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Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles

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We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) to study the (possibly non-linear) response of interacting fermions at thermal equilibrium to perturbations of the external electromagnetic field. This analysis leads to an extension of the results for quasi-free fermions of (Bru et al. Commun Pure Appl Math 68(6):964–1013, 2015; Bru et al. J Math Phys 56:051901-1–051901-51, 2015) to fermion systems on the lattice with short-range interactions. More precisely, we investigate entropy production and charge transport properties of non-autonomous C*-dynamical systems associated with interacting lattice fermions within bounded static potentials and in presence of an electric field that is time and space dependent. We verify the 1st law of thermodynamics for the heat production of the system under consideration. In linear response theory, the latter is related with Ohm and Joule’s laws. These laws are proven here to hold at the microscopic scale, uniformly with respect to the size of the (microscopic) region where the electric field is applied. An important outcome is the extension of the notion of conductivity measures to interacting fermions.

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Authors and Affiliations

  1. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain

    Jean-Bernard Bru

  2. Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    Jean-Bernard Bru

  3. BCAM - Basque Center for Applied Mathematics, Mazarredo, 14., 48009, Bilbao, Spain

    Jean-Bernard Bru

  4. Department of Mathematical Physics, Institute of Physics of the University of São Paulo, São Paulo, Brazil

    Walter de Siqueira Pedra

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  1. Jean-Bernard Bru
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  2. Walter de Siqueira Pedra
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Correspondence to Jean-Bernard Bru.

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Bru, JB., de Siqueira Pedra, W. Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles. Lett Math Phys 106, 81–107 (2016). https://doi.org/10.1007/s11005-015-0806-6

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  • DOI: https://doi.org/10.1007/s11005-015-0806-6

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