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We develop high temperature series expansions for the thermodynamic properties of the honeycomb-lattice Kitaev-Heisenberg model. Numerical results for uniform susceptibility, heat capacity and entropy as a function of temperature for different values of the Kitaev coupling $K$ and Heisenberg exachange coupling $J$ (with $|J|\le |K|$) are presented. These expansions show good convergence down to a temperature of a fraction of $K$ and in some cases down to $T=K/10$. In the Kitaev exchange dominated regime, the inverse susceptibility has a nearly linear temperature dependence over a wide temperature range. However, we show that already at temperatures $10$-times the Curie-Weiss temperature, the effective Curie-Weiss constant estimated from the data can be off by a factor of 2. We find that the magnitude of the heat capacity maximum at the short-range order peak, is substantially smaller for small $J/K$ than for $J$ of order or larger than $K$. We suggest that this itself represents a simple marker for the relative importance of the Kitaev terms in these systems. Somewhat surprisingly, both heat capacity and susceptibility data on Na$_2$IrO$_3$ are consistent with a dominant {\it antiferromagnetic} Kitaev exchange constant of about $300-400$ $K$., Comment: 6 pages, 5 figures, series coefficients in supplementary materials

We investigate the ground states of an Ising model with a transverse field on the square lattice, with additional frustrating second-neighbour Ising interactions, using series expansion methods. The phase boundaries are located to high precision. No strong evidence for the presence of a tricritical point is seen, but it cannot be excluded.

If magnetic frustration is most commonly known for undermining long-range order, as famously illustrated by spin liquids, the ability of matter to develop new collective mechanisms in order to fight frustration is perhaps no less fascinating, providing an avenue for the exploration and discovery of unconventional behaviors. Here, we study a realistic minimal model where a number of such mechanisms converge, which, incidentally, pertain to the perplexing quantum spin ice candidate Yb(2)Ti(2)O(7). Specifically, we explain how thermal and quantum fluctuations, optimized by order-by-disorder selection, conspire to expand the stability region of a degenerate continuous U(1) manifold against the classical splayed ferromagnetic ground state that is displayed by the sister compound Yb(2)Ti(2)O(7). The resulting competition gives rise to multiple phase transitions, in striking similitude with recent experiments on Yb(2)Ti(2)O(7) [Lhotel et al., Phys. Rev. B 89, 224419 (2014)]. By combining a gamut of numerical techniques, we obtain compelling evidence that such multiphase competition is a natural engine for the substantial sample-to-sample variability observed in Yb(2)Ti(2)O(7) and is the missing key to ultimately understand the intrinsic properties of this material. As a corollary, our work offers a pertinent illustration of the influence of chemical pressure in rare-earth pyrochlores.

We give a precise numerical solution for decorated Ising models on the simple cubic lattice which show ferromagnetism, compensation points, and reentrant behaviour. The models, consisting of $S={1\over 2}$ spins on a simple cubic lattice, and decorating S=1 or $S=3/2$ spins on the bonds, can be mapped exactly onto the normal spin-1/2 Ising model, whose properties are well known., Comment: 8 pages, 5 figures

We present a theoretical analysis of the magnetic energy level spectrum of the molecular complex Fe4(OCH3)(dpm)6, which is an example of a "molecular magnet". Using a Heisenberg exchange model, we investigate the energy spectrum and the deviations which result from the slight asymmetry of the cluster. We derive an analytical expression for the crystal field splitting of the ground state as a function of the anisotropy. Finally, we establish the existence of diabolical points in the magnetic-field spectrum and compare to predictions made by semiclassical and perturbation theory approaches, and investigate how the location of these points is affected by higher-order anisotropy terms.

We consider an Ising model on a lattice of squares and hexagons (the `4-6' lattice), which does not appear to have been studied previously. The critical temperature is obtained as the root of a sixth-order equation. An exact expression is obtained for the spontaneous magnetization.

These lectures are intended to give an overview of various aspects of the theory of Bose systems and of Bose-Einstein condensation (BEC) and will, I hope, be useful in their own right as well as providing an introduction, and a basis, for the more advanced lectures to follow. The emphasis is pedagogical. I assume a knowledge of undergraduate statistical mechanics, but not much more. Of course, in these lectures it is impossible to do justice to this fascinating subject, and many topics are either mentioned too briefly or not at all. Roughly speaking my goal is as follows

I give an overview of areas of recent and current research in the theoretical study of strongly interacting lattice systems, in the areas of lattice spin models (magnetism), lattice electron models (superconductivity) and lattice gauge models (elementary particles). The emphasis is on novel and interesting physics, and recent results.

We have obtained the graphs and weights for the twelfth and thirteenth terms of the high temperature multigraph expansion for the susceptibility of general Ising systems. As applications we have computed the coefficients through thirteenth order for the square lattice with first and second neighbor interactions and first and third neighbor interactions.

A lattice model for liquid metals proposed by Nara, Ogawa and Matsubara(1979) is studied using high-temperature series expansions. The grand-canonical potentials are expanded exactly to order beta 9 and the susceptibilities are expanded to order beta 11 on two-, three-, and four-dimensional hypercubic lattices. It is conjectured that, for fixed values of the fermion hopping energy and fermion chemical potential, the system goes critical only at particular values of the lattice-gas chemical potential. The critical values of the lattice-gas chemical potential and the critical temperature are calculated self-consistently and the critical exponent of the inverse isothermal compressibility is estimated using Pade approximants. It is concluded that the critical behaviour of the model is not affected by the presence of the fermions. This conclusion is consistent with the exact solution on a one-dimensional lattice by Thompson, Matsubara and Yang(1994).

We have investigated the phase diagram of the s= 1/2 J1-J2-J3 chain using finite-lattice techniques. The location of the boundary between spin-liquid and dimer phases has been estimated, and the characterization of these phases via correlation functions is presented.

Low-temperature series expansions are used to investigate the phase transition due to oxygen ordering in the high-temperature superconductor YBa2Cu3O6+x. The formulation is based on the lattice-gas model, which has been studied by many authors, and excellent agreement with previous results is obtained. The authors also investigate the effect of a sublattice-dependent chemical potential which plays a role similar to a uniaxial stress in breaking the degeneracy between twinned structures.

We use high-temperature series expansions to obtain thermodynamic properties of the quantum compass model, and to investigate the phase transition on the square and simple cubic lattices. On the square lattice we obtain evidence for a phase transition, consistent with recent Monte Carlo results. On the simple cubic lattice the same procedure provides no sign of a transition, and we conjecture that there is no finite temperature transition in this case., 6 pages, 2 figures

The authors describe the construction of regular lattices in two-dimensional hyperbolic space by means of the action of a discrete subgroups of SU(1,1). They consider an Ising model on such lattices and show how the thermodynamic limit can be handled. They give high- and low-temperature expansions of the free energy, magnetic susceptibility and magnetization and find that these quantities diverge at a critical temperature with mean-field exponents beta =1/2, gamma =1. They also conjecture the long distance behaviour of correlation functions at the critical temperature.

We investigate Ising ferrimagnets on square and simple-cubic lattices with exchange couplings between spins of values S=1/2 and S=1 on neighbouring sites and an additional single-site anisotropy term on the S=1 sites. Based mainly on a careful and comprehensive Monte Carlo study, we conclude that there is no tricritical point in the two--dimensional case, in contradiction to mean-field predictions and recent series results. However, evidence for a tricritical point is found in the three-dimensional case. In addition, a line of compensation points is found for the simple-cubic, but not for the square lattice., Comment: 14 pages, 11 figures

We report results of finite-lattice calculations of the effect of localized bond and spin perturbations on the ground state properties of the spin- 1 2 Heisenberg antiferromagnet on the square lattice. These perturbations are intended to model the effect of a static oxygen hole on the magnetic state of the CuO2 planes in high temperature superconductors.

We report results for phonon dispersion curves and phonon density of states for the material YBa2Cu4O8, based on a shell-model description. We also present a group theoretical analysis of the q=0 modes and obtain good agreement with recently reported Raman Measurements.

The authors use Monte Carlo techniques to study the antiferromagnetic Ising model on the two-dimensional Penrose lattice. Two types of interaction are included, resulting in subtle frustration effects and a rather complex phase diagram.

We use linked-cluster series expansions, both at T=0 and high temperature, to analyze the phase structure of the spin- Heisenberg antiferromagnet with competing first- and second-neighbor interactions on the three-dimensional body-centered-cubic lattice. At zero temperature we find a first-order quantum phase transition at J 2 /J 1 ≃0.705′0.005 between AF 1 (Neel) and AF 2 ordered phases. The high-temperature series yield quite accurate estimates of the bounding critical line for the AF 1 phase, and an apparent critical line for the AF 2 phase, with a bicritical point at J 1 /J 2 ≃0.71, kT/J 1 ≃0.34. The possibility that this latter transition is of first order cannot be excluded.

We estimate, using high-temperature series expansions, the transition temperatures of the spin 1/2, 1 and 3/2 Heisenberg ferromagnet and antiferromagnet in 3-dimensions. The manner in which the difference between Curie and Neel temperatures vanishes with increasing spin quantum number is investigated., Comment: 10 pages

Strong-coupling expansions, to order $(t/J)^8$, are derived for the Kondo lattice model of strongly correlated electrons, in 1-, 2- and 3- dimensions at arbitrary temperature. Results are presented for the specific heat, and spin and charge susceptibilities., revtex4

The large J2 limit of the square-lattice J1-J2 Heisenberg antiferromagnet is a classic example of order by disorder where quantum fluctuations select a collinear ground state. Here, we use series expansion methods and a mean-field spin-wave theory to study the excitation spectra in this phase and look for a finite-temperature Ising-like transition, corresponding to a broken symmetry of the square lattice, as first proposed by Chandra et al. [Phys. Rev. Lett. 64, 88 (1990)]]. We find that the spectra reveal the symmetries of the ordered phase. However, we do not find evidence for a finite-T transition. We suggest a scenario for a T=0 transition based on quantum fluctuations.

Exchange couplings are calculated for Li(2)VOSiO(4) using the local-density approximation (LDA). While the sum of in-plane couplings J(1)+J(2) = 9.5+/-1.5 K and the interplane coupling J( perpendicular) approximately 0.2- 0.3 K agree with recent experimental data, the ratio J(2)/J(1) approximately 12 exceeds the reported value by an order of magnitude. Using geometrical considerations, high temperature expansions and perturbative mean field theory, we show that the LDA-derived exchange constants lead to a remarkably accurate description of the properties of these materials including specific heat, susceptibility, Néel temperature, and NMR spectra.

Many of the rich phenomena and properties of condensed matter physics are utterly unpredictable on a “noninteracting particle” paradigm. This implies that theory must confront the problem of a large number of strongly interacting particles. The usual strategy of theorists is to construct a model Hamiltonian which is simple enough to allow controlled calculations but not so simple that the interesting physics is lost. Computational methods, of the kinds described at this workshop, have an essential role to play.

We study the single-band Hubbard model via high-temperature expansions to order (βt) 9 for the standard two- and three-dimensional lattices. Series are derived for the grand potential and for the ferromagnetic susceptibility, for general values of the Coulomb repulsion parameter U and electron density n. The series are difficult to analyze with any degree of precision, but estimates are obtained for the ferromagnetic critical temperature

Recently derived 10th-order high-temperature expansions for the Hubbard model are used to obtain the ferromagnetic susceptibility and specific heat at high temperatures. Numerical results are obtained for the simple cubic and face-centred cubic lattices by using Pade approximants to sum the series. The results are compared with two solvable limiting cases, namely the non-interacting limit U = 0 and the strongly-correlated or atomic limit t = O.

We report shell model based calculations of the lattice dynamics of YBa2Cu306 and YBa2Cu307. Two models are proposed for YBa2Cu306, the model parameters being obtained by a least squares fit to recent neutron scattering results. The parameters obtained are then used, with modification, to model the lattice dynamics of YBa2Cu307. Since this material has significant carrier concentration, even in the normal state we have used both screened and unscreened versions of the shell model. The screened model gives a better overall description of the lattice dynamics of YBa2Cu307. The phonon dispersion curves along the symmetry directions and the weighted phonon density of states for both YBa2Cu306 and YBa2Cu307 are presented.

The author has derived eleven-term high temperature series for the wavevector dependent susceptibility chi (q) of the ANNNI model in two and three dimensions. In three dimensions the locations of the phase boundaries of the ferromagnetic and modulated phases, and the location of the Lifshitz point, are found to be in good agreement with previous work. In two dimensions the analysis is less precise, but yields results consistent with the currently accepted picture in which the paramagnetic phase extends to zero temperature at the multiphase point.

The authors have investigated the properties of the Ising model in a transverse field for general spin S by the method of high-temperature series. Analysis of the series yields the form of the phase diagram in the temperature field plane. The authors also investigate the variation of critical field with spin, and present evidence for classical-quantum crossover at T=0 for all spin values.

Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J1, J2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J1 0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.

The authors present an ansatz for kink excitations in an easy-plane classical antiferromagnetic chain with a magnetic field in the easy plane. The utility of the ansatz is that it presents the in-plane (XY) and out-of-plane (YZ) kinks as belonging to one continuously connected energy dispersion curve. Linear stability analysis applied to YZ kinks shows that there is a velocity-dependent critical field necessary for stability. The authors also present a set of results of a numerical integration of the equations of motion which verifies the YZ kink stability regimes, as well as showing that XY kinks are stable over a wide range of fields and velocities.

The critical behaviour of an Ising model with competing first- and third-nearest-neighbour interactions ('biaxial next-nearest-neighbour Ising' or BNNNI model) on the square lattice is investigated by high- and low-temperature series.

The critical behaviour of an Ising model with competing interactions along both axes of a square lattice (the BNNNI model) is investigated by Monte Carlo methods. Convincing evidence for the existence of modulated structures near the transition(s) is obtained but the form of the phase diagram remains unclear.

The ground state energy and pair correlations have been computed exactly for the spin [Formula: see text] XY magnet and isotropic Heisenberg antiferromagnet, for a sequence of finite cells on the square and honeycomb lattices. Precise estimates of the ground state energy of the infinite lattices are obtained by extrapolation. It is predicted that the XY magnet has a non-zero transverse magnetization and the Heisenberg antiferromagnet has a non-zero staggered magnetization in the ground state.

A new method of approximation for studying the order-disorder transition in binary alloys is presented. The formulation is in terms of face-centered cubic alloys, although the method can also be applied to other structures. The technique is essentially a generalization of the constant-coupling approximation. We have applied the principles of that method to derive the free energy of the alloy using a tetrahedral cluster of nearest neighbouring sites as the basic unit of the calculation. We only consider the case of nearest neighbour pair interactions, but show how the method can be generalized to include many body and second neighbour interactions. Numerical results are presented and comparison of these results is made with results of cluster variation calculations on the same system and with experimental results on the copper-gold alloy system.

The thermal flattening of a sinusoidally grooved surface described by the discrete Gaussian model is studied using Monte Carlo techniques. An exponential decay of the amplitude of undulations is found both above and below the roughening transition temperature, TR, for particle transport by evaporation-condensation with and without conserving the number of surface atoms. The characteristic relaxation time changes rapidly close to TR. Below TR, trapezoidal profiles are observed.

The classical free‐energy density for both a monatomic and a diatomic chain of particles with harmonic coupling between nearest neighbors is calculated using (i) the transfer integral technique and (ii) real‐space renormalization. It is shown that the results agree with the classical limit of the well‐known quantum‐mechanical free‐energy densities for these chains.

The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. The location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.

Self-avoiding walks of 12 steps and fewer have been enumerated for the square lattice, for the case in which steps can occur between both nearest-neighbour and next-nearest-neighbour lattice sites. The appropriate two-variable generating function diverges along a critical line with a constant exponent gamma approximately 4/3, in agreement with general universality expectations.

The authors obtain high-field series expansions for the anisotropic Ising model on a simple cubic lattice. These series are used to investigate the tricritical point that occurs when the model is used to represent metamagnets. Some new tests of scaling theory for the crossover between two and three dimensions are also undertaken.