker interactions were expected to occur along one of the diagonals of the square plaquettes [65]. The analysis of single-crystal electron paramagnetic resonance spectra [80] indicated the need to revisit the concept of SATL in Cu(en)(H

_{2}O)

_{2}SO

_{4}, which triggered first-principle calculations of exchange couplings [81]. The calculations revealed the formation of a spatially anisotropic zig-zag square lattice (Figure 7a) comprised of 2D array of weakly coupled zig-zag chains with R = J′/J ≈ 0.15. Corresponding quantum Monte Carlo (QMC) calculations of finite-temperature properties of the S = 1/2 HAF on the spatially anisotropic zig-zag square lattice (SAZZSL) including specific heat, susceptibility and magnetization, provided theoretical predictions in a wide range of temperatures and magnetic fields [82]. Subsequent analysis of single-crystal Cu(en)(H

_{2}O)

_{2}SO

_{4}experimental data within the SAZZSL model found the excellent agreement for J/k

_{B}= 3.4 K and R = 0.35 (Figure 8).

_{2}(ClO

_{4})

_{2}was considered as a model system for the realization of the isotropic square lattice [59]. Recent density functional theory (DFT) simulations using structural Cu(pz)

_{2}(ClO

_{4})

_{2}data at 10 K revealed the formation of SAZZSL with R = 0.7 [83] while other authors working with 100 K structural data arrived to the rectangular lattice with R ≈ 0.93 [84]. While it is hard to discriminate between two different results on the basis of magnetic bulk data, as was done in refs. [83,84], specific heat analysis would be subservient to make the decision. As was shown in refs. [79,82], the difference in the maximum specific heat values for R = 1 and 0.7 is easily distinguishable. This approach was applied also in the determination of the 2D magnetic lattice in the verdazyl radical α-2,3,5-Cl

_{3}-V (3-(2,3,5-trichlorophenyl)-1,5-diphenylverdazyl] (Figure 7) where DFT calculations revealed the formation of SAZZSL with R = 0.56 [85].

#### 2.3. The Crossover from 2D to 3D

_{0}is the reduced staggered magnetization at zero temperature, and correlation length is defined by the Equation (5). For the isotropic square lattice (R = 1), (m/m

_{0})

^{2}≈ 0.3 and the spin stiffness in the Equation (5) is ρ

_{s}≈ 0.18 J. Apparently, even a minute amount of J″ is capable to induce a 3D LRO at a finite temperature. Large-scale QMC studies of the S = 1/2 HAF on the spatially anisotropic simple cubic lattice (with R = 1 within layers) provided an empirical formula for T

_{N}which enables the estimation of the inter-layer coupling [88]

_{s}= 0.183 J and b = 2.43.

_{N}. On the other hand, nonzero J″ reduces the strength of the quantum fluctuations, which results in the enhancement of T

_{N}. In highly anisotropic 2D systems (J″« J, J′), most of the entropy is removed above T

_{N}and the effective number of degrees of freedom associated with the 3D LRO is significantly reduced. In such extreme conditions, a λ-like anomaly in the specific heat associated with the onset of 3D LRO completely vanishes. Quantum Monte Carlo studies of the S = 1/2 HAF on the spatially anisotropic simple cubic lattice (R = 1 within layers) showed, that a sharp peak dominates the specific heat behavior for the strong inter-layer coupling, while for J″ < 0.05 J a clear separation of two peaks occurs [87]. Finally, for J″ < 0.015 J, the sharp peak completely vanishes and despite the onset of 3D LRO, the specific heat follows the behavior of the 2D system (Figure 9a).

_{2}(tn = C

_{3}H

_{10}N

_{2}) [89] (Figure 9b) and Cu(pz)

_{2}(pyO)

_{2}(PF

_{6})

_{2}(pyO = pyridine-N-oxide) [90]. Thus, one has to be careful in the declaration of the absence of the 3D LRO on the basis of the specific heat only and other experiments are necessary to confirm the assumption. Besides demanding neutron diffraction experiments, much simpler susceptibility measurements or electron paramagnetic—antiferromagnetic resonance can provide reliable information about the onset of the 3D LRO. Neglecting any other phenomena but J″, the absence or the presence of the λ-like anomaly can be affected by the geometry of inter-layer exchange pathways in real compounds. The comparison of magnetic specific heats of Cu(tn)Cl

_{2}and Cu(en)(H

_{2}O)

_{2}SO

_{4}suggests that the 2D correlations responsible for the appearance of a round maximum have the same character while the manifestation of the 3D correlations is completely different (Figure 9b). The severe weakening of their effect in Cu(tn)Cl

_{2}was ascribed to the combined effect of a geometrical frustration and the large distances (about 10 Å) between Cu(II) ions in the adjacent layers [89].

_{2}O)

_{2}SO

_{4}was estimated to be much lower than 0.015 J [82] thus, according to reference [87], no phase transition should be visible in the specific heat data. In Cu(en)(H

_{2}O)

_{2}SO

_{4}, the adjacent layers are shifted along the b-axis by b/2. As a consequence, a central spin from the magnetic layer has a high number of nearest neighbors (z = 8) from the adjacent layers (Figure 7). It should be noted, that QMC simulations [87] were performed for the geometry of the simple cubic lattice, thus only z = 2 was considered for the central spins in the layer.

_{3}-V with a large inter-layer coupling, J″ ≈ 0.24 J. This is another extreme, since such a high value of J″ should completely suppress the two-dimensional character of the magnetic system in the verdazyl radical. However, thermodynamic properties clearly demonstrate a high measure of magnetic two-dimensionality, comparable to Cu(en)(H

_{2}O)

_{2}SO

_{4}. This contradiction can be explained by very low number of interacting nearest neighbors from the adjacent planes. As can be seen in Figure 7, in verdazyl radical, effectively only z = 0.5 per central spin can be considered, thus reducing the effect of rather strong inter-layer coupling can be expected [85].

## 3. The Effect of the Spin Anisotropy and Magnetic Field in the S = 1/2 HAF on the Square Lattice

#### 3.1. B = 0

^{9}open shell), the interplay of the LS coupling and the crystal field produced by the local surrounding generates the anisotropy of the magnetic moment, observed in the form of the anisotropic g-tensor [92]. In the presence of magnetic interactions between the paramagnetic ions, the exchange anisotropy (symmetric and/or antisymmetric) appears with the relative strength comparable with the order of the g-factor anisotropy [93]. Accordingly, experimental studies of quasi-2D spin 1/2 magnets revealed that the spin anisotropy is very weak, ranging from 10

^{−4}to 10

^{−2}times the intra-layer exchange coupling J [59,80,94].

_{1}, i

_{2}) runs over the sites of the square lattice, d connects the i-th site to the nearest neighbors, J > 0 is antiferromagnetic coupling, ${\Delta}_{\mu}$ and ${\Delta}_{\lambda}$ are the easy-axis and easy-plane anisotropy parameters, respectively. For ${\Delta}_{\mu}$ = ${\Delta}_{\lambda}$= 0, the Equation (8) reduces to the Equation (4) with R = 1 describing the isotropic HAF on the square lattice. The parameters ${\Delta}_{\lambda}$ = 0, $0<{\Delta}_{\mu}\le 1$ define the easy-axis anisotropy, while the easy-plane anisotropy corresponds to ${\Delta}_{\mu}$ = 0, $0<{\Delta}_{\lambda}\le 1$. In the case of the weak easy-axis anisotropy ${\Delta}_{\mu}$ = 10

^{−2}and 10

^{−3}, quantum Monte Carlo studies of finite-temperature properties revealed the existence of a phase transition in the 2D Ising universality class occurring at finite temperatures, k

_{B}T

_{I}≈ 0.28 J and 0.22 J, respectively [97]. In the specific heat, the onset of the 2D LRO was indicated as a small sharp peak superimposed on the left side of a round maximum. As the anisotropy decreased, the sharp peak diminished, moving to low temperatures, while the round maximum converged to that of the HAF on the square lattice. Similarly, the uniform susceptibility follows the prediction for the HAF on the square lattice down to k

_{B}T ≈ 0.4 J for ${\Delta}_{\mu}$ = 10

^{−2}. At lower temperatures, the transverse susceptibility, χ

^{xx}, and longitudinal, χ

^{zz}, separate from the isotropic Heisenberg curve well above the phase transition; at the transition temperature T

_{I}, the χ

^{xx}displays a minimum, while χ

^{zz}monotonically decreases to zero. Apparently, the susceptibility measurement in two different orientations of magnetic field provides a tool for a reliable identification of the phase transition. Similar features of the susceptibility and specific heat were observed in the XXZ model with the easy-plane anisotropy for ${\Delta}_{\lambda}$ = 2 × 10

^{−2}and 10

^{−3}. A crossover temperature, T

_{CO}, from the isotropic Heisenberg to the easy-plane (XY) behavior was estimated [96]

_{s}= 0.214 J. The onset of the XY regime below T

_{CO}is accompanied with the formation of the pairs of vortices and antivortices. Concerning the specific heat, a position of a tiny peak superimposed on the left side of a round maximum, corresponds to the maximum of the temperature derivative of the vortex density, while a phase transition of Berezinskii–Kosterlitz–Thouless type is set at lower temperature. For ${\Delta}_{\lambda}$ = 2 × 10

^{−2}, the uniform susceptibility follows the behavior of the HAF on the square lattice down to 0.4J. At lower temperatures, the transverse and longitudinal susceptibility separate from the isotropic Heisenberg curve well above the transition temperature, T

_{BKT}; at T

_{CO}, the χ

^{zz}component displays a minimum, while χ

^{xx}decreases faster than the isotropic Heisenberg curve, achieving some nonzero value at T = 0. As authors showed, in the experiments with the real quasi-2D quantum magnets, the measurements of a single-crystal uniform susceptibility can help to determine the onset of a phase transition as well as the type of the spin anisotropy. If a minimum in the χ

^{zz}component is observed above T

_{N}(i.e., the temperature of a phase transition to the 3D LRO), this is a signature of the easy-plane anisotropy, while the occurrence of the minimum at the transition temperature suggests the easy-axis anisotropy (Figure 10a).

_{I},

_{BKT}remain finite for any finite easy-plane and easy-axis anisotropy (Figure 10b). Thus, unlike the isotropic Heisenberg model on the square lattice, for the XXZ analogue, the quantum and thermal fluctuations are not able to destroy the phase transitions at finite temperatures.

_{2}CuO

_{2}Cl

_{2}was identified as the first experimental S = 1/2 XXZ square-lattice antiferromagnet with a huge intra-layer coupling J/k

_{B}= 1450 K, extremely weak inter-layer coupling ${J}^{\u2033}/J\cong {10}^{-5}$ and extremely weak easy-plane anisotropy ${\Delta}_{\lambda}\approx {10}^{-3}$ [96].

_{2}(NO

_{3})][PF

_{6}], Cu(pz)

_{2}(ClO

_{4})

_{2}, and Cu(pz)

_{2}(BF

_{4})

_{2}, allowed to determine the presence of the weak easy-plane anisotropy ${\Delta}_{\lambda}\approx {10}^{-3}$ [59]. In these quasi-2D S = 1/2 XXZ square-lattice magnets, the intra-layer coupling is two orders of magnitude lower than in Sr

_{2}CuO

_{2}Cl

_{2}. The authors found that for the compounds with the high degree of the lattice two-dimensionality (i.e., ${J}^{\u2033}/J\le $ 10

^{−3}), the spin anisotropy correlates well with the ratio of the anisotropy field, B

_{A}, and the saturation field, ${\Delta}_{\lambda}\approx {B}_{A}/{B}_{sat}$ while a strong inter-layer coupling disturbs this coincidence [59].

#### 3.2. B ≠ 0

_{SF}, accompanied with the reorientation of spins to be orthogonal to the field and gradually canting in its direction. The spin-flop phase is separated from the paramagnetic one by a critical line of BKT transitions [101,102,103,104]. In the limit ${\Delta}_{\mu}\to 0$, the ordered AF phase gradually vanishes, i.e., B

_{SF}→ 0 and in the isotropic HAF limit, only the spin flop phase remains, separated from the paramagnetic state by the critical line of BKT transitions [101,105].

_{0}is the Hamiltonian described by the Equation (8) for ${\Delta}_{\mu}$ = ${\Delta}_{\lambda}$ = 0 and i = (i

_{1},i

_{2}) runs over the sites of the square lattice), were performed in a wide range of magnetic fields h = gμ

_{B}B/(SJ) from zero to the saturation field h

_{sat}= 8 [106,107]. It was found, that alike in the classical counterpart [101,105], the infinitesimal uniform field induces a BKT transition at a finite temperature. The magnetic phase diagram is characterized by a non-monotonous behavior of the critical temperatures (Figure 11a). In the weak field, the Hamiltonian (Equation (10)) can be mapped on the easy-plane XXZ model in zero magnetic field (Equation (8)) with Δ

_{λ}≈ 0.1 h

^{2}and the transition temperature

^{z}component, resulting in the enhancement of the effective easy-plane anisotropy, which tends to increase T

_{BKT}.

_{BKT}. The effective rotator length goes to zero at the saturation field and corresponding BKT critical temperature vanishes. Apparently, the interplay between the two effects is responsible for the non-monotonous dependence of T

_{BKT}, which for the S = 1/2 achieves maximum values at higher fields than in the classical counterpart (Figure 11a). This shift was ascribed to the effect of quantum fluctuations [106].

_{BKT}, the entropy in h > 0 grows slower due to the presence of the quasi-LRO established in the magnetic field after binding V + AV pairs, while above T

_{BKT}, the growth is much faster due to unbinding V + AV pairs. Finally, at high temperatures, the entropy growth does not depend on the field in fully disordered systems [106]. A uniform magnetization is another bulk quantity, which can be experimentally measured; the QMC calculations showed that for h < 1, the temperature dependence of the uniform magnetization coincides with the uniform susceptibility of the S = 1/2 HAF on the square lattice in h = 0. Deviations appear at low temperatures k

_{B}T < J, displaying a minimum, which indicates a field-induced crossover from the isotropic to the XY regime (Figure 11b).

_{2})(pz)

_{2}]BF

_{4}with J/k

_{B}= 5.9 K and J″/J ≈ 3 × 10

^{−2}. In zero magnetic field, the compound undergoes a phase transition to the 3D LRO at T

_{N}= 1.6 K. The application of the magnetic field up to 8 T led to the enhancement of the transition temperature, and a further field increase resulted in the conventional reduction of T

_{N}(B). While in the presence of the nonzero J″ the finite-temperature critical 2D properties are lost, the QMC studies found that in the quasi-2D magnets with extremely weak inter-layer coupling, this is just the nonmonotonic behavior of T

_{N}(B), which preserves also in the real systems (Figure 12a). On the other hand, the strong J″ will smear even this feature characterizing the ideal 2D magnets and a conventional decrease of the transition temperature will be observed in all magnetic fields. Such behavior was observed in (5CAP)

_{2}CuCl

_{4}, the quasi-2D S = 1/2 HAF on the square lattice with J″/J ≈ 0.25 [64].

_{2}did not allow the formation of a sharp specific heat λ-like anomaly in the zero magnetic field, the application of the magnetic field of 0.75 T was capable to induce a weak anomaly at about 0.7 K. A further increase of the field enhanced the anomaly, shifting its position towards higher temperatures. In the fields above 2 T, the amplitude gradually decreased and the anomaly shifted to lower temperatures [89].

_{N}(B) was also observed in the Cu(pz)

_{2}(ClO

_{4})

_{2}with J″/J < 10

^{−3}. Since the saturation field is very large, the B-T phase diagram was recorded only for the fields lower than B

_{sat}/4. This value corresponds to the fields below which, the BKT temperature in the ideal 2D case of the S = 1/2 HAF on the square lattice grows with the magnetic field [109]. The fact, that the phase diagrams of the Cu(pz)

_{2}(ClO

_{4})

_{2}measured in the fields parallel and perpendicular to magnetic layers were found to be identical, was ascribed to a very weak intrinsic spin anisotropy.

_{2}, no sharp specific heat anomaly was observed down to 0.5 K in [Cu(pz)

_{2}(pyO)

_{2}](PF

_{6})

_{2}, (pyO = pyridine-N-oxide), a quasi-2D S = 1/2 HAF on the square lattice with J/k

_{B}= 8.2 K, J″/J ≈ 2 × 10

^{−4}and the weak easy-plane anisotropy ${\Delta}_{\lambda}\approx 0.007$ [90]. The application of the magnetic field of 1 T induced a weak anomaly at about 2 K, gradually growing and shifting towards higher temperatures. In relatively low fields, the shape of the anomaly depended on the orientation of the magnetic field. When the field was applied perpendicular to the easy plane, in accord with the theory [99], the field plays a role of an effective easy-plane anisotropy, thus fingerprints of the BKT transition in the whole field region up to B

_{sat}can be expected. Considering the field applied within the easy plane, authors expected a field-induced easy-axis anisotropy, established in the fields up to 5 T [90]. In these fields, the formation of tiny λ-like anomalies, typical for the Ising transitions was observed, while at the fields above 5 T, the anomalies evolved to a broad anomaly of the expected BKT type and the system behaved as the isotropic 2D HAF in the field. Authors studied also a temperature dependence of the susceptibility in constant magnetic fields. In both orientations, the susceptibility is characterized by a broad maximum typical for 2D magnets with an upturn below 3 K (Figure 12b). In the field perpendicular to the easy plane, a minimum occurs always above the transition temperature to the 3D LRO, a feature typical for the onset of the XY regime, while in the field applied within the easy plane, such behavior occurs only above 1 T. Experimental studies of other quasi-2D S = 1/2 antiferromagnets with a weak easy-plane anisotropy in the magnetic field applied within the easy plane found, that for the fields lower than the anisotropy field B

_{A}, the temperature dependence of the susceptibility has no upturn and its qualitative behavior follows χ

^{xx}in Figure 10a. On the other hand, in the fields exceeding B

_{A}, the upturn gradually develops and a characteristic minimum forms at temperatures above T

_{N}(B) as a typical sign of the onset of the field-induced XY regime (see Figure 8 of reference [59]). Thus, considering an excellent spatial two-dimensionality of [Cu(pz)

_{2}(pyO)

_{2}](PF

_{6})

_{2}, the application of the aforementioned relation ${\Delta}_{\lambda}\approx {B}_{A}/{B}_{sat}$ provides B

_{A}≈ 0.2 T. Taking into account a combined effect of the inter-layer coupling and the spin anisotropy, some higher external field should compensate both effects, to set the 2D XY regime. The experimental susceptibility data in the fields applied within the easy plane in Figure 12b suggest, that this condition has already been fulfilled at least for B = 1 T.

_{2}(ClO

_{4})

_{2}, the 2D square-lattice magnet with J″/J ≈ 8.8 × 10

^{−4}, T

_{N}= 4.2 K and B

_{sat}≈ 49 T, identified the presence of the easy-plane anisotropy ${\Delta}_{\lambda}\approx 4.6\times {10}^{-3}$ [59]. Subsequent antiferromagnetic resonance experiments and magnetization measurements in the ordered phase refined the character of the spin anisotropy comprising of the out-of-plane (easy-plane) anisotropy ${\Delta}_{\lambda}\approx 3.1\times {10}^{-3}$ and the in-plane anisotropy ${\Delta}_{in}\approx 3.1\times {10}^{-4}$, the latter breaking a continuous symmetry within the easy xy plane [94,110]. Thus, the description of Cu(pz)

_{2}(ClO

_{4})

_{2}within the XYZ model with extremely weak anisotropy is more realistic.

_{2}(ClO

_{4})

_{2}, for the field applied along the easy axis, the AF-SF critical line ended in a bicritical point at 4 K and the field about 0.73 T (Figure 13). For the constant fields B < 0.5 T, the temperature dependence of the corresponding normalized uniform magnetization M(T)/B is characterized by a sharp change of the slope in the vicinity of the transition temperature, separating the collinear AF Néel phase and the paramagnetic (PM) phase. For the fields, 0.5 < B < 0.73 T, the M(T)/B curves cross the SF-AF and AF-PM critical lines. The former crossing is accompanied with a pronounced step in the curves (Figure 13a). For higher fields, the spin-flop phase is stabilized with the XY regime, manifesting by the upturn in the M(T)/B curves with a minimum, ascribed to the 2D Heisenberg-XY crossover. The kinks in the curves were associated with the onset of the 3D LRO. The application of the field along the hard axis z, leads to the behavior of M(T)/B typical for the XY regime, since the magnetic field enforces the effect of the intrinsic easy-plane anisotropy, resulting in the effective easy-plane anisotropy ${\Delta}_{eff}={\Delta}_{\lambda}+a{(g{\mu}_{B}B/J)}^{2}$ [94]. Concerning the middle axis y, authors of reference [16] expected enforcing of the in-plane easy-axis anisotropy, ${\Delta}_{eff}={\Delta}_{in}+a{(g{\mu}_{B}B/J)}^{2}$. However, as was shown in Cu(pz)

_{2}(ClO

_{4})

_{2}, in this orientation, the fields above 2 T introduced the XY regime with the effective anisotropy ${\Delta}_{eff}\approx a{(g{\mu}_{B}B/J)}^{2}$ (Figure 16 in reference [94]). Apparently, alike in the case of the aforementioned easy-plane XXZ model, the theoretical studies of the S = 1/2 2D XYZ model on the square lattice with a weak spin anisotropy are necessary, to verify the persistence of the Ising-like ground state as well as the Ising-XY crossover, induced by the field applied along the middle axis.

_{2}O)

_{2}SO

_{4}can be treated as the realization of the S = 1/2 2D XYZ model [82]. In this respect, both, Cu(pz)

_{2}(ClO

_{4})

_{2}and Cu(en)(H

_{2}O)

_{2}SO

_{4}should represent the same model, i.e., the S = 1/2 XYZ model on the spatially anisotropic zig-zag square lattice, differing in the amount of the spatial and the spin anisotropy. Within the effective S = 1/2 HAF square lattice model, the rescaling B

_{sat}/7 and J

_{eff}/(7k

_{B}) parameters of Cu(pz)

_{2}(ClO

_{4})

_{2}will provide ${B}_{sat}^{\ast}$ ≈ 7 T and ${J}_{eff}^{\ast}/{k}_{B}$ ≈ 2.6 K, the values which are reported for Cu(en)(H

_{2}O)

_{2}SO

_{4}[65]. Concerning the spin anisotropy, applying the relation for the anisotropy-induced energy gaps in the excitation spectrum [94]

_{2}O)

_{2}SO

_{4}[82], the relation provides the in-plane anisotropy ΔJ

^{in}≈ 12 mK, which is about two times larger than ΔJ

^{in}≈ 5.3 mK reported for Cu(pz)

_{2}(ClO

_{4})

_{2}[94]. This result correlates well with the recent antiferromagnetic resonance experiments in Cu(en)(H

_{2}O)

_{2}SO

_{4}[111]. In addition, they determined the ratio of the spin gaps ${E}_{g}^{out}/{E}_{g}^{in}\approx 2.3$, correspondingly ΔJ

^{out}/ΔJ

^{in}≈ 5. In Cu(pz)

_{2}(ClO

_{4})

_{2}, the ratio of the spin gaps and the spin anisotropies is about 3.3 and 10, respectively, which is much closer to the easy-plane XXZ model. Apparently, the size as well as the relative strength of the in-plane anisotropy in Cu(en)(H

_{2}O)

_{2}SO

_{4}is two times larger than in Cu(pz)

_{2}(ClO

_{4})

_{2}. It seems, that this difference correlates well with the difference in the spatial anisotropies; R = 0.7 in Cu(pz)

_{2}(ClO

_{4})

_{2}is two times lower than R = 0.35 in Cu(en)(H

_{2}O)

_{2}SO

_{4}.

_{2}O)

_{2}SO

_{4}already revealed the aforementioned strong rhombic character of the spin anisotropies [80]. It was assumed, that a dipolar coupling between nearest neighbors could provide the main source of the spin anisotropy in Cu(en)(H

_{2}O)

_{2}SO

_{4}, where the exchange pathways are very long, resulting in weak exchange couplings. Consequently, the spin anisotropy could reflect the spatial distribution of Cu(II) magnetic moments. The dipolar coupling was considered as the only origin of the extremely weak easy-axis anisotropy Δ

_{μ}≈ 10

^{−4}, observed in the verdazyl radical [85]. As for Cu(pz)

_{2}(ClO

_{4})

_{2}, the exchange coupling is much stronger than in Cu(en)(H

_{2}O)

_{2}SO

_{4}, but the strength of the in-plane anisotropy is weaker than in Cu(en)(H

_{2}O)

_{2}SO

_{4}. The origin of the in-plane spin anisotropy in Cu(pz)

_{2}(ClO

_{4})

_{2}was ascribed to the weak rhombic distortion of the crystal lattice, being of the same order as the spin anisotropy [94]. Similar symmetry arguments were used to explain the extremely weak spin anisotropy in [Cu(HF

_{2})(pz)

_{2}]SbF

_{6}with a tetragonal structural symmetry in the ordered phase, while stronger anisotropies observed in the Cu(II) magnets with lower crystal symmetry were ascribed to their reduced structural symmetry [84].

_{2}O)

_{2}SO

_{4}, after rescaling magnetic field with the high-temperature g-factors (to remove the effects of the local surrounding of Cu(II) ions introduced via a spin-orbit coupling), the magnetic fields exceeding the B

_{SF}values, produce in all orientations the same behavior of the transition temperatures (Figure 13b). A comparison of the rescaled magnetic diagram with that for the S = 1/2 2D HAF model on the square lattice (R = 1) revealed large discrepancies for B ≈ 0, due to the presence of the inter-layer coupling in Cu(en)(H

_{2}O)

_{2}SO

_{4}. On the other hand, the differences in the vicinity of the saturation field result from the higher measure of the quantum fluctuations in Cu(en)(H

_{2}O)

_{2}SO

_{4}, introduced by the spatial anisotropy of the intra-layer exchange coupling.

## 4. Summary and Concluding Remarks

_{sat}, was theoretically predicted along with the suggestions, how to observe this feature experimentally [112]. Theoretical studies also predict the existence of the roton minimum in the high-energy part of the excitation spectrum of the same model in zero magnetic field [113].

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic representation of a spinon excitation by reversing all spins beyond a certain lattice site. The spinon (domain wall) propagates along the chain moving by two lattice sites.

**Figure 2.**(

**a**) Spectrum of the S = 1/2 HAF chain in zero magnetic field. The thin solid line represents a scattering trajectory for a detector at the scattering angle of 8′ and an incident energy of E

_{o}= 149 meV in KCuF

_{3}. The scattering occurs when the trajectory intersects with the continuum (bold line); (

**b**) Scattering measured in the low-angle detector banks at T = 20 K. (Reproduced with permission from reference [42]).

**Figure 4.**(

**a**) Ground-state staggered magnetization m of the rectangular lattice for different R values calculated by various approaches; the short-dashed lines are fits to the exact diagonalization (ED) data. Cases A, B, C correspond to different choice of parameters in Green’s function approach. The inset demonstrates the deviation of the ED results from the N-1/2 scaling law at R < 0.3 (solid lines are least-squares fits). (Reproduced with permission from reference [72]); (

**b**) The correlation length for the S = 1/2 HAF on the rectangular lattice as a function of the inverse temperature for various R values (α stands for R). Open (filled) symbols denote the correlation length along (perpendicular to) the chain. For small α (α < 0.05), the values of inter-chain correlation length are smaller than one lattice constant and are not shown in the figure. Solid lines are fits to Equation (5), showing the exponential dependence of ξ on 1/T. (Reproduced with permission from reference [75]).

**Figure 5.**(

**a**) Fits of the magnetic susceptibility of Cu(PM)(EA)

_{2}with the model (Equation (4)) for R = 0 (1D) with J/k

_{B}= 10.2 K, g = 2; R = 1 (2D) with J/k

_{B}= 6.8 K, g = 2.05; R = 0.7 (rectangular lattice) with J/k

_{B}= 8.0 K, g = 2.07; (

**b**) Field dependence of the magnetization of Cu(PM)(EA)

_{2}. Lines show the simulations with the same parameters as those used for the susceptibility. (Reproduced with permission from reference [79]).

**Figure 6.**(

**a**) Fits of the magnetic specific heat of Cu(PM)(EA)

_{2}in B = 0 with the model (Equation (4)) for R = 0 (1D), R = 1 (2D) and R = 0.7 (rectangular lattice) with the same parameters as in Figure 5. (Reproduced with permission from reference [79]); (

**b**) Fits of the magnetic specific heat of Cu(en)(H

_{2}O)

_{2}SO

_{4}in B = 0 (open and full circles correspond to powder and single crystal, respectively) with the model (Equation (4)) for R = 0 with J/k

_{B}= 3.6 K, (solid line), R = 1 with J/k

_{B}= 2.8 K (dashed line) and the model of the S = 1/2 HAF on the triangular lattice with J/k

_{B}= 4.4 K (dotted line). (Reproduced with permission from reference [65]).

**Figure 7.**(

**a**) Spatially anisotropic zig-zag square lattice interpolates between the chain (R = J′/J = 0) and the isotropic square lattice (R = 1); (

**b**) Position of adjacent layers (blue) in Cu(en)(H

_{2}O)

_{2}SO

_{4}with respect to the central 2D layer (red). Diamonds denote projections of the positions of blue circles into the central layer; (

**c**) Position of adjacent layers (blue) in the verdazyl radical with respect to the central 2D layer (red). In (

**b**) and (

**c**), the inter-layer exchange couplings J″ are depicted by green lines (see Section 2.3).

**Figure 8.**(

**a**) Temperature dependence of normalized single-crystal Cu(en)(H

_{2}O)

_{2}SO

_{4}(CUEN) susceptibility in the field 10 mT. Solid and dashed lines represent corresponding QMC calculations (L = 128) for the S = 1/2 HAF SAZZSL model with R = 0.35 and 0.15, respectively; (

**b**) Temperature dependence of magnetic specific heat of CUEN single crystal in zero magnetic field. Solid and dashed lines have the same meaning as in (

**a**). (Reproduced with permission from reference [82]).

**Figure 9.**(

**a**) The specific heat of the S = 1/2 HAF model on the spatially anisotropic simple cubic lattice with R = 1 within the layers. Quantum Monte Carlo calculations are performed over a wide temperature range for several different spatial anisotropies (α = J″/J). The system size is 48 × 48 × 12. The separation of the 3D ordering peak from the broad maximum arising from the 2D physics is clearly visible for α < 2

^{−3}. (Reproduced with permission from reference [87]); (

**b**) Temperature dependence of Cu(en)(H

_{2}O)

_{2}SO

_{4}(J/k

_{B}= 2.8 K) and Cu(tn)Cl

_{2}(J/k

_{B}= 3.1 K) magnetic specific heat compared with the theoretical prediction for the S = 1/2 HAF on the square lattice (R = 1).

**Figure 10.**(

**a**) Uniform susceptibility of Sr

_{2}CuO

_{2}Cl

_{2}(thin and thick pluses represent χ

^{zz}, crosses - χ

^{xx}) compared to the theoretical predictions for the S = 1/2 easy-plane XXZ model on the square lattice with ${\Delta}_{\lambda}$ = 10

^{−3}(χ

^{zz}—open diamonds, χ

^{xx}—full diamonds). Stars represent the susceptibility of the isotropic HAF on the square lattice. (Reproduced with permission from reference [96]); (

**b**) Phase diagram of the S = 1/2 XXZ model on the square lattice for weak anisotropies. The curves in the low-anisotropy region, i.e., Δ

_{μ,λ}< 10

^{−1}, represent the behavior of the reduced critical temperatures t

_{I,BKT}= k

_{B}T

_{I,BKT}/J, described by the corresponding expressions. (Reproduced with permission from reference [97]).

**Figure 11.**(

**a**) Magnetic phase diagram of the S = 1/2 HAF on the square lattice (full symbols) in reduced coordinates t

_{BKT}= k

_{B}T

_{BKT}/J and h = gμ

_{B}B/(SJ). Open symbols refer to the classical limit of the model. Inset: t

_{BKT}vs. h for weak fields, ${t}_{s}^{\ast}$ represents a crossover temperature from the isotropic to XY behavior. Shaded area marks the region of disordered XY behavior. (Reproduced with permission from reference [106]); (

**b**) Field-induced uniform magnetization vs reduced temperature for different field values. The stars represent the zero-field uniform susceptibility of the S = 1/2 HAF on the square lattice. The arrows indicate the onset of BKT transition. (Reproduced with permission from reference [106]).

**Figure 12.**(

**a**) Magnetic phase diagram of [Cu(HF

_{2})(pz)

_{2}]BF

_{4}compared with QMC simulations of the S = 1/2 HAF on the spatially anisotropic simple cubic lattice (R = 1 within layers) with the system parameters $({J}_{\left|\right|}\equiv J,{J}_{\perp}\equiv {J}^{\u2033},h=g{\mu}_{B}{\mu}_{0}H)$. For comparison, the results for the pure 2D case (J″ = 0) are also shown. (Reproduced with permission from reference [108]); (

**b**) χ(T) of [Cu(pz)

_{2}(pyO)

_{2}](PF

_{6})

_{2}for the magnetic field applied perpendicular to the easy plane ab (μ

_{0}H‖c) and within the easy plane ab (μ

_{0}H‖ab). The arrows mark the positions of the specific heat peaks. The insets show ∂χ(T)/∂T below 5 T. (Reproduced with permission from reference [90]).

**Figure 13.**(

**a**) (top, bottom) Normalized magnetization M(T)/B of Cu(pz)

_{2}(ClO

_{4})

_{2}for various magnetic fields B ≡ μ

_{0}H applied along the easy (x), middle (y) and hard (z) axis. In all three cases, the cross and droplet symbols mark the minimum and kink, respectively. (Reproduced with permission from reference [94]); (

**b**) (top) Magnetic phase diagram of Cu(pz)

_{2}(ClO

_{4})

_{2}in the field applied along the easy axis. AF, SF and PM denote the collinear antiferromagnetic phase, the spin-flop phase and the paramagnetic phase, respectively. The phase boundaries are constructed from the anomalies in the M(T) and M(H) curves, red and black points correspond to the orientation of the field (ψ = 0° and 12°) with respect to the easy axis. Lines are a guide to the eye. In the inset, an expanded region around the bicritical point is shown. (Reproduced with permission from reference [94]); (

**b**) (bottom) Magnetic phase diagram of Cu(en)(H

_{2}O)

_{2}SO

_{4}. The a, b, c′ axes correspond to the hard, easy and middle axis, respectively. Open symbols represent the theoretical predictions [97] for a field-induced BKT transition in the S = 1/2 HAF square lattice (R = 1, J/k

_{B}≡ J

_{eff}/k

_{B}= 2.8 K, g

_{a}= 2.200, g

_{b}= 2.005 and g

_{c′}= 2.000). Inset: the same diagram, B is replaced by the product B∙g. (Reproduced with permission from reference [82]).

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**MDPI and ACS Style**

Orendáčová, A.; Tarasenko, R.; Tkáč, V.; Čižmár, E.; Orendáč, M.; Feher, A.
Interplay of Spin and Spatial Anisotropy in Low-Dimensional Quantum Magnets with Spin 1/2. *Crystals* **2019**, *9*, 6.
https://doi.org/10.3390/cryst9010006

**AMA Style**

Orendáčová A, Tarasenko R, Tkáč V, Čižmár E, Orendáč M, Feher A.
Interplay of Spin and Spatial Anisotropy in Low-Dimensional Quantum Magnets with Spin 1/2. *Crystals*. 2019; 9(1):6.
https://doi.org/10.3390/cryst9010006

**Chicago/Turabian Style**

Orendáčová, Alžbeta, Róbert Tarasenko, Vladimír Tkáč, Erik Čižmár, Martin Orendáč, and Alexander Feher.
2019. "Interplay of Spin and Spatial Anisotropy in Low-Dimensional Quantum Magnets with Spin 1/2" *Crystals* 9, no. 1: 6.
https://doi.org/10.3390/cryst9010006