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arXiv:1710.08387

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

Jacques Curély

Published Oct 17, 2017• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1710.08387

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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.
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 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.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.
Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where six{{s}_{i}}^{x}si​x, siy{{s}_{i}}^{y}si​y, and siz{{s}_{i}}^{z}si​z are components of the unit vector si{\mathbf{s}}_{i}si​, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions Jijx=Jijy=J{{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=JJij​x=Jij​y=J and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.