[Paper Review] Shift-invariance for FK-DLR states of a 2D quantum bose-gas

Yuri Suhov, Mark Kelbert+1 more|arXiv (Cornell University)|Apr 15, 2013

Cold Atom Physics and Bose-Einstein Condensates10 references3 citations

TL;DR

This paper establishes shift-invariance for all Feynman–Kac–Dobrushin–Lanford–Ruelle (FK-DLR) states of a 2D quantum Bose-gas under general interaction potentials and finite-range conditions. Using a path-integral representation and loop configurations, it proves that any infinite-volume FK-DLR functional—regardless of uniqueness—is invariant under spatial translations, extending the thermodynamic stability of quantum Bose systems to full spatial symmetry.

ABSTRACT

This paper continues the work Y. Suhov, M. Kelbert. FK-DLR states of a quantum bose-gas, arXiv:1304.0782 [math-ph], and focuses on infinite-volume bosonic states for a quantum system (a quantum gas) in a plane. We work under similar assumptions upon the form of local Hamiltonians and the type of the (pair) interaction potential as in the reference above. The result of the paper is that any infinite-volume FK-DLR functional corresponding to the Hamiltonians is shift-invariant, regardless of whether this functional is unique or not.

Motivation & Objective

• To establish the shift-invariance of infinite-volume FK-DLR states for a 2D quantum Bose-gas with hard-core and finite-range interactions.
• To generalize previous results on FK-DLR functionals by proving invariance under spatial translations even when the state is non-unique.
• To extend the framework of FK-DLR measures to loop configurations in the plane, ensuring consistency with thermodynamic limits.
• To validate the structural stability of quantum many-body states under spatial translations in two dimensions.
• To provide a rigorous probabilistic foundation for FK-DLR states using quasi-local C*-algebraic formalism and loop measures.

Proposed method

• Utilizes the Feynman–Kac representation to express density matrices via path and loop configurations in the plane.
• Constructs FK-DLR probability measures μ on loop configuration spaces W*(ℝ²), representing thermodynamic limits of finite-volume Gibbs states.
• Applies shift isomorphisms U^Λ₀(s) between Fock spaces over shifted domains to relate states at different spatial locations.
• Employs a transformation T±(s) on loop configurations that preserves the structure of loops while shifting their positions, ensuring measurable and bijective mappings.
• Implements a perturbative analysis using bounds on loop energy differences and truncated potentials to control deviations in the measure under translation.
• Applies Chebyshev’s inequality to show that the set of 'good' configurations 𝒢_L, where translation effects are negligible, has measure approaching 1 as L → ∞.

Experimental results

Research questions

• RQ1 Are all infinite-volume FK-DLR states of a 2D quantum Bose-gas invariant under spatial translations, even when the state is non-unique?
• RQ2 Can the FK-DLR measure structure be preserved under spatial shifts in the plane, given hard-core and finite-range interactions?
• RQ3 To what extent does the loop representation of quantum states ensure spatial symmetry in the thermodynamic limit?
• RQ4 How do the perturbations in loop configurations affect the invariance of the FK-DLR functional under translation?
• RQ5 What conditions on the interaction potential ensure that the limiting FK-DLR functional remains shift-invariant in two dimensions?

Key findings

• Any FK-DLR functional φ ∈ ℱ(z,β) for a 2D quantum Bose-gas is shift-invariant, regardless of whether the state is unique.
• The FK-DLR probability measure μ on loop configurations is invariant under spatial translations, as shown via the construction of measurable, bijective shift maps T±(s).
• The ratio of Jacobian-like factors J⁺J⁻ in the transformed measure converges uniformly to 1 as system size L → ∞, ensuring measure consistency under translation.
• The set of configurations 𝒢_L for which the translation transformation is well-behaved has measure μ(𝒢_L) ≥ 1−δ for any δ ∈ (0,1), provided L > L₁*.
• The proof relies on uniform bounds on loop energy differences and the decay of interaction terms, ensuring that translation-induced changes vanish in the thermodynamic limit.
• The result holds under the condition ρ̄ = z exp(4βV̄R²/r²) < 1, ensuring convergence of the Gibbs state family and existence of limiting functionals.