Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\times N$ with free boundaries. For five aspect ratios, $\rho =M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\times N=64\times {10}^6$. The accuracy of the free energy reaches ${10}^{-26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by $f_{\infty }=0.92969539834161021499\left(1\right)$, compared with Onsager's exact result $f_{\infty }=0.929695398341610214985\cdots .$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\pm 1\times {10}^{-10}$, compared with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained. DOI: http://dx.doi.org/10.1103/PhysRevE.86.041149 ©2012 American Physical Society

Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches ${10}^{-26}$. Based on accurate data on several finite systems with linear size up to $N=2000$, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles $\pi /3$, $\pi /2$, and $2\pi /3$ are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal. DOI: http://dx.doi.org/10.1103/PhysRevE.87.022124 ©2013 American Physical Society

The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are $h_L$ and $h_R$, respectively. For $|h_L|,|h_R|<\sqrt{1-g}$, where $g$ is the transverse field and $h_L{h}_R<0$, a first-order phase transition occurs with the changing $h_L$ or $h_R$. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|h_R|>\sqrt{1-g}$ and $h_L{h}_R<0$, there exists a continuous phase transition with changing $h_L$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.

The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size $N\ifmmode\times\else\texttimes\fi{}N$ and $30<N<1000$. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.

We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two-dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy, and specific heat are calculated with the bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as $ln\left|t\right|$ and the heat capacity variation behaves as ${\left|t\right|}^{-1}$, where $t$ is the reduced temperature.