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International Conference on Optimization of Electrical and Electronic Equipment

Study of the magnetic properties of two-dimensional (2D) classical square heisenberg antiferromagnets II- spin correlations and susceptibility

Jacques Curély

Published May 1, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.1109/optim.2010.5510535

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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 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 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.
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.
With the use of a rigorous relation of the correlations in the Ising spin system, an approach to the exact magnetic susceptibility of the two-dimensional Ising antiferromagnet has been attempted. The zero-field magnetic susceptibilities for the rectangular as well as the square antiferromagnets are shown to be expressed in terms of finite near-spin correlations. With some decoupling approximations for many-spin correlations, the result for the square antiferromagnet by the present method is proved to be within 0.35% from that of the series expansion.