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arXiv:2210.10697 · Communications in Mathematical Physics

Strict Deformation Quantization and Local Spin Interactions

Nicolò Drago, Christiaan J. F. van de Ven

Published Jan 1, 2024• 4 citations
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Abstract

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.

DOI
10.1007/s00220-023-04887-3

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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.
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.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.
Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,
Abstract We consider the construction of quantum-integrable spin chains with q -deformed long-range interactions by ‘freezing’ integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars–Schneider system. We focus on the elliptic case. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbor to long-range interacting spins. We first define an action of the modular group on the spinless elliptic Ruijsenaars–Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantization to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium while preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane–Shastry chains along with their xxz -like q -deformations (face type), or the antiperiodic Haldane–Shastry chain of Fukui–Kawakami, its elliptic generalization of Sechin–Zotov, and their completely anisotropic q -deformations due to Matushko–Zotov (vertex type). Finally, we show how freezing fits in the setting of ‘hybrid’ integrable systems.
We consider the finite volume mean values of current operators in integrable spin chains with local interactions, and provide an alternative derivation of the exact result found recently by the author and two collaborators. We use a certain type of long range deformation of the local spin chains, which was discovered and explored earlier in the context of the AdS/CFT correspondence. This method is immediately applicable also to higher rank models: as a concrete example we derive the current mean values in the SU(3) S U ( 3 ) -symmetric fundamental model, solvable by the nested Bethe Ansatz. The exact results take the same form as in the Heisenberg spin chains: they involve the one-particle eigenvalues of the conserved charges and the inverse of the Gaudin matrix.