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Spin-boson model through a Poisson-driven stochastic process

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Abstract

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

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Acknowledgments

This work was financially supported by Grant-in-Aid for Science Research (B) 23340032 from JSPS. J.L. thanks IHES, Bures-sur-Yvette, for a visiting fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Okayama University, Okayama , 700-8530, Japan

    Masao Hirokawa

  2. Faculty of Mathematics, Kyushu University, Fukuoka , 819-0395, Japan

    Fumio Hiroshima

  3. School of Mathematics, Loughborough University, Loughborough , LE11 3TU, UK

    József Lőrinczi

Authors
  1. Masao Hirokawa
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  2. Fumio Hiroshima
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  3. József Lőrinczi
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Correspondence to Fumio Hiroshima.

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Hirokawa, M., Hiroshima, F. & Lőrinczi, J. Spin-boson model through a Poisson-driven stochastic process. Math. Z. 277, 1165–1198 (2014). https://doi.org/10.1007/s00209-014-1299-1

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  • DOI: https://doi.org/10.1007/s00209-014-1299-1

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