Abstract
Within the C*-algebraic framework of the resolvent algebra for canonical quantum systems, the structure of oscillating lattice systems with bounded nearest neighbor interactions is studied in any number of dimensions. The global dynamics of such systems acts on the resolvent algebra by automorphisms and there exists a (in any regular representation) weakly dense subalgebra on which this action is pointwise norm continuous. Based on this observation, equilibrium (KMS) states as well as ground states are constructed, which are shown to be regular. It is also indicated how to deal with singular interactions and non-harmonic oscillations.
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Albeverio, S., Kondratiev, Y., Kozitsky, Y., Röckner, M.: The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach. EMS Tracts Math 8 (2009)
Araki H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Pub. RIMS Kyoto Univ. 9, 165–209 (1973)
Bakhrakh V.L., Vetchinkin S.I., Khristenko S.V.: Green’s function of a multidimensional isotropic harmonic oscillator. Theor. Math. Phys. 12, 776–778 (1972)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin (1981)
Buchholz D.: The resolvent algebra: Ideals and dimension. J. Funct. Anal. 266, 3286–3302 (2014)
Buchholz D., Grundling H.: Algebraic supersymmetry: a case study. Commun. Math. Phys. 272, 699–750 (2007)
Buchholz D., Grundling H.: The resolvent algebra: A new approach to canonical quantum systems. J. Funct. Anal. 254, 2725–2779 (2008)
Buchholz, D., Grundling, H.: Quantum systems and resolvent algebras. In: Blanchard, P., Fröhlich, J., (eds.) The Message of Quantum Science: Attempts Towards a Synthesis. Lect. Notes Phys. 899, pp. 33–45. Springer, Berlin (2015)
Dereziński J., Jaksić V., Pillet C.-A.: Perturbation theory of W-*- dynamics: Liouvilleans and KMS-states. Rev. Math. Phys. 15, 447–489 (2003)
Kanda, T., Matsui, T.: KMS states of weakly coupled anharmonic crystals and the resolvent CCR algebra. e-print arXiv:1601.04809
Minlos R.A., Verbeure A., Zagrebnov V.A.: A quantum crystal model in the light-mass limit: Gibbs states. Rev. Math. Phys. 12, 981–1032 (2000)
Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb–Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009)
Nachtergaele, B., Schlein, B., Sims, R., Starr, S., Zagrebnov, V.A.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22, 207–231 (2010)
Nachtergaele, B., Sims, R.: On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian. e-print arXiv:1410.8174v1
Reed M., Simon B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Acacemic Press, New York (1978)
Ruelle D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Mech. 98, 57–75 (2000)
Møller J.S.: Fully coupled Pauli-Fierz systems at zero and positive temperature. J. Math. Phys. 55, 075203 (2014)
Takesaki M.: Disjointness of the KMS-states of different temperatures. Commun. Math. Phys. 17, 33–41 (1970)
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Communicated by H.-T. Yau
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Buchholz, D. The Resolvent Algebra for Oscillating Lattice Systems: Dynamics, Ground and Equilibrium States. Commun. Math. Phys. 353, 691–716 (2017). https://doi.org/10.1007/s00220-017-2869-y
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DOI: https://doi.org/10.1007/s00220-017-2869-y