We start this article by recalling rigorous results previously obtained for 2d square lattices composed of (2N+1)2classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings), in the thermodynamic limit (N→+∞) [4]: (i) the zero-field partition function ZN(0), (ii) the spin correlation which vanishes in the zero-field limit, except at T=0 K so that the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem, (iii) the spin-spin correlation between any two lattice sites, (iv) the correlation length and (v) the static susceptibility. We exclusively focus on the low-temperature behaviors of the correlation length and the static susceptibility. This leads to the determination of a diagram characterized by three magnetic phases. Moreover we show that all the behaviors are in perfect agreement with the corresponding ones derived by using a renormalization method. Finally we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. An experimental test is given for illustrating this theoretical study.
We start this article by recalling rigorous results previously obtained for 2d square lattices composed of (2N+1)2classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings), in the thermodynamic limit (N→+∞) [4]: (i) the zero-field partition function ZN(0), (ii) the spin correlation which vanishes in the zero-field limit, except at T=0 K so that the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem, (iii) the spin-spin correlation between any two lattice sites, (iv) the correlation length and (v) the static susceptibility. We exclusively focus on the low-temperature behaviors of the correlation length and the static susceptibility. This leads to the determination of a diagram characterized by three magnetic phases. Moreover we show that all the behaviors are in perfect agreement with the corresponding ones derived by using a renormalization method. Finally we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. An experimental test is given for illustrating this theoretical study.
In this part labelled II we examine the spin correlations and the susceptibility. We use a similar method which has allowed to derive a closed-form expression of the zero-field partition function ZN(0), for 2D square lattices composed of (2N+1)2classical spins isotropically coupled [1]. We rigorously show that the spin correlation vanishes in the zero-field limit, except at T=0 K. Thus, the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem. For calculating the spin-spin correlation, we show that it is necessary to distinguish a correlation domain in which the correlation path is confined and a wing domain (Theorem 1). In the thermodynamic limit (N→+∞), we give a general closed-form expression for the spin-spin correlation between any two lattice sites. We prove that all the possible paths have the same analytic expression and correspond to the shortest ones in agreement with the classical principle of least action and its quantum version (Theorem 2). As a result and for the first time, we derive the closed-form expression for the susceptibility, without any approximation. We finally test previous experimental fits and we show that the use of a truncated expansion for the susceptibility was totally justified.
We start this article by recalling the main exact results previously obtained for describing the static magnetic properties of 2d square lattices composed of classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings) [4c]. Near the critical temperature TC= 0 K we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. We show that there are three distinct regimes: The Renormalized Classical Regime (RCR), the Quantum Disordered Regime (QDR) and the Quantum Critical Regime (QCR). The static properties give a good image of the spin arrangement and thus remain a good starting point for the study of dynamic properties. As a result, inside each of the three regimes, we may derive the corresponding dynamic behaviors. For practical reasons that we shall detail, we restrict our study to the RCR and QCR cases. An experimental test is given for illustrating this theoretical work. We notably show that it allows one to derive the static correlation length which is an important tool for understanding both static and dynamic properties.
In this part labelled I we rigorously examine 2D square lattices composed of (2N+1)2classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg cou-plings). Each local operator exp(-β Hi,jex) - where Hi,jexis the local exchange Hamiltonian associated with each site (i,j) - is expanded on the basis of spherical harmonics, each one being characterized by the couple of integers (ℓ,m), with ℓ≥0 and m∈[-ℓ,+ℓ]. Using spin lattice symmetries, we derive selection rules over these coefficients, thus allowing to express the zero-field partition function ZN(0). In the thermodynamic limit (N→∞), we show that the value m=0 is selected while ℓ=ℓ0≥0, ∀(i,j). A very simple closed-form expression may be then derived for ZN(0). Finally we report a thermal study of the current term of the characteristic polynomial giving ZN(0). We show that it appears crossovers between two consecutive terms characterized by integers ℓ and ℓ+1, respectively. Coming from high temperatures where the term characterized by ℓ=0 is dominant, near absolute zero, increasing ℓ-values are more and more selected when the temperature is cooling down. But, when T reaches zero, all the successive dominant terms become equivalent and are characterized by the value ℓ=0. We show that this property also prevails at T=0 K rigorously. This confirms the fact that, for a 2D lattice showing Heisenberg couplings, the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem.
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.
The energy, the specific heat, and the magnetic susceptibility of the bond-diluted S=(1/2) Heisenberg ferromagnet on a two-dimensional square lattice are studied using a Monte Carlo sampling of the terms contributing to the partition function. The full lattice is studied in detail for sizes up to 20\ifmmode\times\else\texttimes\fi{}20, and evidence is presented to suggest that the low-temperature correlation length behaves as \ensuremath{\xi}\ensuremath{\sim}${e}^{\ensuremath{\nu}\ensuremath{\beta}J}$ with \ensuremath{\nu}\ensuremath{\sim}1.91\ifmmode\pm\else\textpm\fi{}0.06. The temperature dependence of the susceptibility, however, is not quite a simple exponential in \ensuremath{\beta}J. Knowledge of the correlation length and finite-size scaling allows the determination of the prefactor. Our conclusion is that the low-temperature susceptibility behaves as ${T}^{\mathrm{\ensuremath{-}}1}$${e}^{2\ensuremath{\nu}\ensuremath{\beta}J}$. Square lattices of size ranging from 6\ifmmode\times\else\texttimes\fi{}6 to 12\ifmmode\times\else\texttimes\fi{}12 are studied for kBT/J ranging from 0.1 to 3 and for several values of the bond probability p. Not surprisingly, the most drastic thermodynamic behavior near the critical bond-percolation probability is the sharp increase in the low-temperature susceptibility for p>pc. This corresponds to much larger correlation effects in the percolating lattice than below the percolation threshold. We also compare the thermodynamics at p=pc to the scaling theories of Lubensky and Stanley et al. which suggest that the correlations at the percolation threshold are predominantly one dimensional.
We analyze the thermodynamic properties of the spin-S two-dimensional quantum Heisenberg antiferromagnet on a square lattice with nearest and next-nearest neighbor couplings in the N\'eel phase (J2∕J1<0.4) employing the quantum hierarchical reference theory, a nonperturbative implementation of the renormalization group method to quantum systems. We investigate the staggered susceptibility, the structure factors, and the correlation length at finite temperature and for different values of the frustration ratio. From the finite temperature results, we also extrapolate ground state properties, such as spin stiffness and spontaneous staggered magnetization, providing an estimate of the extent of quantum corrections. The behavior of these quantities as a function of frustration may provide some hints on the breakdown of the N\'eel phase at zero temperature for larger values of J2.