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Steady States and Universal Conductance in a Quenched Luttinger Model

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Abstract

We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian \({H_{\lambda}}\) with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t >  0 via a Hamiltonian \({H_{\lambda'}}\) which differs from \({H_{\lambda}}\) by the strength of the interaction. Asymptotically in time, as \({t \to \infty}\), after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference \({\mu_{+} - \mu_{-}}\) between right- (+) and left- (−) moving fermions obtained from the two-point correlation function. Both I and \({\mu_{+} - \mu_{-}}\) depend on \({\lambda}\) and \({\lambda'}\). Only for the case \({\lambda = \lambda' = 0}\) does \({\mu_{+} - \mu_{-}}\) equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, \({G = I/(\mu_{+} - \mu_{-})}\), has a universal value equal to the conductance quantum \({e^2/h}\) for the spinless case.

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Authors and Affiliations

  1. Department of Theoretical Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden

    Edwin Langmann & Per Moosavi

  2. Departments of Mathematics and Physics, Rutgers University, Piscataway, NJ, 08854, USA

    Joel L. Lebowitz

  3. Dipartimento di Matematica, Università degli Studi di Milano, 20133, Milan, Italy

    Vieri Mastropietro

Authors
  1. Edwin Langmann
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  2. Joel L. Lebowitz
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  3. Vieri Mastropietro
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  4. Per Moosavi
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Correspondence to Per Moosavi.

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Communicated by H. Spohn

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Langmann, E., Lebowitz, J.L., Mastropietro, V. et al. Steady States and Universal Conductance in a Quenched Luttinger Model. Commun. Math. Phys. 349, 551–582 (2017). https://doi.org/10.1007/s00220-016-2631-x

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  • DOI: https://doi.org/10.1007/s00220-016-2631-x

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