High-temperature Series Expansions Research Articles

High temperature series expansions of S = 1/2 Heisenberg spin models: Algorithm to include the magnetic field with optimized complexity

This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin S=1/2S=1/2 in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. HTSE calculations involve two key steps: graph enumeration on the lattice and trace calculations for each graph. The introduction of a non-zero magnetic field adds complexity to the expansion because previously irrelevant graphs must now be considered: bridged graphs. We present an efficient method to deduce the contribution of these graphs from the contribution of sub-graphs, that drastically reduces the time of calculation for the last order coefficient (in practice increasing by one the order of the series at almost no cost). Previous articles of the authors have utilized HTSE calculations based on this algorithm, but without providing detailed explanations. The complete algorithm is publicly available, as well as the series on many lattice and for various interactions.

Finite and high-temperature series expansion via many-body perturbation theory: application to Heisenberg spin-1/2 XXZ chain

We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.

Classical and quantum phases of the pyrochlore S=12 magnet with Heisenberg and Dzyaloshinskii-Moriya interactions

We investigate the ground state and critical temperature phase diagrams of the classical and quantum $S=1/2$ pyrochlore lattice with nearest-neighbor Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider ferromagnetic and antiferromagnetic Heisenberg exchange as well as direct and indirect DMI. Classically, three ground states are found: all-in/all-out, ferromagnetic and a locally ordered $XY$ phase, known as $\Gamma_5$, which displays an accidental classical U(1) degeneracy. Quantum zero-point energy fluctuations are found to lift the classical ground state degeneracy and select the $\psi_3$ state in most parts of the $\Gamma_5$ regime. Likewise, thermal fluctuations treated classically, select the $\psi_3$ state at $T=0^+$. In contrast, classical Monte Carlo finds that the system orders at $T_c$ in the $\psi_2$ state of $\Gamma_5$ for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from $\psi_2$ to $\psi_3$ at a temperature $T_{\Gamma_5} <T_c$. The same method finds that the system orders via a single transition at $T_c$ directly into the $\psi_3$ state for most of the region with ferromagnetic Heisenberg exchange and indirect DMI. Such ordering behavior at $T_c$ for the $S=1/2$ quantum model is corroborated by high-temperature series expansion. To investigate the $T=0$ quantum ground states, we apply the pseudo-fermion functional renormalization group (PFFRG). The quantum paramagnetic phase of the pure antiferromagnetic $S=1/2$ Heisenberg model is found to persist over a finite region in the phase diagram for both direct or indirect DMI. We find that near the boundary of ferromagnetism and $\Gamma_5$ antiferromagnetism the system may potentially realize a quantum ground state lacking conventional magnetic order. Otherwise, for the largest portion of the phase diagram, PFFRG finds the same ordered phases as in the classical model.

Interaction models and configurational entropies of binary MoTa and the MoNbTaW high entropy alloy

We introduce a simplified method to model the interatomic interactions of high entropy alloys based on a lookup table of cluster energies. These interactions are employed in replica exchange Monte Carlo simulations with histogram analysis to obtain thermodynamic properties across a broad temperature range. Kikuchi's cluster variation method entropy formalism is applied to directly calculate entropy from statistics on short- and long-range chemical order, and we discuss the convergence of the entropy as clusters of differing size are included. A high temperature series expansion aids in our understanding of the convergence. Computer codes implementing these methods, and supporting data, are freely available on the internet.

Thermal features of Heisenberg antiferromagnets on edge- versus corner-sharing triangular-based lattices: a message from spin waves

We propose a new scheme of modifying spin waves so as to describe the thermodynamic properties of various noncollinear antiferromagnets with particular interest in a comparison between edge- versus corner-sharing triangular-based lattices. The well-known modified spin-wave theory for collinear antiferromagnets diagonalizes a bosonic Hamiltonian subject to the constraint that the total staggered magnetization be zero. Applying this scheme to frustrated noncollinear antiferromagnets ends in a poor thermodynamics, missing the optimal ground state and breaking the local U(1) rotational symmetry. We find such a plausible double-constraint condition for spin spirals as to spontaneously go back to the traditional single-constraint condition at the onset of a collinear Néel-ordered classical ground state. We first diagonalize only the bilinear terms in Holstein-Primakoff boson operators on the order of spin magnitude S and then bring these linear spin waves into interaction in a perturbative rather than variational manner. We demonstrate specific-heat calculations in terms of thus-modified interacting spin waves on various triangular-based lattices. In zero dimension, modified-spin-wave findings in comparison with finite-temperature Lanczos calculations turn out so successful as to reproduce the monomodal and bimodal specific-heat temperature profiles of the triangular-based edge-sharing Platonic and corner-sharing Archimedean polyhedral-lattice antiferromagnets, respectively. In two dimensions, high-temperature series expansions and tensor-network-based renormalization-group calculations are still controversial especially at low temperatures, and under such circumstances, modified spin waves interestingly predict that the specific heat of the kagome-lattice antiferromagnet in the corner-sharing geometry remains having both mid-temperature broad maximum and low-temperature narrow peak in the thermodynamic limit, while the specific heat of the triangular-lattice antiferromagnet in the edge-sharing geometry retains a low-temperature sharp peak followed by a mid-temperature weak anormaly in the thermodynamic limit. By further calculating one-magnon spectral functions in terms of our newly developed double-constraint modified spin-wave theory, we reveal that not only the elaborate modification scheme but also quantum corrections, especially those caused by the O(S 0) primary self-energies, are key ingredients in the successful description of triangular-based-lattice noncollinear antiferromagnets over the whole temperature range of absolute zero to infinity.

Random-bond Ising model and its dual in hyperbolic spaces.

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature T and the fraction of antiferromagnetic bonds p. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.

On general-n coefficients in series expansions for row spin–spin correlation functions in the two-dimensional Ising model

We consider spin–spin correlation functions for spins along a row, R n = ⟨σ 0,0 σ n,0⟩, in the two-dimensional Ising model. We discuss a method for calculating general-n expressions for coefficients in high-temperature and low-temperature series expansions of R n and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for a nonlinear ordinary differential equation whose solution would determine R n , analogous to the known nonlinear ordinary differential equation whose solution determines the diagonal correlation function ⟨σ 0,0 σ n,n ⟩ in this model.

Slow spin dynamics of cluster spin-glass spinel Zn(FeRu x )2O4: role of Jahn–Teller active spin-1/2 Cu2+ ions at B-sites

We report the slow spin dynamics of cluster spin-glass (SG) spinel Zn(FeRu x )2O4 by means of detailed -magnetization and -susceptibility studies combined with the heat capacity analysis. Two specific compositions (x = 0.5, 0.75) have been investigated in detail along with the substitution of Jahn–Teller (JT) active spin-1/2 Cu2+ ions at B-sites. Measurements based on the frequency and temperature dependence of -susceptibility () and the subsequent analysis using the empirical scaling laws such as: (a) Vogel–Fulcher law and (b) Power law reveal the presence of cluster SG state below the characteristic freezing temperature (17.77 K (x = 0.5) and 14 K (x = 0.75)). Relaxation dynamics of both the compositions follow the non-mean field de Almeida–Thouless (AT)-line approach , with an ideal value of φ = 3. Nevertheless, the analysis of temperature dependent high field -susceptibility, (, T) provides evidence for Gabay–Toulouse type mixed-phase (coexistence of SG and ferrimagnetic (FiM)) behaviour. Further, in the case of Cu0.2Zn0.8FeRuO4 system, slowly fluctuating magnetic clusters persist even above the short-range FiM ordering temperature () and their volume fraction vanishes completely across ∼6. This particular feature of the dynamics has been very well supported by the time decay of the thermoremanent magnetization and heat-capacity studies. We employed the high temperature series expansion technique to determine the symmetric exchange coupling () between the spins which yields eV for Cu0.2Zn0.8FeRuO4 representing the dominant intra-sublattice ferromagnetic interactions due to the dilute incorporation of the JT active Cu2+ ions. However, the antiferromagnetic coupling is predominant in ZnFeRuO4 and Cu0.2Zn0.8Fe0.5Ru1.5O4 systems. Finally, we deduced the magnetic phase diagram in the plane using the characteristic parameters obtained from the field variations of both - and -magnetization measurements.

Thermodynamic perturbation theory coefficients for hard spherocylinders and cylinders

Thermodynamic perturbation theories are indispensable in the development of equations of state based on Statistical Mechanics. Recently, a new statistical thermodynamic model has been proposed in which molecules are represented as cylindrically symmetric hard bodies (ellipsoids of revolution, spherocylinders, and cylinders) interacting with identical neighboring molecules through a spherically symmetric square-well potential. This novel equation of state embeds at least two approximations to the Onsager’s second virial theory: one restrains the equation of state to systems with an isotropic (disordered) orientation distribution, while the other decouples the translational and orientational degrees of freedom, being only valid at low densities. Using canonical Monte Carlo simulations, we investigated the feasibility of such approximations by determining the uniaxial nematic order parameter of the systems and calculating the first- and second-order perturbation coefficients of the high-temperature series expansion (HTSE) of the Helmholtz free energy. Our results indicate that both approximations are reasonable for a certain range of packing fractions and molecular anisotropies. We have also calculated the full Helmholtz free energy of the perturbed system, which can be used for determining high-order perturbation terms to improve the predictive capability of the equation of state.

Ground-state and thermodynamic properties of the spin-$\frac{1}{2}$ Heisenberg model on the anisotropic triangular lattice

The spin-\frac1212 triangular lattice antiferromagnetic Heisenberg model has been for a long time the prototypical model of magnetic frustration. However, only very recently this model has been proposed to be realized in the Ba_88CoNb_66O_{24}24 compound. The ground-state and thermodynamic properties are evaluated from a high-temperature series expansion interpolation method, called ``entropy method’’, and compared to experiments. We find a ground-state energy e_0 = -0.5445(2)e0=−0.5445(2) in perfect agreement with exact diagonalization results. We also calculate the specific heat and entropy at all temperatures, finding a good agreement with the latest experiments, and evaluate which further interactions could improve the comparison. We explore the spatially anisotropic triangular lattice and provide evidence that supports the existence of a gapped spin liquid between the square and triangular lattices.

Collinear order in the spin- 52 triangular-lattice antiferromagnet Na3Fe(PO4)2

We set forth the structural and magnetic properties of the frustrated spin-$5/2$ triangle lattice antiferromagnet ${\mathrm{Na}}_{3}\mathrm{Fe}{({\mathrm{PO}}_{4})}_{2}$ examined via x-ray diffraction, magnetization, heat capacity, and neutron diffraction measurements on the polycrystalline sample. No structural distortion was detected from the temperature-dependent x-ray diffraction down to 12.5 K, except a systematic lattice contraction. The magnetic susceptibility at high temperatures agrees well with the high-temperature series expansion for a spin-$5/2$ isotropic triangular lattice antiferromagnet with an average exchange coupling of $J/{k}_{\mathrm{B}}\ensuremath{\simeq}1.8$ K rather than a one-dimensional spin-$5/2$ chain model. This value of the exchange coupling is consistently reproduced by the saturation field of the pulse field magnetization data. It undergoes a magnetic long-range order at ${T}_{\mathrm{N}}\ensuremath{\simeq}10.4$ K. Neutron diffraction experiments elucidate a collinear antiferromagnetic ordering below ${T}_{\mathrm{N}}$ with the propagation vector $k=(1,0,0)$. An intermediate value of frustration ratio ($f\ensuremath{\simeq}3.6$) reflects moderate frustration in the compound which is corroborated by a reduced ordered magnetic moment of $\ensuremath{\sim}1.52{\ensuremath{\mu}}_{\mathrm{B}}$ at 1.6 K, compared to its classical value ($5{\ensuremath{\mu}}_{\mathrm{B}}$). Magnetic isotherms exhibit a change of slope envisaging a field induced spin-flop transition at ${H}_{\mathrm{SF}}\ensuremath{\simeq}3.2$ T. The magnetic field vs temperature phase diagram clearly unfold three distinct phase regimes, reminiscent of a frustrated magnet with in-plane ($XY$-type) anisotropy.

Universal large-order asymptotic behavior of the strong-coupling and high-temperature series expansions

For theories that exhibit second-order phase transition, we conjecture that the large-order asymptotic behavior of the strong-coupling (high-temperature) series expansion takes the form ${\ensuremath{\sigma}}^{n}{n}^{b}$ where $b$ is a universal parameter. The associated critical exponent is then given by $b+1$. The series itself can be approximated by the hypergeometric approximants ${_{p}F}_{p\ensuremath{-}1}$ which can mimic the same large-order behavior of the given series. Near the tip of the branch cut, the hypergeometric function ${_{p}F}_{p\ensuremath{-}1}$ has a power-law behavior from which the critical exponent and critical coupling can be extracted. We test the conjecture in this work for the perturbation series of the ground state energy of the Yang-Lee model as a strong-coupling form of the $\mathcal{P}\mathcal{T}$-symmetric $i{\ensuremath{\phi}}^{3}$ theory and the high-temperature expansion within the Ising model. From the known $b$ parameter for the Yang-Lee model, we obtain the exact critical exponents, which reflects the universality of $b$. Very accurate prediction for $b$ has been obtained from the many orders available for the high-temperature series expansion of the Ising model, which in turn predicts accurate critical exponents. Apart from critical exponents, the hypergeometric approximants for the Yang-Lee model show almost exact predictions for the ground state energy from low orders of perturbation series as input.

Specific Heat of the Kagome Antiferromagnet Herbertsmithite in High Magnetic Fields

Measuring the specific heat of herbertsmithite single crystals in high magnetic fields (up to $34$ T) allows us to isolate the low-temperature kagome contribution while shifting away extrinsic Schottky-like contributions. The kagome contribution follows an original power law $C_{p}(T\rightarrow0)\propto T^{\alpha}$ with $\alpha\sim1.5$ and is found field-independent between $28$ and $34$ T for temperatures $1\leq T\leq4$ K. These are serious constrains when it comes to replication using low-temperature extrapolations of high-temperature series expansions. We manage to reproduce the experimental observations if about $10$ % of the kagome sites do not contribute. Between $0$ and $34$ T, the computed specific heat has a minute field dependence then supporting an algebraic temperature dependence in zero field, typical of a critical spin liquid ground state. The need for an effective dilution of the kagome planes is discussed and is likely linked to the presence of copper ions on the interplane zinc sites. At very low temperatures and moderate fields, we also report some small field-induced anomalies in the total specific heat and start to elaborate a phase diagram.

Logarithmic divergent specific heat from high-temperature series expansions: Application to the two-dimensional XXZ Heisenberg model

We present an interpolation method for the specific heat $c_v(T)$, when there is a phase transition with a logarithmic singularity in $c_v$ at a critical temperature $T=T_c$. The method uses the fact that $c_v$ is constrained both by its high temperature series expansion, and just above $T_c$ by the type of singularity. We test our method on the ferro and antiferromagnetic Ising model on the two-dimensional square, triangular, honeycomb, and kagome lattices, where we find an excellent agreement with the exact solutions. We then explore the XXZ Heisenberg model, for which no exact results are available.

Thermodynamic perturbation theory coefficients for ellipsoidal molecules

Perturbation theory has been the center of the development of the new generation of equations of state. First- and second-order perturbation theories are very common, but require approximations for obtaining an analytical form. Recently, a new equation of state has been proposed in which the reference fluid is based on the hard Gaussian overlap approach, and the perturbed potential is defined as a spherically symmetric square well. In such an approach, the first- and second-order coefficients were considered the same as the ones applied for a system in which the reference term is spherical. Using Monte Carlo simulations, we investigated the validity of such an approximation by calculating the first- and second-order coefficients of the high-temperature expansion series of the Helmholtz free energy. With our findings, this approximation seems to be quite reasonable for a certain range of anisotropies. We also present a calculation of the perturbed molar Helmholtz free energy using Monte Carlo simulations, which could in principle be used for improving the equation of state.

Magnetic and electronic properties of Zn-Ni ferrites: First principle calculations, mean-field theory, high-temperature series expansions and Monte Carlo study

The structural, electronic, and magnetic properties of the ZnxNi1-xFe2O4 compounds are studied using several theoretical methods such as first-principle calculations based on density functional theory (DFT+U), Monte Carlo simulations, High temperature series expansions(HTSE) combined with the Padé approximants(PA) and mean-field theory. The total magnetic moment increases and the Curie temperature decreases with Zn substitution. The gap energy results of 1.3 eV obtained using GGA+U are reasonable compared to the available experimental data. The ZnFe2O4 possesses the insulating ground state and antiferromagnetic ordering. The critical exponent γ associated with the magnetic susceptibility is obtained. The dependence of the magnetization and the magnetic susceptibility on temperature is established for ZnxNi1−xFe2O4 using Monte Carlo study.

Magnetic properties of spinels Co x Zn1−x Cr2O4 systems: Green’s functions, high-temperature series expansions technique and mean-field theory

ABSTRACT Many-body Green’s function and mean-field theories have been developed for magnetic systems Co x Zn1−x Cr2O4. We apply these theories to evaluate the thermal magnetization and magnetic susceptibility for different values of magnetic field and dilution x, by considering all components of the magnetization when an external magnetic field is applied in (x-z) plane. The critical temperature of Co x Zn1−x Cr2O4 spinels have been calculated in the range 0 ≤ x ≤ 1. The Green’s function results are compared with the results of high-temperature series expansion technique and experimental magnetic measurements.

Frustrated quantum spins at finite temperature: Pseudo-Majorana functional renormalization group approach

The pseudofermion functional renormalization group (PFFRG) method has proven to be a powerful numerical approach to treat frustrated quantum spin systems. In its usual implementation, however, the complex fermionic representation of spin operators introduces unphysical Hilbert-space sectors which render an application at finite temperatures inaccurate. In this work we formulate a general functional renormalization group approach based on Majorana fermions to overcome these difficulties. We, particularly, implement spin operators via an $\text{SO}(3)$ symmetric Majorana representation which does not introduce any unphysical states and, hence, remains applicable to quantum spin models at finite temperatures. We apply this scheme, dubbed pseudo-Majorana functional renormalization group (PMFRG) method, to frustrated Heisenberg models on small spin clusters as well as square and triangular lattices. Computing the finite-temperature behavior of spin correlations and thermodynamic quantities such as free energy and heat capacity, we find good agreement with exact diagonalization and the high-temperature series expansion down to moderate temperatures. We observe a significantly enhanced accuracy of the PMFRG compared to the PFFRG at finite temperatures. More generally, we conclude that the development of functional renormalization group approaches with Majorana fermions considerably extends the scope of applicability of such methods.

Honeycomb rare-earth magnets with anisotropic exchange interactions

We study the rare-earth magnets on a honeycomb lattice, and are particularly interested in the experimental consequences of the highly anisotropic spin interaction due to the spin-orbit entanglement. We perform a high-temperature series expansion using a generic nearest-neighbor Hamiltonian with anisotropic interactions, and obtain the heat capacity, the parallel and perpendicular spin susceptibilities, and the magnetic torque coefficients. We further examine the electron spin resonance linewidth as an important signature of the anisotropic spin interactions. Due to the small interaction energy scale of the rare-earth moments, it is experimentally feasible to realize the strong-field regime. Therefore, we perform the spin-wave analysis and study the possibility of topological magnons when a strong field is applied to the system. The application and relevance to the rare-earth Kitaev materials are discussed.

Structural and magnetic properties of Cd-Ni spinel ferrites: density functional theory calculations and high-temperature series expansions

ABSTRACT First-principles calculations of density functional theory (DFT) using the generalized gradient approximation (GGA) and generalized gradient approximation with the Hubbard U (GGA + U) for the exchange-correlation energy functional are used to investigate the structural, electronic, and magnetic properties of the spinel system in the range . High-temperature series expansions (HTSE) combined with the Padé approximants (PA) method and mean field theory (MFT) are applied to the systems. The lattice parameter and magnetic moment of Fe and Ni ions increase when Cd content x increases. Density of States (DOS) calculations show that the has a semiconductor behaviour. The obtained theoretical results show that the Curie temperature decreases with x. These obtained results are in agreement with experimental ones.