Two-spin Cluster Research Articles

Calculation of partition function of Ising model on quantum computer

We study the partition function of the Ising model on a graph with the help of quantum computing. The Boltzmann factor is modeled on a quantum computer as a trace of some evolution operator with effective Hamiltonian over ancilla spins (qubits) corresponding to graph links. We propose two methods for this which are based on effective Hamiltonian with three-spin interaction and on two-spin interaction. The limit of small temperatures allows us to find the ground state of the system that is related to the discrete combinatorial optimization problem. The partition function of the Ising model for two-spin clusters is calculated on IBM's quantum computer. The possibility of finding ground state is also demonstrated for two-spin clusters.

An Influence of Fluorinated Alkyl Substituents on Structure and Magnetic Properties of Mn(II) Complexes with Pyrazolyl-Substituted Nitronyl Nitroxides

New complexes of manganese(II) hexafluoroacetylacetonate [Mn(hfac)2] with 2-(1-R-3-pyrazol-4-yl)-4,4,5,5-tetramethyl-2-imidazoline-3-oxide-1-oxyl (R = CHF2, CH2CH2F, CH2CHF2 or CH2CF3) were synthesised and characterised structurally and magnetically. All complexes were prepared under similar conditions. Nonetheless, their crystal structures were considerably different. Depending on the structure of fluorinated alkyl substituent R, the complexation reaction led to complexes of three types: chain-polymeric complexes with the head-to-head or head-to-tail motif and complexes of molecular structure. All complexes show strong antiferromagnetic behaviour in a high-temperature region (150–300 K) and weak ferro- or antiferromagnetic exchange interactions at low temperatures. The stronger antiferromagnetic exchange, −101.7 ± 1.5 or −136 ± 10 cm−1, −82.3 ± 1.3 cm−1 and −87.4 ± 1.3 cm−1, was attributed to the magnetic interaction in three- or two-spin clusters: {>N∸O–Mn2+–O∸N<} or {>N∸O–Mn2+}, respectively. The weaker antiferromagnetic interaction, −0.005, between three-spin clusters or ferromagnetic interactions, 0.18–0.81 cm−1, between two-spin clusters are realised through the pyrazole ring or intermolecular contacts.

Critical properties of the spin-1 Heisenberg model with random magnetic field and Dzyaloshinskii–Moriya interaction

The spin-1 Heisenberg model with Dzyaloshinskii–Moriya interaction and in presence of a random magnetic field is studied by using the framework of the two-spin cluster approximation. The formalism is applied to the simple cubic lattice (q=6). At finite temperature, the phase diagrams of the system reveal the presence of tricritical points, implying the presence of second and first-order phase transitions. In addition, the occurrence of fourth-order critical points is also investigated.

Investigation of spin-phonon coupling and local magnetic properties in magnetoelectric Fe2TeO6

Spin-phonon coupling originated from spin-lattice correlation depends upon different exchange interactions in transition metal oxides containing 3d magnetic ions. Spin-lattice coupling can influence the coupling mechanism in magnetoelectric material. To understand the spin–lattice correlation in inverse trirutile Fe2TeO6 (FTO), magnetic properties and phonon spectra are studied. Signature of short-range magnetic correlation induced by 5/2–5/2 dimeric interaction and magnetic anomaly at 150 K is observed apart from the familiar sharp transition (TN ∼ 210 K) corresponding to long-range order by magnetization and heat capacity measurements. The magnetic transitions and the spin dynamics are further locally probed by muon spin resonance (μSR) measurement in both zero fields (ZF) and longitudinal field (LF) mode. Three dynamically distinct temperature regimes; (i) T > TN, (ii) TN > T > 150 K, and (iii) T < 150 K, are observed. A change in the spin dynamics is realized at 150 K by μSR, though previous studies suggest long-range antiferromagnetic order. The renormalization of phonon frequencies observed in Raman spectra below 210 K suggests the existence of spin-phonon coupling in the material. The coupling strength is quantified as in the range 0.1–1.2 cm−1 following the mean-field and two-spin cluster approximations. We propose that the spin-phonon coupling mediated by the Fe-O2-Fe interbilayer exchange play a significant role in magnetoelectric ME coupling observed in the material.

Tricritical behavior of the spin-3/2 anisotropic Heisenberg model with Dzyaloshinskii–Moriya interaction

The spin-3/2 anisotropic Heisenberg model with Dzyaloshinskii–Moriya interaction is studied by using the framework of the two-spin cluster approximation. The formalism is developed for lattices with general coordination number. However, we address our attention to the simple cubic lattice (q=6). The phase diagram of the system reveals the presence of tricritical points and the first and second-order phase transitions are obtained. The critical behavior is discussed and compared with the spin-1/2 and spin-1 Heisenberg systems.

Phase diagram of the classical Heisenberg model in a trimodal random field distribution

Abstract The classical spin 1 / 2 Heisenberg model on a simple cubic lattice, with fluctuating bond interactions between nearest neighbors and in the presence of a random magnetic field, is investigated by effective field theory based on two-spin cluster. The random field is drawn from the asymmetric and anisotropic trimodal probability distribution. The fluctuating bond is extracted from the symmetric and anisotropic bimodal probability. We estimate the transition temperatures, and the phase diagram in the T c - h , T c - p and T c − α planes. We observe that the temperature of the tricritical point decreases with the increase of disorder in exchange interactions until the system ceases to display tricritical behavior. The disorder of the interactions and reentrant phenomena depends on the trimodal distribution of the random field.

Exchange coupling transformations in Cu (II) heterospin complexes of “breathing crystals” under structural phase transitions

Family of “breathing crystals” is the polymer-chain complexes of Cu(hfac)2 with nitroxides. The polymer chains consist of one-, two- or three-spin clusters. The “breathing crystals” experience simultaneous magnetic and Jahn-Teller type structural phase transitions with change of total cluster spin and drastic change of bond lengths (ca. 10-12%). For the first time the intra-cluster magnetic couplings in ”breathing crystals” have been calculated both by band structure methods GGA + U and hybrid DFT (B3LYP and PBE0) for the isolated exchange clusters. The temperature dependence of the magnetic coupling constant was calculated for two polymer-chain compounds of the “breathing crystal” family - C21H19CuF12N4O6 with the chains containing two-spin clusters and C22H21CuF12N4O6 with the chains of alternating three-spin clusters and one-spin sites. It was found that adding a Hubbard-like parameter not only to the copper 3d electrons but also to the oxygen 2p electrons (GGA + Ud + Up approach) results in an improved description of exchange coupling in the “breathing crystal” compounds. At the same time treatment of the isolated clusters by a large basis hybrid DFT with high computational cost provides a similar quality fit of the experimental magneto-chemical data as that for the GGA + Ud + Up band structure calculation scheme. Our calculations also showed that in spite of the abrupt transformation of the magnetic coupling constant under the phase transition, the band gap in the “breathing crystals” remains about the same value with temperature decrease.

Ornstein-Zernike correlations and magnetic ordering in nanostructures

In this paper we propose a Heisenberg variational approach to study pseudo-critical phenomena on ferromagnetic nanostructures. We combine a two-spin cluster 3-dimensional Heisenberg Hamiltonian with Orstein-Zernike correlations and consider several geometries and crystalline lattices to explore the relationship among these factors and the effective number of nearest neighbors defined in several kind of nanometric structures. With this method we examine the size at which the pseudo-critical temperature of a magnetic nanoparticle reaches its bulk value. Our results shed light on the nanoscopic-microscopic limit, evidencing in particular that when one dimension is very small, independently of how big the other dimensions become, it is not possible for the structure to reach the bulk-like behavior. The results of our model are in good agreement with experimental data and other available analytical models.

Random field effects on the anisotropic quantum Heisenberg model with Gaussian random magnetic field distribution

The effect of a zero-centered Gaussian random magnetic field distribution on the phase transition properties of the anisotropic quantum Heisenberg model has been investigated on a honeycomb lattice within the framework of effective field theory (EFT) for a two-spin cluster (which is abbreviated as EFT-2). Particular attention has been devoted to investigation of the effect of the anisotropy in the exchange interaction on a system with Gaussian random magnetic field distribution. The variation of the critical temperature with the randomness parameter (i.e., the width of the distribution) has been obtained for several anisotropy parameters. Critical Gaussian distribution width values, which make the critical temperature zero, have been obtained. Moreover, it has been concluded that all critical temperatures are of second order, and that reentrant behavior does not exist in the phase diagrams.

Quantum many-body theory of qubit decoherence in a finite-size spin bath

Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. We develop a cluster-correlation expansion (CCE) theory for the spin-bath dynamics relevant to the qubit decoherence problem. A cluster-correlation term is recursively defined as the evolution of a group of bath spins divided by the cluster correlations of all the subgroups. This correlation accounts for the authentic (nonfactorizable) collective excitations within a given group. The bath propagator is the product of all possible cluster correlation terms. For a finite-time evolution as in the qubit decoherence problem, a convergent result can be obtained by truncating the expansion up to a certain cluster size. The two-spin cluster truncation of the CCE corresponds to the pair-correlation approximation developed previously [W. Yao et al., Phys. Rev. B 74, 195301 (2006)]. In terms of the standard linked cluster expansion, a cluster-correlation term is the infinite summation of all the connected diagrams with all and only the spins in the group flip-flopped, and thus the expansion is exact whenever convergence occurs. When the individual contribution of each higher-order correlation term to the decoherence is small (while all the terms combined in product could still contribute substantially), as the usual case for relatively large baths, where the decoherence could complete well within the bath spin flip-flop time, the CCE coincides with the cluster expansion [W. M. Witzel and S. Das Sarma, Phys. Rev. B 74, 035322 (2006)]. For small baths, however, the qubit decoherence may not complete within the bath spin flip-flop time scale and thus individual higher-order cluster correlations could become significant. In such cases, only the CCE converges to the exact coherent dynamics of multispin clusters. We check the accuracy of the CCE in an exactly solvable spin-chain model.

Quantum phase transition in the three-dimensional anisotropic frustrated Heisenberg antiferromagnetic model

The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest $({J}_{1})$- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state $(T=0)$ and the finite-$T$ phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.

Quantum Model for Ferromagnetic Thin Films with an Alternating Crystal Field

The ferromagnetic Blume–Capel–Ising (BCI) model has been studied within the mean field approximation [1], the effective field theory [2], the two-spin cluster approximation in the cluster expansion method [3, 4], Monte Carlo simulations [5], a thermodynamically self-consistent theory based on an Ornstein–Zernike approximation [6], the exact solution based on the Bethe lattice by means of the exact recursion relations [7]. Most of the studies mentioned above displays also the existence of a tricritical point at which the phase transition changes from second order to first order when the value of K2 becomes negative. Our work represents the first atempt to consider a quantum version of BCI model. Within quantum Blume–Capel (QBC) model we will study the influence of the enhancement of the surface exchange coupling and the alternative single-ion anisotropy K2(1) on the odd atomic layers and K2(2) on the even ones on the critical behaviour of thin ferromagnetic films. The Hamiltonian of the considered system consists a Heisenberg exchange interaction with strength Jij > 0 between nearest neighbour lattice sites, an exchange anisotropy with strength D > 0, and a second-order single-ion anisotropy with strength K2 > 0:

Phase diagram and tricritical behavior of the spin-1 Heisenberg model with Dzyaloshinskii–Moriya interactions

Using the two-spin cluster mean-field method, the spin-1 Heisenberg model with Dzyaloshinskii–Moriya (DM) interactions is studied for the simple cubic lattice. For the case of the DM vector coupling D ⇒ = D z ^ ( D is the DM interaction parameter and z ^ is the unit vector of the z-axis direction), the phase diagram of this system and the thermal behavior of the magnetization are obtained, and it is found that the system exhibits the tricritical point. The critical behavior of the system may be interpreted as a result of a competition between the exchange interaction and the DM interaction.

Tricritical behavior of a mixed ising spin system within the effective-field approximation

We investigate the ability of the effective-field theory, within the one- and two-spin cluster approximation, to predict the absence of tricritical behavior, for a mixed spin- 1 2 and spin-1 Ising system on the honeycomb and the square lattice.

Two-spin cluster theory applied to the ferromagnetic spinel CdCr 2pIn 2–2pS 4

The magnetic properties of the B-spinel ferromagnetic CdCr 2pIn 2–2pS 4 compound are studied via a cluster series expansion approximation with nearest and next-nearest exchange integrals J 1 and J 2, respectively. Using the minimization of the free-energy expressions, the magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat are obtained and computed numerically as a function of temperature and for each composition of the system. The magnetization curves are used to determine the critical temperatures T c. Using the power laws in the vicinity of the critical regions the critical exponents β, γ and α associated, respectively, with the magnetization, the magnetic susceptibility and the specific heat are numerically calculated. The critical temperatures T c obtained are in very good agreement with those predicted by the magnetic measurements and the values of the critical exponents may be compared with other theoretical results based on the 3D Heisenberg model.

Néel temperature and sublattice magnetiztion of a three-dimensional classical and quantum Heisenberg antiferromagnet

The effective-field theory (EFT) of a two-spin cluster is applied to study the critical behavior of a three-dimensional classical and quantum Heisenberg antiferromagnet (AF). Néel temperature ( T N ) and sublattice magnetization are calculated numerically and compared with the results of these models with ferromagnetic (F) interaction. We find that our results, for the quantum model T N is higher than T c (Curie temperature, F case) but for the classical model T N = T c . The difference between F and AF results for the quantum model obtained here by EFT is in accordance with Monte Carlo simulation and high-temperature expansion.

Spin dynamics in solid dilute paramagnets

The free-induction decay signals for the two-spin clusters, the “matrix,” and the entire spin system as a whole are calculated for a real dipole-dipole interaction on the basis of a method for the cluster expansion of correlation functions previously developed by the authors. The intracluster interactions, which generate a discrete spectrum, are taken into account exactly, and the interactions outside the clusters are taken into account on the basis of the Anderson-Weiss-Kubo (AWK) theory of phase relaxation. The corresponding shape functions of the resonance line are calculated. Before the clusters are separated, an AWK analysis in the limit of slow fluctuations of the local fields is in satisfactory agreement with the exact results obtained in the Anderson model of dipolar interactions, but a similar analysis for realistic fluctuation rates and Dt?1 leads to a significant qualitative difference: exp(−Dt) transforms into \(\exp ( - \sqrt {D_1 t} )\). After the clusters are separated, this difference is obliterated. The free-induction decay is almost exponential up to Dt∼10, and nonexponential behavior of the long-time asymptote appears sooner. This finding is not altered significantly when the analysis is complicated by raising the maximum rank of the clusters to 3 or more.

Critical behavior of [formula omitted] anisotropic Heisenberg ferromagnet with a transverse field

The spin- 1 2 anisotropic Heisenberg ferromagnet with a transverse field is studied by the use of a new method, which combines the two-spin cluster expansion with the perturbation theory. The phase diagrams in which the transition temperature is shown as a function of the transverse field as well as the exchange anisotropic parameter are obtained for a two-dimensional square lattice and a three-dimensional simple cubic lattice. The results can be compared with that of mean field renormalization group (MFRG) and variational-cumulant expansion (VCE).

Cluster expansion for the [formula omitted] Blume-Capel model

The two-spin cluster approximation in the cluster expansion method is applied for the spin- 3 2 Blume-Capel model on the simple cubic lattices. Within this framework the phase diagram and the temperature dependence of the magnetization are obtained. We have found in the ordered phase a first-order line ending at an isolated critical point in the midst of an ordered phase (coexistence of two critical phases). The results are in qualitative agreement with the results of the molecular field approximation.

Two-spin cluster theory for the Blume-Capel model with arbitrary spin

A two-spin cluster effective field theory for the Blume-Capel model with arbitrary spin S is presented by making use of exact spin identities and taking advantage of the differential operator technique. The dependence of the transition temperature is studied as a function of the single-ion anisotropy field strength for the particular cases S = 1, 3 2 and 2 . The results are compared to those of the single-spin cluster theory recently reported in the literature as well as to other methods. In particular, it is shown that the present approach correctly distinguishes the geometry of the lattice structure beyond its coordination number.