Journal of Statistical Physics

, Volume 167, Issue 5, pp 1143–1163 | Cite as

Step Density Profiles in Localized Chains

  • Wojciech De RoeckEmail author
  • Abhishek Dhar
  • François Huveneers
  • Marius Schütz


We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length L. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. From theoretical arguments, we find that the width of the step grows not faster than \(\sqrt{L}\), and we confirm this numerically for harmonic oscillators. In this case, we also observe a drastic breakdown of local equilibrium at the step, resulting in a heavily oscillating temperature profile.


Nonequilibrium Transport Localization 



This work grew out of discussions of F. Huveneers with J. Lebowitz and D. Huse. We are most grateful to both of them. Moreover F. H. thanks J. Lebowitz for his kind invitation at IAS (Princeton), where this work started. F. H. thanks the IAS (Princeton), the ANR grant JCJC, and the CNRS InPhyNiTi Grant (MaBoLo) for financial support. AD acknowledges support from the Indo-Israel joint research project No. 6-8/2014(IC) and from the French Ministry of Education through the Grant ANR (EDNHS). W.D.R. and M.S. are thankful to the DFG (German Research Fund) and the InterUniversity Attraction Pole DYGEST (Belspo, Phase VII/18) for funding.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Wojciech De Roeck
    • 1
    Email author
  • Abhishek Dhar
    • 2
  • François Huveneers
    • 3
  • Marius Schütz
    • 1
  1. 1.Instituut voor Theoretische FysicaKU LeuvenLouvainBelgium
  2. 2.International Centre for Theoretical SciencesTIFRBengaluruIndia
  3. 3.PSL Research University, CNRS, CEREMADEUniversité Paris-DauphineParisFrance

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