arXiv subjects

Jacques Curély

Publications and source records attributed to Jacques Curély.

3 recordsLinked to original sources

arXiv2020

The two-dimensional infinite Heisenberg classical square lattice: zero-field partition function and correlation length

We rigorously examine 2d-square lattices composed of classical spins isotropically coupled between first-nearest neighbours. A general expression of the characteristic polynomial associated with the zero-field partition function Zinf{N}(0) is established for any lattice size. In the infinite-lattice limit a numerical study allows to select the dominant term: it is written as a l-series of eigenvalues, each one being characterized by a unique index l whose origin is explained. Surprisingly Zinf{N}(0) shows a very simple exact closed-form expression valid for any temperature. The thermal study of the basic l-term allows to point out crossovers between l- and (l+1)-terms. Coming from high temperatures where the l=0-term is dominant and going to 0 K, l-eigenvalues showing increasing l-values are more and more selected. At T = 0 K l tends to infinity and all the successive dominant l-eigenvalues become equivalent. As the z-spin correlation is null for T greater than 0 K but equal to 1 (in absolute value) for T = 0 K the critical temperature is Tinf{c} = 0 K. Using an analytical method similar to the one employed for Zinf{N}(0) we also give an exact expression valid for any temperature for the spin-spin correlations as well as for the correlation length xsi. In the T=0-limit we obtain a diagram of magnetic phases which is similar to the one derived through a renormalization approach. By taking the low-temperature limit of xsi we obtain the same expressions as the corresponding ones derived through a renormalization process, for each zone of the magnetic phase diagram, thus bringing for the first time a strong validation to the full exact solution of the model valid for any temperature.

cond-mat.stat-mech↗

arXiv2019

The Two-Dimensional Infinite Heisenberg Classical Square Lattice: Exact Theory and Experimental Results

We rigorously examine 2d-infinite square lattices composed of classical spins isotropically coupled between first-nearest neighbors. Each local exchange Hamiltonian is expanded on the basis of its eigenfunctions played by spherical harmonics Yinf{l,m}. The corresponding eigenvalues are modified Bessel functions of the first kind. In the thermodynamic limit a numerical study allows one to select the higher-degree term of the characteristic polynomial associated with the zero-field partition function Zinf{N}(0). A very simple exact closed-form expression is derived, thus permitting to express the free energy F and the specific heat Cinf{V}, for any temperature. We report a thermal study of the basic term appearing in the higher-degree term of Zinf{N}(0). We show that it appears crossovers between two consecutive terms. Coming from high temperatures where the l= 0-term is dominant, near the critical temperature Tinf{c}= 0 K, eigenvalues showing increasing l-values are more and more selected. We derive an exact expression for the spin-spin correlations, the correlation length ksi and the susceptibility khi. Near Tinf{c}= 0 K we obtain a diagram of magnetic phases. We derive the same expressions for xsi, F, Cinf{V} and khi as the corresponding ones derived through a renormalization process. We show that, near 0 K, the lattice is composed of quasi rigid quasi independent Kadanoff blocks of length ksi and magnetic moment M(T), the unit cell moment, so that khi.kinf{B}T=ksi^2.M(T)^2. Finally we compare experimental susceptibilities to the theoretical expression of khi, for two types of 2d-compounds (showing or not organic ligands inside and between sheets of Mn2+ ions). We obtain a remarkable good agreement between the J-values of the exchange energy derived from the fits and the corresponding ones previously measured as well as a value of the Landé factor close to the theoretical one.

cond-mat.stat-mech↗

arXiv2018

Zero-field Partition Function and Free Energy Density of the Two-Dimensional Heisenberg Classical Square Lattice

We rigorously examine 2d square lattices composed of Ninf{S} classical spins isotropically coupled. If Hsup{ex},inf{i,j} is the local exchange Hamiltonian each operator exp(-beta.Hsup{ex},inf{i,j}) is expanded on the basis of spherical harmonics Yinf{linf{ i,j }, minf{ i,j }}. We derive selection rules for the linf{ i,j}'s and minf{ i,j }'s. For infinite Ninf{S} the value m = 0 is selected. We obtain an exact l-polynomial for the zero-field partition function, valid for any temperature. Its thermal study allows to point out crossovers between the l-eigenvalues. Near Tinf{c} = 0 K we derive a diagram showing three magnetic phases, each one being characterized by the low-temperature behavior of the correlation length. At Tinf{c}= 0 K, we retrieve the critical exponent nu = 1. We identify three regimes: the renormalized classical, the quantum disordered and the quantum critical regimes. We exactly express the free energy density F. For each result we retrieve the corresponding one derived from the renormalization approach.

cond-mat.stat-mech↗