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arXiv:2007.03390 · Journal of Mathematical Physics

The classical limit of mean-field quantum spin systems

Christiaan J. F. van de Ven

Published Dec 1, 2020• 10 citations
arXivHTMLPDF

Abstract

The theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I={0}∪1/N*⊂[0,1], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q1/N:Ã0→A1/N, with Ã0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω0(f)=1n∑i=1nf(Ωi), where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2.

DOI
10.1063/5.0021120

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The theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I={0}∪1/N*⊂[0,1], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q1/N:Ã0→A1/N, with Ã0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω0(f)=1n∑i=1nf(Ωi), where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2.
The algebraic properties of a strict deformation quantization are analyzed on the classical phase space R2. The corresponding quantization maps enable us to take the limit for ħ→0 of a suitable sequence of algebraic vector states induced by ħ-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on R2, defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of ∗ algebras) parametrized by ħ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in R2 depending on the symmetry of the potential. In addition, since this ∗-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off ħ. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.
The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off [Formula: see text]. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.
Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.
Abstract We define a strict deformation quantization which is compatible with any Hamiltonian with local spin interaction (e.g. the Heisenberg Hamiltonian) for a spin chain. This is a generalization of previous results known for mean-field theories. The main idea is to study the asymptotic properties of a suitably defined algebra of sequences invariant under the group generated by a cyclic permutation. Our point of view is similar to the one adopted by Landsman, Moretti and van de Ven (Rev Math Phys 32(10):2050031, 2020, https://doi.org/10.1142/S0129055X20500312 ), who considered a strict deformation quantization for the case of mean-field theories. However, the methods for a local spin interaction are considerably more involved, due to the presence of a strictly smaller symmetry group.
We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number sss. It is known that quantum states and observables can be represented by functions defined on the phase space S{\mathcal S}S, which in our case is the NNN-fold product of unit spheres. Therefore, the classical limit s→∞s\to\inftys→∞ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on S{\mathcal S}S. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree nnn and thus have to deal with the problem of the limit of double sequences (depending on nnn and sss) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.