Transfer-matrix Finite-size Scaling Research Articles

Finite-size-scaling study of the anisotropic spin- [formula omitted] Ashkin–Teller model

Using transfer-matrix finite-size-scaling (TMFSS), we investigate the anisotropic spin- 1 2 Ashkin–Teller model (ATM) where the two bilinear interactions are different ( L 2≠ J 2) and with a four coupling interaction K 4. Conclusive numerical evidence is obtained, indicating the presence of another partially ordered phase in the phase diagram. Also, from exponents calculations we show that all the transitions are in the Ising universality except for the multicritical point B 4 which is nonuniversal.

Numerical study of a lattice-gas model for micellar binary solutions

Two nonperturbative methods such as transfer-matrix finite-size scaling and Monte Carlo simulations are used to investigate the multicritical behavior of a lattice-gas model proposed by Shnidman and Zia [J. Stat. Phys 50, 839 (1988)] for studying the micellar binary solutions of water and amphiphile. The phase diagrams are obtained in both the magnetic field-temperature (H,T) and amphiphile density temperature $({\ensuremath{\rho}}_{s},T)$ for different values of competing interactions ${(K}_{0}$ and ${K}_{1})$ and in the presence of the attraction interaction intermicellar parameter (J). Our nonperturbative results are compared with previous mean-field ones. Both methods confirm the absence of the rarefied micelle at high temperature as found previously by mean-field calculations. Also our phase diagrams present transitions of first- and second-order transitions linked by tricritical and multicritical points of higher order. Finally, with the use of finite-size-scaling ideas, the critical exponents have been calculated. Our results show that this model has a nonuniversal behavior that belongs to the XY model with cubic anisotropy for a certain range of interactions parameters and a universal behavior that belongs to the $d=2$ Ising model.

Phase transition in the Blume-Capel model with second neighbour interaction

A two dimensional antiferromagnetic spin-1 Ising model with negative next- nearest neighbour interaction (J2<0) and under an external magnetic field is investigated by two methods: The mean-field theory and Finite-Size-Scaling based on transfer matrix (TMFSS) calculations. The ground state diagrams exhibit several new phases including frustrated ones. At finite temperature we obtain by these two methods quite rich phase diagrams, with several multicritical points. While Mean field approximation yields phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from transfer matrix finite size scaling calculations. For a certain range of interaction parameters, the model is shown to violate the ordinary universality hypothesis.

Structural phase transitions and oxygen-oxygen interaction energies in YBa2Cu3O6+x.

We investigate the effects of varying the interaction constants in a lattice-gas model for oxygen ordering in the high-temperature superconductor ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6+\mathit{x}}$. A detailed ground-state calculation yields seven topologically distinct ground-state diagrams, which are realized for different values of the interactions. For interactions that produce the experimentally observed ordered phases, we present transfer-matrix finite-size scaling (TMFSS) results which, together with the gound-state diagram, can be used to approximate the global shape of the finite-temperature phase diagram. The effective oxygen-oxygen interactions in ${\mathrm{YBa}}_{2}$Cu $_{3}\mathrm{O}_{6+\mathit{x}}$ are estimated by a nonlinear least-squares fit to newly available experimental data of the numerical lattice-gas phase diagram obtained by the TMFSS method with strip widths 8 and 12. An interesting feature of the fitted phase diagram is that a narrow region of the ortho-1/2 phase is found to extend between the tetragonal and ortho-1/4 phases at low oxygen concentration. The fitted interactions and the resulting phase diagram are compared with those recently obtained by Sterne and Wille, using the linear muffin-tin orbital method. We find the reasonable agreement achieved between the effective interactions predicted by these two fundamentally different methods highly encouraging.

First-order structural phase transitions in a lattice-gas model for YBa2Cu3O6+x.

Recent experiments indicate that the structural phase transitions in the high-temperature superconductor ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6+\mathit{x}}$ may be first order at low temperatures. The oxygen ordering in the Cu-O basal planes of this material may be described by a lattice-gas model introduced by Wille et al. B [Phys. Rev. Lett. 60, 1065 (1988)]. However, as recent numerical studies show, this model does not exhibit first-order transitions. Here, we extend the model to include weak attractive interactions (2% of the nearest-neighbor interactions) between next-nearest-neighbor oxygen chains, and we study the extended model using transfer-matrix finite-size scaling. We find that these weak interactions produce tricritical points and first-order transitions at low temperatures, which are in good agreement with experiments for the structural order-order phase transitions. Furthermore, we predict a tricritical point and first-order transitions at low temperatures on the order-disorder branch of the phase diagram. We also present the phase diagram.

Phase diagram and critical properties of a two-dimensional lattice-gas model of oxygen ordering in YBa2Cu3Oz.

Using transfer-matrix finite-size scaling, we investigate a square lattice-gas model for the oxygen ordering in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{\mathrm{z}}$, which reproduces the experimentally observed orthorhombic and double-cell ordered phases. Contrary to the results of cluster-variational methods of calculation, the phase boundary between the (disordered) tetragonal and double-cell phases is continuous, as is the boundary between the orthorhombic and double-cell phases. Critical properties of the model can be understood by drawing an analogy with the Ashkin-Teller model; numerical results are consistent with this picture. At low temperature and low oxygen concentration, there is complicated behavior, possibly indicative of a second orthorhombic phase.

Phase diagram and critical properties of oxygen ordering in YBa2Cu3Oz

Using transfer-matrix finite-size scaling, we reinvestigate a square lattice-gas model Hamiltonian to describe the order-disorder transition in the CuO basal plane YBa2Cu3Oz. The model contains strong nearest-neighbor repulsions and diagonal (2nd neighbor) interactions that alternate in sign. These interactions ensure the stability (at zero temperature) of two experimentally observed structures: a single-cell orthorhombic phase near z= 7 and a double-cell orthorhombic phase near z= 6.5. Critical properties can be understood by recasting the model as an Ashkin-Teller model; numerical results are consistent with this picture.

Multi-adatom interaction effects in a lattice gas model for Cu and Au adsorption on W(110)

A lattice-gas model for Cu and Au adsorption on W(110) is proposed and discussed in the light of recent cohesive energy calculations for these systems. Three types of lateral, two types of trio, and a quarto interaction were considered. It is argued that the unusual shape of the binding hollows for these systems results in anomalously large effective trio interactions. Phase diagrams were calculated via a new transfer matrix finite-size scaling method. Repulsive trio interactions were found to increase the degree of asymmetry in the T−θ phase diagram and reduce critical temperatures. Although accurate reproduction of experimental phase diagrams was not attempted, our model seems able to do so and has important advantages over previously proposed models.

Transfer-matrix study of the adsorption of a flexible self-avoiding polymer chain in two dimensions

The authors model the polymer as a self-avoiding walk on a semi-infinite square lattice with an energy contribution - in from each step along the lattice boundary. The critical adsorption temperature Ta and the 'ordinary' and 'special' surface exponents of the walk above and at the adsorption temperature, respectively, are estimated with transfer-matrix finite-size scaling methods. For the adsorption temperature they obtain exp( in /kTa)=2.041+or-0.002, and for the crossover and susceptibility exponents of the special transition, phi sp=0.501+or-0.003 and gamma 1sp=1.454+or-0.004.

Wavevector scaling, surface critical behavior, and wetting in the 2-d, 3-state chiral clock model

The controversial 2-d, 3-state chiral Potts model is studied using transfer matrix finite size scaling. at Δ=0, we find dq N/dΔ∝N −4/5, whereq is the “wavevector”, Δ the chiral field, andN the strip width (N=4−10). The result is consistent with den Nijs's crossover exponent ϕ=1/6. With surface fields on the infinite free boundaries, exponents associated with bulk magnetizationy H, surface magnetizationy′ H, and surface susceptibility χ are computed vs. Δ; results are similar for $$\hat \Delta$$ ‖ or ⊥ to the infinite direction. Preliminary results are given for the bulk specific heat critical amplitudes, to test the universality of amplitude ratios. The interface wetting line is located for 0≦Δ≦1/4 using simple transfer matrix calculations of surface tensions in the solid-on-solid approximation. Overhangs or bubbles seem relatively unimportant at all temperatures.

Convergence of finite-size scaling renormalisation techniques

Abstract The rate of convergence of the transfer matrix finite-size scaling (or phenomenological renormalisation) method is studied. It is shown both heuristically and numerically that the convergence of estimates for exponents, etc, is governed asymptotically by the leading irrelevant-variable scaling exponent. The more rapid apparent convergence rates observed in many practical calculations for two-dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms.