Magnetic Properties of 2 D Nano-Islands Subject to Anisotropy and Transverse Fields : EFT Ising Model

An Ising effective field theory (EFT) is presented to calculate the characteristic magnetic properties of a 2D nano-island presenting an out-of-plane magnetization, and subject to an applied in-plane transverse magnetic field. A non-diagonal Ising Hamiltonian with nearest neighbor exchange, single-atom magnetic anisotropy, and a transverse Zeeman term, defines the ground state of the system. We investigate the effects due to the transverse field acting on the magnetic order, in conjunction with those due to the reduced dimensionalities of the core and periphery domains of the nano-island. The choice of a model spin S ≥ 1 for the atoms permits the analysis of spin fluctuations via the single-atom spin correlations. A numerical method is developed to avoid approximations inherent to analytical treatments of the non-diagonal Hamiltonian for spin S ≥ 1 systems. It is applied successfully for nano-island spin S = 1 and 2 systems, generating accurate EFT results. Detailed computations are made for the characteristic magnetic properties of the nano-island over its hexagonal lattice, and applied numerically to calculate the properties of the 2D Co nano-island on an fcc(111) surface. It is shown how the transverse magnetic field perturbs the magnetic order, generating spin correlations and magnetizations for the core and periphery domains that are fundamentally different along the longitudinal and transverse directions. The transverse field drives the system Curie temperature to lower values with increasing strength. The isothermal susceptibilities are shown to be exchange dominated along the out-of-plane direction and quasi-paramagnetic in the inplane. A characteristic thermodynamic function that scales directly with the spin and the transverse field is derived for the correlations of the longitudinal and transverse spin components on the nano-island atomic sites.


Introduction
The study of the magnetic properties of nanostructures on surfaces is of current interest because of the potential use of these structures as basic elements in information storage technology.The 2D magnetically ordered nano-islands on metallic and buffer layer substrates, as Co nano islands on Pt, Au, and Ag (Rasponi et al., 2003;Gambardella et al., 2005;Weiss, 2004;Sessi et al., 2010), constitute a promising class.These nano-island systems present out-of-plane magnetic order.
Previously, we gave the Ising spin effective field theory (EFT) for the characteristic magnetic properties of the 2D nano-islands free from externally applied magnetic fields (Khater & Abou Ghantous, 2011), and also model calculations for such nano-island systems when subject to magnetic fields applied out-of-plane (Abou Ghantous & Khater, 2011).
The purpose of the present paper is to extend the EFT theory to investigate the consequences to the nano-island systems of magnetic fields applied in-plane.The present Ising spin model calculations, furthermore, are given for spins S ≥ 1.These model calculations are applied in particular to compute the characteristic magnetic properties of 2D Co nano-islands on Pt(111) surface.
The influence of an in-plane or transverse field may be traced back to the work of de Gennes (1963) who introduced this term to take into account the tunneling between two potential wells in ferroelectric crystals.Blinc (1960) introduced the notion of a pseudo-spin 1/2, and Stinchcombe (1973) is the first to use the field theoretical approach to treat the transverse field in an Ising spin 1/2 model.Kaneyoshi et al. (1993) have considered this transverse field for some cases of higher spin in bulk systems, but neglected the anisotropy.
In the present paper we develop a novel approach which avoids analytical approximations inherent to previous theoretical treatments of Ising spin Hamiltonians which present both diagonal and off-diagonal terms (Kaneyoshi et al., 1993;Htoutou et al., 2004Htoutou et al., , 2005;;Yüksel, 2010).This novel approach is general for any spin system.Significantly, our results go to the appropriate limits when we suppress selectively the single-atom magnetic anisotropy or the transverse magnetic field.The model computations are carried out in particular for the hexagonal lattice symmetry, and for the spin S = 1 system, as this is the symmetry of Co nano-islands on fcc(111) metallic surfaces, and for the spin S = 2 system to illustrate the robustness of the approach.Lower lattice symmetries such as square or honeycomb are treated in a straightforward manner within the present model.
The characteristic magnetic properties are then calculated using the nondiagonal Ising spin Hamiltonian which contains the nearest neighbors exchange, single-atom magnetic anisotropies, and transverse applied magnetic fields.The choice of the spin S = 1 for the system corresponds to the Co spin, and permits the analysis of local spin fluctuations via the single site spin correlations, which are responsible for many of the magnetic properties of the nano-island systems (Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).
Our present computations yield in particular the longitudinal and transverse components of the single-atom spin correlations, and the domain magnetizations and isothermal susceptibilities, for both the core and periphery domains.The distinction between these domains on the nano-islands is based on their structural differences due to different dimensionalities underlined by an order of magnitude ratio between their corresponding local anisotropies.The magnetic exchange in contrast is the same throughout the system.
The paper is organized as follows.The Ising spin Hamiltonian and required information related to our new EFT theoretical and numerical model approach are presented in section 2. In the following section 3, we apply this model to obtain numerical results for the characteristic magnetic properties for the core and the periphery domains of the nano-island spin S = 1 system on a hexagonal lattice, subject to an applied transverse magnetic field.Since higher spin systems can also be treated within the framework of this model, we give elements related to the nano-island spin S = 2 system as an example of the new numerical approach validity range.Finally, the conclusions and discussion are presented in section 4.

Numerically Generated Ising EFT Model
The general Ising spin Hamiltonian H for a spin S, of an ordered magnetic nano-island system which contains exchange, local magnetic anisotropy, and applied transverse magnetic fields, may be expressed as 2 , The local Hamiltonian may then be written as ⟨i, j⟩ denotes the sum over all nearest neighbours.J is the magnetic exchange, D i the single-atom anisotropy, different for core atoms, D c , and periphery atoms, D p , and Ω the transverse magnetic field, all in meV energy units.The parametric variables for the core atoms will not carry a subscript in general, whereas those for the periphery atoms shall carry systematically the subscript p.Note that the core and periphery atoms of a nano-island have different coordination numbers z c > z p , for any considered lattice, and hence different dimensionalities.Note also that the core domain of a nanoisland has a much greater number of atoms than the periphery domain on the nano-island outer boundary; for example, the number of Co periphery atoms does not exceed 15% of the total Co nano-island atoms, as has been experimentally verified (Weiss, 2004).
The exchange energy J induces the spin alignment along the z-direction.In contrast, the off-diagonal applied magnetic Ω field tends to perturb this magnetic order, generating a thermodynamic average for the spin component along the x-direction, and may also induce quantum-mechanical tunneling, flipping spin orientations.
The local Hamiltonian in Equation (3) remains however invariant when S z goes to −S z on a site, leaving S x unchanged under a Z 2 symmetry.
The spin operators of the local Hamiltonians may be correlated in principle for neighboring sites.This problem is considered for the spin S = 1/2 system by Sá Barretto et al. (1985) using the EFT approach.However, for higher spin values, as those considered in the present investigation, the approach in the cited reference does not apply.To be able to reach a point where numerical calculations become useful, a decoupling approximation is hence considered in the present work which neglects neighboring site correlations but retains single-site correlations.This maintains the invariance of the thermodynamic magnetic properties under spatial translation for each domain over its dimensionality.It permits the computation of the magnetic properties of the nano-island system for arbitrary temperatures except in the neighborhood of the order-disorder transition temperature T c when critical fluctuations render inapplicable the EFT approach.
In the mixed Hamiltonian of Equation ( 1), the single-atom magnetic anisotropy term is diagonal but the transverse magnetic field term is off-diagonal, in the spin states.Kaneyoshi et al. (1993) considered the problem of a bulk system with arbitrary spin subject to a transverse field but neglected the magnetic anisotropy.In recent work (Jiang et al., 1993(Jiang et al., , 1994;;Elkouraychi et al., 1993;Htoutou et al., 2004Htoutou et al., , 2005;;Yüksel, 2010), attempts are made to elaborate analytical approximations towards the diagonalization of the mixed Hamiltonian of Equation (1) for bulk spin S = 1 systems.Despite their usefulness these treatments are limited and inapplicable for higher spins.
In contrast in the present model calculations, we fully treat both the diagonal and off-diagonal terms of the mixed Hamiltonian for the nano-island spin S = 1 and 2 systems, using a numerically generated EFT method to obtain results directly.This approach avoids the previous analytical approximations, which is a necessary condition because the magnetic properties of the nano-island systems depend in a sensitive manner on the domain and periphery dimensionalities and on their spin correlations (Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).This numerical approach yields accurate EFT results valid for a wide range of D i and Ω values.
Using quantum mechanics, the EFT generating functions f Op (x, D i , Ω) are evaluated.These are necessary to formulate the thermodynamic averages for the spin operators Op on the domain sites.For example, the matrix representations for S iz and S ix for spin S = 1 yield following Equation (2) a local spin Hamiltonian matrix in the form 2 2 2 0 ( , , 2 ) 0 The thermodynamic average for a given spin operator Op is calculated as the trace of the product of this operator with the density matrix.Since H i (x, D i , Ω) of Equation ( 3) is not diagonal, it is necessary to develop an appropriate product.We have consequently developed a symbolic and numerical procedure to treat the matrix function Exp[−βH i ] without analytical approximations.We have extensively checked this symbolic and numerical approach, retrieving exact well known results when the transverse field or the anisotropy are selectively suppressed.This numerical formulation can be applied successfully to higher spin values provided the appropriate decoupling approximations for higher order single site spin operators are adopted.
We present below the general expression, written in the Mathematica code, for the characteristic generating functions f Op , as used in our manipulations of the symbolic and numerical package, where β = 1/kT with temperature kT in units of meV Tr MatrixExp ( , , ) In the limit 0   , Equation (4) reduces appropriately to the generating functions derived in previous work for the corresponding systems (Kaneyoshi et al., 1993;Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).The operation of Exp in Equation ( 4) is defined in the Mathematica package as the code MatrixExp.To illustrate these generating functions, we give in the appendix the exact Van der Wearden identity for S = 2 and its characteristic functions as deduced from Equation (4), omitting the anisotropy therein for simplicity.
The thermodynamic canonical averages in EFT concern the appropriate set of spin operators in the system.For the present calculation, these are the longitudinal and transverse magnetizations, namely m z = <S zi > and m x = <S xi >, the single site component correlations , and the single site cross correlations , where i denotes a given site on the core and periphery domains.The computations of these variables are developed using the exact Van der Wearden identities.For spin S = 1, for example, this identity is where ∇ = ∂/∂x is the differential operator.In Equation ( 5) the operators for the magnetizations S z and the single site spin correlations 2 z S are clearly identified under the identity.The EFT approach is general and may be applied to higher spin values with corresponding Van Der Wearden identities, but for these systems one needs to make appropriate decoupling approximations for the multiple-spin correlations as shown for the S = 2 system, for example, in the appendix.
The developed Mathematica package that pilots the code to calculate Equation ( 4) consists of a number of symbolic and numerical procedures.To give an idea of its scale, one manipulates ∼ 500 and ∼ 1500 terms for the square and hexagonal lattices, respectively.These terms are necessary to compute the canonical averages for each magnetic variable, and their number increases considerably with increasing z c , reaching ∼ 20000 terms for an fcc lattice for S = 5/2 systems (Abou Ghantous et al., 2012a, 2012b).The developed package solves these symbolic problems for the needed expressions in a direct way.
For a core site pertaining to the nano-island core domain, modeled by a 2D lattice with coordination number z c , the generating equations that yield the required thermodynamic averages for a given spin operator, ⟨Op⟩, are expressed generically as A core site interfacing the periphery domain has nearest-neighbor atoms in the core as well as on the periphery domain, which necessitates writing separate generating equations.However, since the core domain is significantly larger than the periphery domain, we neglect the periphery-core exchange interactions for the core Hamiltonian.
All atomic sites pertaining to the periphery domain interface naturally the core domain, and have each r nearest-neighbor atoms on the periphery and s nearest neighbors in the core domain.The required generating equations for a periphery site are hence given by The exponential operators in Equations ( 6), (7), and (8) are replaced by the Van der Wearden identity of Equation ( 5) for spin S = 1, and should be replaced by the appropriate Van derWearden identities for any other given spin system.Note that z c = 6 > z p = (r + s) for a hexagonal lattice, and that the differential operator has the following property Equations ( 6), ( 7), and ( 8) are necessary and sufficient to compute the magnetic properties for the core and periphery domains as detailed as follows.
Equation ( 6) and Equation ( 8) for the core domain sites generate in their applications to the site spin operators and single-atom spin correlations, a system of nonlinear polynomial equations in the longitudinal and transverse magnetizations, m z (kT; J, D c , Ω) and m x (kT; J, D c , Ω), and in the single-atom spin correlations q zz (kT; J, D c , Ω), q xx (kT; J, D c , Ω), and q zx (kT; J, D c , Ω).They are self-consistent equations which can be used to solve successively for the magnetic variables of interest.For example, the magnetization m for a core site can be calculated from its its characteristic reduced polynomial 3 5 (1) (3) (5) and the single-atom spin correlations q zx for a core site may then be calculated from its characteristic reduced polynomial Equation ( 7) and Equation ( 8) for the periphery domain sites generate in their applications, using r = 2 and s = 2 for the hexagonal lattice, the set of nonlinear polynomials per periphery site for the longitudinal and transverse magnetizations, m zp (kT; Ω, D p ; m z , q zz ) and m xp (kT; Ω, D p ; m z , m x , q zz ), and single atom spin correlations q zzp (kT; Ω, D p ; m z , m x , q zz ), q xxp (kT; Ω, D p ; m z , m x , q zz ), and q zxp (kT; Ω, D p ; m z , m x , q zz ).We give here only those for m zp and q zxp as a sample representation The coefficients a(n), e(n),..., and u(n), p(n),..., in Eqs.(9, 10,...; 11, 12...) are in general extensive polynomial functions of the temperature kT, core variable q zz , and system constants (Ω, J, D c , D p ).They are determined exactly from the numerical results obtained by applying the symbolic and numerical procedures of the Mathematica package code.It is in this sense that Equations (9, 10, ...; 11, 12...) are accurate numerically generated results.The ensembles of the nonlinear polynomial equations are obtained directly in our procedure; however, they will not be detailed here because of their extensive symbolic and numerical character.
Higher nano-island spin S > 1 systems generate, for the same lattice, higher-order nonlinear polynomial equations, characteristic of the corresponding z c core, and r and s periphery, coordination numbers.
To obtain unique solutions for the magnetic variables for the core domain, the nonlinear polynomials for m z and q zz must be satisfied simultaneously at all temperatures.Note that Equation ( 9) admits simple solutions for mz, which correspond to four for the hexagonal lattice.We have developed a specific procedure to solve for the unique solution of this variable in the entire temperature range, see details in (Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).Substituting for the physically appropriate root m z from Equation ( 9) into the nonlinear polynomial for q zz , we are able to reduce the latter to a polynomial identity in the single-atom spin correlations q zz (kT; D c , Ω).The set of the calculated unique physical solutions for m z and q zz permit in turn the determination of the transverse magnetization m x (kT; D c , Ω), and the single-atom spin correlations q xx (kT; D c , Ω) and q zx (kT; D c , Ω), for the core domain from Equations (9, 10, ...).
The core domain results are injected in the periphery polynomial equations Equations (11, 12, ...) to obtain the corresponding variables q zzp and m zp for the periphery atoms.These permit in turn the determination of the transverse magnetization m xp , and the correlation variables q xxp and q zxp .
The above results give direct access to the isothermal longitudinal and transverse susceptibilities, χ z (kT; Ω) and χ x (kT; Ω), for the core and periphery domains, under the influence of the transverse magnetic field.The susceptibilities are calculated in the normalized form where α = x, z.The longitudinal and transverse susceptibilities, for a core site, χ z and χ x , and for a periphery site, χ zp and χ xp , are consequently calculated using Equation (13).
The characteristic magnetic properties for a nano-island system are consequently computed in a comprehensive manner.The model calculations for the spin S = 1 system are applied numerically to compute the magnetic properties for the 2D mono-layer Co nano-islands on nonmagnetic fcc(111) metal surfaces.Since this nano-island system presents a hexagonal lattice symmetry, it is of interest to specifically consider this lattice.The numerical results are computed using the magnetic exchange and core anisotropy constants, J = 2 meV and D c = −0.1 meV respectively, characteristic of the 2D Co nano-islands on Pt(111) substrates.The present computations consider in particular the influence of low and high transverse magnetic fields, namely for Ω/J = 1/2 and Ω/J = 3, in comparison with the nearest neighbor magnetic exchange.The EFT model computations for the honeycomb, square and hexagonal lattices show common features for the magnetic properties of the nano-island under an applied transverse field.However, the results presented here are only for the hexagonal lattice.

Magnetic Properties of the Hexagonal Nano-Island Under the Influence of a Transverse Magnetic Field
The calculated results for q zz and q xx for the core domain of the Co nanoislands are presented in Figure 1 as a function of temperature, for the considered transverse fields.
The calculated longitudinal and transverse magnetizations, m z and m x , for sites in the core domain of the 2D Co nano-island, are presented in Figure 2 as a function of temperature, for the relatively low and high transverse magnetic fields, Ω/J = 1/2 and Ω/J = 3.They are seen to satisfy the required relation 2 0K for both fields.The order parameter m z decreases in the ordered phase going to zero at a T c that corresponds in each case to the selected transverse field.In general m x increases slightly with temperature in the ordered phase before decreasing monotonically for temperatures T > T c .
Further, it is observed in Figure 2 that T c decreases with increasing Ω.This is a direct measure of the ability of the applied transverse field term to disrupt and reduce the magnetic order.A potential outcome under a critical transverse field c    is to drive the order-disorder transition temperature 0 c T  , provided the magnitude of the single-atom magnetic anisotropy D c is sufficient to quench the tri-critical point in the phase diagram (kT c /J, Ω/J) of the core domain (Miao et al., 2009;Yüksel et al., 2010).The value D c = −0.1 meV for the 2D Co nano-island core domain satisfies these criteria.This potential outcome marks the possibility of a quantum phase transition for the core domain and the nano-island at T c ≈ 0, that may be accessed by varying the transverse magnetic field up to Ω c .Note that the Hamiltonian of Equation ( 1) is similar to that used for the investigation of quantum critical points (Sachdev, 2001), when Ω is large and the system is near its critical temperature.Figure 3 and Figure 4 present our calculated results for the core domain susceptibilities, χ z and χ x respectively, as a function of temperature, for the relatively low and high transverse magnetic fields.In Figure 3 the core domain longitudinal susceptibility preserves its character of a second order phase transition, as has been observed for a nano-island spin system free from the applied transverse magnetic field (Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).An important feature, however, is observed for χ z at very low temperatures, namely that it starts from large non zero values.The transverse field provokes this, effectively, by reducing q z and 2 z m in unequal proportions.The model calculations yield equally the correlations for the longitudinal and transverse components of the spin on a single atom, namely q zx (kT; Ω) = ⟨S zc S xc ⟩.This prompts us to consider the function ϕ zx (kT; Ω) defined by the following equation Equation ( 14) introduces the characteristic thermodynamic function ϕ zx as a measure for the random phase decoupling approximation for the correlations of the longitudinal and transverse spin components on a single atom.We have verified numerically that q zx is symmetric, q zx = ⟨S z S x ⟩ = ⟨S x S z ⟩ = q xz , commuting x and z.See also Equation ( 4).The calculated results for q zx and ϕ zx for the spin S = 1 system are presented in Figure 5.Note that ϕ zx is not identical to unity, and that it varies monotonically with temperature in the magnetically ordered phase, T ≤ T c .In particular ϕ zx (0; Ω) = 0.5 with varying transverse fields.
The numerical calculations are carried out using the same constants for the exchange, the core and the periphery anisotropies, characteristic of the 2D Co nano-islands on Pt(111) substrates.In these calculations we consider again the influence of the relatively low and high transverse fields in comparison with the magnetic exchange, namely Ω/J = 1/2 and Ω/J = 3. Figure 6a and Figure 6b present, for respectively low and high transverse fields, the calculated results for the longitudinal and transverse single-atom spin correlations for a periphery site, q zzp and q xxp , in comparison with the longitudinal q zz for a core site.As is physically expected the q zzp decreases whereas the qxxp increases with the transverse field.For a sufficiently high transverse field a crossover temperature exists above which qxxp may become greater than q zz on a periphery site, as in Figure 6b.
The calculated results for the periphery longitudinal and transverse magnetizations, m zp and m xp , are presented in Figure 7, and seen to satisfy the required relation 2   at T = 0 K for the considered transverse fields, as is observed for the core magnetizations.However, contrary to the latter, m xp and m zp decrease monotonically at the periphery in the ordered phase going together to zero at T c .For a given transverse field there is a crossover temperature above which the transverse magnetization becomes greater than the longitudinal magnetization on a periphery site, this is particularly clear for the high transverse magnetic field.
Further, Figure 5a, Figure 5b, and Figure 6 show that the longitudinal and transverse single-atom spin correlations, q zzp and q xxp , and the longitudinal and transverse magnetizations, m zp and m xp , are well defined for the periphery domain in the ordered magnetic phase, T ≤ T c , but vanish however in the disordered phase for T > T c .This thermodynamic behavior is a general consequence of the lower dimensionality of the periphery domain in comparison with the core domain, which has been observed previously in the theoretical analysis of the magnetic properties of the the 2D Co nano-islands in two other specific cases, namely in the absence of an externally applied magnetic field (Khater & Abou Ghantous, 2011), and under the influence of an applied longitudinal field (Abou Ghantous & Khater, 2011).Figure 6.Calculated single-atom spin correlations q zzp and q xxp for the Co nano-island periphery domain, for the relatively low and high transverse fields Ω/J = 1/2 and Ω/J = 3.The q zz for the Co nano-island core domain is indicated for comparison.The hexagonal lattice symmetry and system constants D c = −0.1,D p = 0.9, and J = 2 in meV, correspond to Co nano-islands on the Pt(111) surface The transition temperature T c is common to the core and periphery nanoisland domains and decreases with increasing transverse field.This implies that under a sufficiently strong critical transverse field c    , the nanoisland domains approach simultaneously a quantum phase transition at T = 0 in the absence of the tri-critical point, as observed previously using the same detailed numerical technique (Khater & Abou Ghantous, 2011).
The isothermal longitudinal susceptibility χ zp (kT; Ω) for the periphery domain of the Co nano-island in the presence of the transverse field Ω, is calculated in the normalized form using Equation ( 13).In Figure 8(left) and Figure 9(left) we present our calculated results for the χ zp for a periphery site as a function of temperature, for respectively low and high transverse fields, in comparison with the χ z for a core site.Further, it is known that the experimentally measured isothermal longitudinal susceptibility for an ensemble of nano-islands, with statistical variations of the sizes of the core and periphery domains, is effectively a statistical average (Khater & Abou Ghantous, 2011;Abou Ghantous & Khater, 2011).We present hence in Figure 8(right) and Figure 9(right) the calculated total longitudinal susceptibility χ z;total for an average nano-island configuration putting 15% of its atoms on the periphery, and 85% in the core.Though the core sites present a second order phase transition, it is observed that the nano-island χ z;total does not strictly present the character of this transition.A discontinuity occurs at T = T c for low transverse field, while it is absent for relatively high transverse fields.11), as in the previous section.In Figure 11(right) we present our calculated results for the function ϕ zxp for two transverse fields values 2 and 6 meV grouping each the periphery sites and core sites curves for comparison.Despite the inflexion in ϕ zxp for periphery sites it is quite comparable to ϕ zx for core sites, for low and high transverse fields.
Figure 12.Calculated core magnetic variables m, q, m x , and χ T , for the spin S = 2 system, showing similarity with the same calculated magnetic variables for the spin S = 1 system as presented in previous figures.The system parameters are indicated Although they are not presented here, we have also calculated the principal magnetic properties for the hexagonal lattice for the spin S = 2 nanoisland system, to test the robustness of the method.Comparable forms are derived for the magnetic variables including q zx , ϕ zx , q zxp , and ϕ zxp .In this case however ϕ zx (0; Ω) = 0.8.This confirms the observation that the ϕ function presents attributes of a general form scaling directly and simply with the spin and the applied transverse field.In particular, Figure 12 presents the results for the spin S = 2.The Van Der Wearden identity and the decoupling approximation are given in the appendix.The same behavior, apart the change of scale with the spin S = 1 system, is observed for the q variable.Given the importance of this variable to other quantities, we can use this observation to verify that one can get the result for the spin S = 2 system directly from that of the spin S = 1 system by scaling the results under the condition that both must have the same critical temperature but with different values of exchange constants J.The verification was better than 10 −4 .Higher spin systems, for spin S = 5/2, and 9/2, have been investigated in this respect (Abou Ghantous, 2012a, 2012b) related to the island of 60 C  adsorbed on buffer surfaces, where the transverse field describes the tunneling between the JT wells in the fullerene anion.

Summary and Conclusions
In this work we present an Ising spin EFT model to investigate the consequences of an applied transverse field for the magnetic properties of a 2D mono-layer nano-island.A non-diagonal Ising Hamiltonian with nearest neighbor exchange, single-atom magnetic anisotropy, and a transverse magnetic field term, defines the ground state of the system.The choice of Ising spin S = 1 and S = 2 systems for the nano-island, permits the analysis of the effects of spin fluctuations via the single-atom spin correlations.
To avoid approximations inherent to previous analytical treatments of the Ising spin Hamiltonian which presents diagonal and off-diagonal terms, a novel symbolic and numerical approach is developed.The procedure consists of using defined codes in Mathematica package for the density matrix over any given characteristic spin operator.It is this numerical property which permits the generation of numerically exact EFT results, valid for a wide range of values for the local anisotropy and applied magnetic transverse fields.This novel approach is general.It can be applied successfully to spin systems higher than S = 1, and for diverse 2D lattices such as the honeycomb, square, and hexagonal.Compared to other theoretical and numerical treatments, such as the Monte Carlo simulations, the Ising EFT model remains a robust and useful approach for the study of 2D nano-islands which goes beyond mean field theory.
In particular, we investigate the consequences of an applied transverse magnetic field in conjunction with the structural effects inherent to the different reduced dimensionalities and their anisotropies on the nano-island core and periphery domains.The principal longitudinal and transverse magnetic properties of the domains of a magnetic nano-island, are computed for relatively low and high transverse fields.Though both the core and periphery domains have a common order-disorder transition temperature, the properties of each are very different, they are also different along the longitudinal and transverse directions.Our model calculations are systematically applied to calculate the charactersitic magnetic properties of the magnetically ordered 2D mono-layer Co nano-islands on the Pt(111) surface.
We have primarily developed and presented in this work the model calculations for a hexagonal lattice nano-island, with spin S = 1.To test the robustness of our model, we have also applied our method to calculate the magnetic properties of a square lattice nano-island with spin S = 2, although these latter results are not presented here.

Figure 1 .
Figure 1.Calculated single-atom spin correlations, q zz and q xx , along the longitudinal and transverse directions, for the core domain of the Co nano-island, as a function of temperature in meV, for relatively low and high transverse fields Ω/J = 1/2 and Ω/J = 3.The hexagonal lattice symmetry and system constants D c = −0.1 and J = 2 in meV correspond to Co nano-islands on the Pt(111) surface.The discontinuities of the qxx derivative mark the system T c for the corresponding Ω values

Figure 4 (
Figure4(right) gives in particular the calculated results for the inverse susceptibility 1/χ x of the transverse spin component in the core domain, showing an approximate paramagnetic behavior in the ordered phase for temperatures up to T ∼ T c /2.In the region about T c , it has a complicated behavior due to the decreasing magnetic order.Above T c the linear slopes of 1/χ x with temperature point clearly a paramagnetic regime for the transverse spin component in the disordered phase.

Figure 4 .
Figure 4. Calculated isothermal transverse susceptibility χ x (left), and its inverse 1/χ x (right), for the core domain of the Co nano-island, as in Figure 1.The reciprocal 1/χ x indicates clearly two limiting paramagnetic regimes for the transverse components of spin in the ordered and disordered phases above T c

Figure 5 .
Figure 5.As in Figure 1, (left) single-atom spin correlations, q zx , and (right) characteristic thermodynamic function ϕ zx , for the Co nano-island domain

Figure 7 .
Figure 7.As in Figure 6, for the longitudinal and transverse magnetizations, m zp and m xp , for the Co nano-island periphery domain subsumed in the generating functions f op .used with the decoupling procedure