Abstract
In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample \({\mathcal{S}}\) connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer–Büttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of \({\mathcal{S}}\). In this spirit, we define Landauer–Büttiker crystalline currents by extending the Landauer–Büttiker formula to a setup where the sample \({\mathcal{S}}\) is replaced by a periodic structure whose unit cell is \({\mathcal{S}}\). We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.
Similar content being viewed by others
References
Abrahams E., Anderson P.W., Licciardello D.C., Ramakrishnan T.V.: Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979)
Aschbacher W., Jakšić V., Pautrat Y., Pillet C.-A.: Transport properties of quasi-free fermions. J. Math. Phys. 48, 032101 (2007)
Ben Sâad, R., Pillet, C.-A.: A geometric approach to the Landauer–Büttiker formula. J. Math. Phys. 55, 075202 (2014)
Busch, P.: The time-energy uncertainty relation. In: Muga J.G., Sala Mayato R., Egusquiza Í.L. (eds.) Time in Quantum Mechanics, 2nd edn. Springer, Berlin (2008)
Bruneau, L., Jakšić, V., Pillet, C.A.: Landauer–Büttiker formula and Schrödinger conjecture. Commun. Math. Phys. 319, 501–513 (2013)
Cornean, H.D., Jensen, A., Moldoveanu, V.: A rigorous proof of the Landauer–Büttiker formula. J. Math. Phys. 46, 042106 (2005)
Edwards J.T., Thouless D.J.: Numerical studies of localization in disordered systems. J. Phys. C: Solid State Phys. 5, 807–820 (1972)
Fröhlich J., Pfeifer P.: Generalized time-energy uncertainty relations and bounds on lifetimes of resonances. Rev. Mod. Phys. 67, 759–779 (1995)
Grech P., Jakšić V., Westrich M.: The spectral structure of the electronic black box Hamiltonian. Lett. Math. Phys. 103, 1135–1147 (2013)
Gesztesy F., Nowell R., Potz W.: One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics. Differ. Integr. Equ. 10, 521–546 (1997)
Gesztesy, F., Simon, B.: Inverse spectral analysis with partial information on the potential I. The case of an a.c. component in the spectrum. Helv. Phys. Acta 70, 66–71 (1997)
Jakšić, V.: Topics in spectral theory. In: Open Quantum Systems I. The Hamiltonian Approach. Lecture Notes in Mathematics, vol. 1880, pp. 235–312. Springer (2006)
Jakšić, V., Landon, B., Panati, A.: A note on reflectionless Jacobi matrices. Commun. Math. Phys. 332, 827–838 (2014)
Jakšić, V., Landon, B., Pillet, C.-A.: Entropic fluctuations in XY chains and reflectionless Jacobi matrices. Ann. Henri Poincaré 14, 1775–1800 (2013)
Last, Y.: Conductance and spectral properties. Ph.D. Thesis, Technion (1994)
Nenciu, G.: Independent electrons model for open quantum systems: Landauer–Büttiker formula and strict positivity of the entropy production. J. Math. Phys. 48, 033302 (2007)
Mandelstam L., Tamm I.: The uncertainty relation between energy and time in non-relativistic quantum mechanics. J. Phys. (Moscow) 9, 249–254 (1945)
Reed M., Simon B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York (1978)
Simon, B.: Szegö’s Theorem and Its Descendants. Spectral Theory for L 2 Perturbations of Orthogonal Polynomials. M.B. Porter Lectures. Princeton University Press, Princeton (2011)
Thouless D.J.: Maximum metallic resistance in thin wires. Phys. Rev. Lett. 39, 1167–1169 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Dedicated to the memory of Markus Büttiker
Rights and permissions
About this article
Cite this article
Bruneau, L., Jakšić, V., Last, Y. et al. Landauer–Büttiker and Thouless Conductance. Commun. Math. Phys. 338, 347–366 (2015). https://doi.org/10.1007/s00220-015-2321-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2321-0