In this paper we investigate the behaviour of the specific heat around the critical point of the Ising model in dimension 5 to 7. We find a specific heat discontinuity, like that for the mean field Ising model, and provide estimates for the left and right hand limits of the specific heat at the critical point. We also estimate the singular exponents, describing how the specific heat approaches those limits. Additionally, we make a smaller scale investigation of the same properties in dimension 6 and 7, and provide strongly improved estimates for the critical termperature $K_c$ in $d=5,6,7$ which bring the best MC-estimate closer to those obtained by long high temperature series expanions.
We consider the zero-temperature random-field Ising model in the presence of an external field, on ladders and in one dimension with finite range interactions, for unbounded continuous distributions of random fields, and show that there is no jump discontinuity in the magnetizations for any quasi-one dimensional model. We show that the evolution of the system at an external field can be described by a stochastic matrix and the magnetization can be obtained using the eigenvector of the matrix corresponding to the eigenvalue one, which is continuous and differentiable function of the external fi…
We produce the discontinuity in the specific heat of a homogeneous, dilute, and weakly interacting Bose gas in a short-wavelength range with a simple statistical method. The magnitude of the discontinuity at the phase transition temperature is obtained as a function of the density and scattering length of the Bose particles.
We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay $P(K)\sim K^{-λ}$. It is well established that the model is characterized by classical mean-field exponents for $λ>5$. In this note we show that the specific-heat discontinuity $δc_h$ at the critical point remains $λ$-dependent even for $λ>5$: $δc_h=3(λ-5)(λ-1)/[2(λ-3)^2]$ and attains its mean-field value $δc_h=3/2$ only in the limit $λ\to \infty$. We compare this behaviour with recent measurements of the $d$ dependency of $δc_h$ made for the Ising model on lattices wi…
We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from minus infinity to plus infinity by setting up the self-consistent field equations, which we show are exact in this case. We find that for a 3-coordinated Bethe lattice, there is no jump discontinuity in magnetization for arbitrarily small gaussian disorder, but the discontinuity is present for larger coordination numbers. We have checked our results by Monte Carlo…
We investigate the relationship between a diffusive motion of an interface, heat conduction, and the roughening transition in the three-dimensional Ising model. We numerically compute the thermal conductivity and the diffusion constant and find that the diffusion constant shows a crossover in its temperature dependence. The crossover temperature is equal to the roughening transition temperature in equilibrium and deviates from it when heat flows in the system. From these results, we discuss the possibility that heat conduction causes a shift of the roughening transition temperature.
Monte Carlo simulations of Ising models coupled to heat baths at two different temperatures are used to study a fluctuation relation for the heat exchanged between the two thermostats in a time $τ$. Different kinetics (single--spin--flip or spin--exchange Kawasaki dynamics), transition rates (Glauber or Metropolis), and couplings between the system and the thermostats have been considered. In every case the fluctuation relation is verified in the large $τ$ limit, both in the disordered and in the low temperature phase. Finite-$τ$ corrections are shown to obey a scaling behavior.
The specific heat of regular Ising polyhedra is investigated in detail as a function of temperature and magnetic field. It is shown that the regular Ising polyhedra display diverse double-peak temperature dependences of the specific heat whenever the magnetic field approaches a level-crossing field. The Schottky theory of a two-level system often provides a plausible explanation of a height and position of low-temperature peak, which emerges in the specific heat of a regular Ising polyhedron due to low-lying excitations from a ground state to a first-excited state. The height and position of S…
Based on a high temperature expansion, we compute the two-point correlation function and the critical line of an Ising lattice gas driven into a non-equilibrium steady state by a uniform bias E. The lowest nontrivial order already reproduces the key features, i.e., the discontinuity singularity of the structure factor and the (qualitative) E-dependence of the critical line. Our approach is easily generalized to other non-equilibrium lattice models and provides a simple analytic tool for the study of the high temperature phase and its boundaries.
The spin-1/2 Ising-Heisenberg two-leg ladder accounting for alternating Ising and Heisenberg inter-leg couplings in addition to the Ising intra-leg coupling is rigorously mapped onto to a mixed spin-(3/2,1/2) Ising-Heisenberg diamond chain with the nodal Ising spins $S=3/2$ and the interstitial spin-1/2 Heisenberg dimers. The latter effective model with higher-order interactions between the nodal and interstitial spins is subsequently exactly solved within the transfer-matrix method. The model under investigation exhibits five different ground states: ferromagnetic, antiferromagnetic, superant…
Specific heat per particle ($c_v$) of an ideal gas, in many occasions, is interpreted as energy fluctuation per particle ($\triangleε^2$) of the ideal gas through the relation: $\triangleε^2=kT^2c_v$, where $k$ is the Boltzmann constant and $T$ is the temperature. This relationship is true only in the classical limit, and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3-D free Bose and Fermi gases. We also have explored the same for harmonically trapped cases. We have obtained a hump…
We derive a direct connection between Monte Carlo time and physical time in terms of physical parameters, using a quantum Hamiltonian with a $d$-dimensional phonon heat bath interacting with a square-lattice Ising ferromagnet. Based on the calculated transition rates, we perform dynamic Monte Carlo simulations using absorbing Markov chains to measure the lifetimes of the metastable state at low temperatures. We also calculate the lifetimes analytically using absorbing Markov chains. The phonon dynamic gives field-dependent prefactors in the lifetimes at low temperatures, that are different fro…
We study a two dimensional Ising model between thermostats at different temperatures. By applying the recently introduced KQ dynamics, we show that the system reaches a steady state with coexisting phases transversal to the heat flow. The relevance of such complex states on thermodynamic or geometrical observables is investigated. In particular, we study energy, magnetization and metric properties of interfaces and clusters which, in principle, are sensitive to local features of configurations. With respect to equilibrium states, the presence of the heat flow amplifies the fluctuations of both…
In this paper, we use the dimer method to obtain the free energy of Ising models consisting of repeated horizontal strips of width $m$ connected by sequences of vertical strings of length $n$ mutually separated by distance $N$, with $N$ arbitrary, to investigate the effects of connectivity and proximity on the specific heat. The decoration method is used to transform the strings of $n+1$ spins interacting with their nearest neighbors with coupling $J$ into a pair with coupling $\bar J$ between the two spins. The free energy per site is given as a single integral and some results for critical t…
The partition functions for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for finite square lattices with the help of graph theoretical procedures, show-bit algorithm, enumeration of configurations and transfer matrix methods. A new recurrence relation valid for all lattice sizes is proposed. A novel method of computing the critical temperature has been demonstrated. The magnetization and susceptibility for infinite lattices in the presence of magnetic field is estimated. The critical exponent δ pertaining to the magnetization has also been comput…
We apply the recently developed critical minimum energy subspace scheme for the investigation of the random-field Ising model. We point out that this method is well suited for the study of this model. The density of states is obtained via the Wang-Landau and broad histogram methods in a unified implementation by employing the N-fold version of the Wang-Landau scheme. The random-fields are obtained from a bimodal distribution ($h_{i}=\pm2$), and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from $L=4$ to $L=32$. Observing the finite-size scaling behavio…
In this paper we investigate the nature of the singularity of the Ising model of the 4-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent $α=0$ but a non-rigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at $T_c$. We find that within the given accuracy the canonical ensemble data is consistent both with a logarithmic singularity and a bounded specific heat, but that the micro-canonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are …
In this study, the internal energy and the specific heat of the one- dimensional Ising model obtained in the frame of the generalized statistical thermodynamics by Andrade [Physica A175 (1991) 285], have been extended somewhat and compared with the internal energy and the specific heat which have been calculated in the standard statistical thermodynamics.
The mixed spin-(1,1/2) Ising-Heisenberg model on a distorted diamond chain with the spin-1 nodal atoms and the spin-1/2 interstitial atoms is exactly solved by the transfer-matrix method. An influence of the geometric spin frustration and the parallelogram distortion on the ground state, magnetization, susceptibility and specific heat of the mixed-spin Ising-Heisenberg distorted diamond chain are investigated in detail. It is demonstrated that the zero-temperature magnetization curve may involve intermediate plateaus just at zero and one-half of the saturation magnetization. The temperature de…
The bond-propagation (BP) algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works [X.-T. Wu {\it et al} Phys. Rev. E {\bf 86}, 041149 (2012) and Phys. Rev. E {\bf 87}, 022124 (2013)], we reach much higher accuracy ($10^{-26}$) of the internal energy and specific heat, compared to the accuracy $10^{-11}$ of the internal energy and $10^{-9…
Let f, U and C represent, respectively, the free energy, the internal energy and the specific heat of the critical Ising model on the square M x N lattice with periodic boundary conditions. We find that N f and U are well-defined odd function of 1/N. We also find that ratios of subdominant (N^(-2 i - 1)) finite-size corrections amplitudes for the internal energy and the specific heat are constant. The free energy and the internal energy at the critical point are calculated asymtotically up to N^(-5) order, and the specific heat up to N^(-3) order.
The ground state and the thermodynamics of a spin-1/2 asymmetric diamond Ising--Heisenberg chain are considered. For the $XYZ$ anisotropic Heisenberg interaction, the exact calculations of the free energy, entropy, heat capacity, magnetization, and magnetic susceptibility are performed using the method of decoration-iteration transformation. In the case of antiferromagnetic interactions (Ising and $XXZ$ anisotropic Heisenberg ones), the ground state, magnetization process, temperature dependence of the magnetization, magnetic susceptibility, and heat capacity are investigated. The influence of…
We introduce new fast canonical local algorithms for discrete and continuous spin systems. We show that for a broad selection of spin systems they compare favorably to the known ones except for the Ising +/-1 spins. The new procedures use discretization scheme and the necessary information have to be stored in computer memory before the simulation. The models for testing discrete spins are the Ising +/-1, the general Ising S or Blume-Capel model, the Potts and the clock models. The continuous spins we examine are the O(N) models, including the continuous Ising model (N=1), the ϕ^4 Ising model …
We provide an expression quantitatively describing the specific heat of the Ising model on the simple-cubic lattice in the critical region. This expression is based on finite-size scaling of numerical results obtained by means of a Monte Carlo method. It agrees satisfactorily with series expansions and with a set of experimental results. Our results include a determination of the universal amplitude ratio of the specific-heat divergences at both sides of the critical point.
Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.
Bond propagation and site propagation algorithm are extended to the two dimensional Ising model with a surface field. With these algorithms we can calculate the free energy, internal energy, specific heat, magnetization, correlation function, surface magnetization, surface susceptibility and surface correlation. To test these algorithms, we study the Ising model for wetting transition, which is solved exactly by Abraham. We can locate the transition point accurately to $10^{-8}$. We carry out the calculation of the specific heat, surface susceptibility on the lattices with the sizes are up to …
Part I of this article studied the specific heats of planar alternating layered Ising models with strips of strong coupling $J_1$ sandwiched between strips of weak coupling $J_2$, to illustrate qualitatively the effects of connectivity, proximity, and enhancement in analogy to those seen in extensive experiments on superfluid helium by Gasparini and coworkers. It was demonstrated graphically that finite-size scaling descriptions hold in a variety of temperature regions including in the vicinity of the two specific heat maxima. Here we provide exact theoretical analyses and asymptotics of the s…
A frustrated Ising model on a diamond hierarchical lattice is studied. We obtain the exact partition function of this model and calculate the transition temperature, specific heat, entropy, magnetization, and ferromagnetic correlation function. Depending on the magnitude of a parameter giving the frustration, there exist three types of ground states: ferromagnetic, classical spin-liquid with highly developed short-range order, and paramagnetic. The dependence of the zero-temperature entropy on the frustration parameter has an infinite number of steps. The temperature dependence of the specific…
We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A+/A- in the three-dimensional Ising model via the impact angle ϕof complex temperature zeros. We also measure the correlation-length critical exponent νfrom finite-size scaling, and the specific-heat exponent αthrough hyperscaling. Extrapolations to the thermodynamic limit yield ϕ= 59.2(1.0) degrees, A+/A- = 0.56(3), ν= 0.63048(32) and α= 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
The mixed spin-1/2 and spin-S (S>1/2) Ising model on a rope ladder is examined by combining two exact analytical methods. By the decoration-iteration mapping transformation, this mixed-spin system is firstly transformed to a simple spin-1/2 Ising model on the two-leg ladder, which is then exactly solved by the standard transfer-matrix method. The thermal variations of the zero-field specific heat are discussed in particular.
The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained the free energy, internal energy and specific heat numerically on square lattices with square shape and various combinations of the four types boundary conditions. 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…
Fluctuation relations for the entropy production in non equilibrium stationary states of Ising models are investigated by Monte Carlo simulations. Systems in contact with heat baths at two different temperatures or subject to external driving will be studied. In the first case, by considering different kinetic rules and couplings with the baths, the behavior of the probability distributions of the heat exchanged in a time $τ$ with the thermostats, both in the disordered and in the low temperature phase, are discussed. The fluctuation relation is always verified in the large $τ$ limit and devia…
Using the 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: triangle, rhombus, trapezoid, hexagon and rectangle. 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 predi…
Two conditions are derived for Ising models to show non-universal critical behaviour, namely conditions concerning 1) logarithmic singularity of the specific heat and 2) degeneracy of the ground state. These conditions are satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising models with short- or long-range interactions and even Ising systems without the translational or the rotational invariance.
We investigate the phase transitions in a coupled system of Ising spins and a fluctuating network. Each spin interacts with $q$ neighbors through links of the rewiring network. The Ising spins and the network are in thermal contact with the heat baths at temperatures $T_S$ and $T_L$, respectively, so that the whole system is driven out of equilibrium for $T_S \neq T_L$. The model is a generalization of the $q$-neighbor Ising model, which corresponds to the limiting case of $T_L=\infty$. Despite the mean field nature of the interaction, the $q$-neighbor Ising model was shown to display a discon…
The specific heat of the two-dimensional $\pm J$ Ising model has been investigated by the numerical transfer matrix method and Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes the negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. The region mentioned above is found to be fairly large in the $p-T$ plane ($p$ is the concentration of the ferromagnetic bond and $T$ is the temperature). Moreover, it includes the Nishimori line. Namely, it includes a part…
We consider the conformal bootstrap for spacetime dimension $1<d<2$. We determine bounds on operator dimensions and compare our results with various theoretical and numerical models, in particular with resummed $ε$-expansion and Monte Carlo simulations of the Ising model on fractal lattices. The bounds clearly rule out that these models correspond to unitary conformal field theories. We also clarify the $d\to 1$ limit of the conformal bootstrap, showing that bounds can be - and indeed are - discontinuous in this limit. This discontinuity implies that for small $ε=d-1$ the expected critical exp…
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 $ρ=M/N=1,2,4,8,16$, 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 $10^{-26}$. Basing 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 free energy, internal energy and specific heat. The fit…
We analyse Monte Carlo data for the energy and specific heat at and close to the critical point of the 3D cubic Ising model. From the finite size scaling of the energy E and the specific heat C at criticality we obtain the estimate nu = 0.6308(10). Furthermore, one obtains precise estimates for the ``backgrounds'' (nonsingular parts) E_ns and C_ns. Fitting solely off critical energy estimates to a scaling law, we find depending on the choice of the reduced temperature, either A_+/A_- = 0.550(12) and alpha=0.1115(37), or A_+/A_- = 0.567(16) and alpha=0.1047(48). Including information from the d…
The spin-1 Ising-Heisenberg diamond chain with the second-neighbor interaction between the nodal spins is rigorously solved using the transfer-matrix method. Exact results for the ground state, magnetization process and specific heat are presented and discussed in particular. It is shown that the further-neighbor interaction between the nodal spins gives rise to three novel ground states with a translationally broken symmetry, but at the same time, it does not increases the total number of intermediate plateaus in a zero-temperature magnetization curve compared with the simplified model withou…
Based on the analytic expression of free energy for infinitely long Ising strip with finite width joined antiperiodically on a variety of planar lattices, we show the existence of first-order phase transition at the critical point of Ising transition. The critical gaps of the transition are also calculated analytically by measuring the discontinuities in the internal energy and the specific heat.
The Ashkin-Teller (AT) model is a generalization of Ising 2-d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four-spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we…
We study the warming process of a semi-infinite cylindrical Ising lattice initially ordered and coupled at the boundary to a heat reservoir. The adoption of a proper microcanonical dynamics allows a detailed study of the time evolution of the system. As expected, thermal propagation displays a diffusive character and the spatial correlations decay exponentially in the direction orthogonal to the heat flow. However, we show that the approach to equilibrium presents an unexpected slow behavior. In particular, when the thermostat is at infinite temperature, correlations decay to their asymptotic …
We study the critical behavior of the entropy production of the Ising model subject to a magnetic field that oscillates in time. The mean-field model displays a phase transition that can be either first or second-order, depending on the amplitude of the field and on the frequency of oscillation. Within this approximation the entropy production rate is shown to have a discontinuity when the transition is first-order and to be continuous, with a jump in its first derivative, if the transition is second-order. In two dimensions, we find with numerical simulations that the critical behavior of the…
We study the frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice, taking into account the exchange interactions of atomic spins at the sites of the nearest, next-nearest, and third neighbors. The exact analytical expressions for the thermodynamic functions of the system are obtained using the Kramers--Wannier transfer matrix technique. Criteria for the emergence of magnetic frustrations in the presence of competition between the energies of exchange interactions are formulated. The points and intervals of the existence of frustrations, which depend on …
The mixed spin-1/2 and spin-S Ising model on a decorated planar lattice accounting for lattice vibrations of decorating atoms is treated by making use of the canonical coordinate transformation, the decoration-iteration transformation, and the harmonic approximation. It is shown that the magnetoelastic coupling gives rise to an effective single-ion anisotropy and three-site four-spin interaction, which are responsible for the anomalous spin frustration of the decorating spins in virtue of a competition with the equilibrium nearest-neighbor exchange interaction between the nodal and decorating …
Magnetic properties, such as magnetizations, internal energy, specific heat, entropy, Helmholtz free energy and phase diagrams of the spin-3/2 Blume-Capel model on hexagonal Ising nanowire (HIN) with core-shell structure are studied by utilizing the effective-field theory (EFT) with correlations. Moreover, the hysteresis behaviors of the system are also investigated and the effects of Hamiltonian parameters on hysteresis behaviors are discussed, in detail. Finally, the obtained results are compared with some experimental and theoretical results and a qualitatively good agreement is found.
The specific heats of exactly solvable alternating layered planar Ising models with strips of width $m_1$ lattice spacings and ``strong'' couplings $J_1$ sandwiched between strips of width $m_2$ and ``weak'' coupling $J_2$, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or `bulk' critical temperature, $T_c(J_1,J_2;m_1,m_2)$, arising from the collective effects reflect the observations of Gasparini and coworkers in experiments on confined superfluid helium. Explic…
In this paper the three dimensional random field Ising model is studied at both zero temperature and positive temperature. Critical exponents are extracted at zero temperature by finite size scaling analysis of large discontinuities in the bond energy. The heat capacity exponent $α$ is found to be near zero. The ground states are determined for a range of external field and disorder strength near the zero temperature critical point and the scaling of ground state tilings of the field-disorder plane is discussed. At positive temperature the specific heat and the susceptibility are obtained usin…
The critical properties of an infinitely long Ising strip with finite width L joined periodically or antiperiodically are investigated by analyzing the distribution of partition function zeros. For periodic boundary condition, the the leading finite-size scaling of partition function zeros and its corrections are given. For antiperiodic boundary condition, the critical point of 2D Ising transition is one of the loci of the zeros, and the associated non-analyticity is identified as a first-order phase transition. The exact amount of the latent heat released by the transition is 4/L.
The Ising model on an alternating triangular lattice with the nearest-neighbor interaction in a magnetic field is presented. Exact solution of this model is found. The thermodynamic quantities, like free energy, specific heat a finite temperature and function correlaction, are calculated by treatment based on the transfer matrix method. Our results show that the model proposed does not have phase transition to a finite temperature, but the curve of specific heat presented a Schottky anomaly.
We analyze the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L^{-3} to the energy and the corrections of order L^{-2} and L^{-3} to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L^{-1} occur, unmodified by logarithms (except of course for the leading $\log L$ term in the specific heat); and the energy expansion contains only odd powers of L^{-1}. In …
The three-dimensional bimodal random-field Ising model is investigated using the N-fold version of the Wang-Landau algorithm. The essential energy subspaces are determined by the recently developed critical minimum energy subspace technique, and two implementations of this scheme are utilized. The random fields are obtained from a bimodal discrete $(\pmΔ)$ distribution, and we study the model for various values of the disorder strength $Δ$, $Δ=0.5, 1, 1.5$ and 2, on cubic lattices with linear sizes $L=4-24$. We extract information for the probability distributions of the specific heat peaks ov…
In the present paper, a study of the magnetic properties of a spin-1/2 Ising-Heisenberg Cairo pentagonal structure is presented. The model has been investigated in Ref. [34] in the absence of external magnetic field. Here, we consider the effects of an external tunable magnetic field. By using the transfer matrix approach, we investigate the magnetic ground-state phase transition, the low-temperature magnetization process, and how the magnetic field influences the various thermodynamic parameters such as entropy, internal energy and specific heat. It is shown that the model exhibits intermedia…
The two-dimensional Ising model with Brascamp-Kunz boundary conditions has a partition function more amenable to analysis than its counterpart on a torus. This fact is exploited to exactly determine the full finite-size scaling behaviour of the Fisher zeroes of the model. Moreover, exact results are also determined for the scaling of the specific heat at criticality, for the specific-heat peak and for the pseudocritical points. All corrections to scaling are found to be analytic and the shift exponent $λ$ does not coincide with the inverse of the correlation length exponent $1/ν$.
