Low-temperature Series Research Articles

Universal amplitude ratios in the three-dimensional Ising model

We present a high precision Monte Carlo study of various universal amplitude ratios of the three dimensional Ising spin model. Using state of the art simulation techniques we studied the model close to criticality in both phases. Great care was taken to control systematic errors due to finite size effects and correction to scaling terms. We obtain $C_+/C_-=4.75(3)$, $f_{+,2nd}/f_{-,2nd}=1.95(2)$ and $u^*=14.3(1)$. Our results are compatible with those obtained by field theoretic methods applied to the $\phi^4$ theory and high and low temperature series expansions of the Ising model. The mismatch with a previous Montecarlo study by Ruge et al. remains to be understood.

Low-temperature series for the square lattice Potts model using the improved finite-lattice method

The low-temperature series are calculated for the free energy, magnetization and susceptibility in the Q-state Potts model on the square lattice, using the improved algorithm of the finite lattice method. The series are obtained to the order of $z^{41}$ for each of Q=5-50, and the result of their Pad\'e type analysis is compared with those of the large-Q expansion and the Monte Carlo simulations.

Complex-temperature singularities in Potts models on the square lattice.

We report some results on the complex-temperature (CT) singularities of $q$-state Potts models on the square lattice. We concentrate on the problematic region $\mathrm{Re}(a)<0$ (where $a={e}^{K}$) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for $3<~q<~8$, we establish the existence, in this region, of complex-conjugate CT singularities at which the magnetization and susceptibility diverge. From calculations of zeros of the partition function, we obtain evidence consistent with the inference that these singularities occur at endpoints ${a}_{e}$,${a}_{e}^{*}$ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for $q=3$, we find ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, consistent with ${\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}$. By duality, these results also imply associated arcs extending into the (CT extension of the) symmetric paramagnetic phase. Analytic expressions are suggested for the positions of some of these singularities; e.g., for $q=5$, our finding is consistent with the exact value ${a}_{e}$, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.

Low-temperature series expansions for the square lattice Ising model with spin

We derive low-temperature series (in the variable $u = \exp[-\beta J/S^2]$) for the spontaneous magnetisation, susceptibility and specific heat of the spin-$S$ Ising model on the square lattice for $S=\frac32$, 2, $\frac52$, and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with $S$. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of $S$. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of $S$, though consistent with universality.

Extrapolation procedure for low-temperature series for the square lattice spin-1 Ising model

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetization, and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable and using the new extrapolation procedure to calculate an additional 13 - 14 terms.

Modelling of 2d first-order phase transitions in adsorption layers at the metal/electrolyte interface

Several organic substances exhibit 2D first-order phase transitions in their adsorption at the metal/electrolyte interface. This phenomenon is discussed on the basis of the Ising lattice gas model. Firstly, its predictions are summed up concerning the critical behaviour of adsorption layers (phase diagrams, critical exponents). Secondly, the mean-field approximation and the quasi-chemical approximation for condensed adsorption layers are investigated critically. In conclusion, low-temperature series expansions of the degree of coverage are found to be adequate isotherms.

Complex-temperature singularities of Ising models

We report new results on complex-temperature properties of Ising models. These include studies of the $s=1/2$ model on triangular, honeycomb, kagom\'e, $3 \cdot 12^2$, and $4 \cdot 8^2$ lattices. We elucidate the complex--$T$ phase diagrams of the higher-spin 2D Ising models, using calculations of partition function zeros. Finally, we investigate the 2D Ising model in an external magnetic field, mapping the complex--$T$ phase diagram and exploring various singularities therein. For the case $\beta H=i\pi/2$, we give exact results on the phase diagram and obtain susceptibility exponents $\gamma'$ at various singularities from low-temperature series analyses.

Low-temperature series for the Potts model by finite-lattice method

The low-temperature series for the two-dimensional Potts model are generated for free energy, magnetization and susceptibility using an improved algorithm of the finite-lattice method. The algorithm enables us to obtain longer series than the standard algorithm of the finite-lattice method and it is more efficient for higher-state Potts model. The obtained series are also analyzed.

Low-temperature expansion of the surface width in the D = 3 Ising model

The low-temperature series for the interface width in the three-dimensional Ising model and the SOS model are generated using the finite-lattice method. The series obtained are about two or three times longer than the previous ones calculated by the standard graphical method. The series are analyzed to estimate the roughening transition point and the critical exponents.

Complex-temperature singularities in thed= 2 Ising model: triangular and honeycomb lattices

We investigate complex-temperature singularities in the Ising model on the triangular lattice. Extending an earlier analysis of the low-temperature series expansions for the (zero-field) susceptibility $\bar\chi$ by Guttmann \cite{g75} to include the use of differential approximants, we obtain further evidence in support of his conclusion that the exponent describing the divergence in $\chi$ at $u=u_e=-1/3$ (where $u = e^{-4K}$) is $\gamma_e'=5/4$ and refine his estimate of the critical amplitude. We discuss the remarkable nature of this singularity, at which the spontaneous magnetisation diverges (with exponent $\beta_e=-1/8$) and show that it lies at the endpoint of a singular line segment constituting part of the natural boundaries of the free energy in the complex $u$ plane. Using exact results, we find that the specific heat has a divergent singularity at $u=-1/3$ with exponent $\alpha_e'=1$, so that the relation $\alpha_e'+2\beta_e+\gamma_e'=2$ is satisfied. We also study the singularity at $u=u_s=-1$, where $M$ vanishes (with $\beta_s=3/8$) and $C$ diverges logarithmically (with $\alpha_s' = \alpha_s = 0$).

Thermodynamic properties of particles with intermediate statistics.

Analytic expressions for the distribution function of an ideal gas of particles (exclusons) which have statistics intermediate between Fermi-Dirac and Bose-Einstein are obtained for all values of the Haldane statistics parameter \ensuremath{\alpha}\ensuremath{\in}[0,1]. The analytic structure of the distribution function is investigated and found to have no singularities in the physical region when the parameter \ensuremath{\alpha} lies in the range 0\ensuremath{\le}1. High- and low-temperature series are also derived for the internal energy E and heat capacity ${\mathit{C}}_{\mathit{V}}$ of the D-dimensional excluson gas. The low-temperature series for the thermodynamic properties illustrate the pseudofermion nature of exclusons. \textcopyright{} 1996 The American Physical Society.

Exact distribution of energies in the two-dimensional ising model.

The low-temperature series expansion for the partition function of the two-dimensional Ising model on a square lattice can be determined exactly for finite lattices using Kaufman's generalization of Onsager's solution. The exact distribution function for the energy can then be determined from the coefficients of the partition function. This provides an exact solution with which one can compare energy histograms determined in Monte Carlo simulations. This solution should prove useful for detailed studies of statistical and systematic errors in histogram reweighting.

Complex-temperature properties of the two-dimensional Ising model for nonzero magnetic field.

We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.

HOLE–SPIN COUPLING AND SPIN DISTORTION IN HIGH Tc SUPERCONDUCTORS

The effect of hole–spin coupling together with spin distortion on the energy and hole correlation function have been studied in detail. Standard Green function theory and Low Temperature Series Expansion method have been utilised to get analytical results.

CRITICAL TEMPERATURE AND AMPLITUDE RATIOS FROM A FINITE TEMPERATURE RENORMALIZATION GROUP

We study λφ4 theory using an environmentally friendly finite temperature renormalization group. We derive flow equations, using a fiducial temperature as flow parameter, develop them perturbatively in an expansion free from ultraviolet and infrared divergences, then integrate them numerically from zero to temperatures above the critical temperature. The critical temperature, at which the mass vanishes, is obtained by integrating the flow equations, and is determined as a function of the zero temperature mass and coupling. We calculate the field expectation value and the minimum of the effective potential as functions of temperature and derive some universal amplitude ratios which connect the broken and symmetric phases of the theory. The latter are found to be in good agreement with those of the three-dimensional Ising model obtained from high and low temperature series expansions.

Complex-temperature properties of the 2D Ising model with beta H=+or-i pi /2

We study the complex-temperature properties of a rare example of a statistical mechanical model which is exactly solvable in an external symmetry-breaking field, namely, the Ising model on the square lattice with $\beta H = \pm i \pi/2$. This model was solved by Lee and Yang \cite{ly}. We first determine the complex-temperature phases and their boundaries. From a low-temperature, high-field series expansion of the partition function, we extract the low-temperature series for the susceptibility $\chi$ to $O(u^{23})$, where $u=e^{-4K}$. Analysing this series, we conclude that $\chi$ has divergent singularities (i) at $u=u_e=-(3-2^{3/2})$ with exponent $\gamma_e'=5/4$, (ii) at $u=1$, with exponent $\gamma_1'=5/2$, and (iii) at $u=u_s=-1$, with exponent $\gamma_s'=1$. We also extract a shorter series for the staggered susceptibility and investigate its singularities. Using the exact result of Lee and Yang for the free energy, we calculate the specific heat and determine its complex-temperature singularities. We also carry this out for the uniform and staggered magnetisation.

Low-temperature series for renormalized operators: The ferromagnetic square-lattice Ising model

A method for computing low--temperature series for renormalized operators in the two--dimensional Ising model is proposed. These series are applied to the study of the properties of the truncated renormalized Hamiltonians when we start at very low temperature and zero field. The truncated Hamiltonians for majority rule, Kadanoff transformation and decimation for $2 \times 2$ blocks depend on the how we approach the first--order phase--transition line. These Renormalization Group transformations are multi--valued and discontinuous at this first--order transition line when restricted to some finite--dimensional interaction space.

Low-temperature expansion of the free energy in the ASOS model

We calculate the low-temperature series of the free energy in the absolute-value solid-on-solid (ASOS) model to order u 23 using the finite-lattice method. The property of the obtained series and the behavior of their Padé approximants confirms us that the roughening transition in the ASOS model is of the Kosterlitz-Thouless type.

Activity and Characterization of Self-Supported Model Catalysts Derived from Cobalt-Based Clusters of Clusters: Hydrogenation of 1,3-Butadiene

Complex coalt-carbonyl ligand-based clusters of clusters are used as molecular precursors for self-supported model catalysts. These precursors present two metal layers. The outer layer is associated with the complex Co-carbonyl ligands, and the core layer, which is either cobalt or zinc. These two precursors are identified as CoCo and ZnCo, respectively. In situ DRIFTS, TGA, TPD-MS, XPS, and catalytic measurements show two distinct forms of the catalysts. Partial thermolysis at low temperature (LT) under hydrogen results in almost completely decarbonylated catalysts, which appear to posses a mainly unchanged carboxylate metal core. Complete pyrolysis at higher temperatures (HT) in hydrogen leads to a mixed metal environment. The activities of the catalysts are studied in the hydrogenation of 1,3-butadiene. The LT series show similar catalytic behavior during the hydrogenation of 1,3-butadiene. However, significantly different catalytic behavior is observed between the HT catalysts. The effect of the core metal on the d-electron density appears to be the factor leading to very different catalytic behavior patterns.

Low-temperature series for Ising model by finite-lattice method

We have calculated the low-temperature series for the second moment of the correlation function in $d=3$ Ising model to order $u^{26}$ and for the free energy of Absolute Value Solid-on-Solid (ASOS) model to order $u^{23}$, using the finite-lattice method.