A quest for microcanonical functionals describing near-critical behavior has led to an exact local variational functional for square lattice Ising models in zero field with varying bond strengths along the z axis. In the continuum limit near
TC this yields a second-order differential equation for the transverse Fourier coefficients of the energy density. Applications to a ``grain boundary'' between differing regions and to a wall with a surface field
h1 give bulk and surface exponents and scaling-limit energy-density profiles which exhibit rich crossovers and nonmonotonic variation.
We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate
τ=logsinh2K, so that the reduced free energy decomposes as
Φ(τ)=τ/2+G(∣τ∣). The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of
1.915×10−5 on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying
L−2,L−4,…. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy
Φmag(η),
η=m/∣h∣8/15. The low-temperature input contains the
∣h∣ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm
M∝∣H∣1/15, and the two one-sided limits
M(H→0±)=±M0 without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables
xt=Lm and
xh=L15/8h. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface
Yglue(xt,xh). The resulting lattice free energy closes in the compact form
ΦLglue(K,H)=Φlatglue(K,H)+L2Yglue(Lm,L15/8h)−Ybulk(Lm,L15/8h)+O(L−4) where
m=m(r,H) and
h=h(r,H) are the nonlinear scaling fields and
Ybulk is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess
Yglue−Ybulk, while the thermodynamic lattice free energy remains the previously constructed
Φlatglue. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error
3.90×10−4 and maximum absolute error
9.55×10−4 in
Y. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.
The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of
n parallel layers with free boundary conditions and a surface magnetic field
H1=kBTh1 imposed on the last layer, is analyzed for large
n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions
Wi(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where
a is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant
p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.
Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.
In this thesis, we study the critical behaviour of the two-dimensional Ising model on the regular lattices. Using the numerical solution of the model on the square, triangular and honeycomb lattices we compute the universal scaling function, which turns out to be identical on each of the lattices, in addition to being identical to the scaling function of the Ising Field Theory, computed previously by Fonseca and Zamolodchikov. To cope with the lattice contributions we carefully examined series expansions of the lattice free energy derivatives. We included the non-scaling regular part of the free energy as well as non-linear Aharony-Fisher scaling elds, which all have non-universal expansions. Using as many of the previously known exact results as possible, we were able to t the unknown coe cients of the scaling function expansion and obtain some non-universal coe cients. In contrast to the IFT approach of Fonseca and Zamolodchikov, all coe cients were obtained independently from separate datasets, without using dispersion relations. These results show that the Scaling and Universality hypotheses, with the help of the Aharony-Fisher corrections, hold on the lattice to very high precision and so there should be no doubt of their validity. For all numerical computations we used the Corner Transfer Matrix Renormalisation Group (CTMRG) algorithm, introduced by Nishino and Okunishi. The algorithm combines Baxter's variational approach (which gives Corner Transfer Matrix (CTM) equations), and White's Density Matrix Renormalisation Group (DMRG) method to solve the CTM equations e ciently. It was shown that given su cient distance from the critical point, the algorithmic precision is exceptionally good and is unlikely to be exceeded with any other general algorithm using the same amount of numerical computations. While performing tests we also con rmed several critical parameters of the three-state Ising and Blume-Capel models, although no extra precision was gained, compared to previous results from other methods. In addition to the results presented here, we produced an efficient and reusable implementation of the CTMRG algorithm, which after minor modifications could be used for a variety of lattice models, such as the Kashiwara-Miwa and the chiral Potts models.