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A topographic exploration of the von Neumann entropy surface for a spin-1 (three-level) nuclear paramagnet

David J. Siminovitch

Published Jul 1, 2025• 0 citations
Publisher

Abstract

Paramagnetic spin systems are typically analyzed using classical statistical mechanics, and the discussion is usually limited to the simplest, two-level, isolated spin-1/2 system. Here, I extend the analysis of these paramagnetic spin systems in two significant ways: first, I analyze a three-level, isolated spin-1 system, and second, I use quantum statistical mechanics for the analysis. In this way, I provide an introduction to quantum statistical mechanics, including the density operator and von Neumann entropy, but I do so using paramagnetic spin systems, which are quite familiar to students from their study of classical statistical mechanics. The high-field equilibrium density operator ρ̂eq of an isolated spin-1 system can be reincarnated as an angle-axis {Θ,ẑ} parameterized rotation operator ρ̂eq≃exp(βIz)=D̂(j)(Θ,ẑ)=exp(−iΘIz), where the modulus of the imaginary rotation angle Θ = iβ is the normalized inverse spin temperature |Θ| = β = ℏω0/(kBTS). Using the Chebyshev polynomial operators of a discrete real variable fλ(j)(Iz) as the density operator expansion basis, I show that all of the most significant quantum statistical mechanical properties of this paramagnetic spin-1 system, the quantum partition function Z=Tr[exp(βIz)], the average energy ⟨H⟩, the equilibrium von Neumann entropy SνNeq, the spin temperature TS, and the vector (dipole) and tensor (quadrupole) polarizations, can all be expressed in terms of the generalized characters χλ(j)(Θ)≡χλ(j)(iβ) of irreducible representations of the rotation group SO(3). Expanding the density operator ρ̂ in spin polarization moments (or statistical tensors) ρ10(1) and ρ20(1) defines the von Neumann entropy surface SνN(ρ10, ρ20), whose topography defined on the two-dimensional polarization moments landscape is used (i) to elicit the relation between the thermal equilibrium polarization moments ρ10(β) and ρ20(β), and (ii) to visualize the connection between entropy and spin temperature.

DOI
10.1063/5.0234676

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arXiv:0910.1568•The Journal of Chemical Physics•Jul 20, 2010•17 citations

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We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model. We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable ( s =±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem ( σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix . The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom. By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex! In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t = –iτ , where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics .

Cambridge University Press eBooks•Aug 31, 2017•0 citations

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arXiv:1904.03145•Apr 5, 2019•0 citations

A Short Note on Quantum Entropies

Amelia Carolina Sparavigna

In quantum statistical mechanics, the extension of the classical Gibbs entropy is the von Neumann entropy, obtained from a quantum-mechanical system described by means of its density matrix. Here we shortly discuss this entropy and the use of generalized entropies instead of it

•Jan 11, 2016•1 citation
Paramagnetic spin systems are typically analyzed using classical statistical mechanics, and the discussion is usually limited to the simplest, two-level, isolated spin-1/2 system. Here, I extend the analysis of these paramagnetic spin systems in two significant ways: first, I analyze a three-level, isolated spin-1 system, and second, I use quantum statistical mechanics for the analysis. In this way, I provide an introduction to quantum statistical mechanics, including the density operator and von Neumann entropy, but I do so using paramagnetic spin systems, which are quite familiar to students from their study of classical statistical mechanics. The high-field equilibrium density operator ρ̂eq of an isolated spin-1 system can be reincarnated as an angle-axis {Θ,ẑ} parameterized rotation operator ρ̂eq≃exp(βIz)=D̂(j)(Θ,ẑ)=exp(−iΘIz), where the modulus of the imaginary rotation angle Θ = iβ is the normalized inverse spin temperature |Θ| = β = ℏω0/(kBTS). Using the Chebyshev polynomial operators of a discrete real variable fλ(j)(Iz) as the density operator expansion basis, I show that all of the most significant quantum statistical mechanical properties of this paramagnetic spin-1 system, the quantum partition function Z=Tr[exp(βIz)], the average energy ⟨H⟩, the equilibrium von Neumann entropy SνNeq, the spin temperature TS, and the vector (dipole) and tensor (quadrupole) polarizations, can all be expressed in terms of the generalized characters χλ(j)(Θ)≡χλ(j)(iβ) of irreducible representations of the rotation group SO(3). Expanding the density operator ρ̂ in spin polarization moments (or statistical tensors) ρ10(1) and ρ20(1) defines the von Neumann entropy surface SνN(ρ10, ρ20), whose topography defined on the two-dimensional polarization moments landscape is used (i) to elicit the relation between the thermal equilibrium polarization moments ρ10(β) and ρ20(β), and (ii) to visualize the connection between entropy and spin temperature.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
The statistical thermodynamics of the quantum spin- s XXZ Heisenberg model in the Weiss (mean-field) limit rests on the outer multiplicities of the irreducible representations in the Clebsch–Gordan decomposition of the n -fold spin- s tensor product, which determine the energy-level degeneracies and hence the partition function. Using known closed forms for these multiplicities in terms of generalized binomial coefficients of order 2 s + 1 , we write the exact finite-size partition function of the effective collective Hamiltonian and evaluate its thermodynamic limit by a large-deviation (Laplace) analysis validated against direct summations for systems of up to n = 10 4 spins. The free energy then follows from a two-parameter variational principle in the total spin and the magnetization per site. The ferromagnetic ordering is driven by the rotationally invariant part of the Hamiltonian, so that the Curie temperature T c = 1 3 J s ( s + 1 ) is independent of the anisotropy Δ, and the transition is of second order for every spin s , with classical mean-field exponents. The anisotropy reshapes the phase diagram: for Δ < 1 the field-driven first-order line of the isotropic ( Δ = 1 ) model opens into a dome of easy-plane ferromagnetic order, bounded by a continuous transition that restores the residual U(1) symmetry, and no first-order transitions or metastability occur throughout − 1 ≤ Δ < 1 ; the isotropic Curie-Weiss ferromagnet is recovered as Δ → 1. In two appendices we detail the asymptotics of the outer multiplicities down to the Gaussian-fluctuation prefactor and discuss the q -number and Jacobi-theta structures underlying the model, including the partial-theta-type structure that obstructs a standard theta-function reduction of the residual total-spin sum.
We are concerned with an information theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase space probability (z,m I rho Iz,m) which is known as Husimi function, where rho is a density operator and Iz,m) are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Qmbeta of the thermal density operator rhobeta of the harmonic oscillator, which leads, by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of Qmbeta such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions Qmbeta, m=0,1,... For m=1,2,..., we establish an exact expression for the Wehrl entropy of the density operator rhobeta and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/beta.
We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model. We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable ( s =±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem ( σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix . The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom. By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex! In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t = –iτ , where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics .
In this work, we consider the phase-space picture of quantum mechanics. We then introduce non-negative Wigner-like (operational) distributions \widetilde{\mathcal W}_{rho;alpha}(x,p) corresponding to the density operator \hat{rho} and being proportional to {W_{rho^(alpha/2)}(x,p)}^2, where W_{rho}(x,p) denotes the usual Wigner function. In doing so, we utilize the formal symmetry between the purity measure Tr(rho^2) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho}(x,p)}^2 and then consider, as a generalization, such symmetry between the fractional moment Tr(\hat{rho}^{alpha}) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho^{alpha/2}}(x,p)}^2. Next, we create a framework that enables explicit evaluation of the Renyi-Wigner entropies for the classical-like distributions \widetilde{\mathcal W}_{rho;alpha}(x,p). Consequently, a better understanding of some non-Gaussian features of a given state rho will be given, by comparison with the Gaussian state rho_G defined in terms of its Wigner function W_{rho_G}(x,p) and essentially determined by its purity measure T(rho_G)^2 alone. To illustrate the validity of our framework, we evaluate the distributions \widetilde{\mathcal W}_{beta;alpha}(x,p) corresponding to the (non-Gaussian) thermal state rho_β of a single particle confined by a one-dimensional infinite potential well with either the Dirichlet or Neumann boundary condition and then analyze the resulting Renyi entropies. Our phase-space approach will also contribute to a deeper understanding of non-Gaussian states and their properties either in the semiclassical limit (hbar \to 0) or in the high-temperature limit (beta \to 0), as well as enabling us to systematically discuss the quantal-classical Second Law of Thermodynamics on the single footing.