Showing papers on "Ising model published in 1970"
TL;DR: In this paper, the one-dimensional Ising model with a transverse field is solved exactly by transforming the set of Pauli operators to a new set of Fermi operators.
1,266 citations
TL;DR: In this paper, the simplest Kondo problem is treated exactly in the ferromagnetic case, and given exact bounds for the relevant physical properties in the antiferromagnetic cases, by use of a scaling technique on an asymptotically exact expression for the ground-state properties given earlier.
Abstract: The simplest Kondo problem is treated exactly in the ferromagnetic case, and given exact bounds for the relevant physical properties in the antiferromagnetic case, by use of a scaling technique on an asymptotically exact expression for the ground-state properties given earlier. The theory also solves the $n=2$ case of the one-dimensional Ising problem. The ferromagnetic case has a finite spin, while the antiferromagnetic case has no truly singular $T\ensuremath{\rightarrow}0$ properties (e.g., it has finite $\ensuremath{\chi}$).
403 citations
TL;DR: In this article, a general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed, besides the well-known Ising and "ice" models, other soluble cases are also considered.
Abstract: A general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed. Besides the well-known Ising and "ice" models, other soluble cases are also considered. After discussing some general symmetry properties of this model, we consider in detail a particular class of the soluble cases, the "free-fermion" model. The explicit expressions for all thermodynamic functions with the inclusion of an external electric field are obtained. It is shown that both the specific heat and the polarizability of the free-fermion model exhibit in general a logarithmic singularity. An inverse-square-root singularity results, however, if the free-fermion model also satisfies the ice condition. The results are illustrated with a specific example.
335 citations
TL;DR: In this article, the Griffiths inequalities on the correlations of ferromagnetic spin systems appear as natural consequences of general assumptions, such as the Ising model with arbitrary spins, and the plane rotator model.
Abstract: We present a general framework in which Griffiths inequalities on the correlations of ferromagnetic spin systems appear as natural consequences of general assumptions. We give a method for the construction of a large class of models satisfying the basic assumptions. Special cases include the Ising model with arbitrary spins, and the plane rotator model. The general theory extends in a straightforward way to the non-commutative (quantum) case, but non-commutative examples satisfying all the assumptions are lacking at the moment.
292 citations
TL;DR: In this paper, an inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0.
Abstract: An inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0. The concavity of magnetization, which holds not only for spin‐½ but also for arbitrary‐spin Ising ferromagnets, provides a basis for certain thermodynamic inequalities near the ferromagnetic critical point, including one involving the ``high temperature'' indices α and γ.
257 citations
TL;DR: In this article, the Kondo problem is shown to be equivalent to the thermodynamics of charged rods moving on a circle, or to that of an Ising model with inverse-square interaction.
Abstract: Nozi\`eres and De Dominicis's one-body theory of the x-ray singularity is extended to the Kondo effect, and also to the finite-etmperature case. The Kondo problem is shown to be equivalent to the thermodynamics of charged rods moving on a circle, or to that of an Ising model with inverse-square interaction.
197 citations
TL;DR: In this article, a detailed discussion of pair correlations ω2(r) = 〈σ 0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given.
Abstract: A detailed discussion of pair correlations ω2(r) = 〈σ0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω2(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long‐range order below the Curie point TC and short‐range order above. It is shown that correlations on the anisotropic antiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Neel point TN (antiferromagnetic critical point), the completely anisotropic lattice exhibits antiferromagnetic long‐range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between TN and a uniquely located temperature TD, there is antiferro...
139 citations
TL;DR: The phase transition in the Ising model has been studied as a function of an applied transverse field, by Green's function and series-expansion methods as discussed by the authors, and the critical indices are probably independent of the applied field except at $T=0$ where they appear to be related to those of the Ised model in one higher dimension.
Abstract: The phase transition in the Ising model has been studied as a function of an applied transverse field, by Green's function and series-expansion methods. The critical indices are probably independent of the applied field except at $T=0$ where they appear to be related to those of the Ising model in one higher dimension.
124 citations
TL;DR: In this article, Wu et al. investigated the asymptotic behavior of the pair correlation ω2(r) = 〈σ 0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice.
Abstract: The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero‐point behavior of the correlation is of the form ω2(r) ∼ e0r−½ cos ⅔πr, where r = |r| is the spin separation and e0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth‐order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.
123 citations
TL;DR: In this paper, pair correlations on two one-dimensional Ising models with next-nearest-neighbor interactions are calculated exactly, and the zero-field susceptibilities were calculated exactly.
Abstract: Pair correlations on two one-dimensional Ising models with next-nearest-neighbor interactions are calculated exactly. For an appropriate range of values of the interaction energies, each of these models exhibits a disorder point, at which the nature of the short-range order changes abruptly, and at which there is a cusp in the graph of range of order vs. temperature. The zero-field susceptibilities are calculated exactly, and are found to have quite smooth temperature dependence.
96 citations
TL;DR: In this article, it was shown that all the zeros of the partition function lie on the unit circle in the complex ''fugacity'' plane and the first inequality of Griffiths, Kelly and Sherman for the spin correlation functions also holds for the anisotropic Heisenberg ferromagnets as in the Ising model.
Abstract: It is shown that all the zeros of the partition function lie on the unit circle in the complex `fugacity' plane and the first inequality of Griffiths, Kelly and Sherman for the spin correlation functions also holds for the anisotropic Heisenberg ferromagnets as in the Ising model. The isotropic Heisenberg ferromagnets and the Ising ferromagnets are involved as special cases and thus the methods of this article provide alternative proof of the corresponding theorems in the Ising model.
TL;DR: In this paper, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature, and coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two-and three-dimensional lattices.
Abstract: First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
TL;DR: Conformational transitions in proteins, nucleic acids, and other biopolymers evidently play a decisive role in many biological processes, particularly in control processes, and the resulting equilibrium properties of cooperative systems can be quantitatively explained for linear systems by the linear Ising model.
Abstract: Conformational transitions in proteins, nucleic acids, and other biopolymers evidently play a decisive role in many biological processes, particularly in control processes. They often proceed cooperatively, i.e. the elementary process of the transition of an individual segment of these macromolecules in influenced by the state of other segments via intramolecular interactions. In general, the segments favor the same state as their neighbours. The resulting equilibrium properties of cooperative systems, e.g. the sharpness of the transitions and their dependence on the chain length, can be quantitatively explained for linear systems by the linear Ising model. The molecular causes of the cooperativity can be explained for simple model polymers.
TL;DR: In this paper, the isothermal elastic constants and the coefficient of anomalous thermal expansion of a magnetic lattice are discussed and the spin system is described by the Ising model with an exchange coupling depending on lattice spacing.
Abstract: The isothermal elastic constants and the coefficient of anomalous thermal expansion of a magnetic lattice are discussed. The spin system is described by the Ising model with an exchange coupling depending on lattice spacing. A behavior of the elastic constants and the coefficient of thermal expansion is found which is in qualitative agreement with experiments. The isothermal compressibility remains positive nearTc and no thermo-mechanical instability occurs (which would lead to a first-order phase transitions), in contrast to earlier theories.
TL;DR: In this paper, the existence of disorder points for one and two-dimensional Ising lattices with antiferromagnetic next-nearest-neighbor interactions was established.
Abstract: The existence of a disorder point ${T}_{D}$ is established for some one- and two-dimensional Ising lattices with antiferromagnetic next-nearest-neighbor interactions. Within the disordered phase, the decay of axial next-nearest-neighbor pair correlations with increasing spin separation is positive monotonic exponential below ${T}_{D}$ and oscillatory with exponential envelope above ${T}_{D}$. A general definition of a disorder point is formulated, and a method of estimating ${T}_{D}$ described.
TL;DR: In this article, the authors studied the asymptotic form of the coefficients in high temperature series expansions for the Ising model of spin 1/2, the classical Heisenberg and planar classical heisenberg models.
Abstract: Analysis is made of the asymptotic form of the coefficients in high temperature series expansions for the Ising model of spin 1/2, the classical Heisenberg and planar classical Heisenberg models. For these models, if suitable expansion variables are chosen, only lattice constants of multiply connected graphs need be considered. Numerical investigations indicate that in the initial stages of each expansion the largest contribution is due to one simple basic graph. For the specific heat this graph is a polygon, and for the pair-correlation function and susceptibility it is a chain. If all other contributing graphs are ignored, critical indices are related to the geometrical properties of a self-avoiding walk on the lattice; hence this is termed the 'self- avoiding walk approximation'. Critical indices of higher derivatives with respect to magnetic field are then related to the virial coefficients of chains.
TL;DR: In this paper, the leading coefficients in the series expansions of hI(x) about x= infinity (critical 'isochore'), x=0 (critical isotherm) and x=-x0 (phase boundary) were estimated with the aid of Pade approximants.
Abstract: The function h(x) appearing in the magnetic equation of state near the critical point, namely H=Mdelta h(x) has been constructed for the two- and three-dimensional Ising model. (h(x) is denoted by hI(x) in this case.) This has been achieved by first estimating the leading coefficients in the series expansions of hI(x) about x= infinity (critical 'isochore'), x=0 (critical isotherm) and x=-x0 (phase boundary), and then extrapolating these series with the aid of Pade approximants. hJ(x) is used to test various approximate forms for h(x), and is compared with experimental results for real ferromagnets and fluids. Finally hI(x) is used to compute the function mI( theta ) appearing in Schofield's parametric representation of the equation of state.
TL;DR: In this paper, the one-dimensional spin-textonehalf-ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength.
Abstract: The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength ${J}_{\mathrm{LR}}$ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} gR=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{
e}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2l\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}lR$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0l\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}l2$ there is no phase transition in zero field, although there are classical critical points in a field.
TL;DR: In this paper, the exact temperature dependence of the ralaxation time of the polarization in NaNO 2 has been given through a new approach, in which the relaxation time is derived from the imaginary part of the complex dielectric constant in the low frequency region.
Abstract: The precise temperature dependence of the ralaxation time of the polarization in NaNO 2 has been given through a new approach, in which the relaxation time is derived from the imaginary part of the complex dielectric constant in the low frequency region. On the way of this experimental procedure, it was found that the measurement of the static electric susceptibility should be made at around 1 MHz, because another dielectric dispersion occurs below this frequency. The critical index for the static susceptibility is determined as γ=1.11±0.05. Throughout the paraelectric and sinusoidal antiferroelectric phases, the temperature dependence of the relaxation time has been explained qualitatively by simple equations in molecular field approximation for the kinematical Ising model. The critical index of the kinematical slowing down is found to be positive and smaller than 0.20. This value is consistent with that expected theoretically.
TL;DR: The site percolation problem when atoms interact and the atomic distribution is not random can be treated using the cluster variation method by placing a + or a − tag on the atomic species whose connectively is being sought, with the restriction that no (+ −) connection occurs as mentioned in this paper.
Abstract: It is proved that the dilute alloy ferromagnet (Ising model) at the absolute zero and the site percolation problem are mathematically identical by introducing the uniqueness theorem (there is only one infinitely extended connected cluster of one atomic species) and the symmetry theorem (the number of clusters of a certain finite shape and size with a plus spin is the same as that with a mimus spin). It is then shown that the long‐range order parameter in the site problem is the density of atoms belonging to the infinitely extending cluster. It is suggested that the site percolation problem when atoms interact and the atomic distribution is not random can be treated using the cluster variation method by placing a + or a − tag on the atomic species whose connectively is being sought, with the restriction that no (+ −) connection occurs and by maximizing the entropy (not minimizing the free energy) of the system. These signs are mathematical devices and have no physical meaning. Some results of higher order ...
TL;DR: In this paper, it was shown that the partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is exactly equal to that of a one-dimensional gas.
Abstract: The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.
01 Jan 1970
TL;DR: In this paper, the first series of magnetic measurements on small, hydrothermally grown single crystals (1 to 10 mg) were reported, which show that Tb(OH)3, Dy(OH), and Ho(OH)-3 order ferromagnetically at 3.72°, 3.50°, and 2.55°K, respectively.
Abstract: The rare‐earth hydroxides, R(OH)3 with R–La to Yb and Y form a series of simple magnetic crystals isostructural with the hexagonal rare‐earth trichlorides. Compared with most ionic rare‐earth crystals they have relatively small lattice parameters, and one may therefore expect magnetic cooperative effects at readily accessible temperatures. In this paper we report the first series of magnetic measurements on small, hydrothermally grown single crystals (1 to 10 mg) which show that Tb(OH)3, Dy(OH)3, and Ho(OH)3 order ferromagnetically at 3.72°, 3.50°, and 2.55°K, respectively, while Nd(OH)3 and Gd(OH)3 undergo more complex antiferromagnetic transitions near 1.7° and 2.0°K. Er(OH)3 remains paramagnetic down to 1.2°K. Magnetization measurements on Tb(OH)3, Dy(OH)3, and Ho(OH)3 in fields up to 14 kG give saturation moments (corrected for Van Vleck temperature‐independent paramagnetism) of 1350 emu/cc (9.0 μB/ion), 1418 emu/cc (9.6 μB/ion), and 1121 emu/cc (7.6 μB/ion) parallel to the c axis and almost zero perp...
TL;DR: In this article, the authors studied the time evolution of generalized quantum Ising models from the algebraic point of view and showed that all of them are weakly almost periodic in time.
Abstract: The time evolution of a class of generalized quantum Ising models (with various long‐range interactions, including Dyson's 1/rα) has been studied from the C*‐algebraic point of view. We establish that: (1) All 〈A〉t are weakly almost periodic in time; (2) there exists a unique averaging procedure over time; (3) the time evolution in the thermodynamical limit can be locally implemented by effective Hamiltonians in the algebra of quasilocal observables; (4) there exists a specific connection between the spectral properties of the time evolution of the initial state and the approach to equilibrium; (5) there are examples in which the time evolution is not G‐Abelian.
TL;DR: In this paper, a configurational analysis of low-temperature series for the Ising model with Cayley tree configurations with proper volume exclusion is presented, and good approximations are obtained for all the singularities in zero magnetic field.
Abstract: Low-temperature series for the Ising model show a marked lattice dependence and a recent analysis by Guttmann (1969) has revealed a pattern of spurious singularities which are sensitive to the coordination number of the lattice. The present paper attempts a configurational analysis of these series. Taking account of Cayley tree configurations with proper volume exclusion, it is found that good approximations are obtained for all the singularities in zero magnetic field. The form of the singularities in a nonzero field is also discussed.
TL;DR: In this article, the solvent-induced conformational transition between the two helical forms of poly-L-proline is studied as a model for cooperative order ⇌ order transitions.
Abstract: The solvent-induced conformational transition between the two helical forms of poly-L-proline is studied as a model for cooperative order ⇌ order transitions. The chain length dependent equilibrium data in two solvent systems are described by Schwarz's theory, which is based upon the most general formulation of the linear Ising model with nearest neighbor interactions. The parameter σ which describes the difficulty of nucleation of a I (II) residue in an uninterrupted II (I)-helix is 10−5 in both solvent systems. The ratios of the nucleation difficulties of states I and II at the ends of the chains β′ and β″ are very different in the two systems. Nucleation difficulty within the chain is interpreted as being due to unfavorable excess interaction energies at the I–II and II–I junctions, which add up to 7 kcal/mole of nuclei as calculated from the σ value. A similar value is computed from the atomic interactions at the junctions. In contrast to this intrinsic properly of poly-L-proline, the energies of I and II residues at the ends are heavily influenced by interactions of the endgroups with the solvent. The above values of the nucleation parameters are determined by a new least-square fitting procedure which does not necessitate the assumption of the dependence of the equilibrium constant s for propagation upon the external parameters, but yields this function from the experimental transition data. A quantitative explanation of this experimental s function through the binding of solvent is attempted. In the transition region a very small free energy change (about 0.1 kcal/mole), arising from a preferential binding of solvent molecules to one of the conformational states, is sufficient for a complete conversion from one helical form to the other.
TL;DR: In this paper, the authors generalize this conclusion to an arbitrary distribution of impurity bonds and show that the specific heat fails to be analytic at such a distribution, although it is infinitely differentiable there.
Abstract: In the first paper of this series, we located ${T}_{c}$ for an arbitrary distribution of impurity bonds $P({E}_{2})$ and, for a particular $P({E}_{2})$ with a narrow width, we found that the specific heat fails to be analytic at ${T}_{c}$, although it is infinitely differentiable there. In this paper, we generalize this conclusion to an arbitrary distribution $P({E}_{2})$.
TL;DR: In this paper, the powder susceptibility of CsNiCl3 has been measured from 1.3 to 130°K. The results, indicative of one-dimensional antiferromagnetic interactions, were used to test the onedimensional Ising model for spin unity, and the Fisher model utilizing the classical Hamiltonian for infinite spin.
Abstract: The powder susceptibility of CsNiCl3 has been measured from 1.3 to 130°K. The results, indicative of one‐dimensional antiferromagnetic interactions, are used to test the one‐dimensional Ising model for spin unity, and the Fisher model utilizing the classical Hamiltonian for infinite spin. The Fisher model with J = − 14.2k is found to give the better fit, but both models are found to deviate appreciably from the experimental results for kT less than J.
TL;DR: In this article, the density, pressure and temperature state surface of a "bonded" fluid on a plane triangular lattice was studied using a first-order statistical approximation based on a triangular group of sites.
Abstract: The density, pressure and temperature state surface is studied for a 'bonded' fluid on a plane triangular lattice. Calculations were made using a first-order statistical approximation based on a triangular group of sites. Separation between liquid and vapour phases occurs below a critical pressure pc. If the interaction energy of an unbonded nearest-neighbour pair of molecules is zero the properties of the model can be related to those of an Ising ferromagnet on a honeycomb sublattice and accurate values deduced for the critical density, pressure and temperature.