Abstract
We consider the macroscopic system of free lattice fermions in one dimension assuming that the one-body Hamiltonian of the system is the one dimensional discrete Schrödinger operator with independent identically distributed random potential. We show analytically and numerically that the variance of the entanglement entropy of the segment \([-M,M]\) of the system is bounded away from zero as \(M\rightarrow \infty \). This manifests the absence of the selfaveraging property of the entanglement entropy in our model, meaning that in the one-dimensional case the complete description of the entanglement entropy is provided by its whole probability distribution. This also may be contrasted the case of dimension two or more, where the variance of the entanglement entropy per unit surface area vanishes as \( M\rightarrow \infty \) [8], thereby guaranteeing the representativity of its mean for large M in the multidimensional case.
Similar content being viewed by others
References
Abdul-Rahman, H., Stolz, G.: A uniform area law for the entanglement of the states in the disordered XY chain. J. Math. Phys. 56, 121901 (2015)
Aizenman, M., Warzel, S.: Random Operators: Disorder Effects on Quantum Spectra and Dynamics. American Mathematical Society, Providence (2015)
Ares, F., Esteve, J.G., Falceto, F., Sanchez-Burillo, E.: Excited state entanglement in homogeneous fermionic chains. J. Phys. A: Math. Theor. 47, 245301 (2014)
Calabrese, P., Cardy, J., Doyon, B.: Entanglement entropy in extended systems. J. Phys. A. Math. Theor. 42, 500301 (2009)
Dahlsten, O.C.O., Lupo, C., Mancini, S., Serafini, A.: Entanglement typicality. J. Phys. A. Math. Theor. 47, 363001 (2014)
Dittrich, T., Hänggi, P., Ingold, G.-L., Kramer, B., Schön, G., Zwerger, W.: Quantum Transport and Dissipation. Willey, Weinheim (1998)
Eisert, J., Cramer, M., Plenio, M.B.: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010)
Elgart, A., Pastur, L., Shcherbina, M.: Large block properties of the entanglement entropy of free disordered fermions. J. Stat. Phys. 166, 1092–1127 (2017)
Lehmann, E.L., Casella, G.: Theory of Point Estimation. Springer, Berlin (1998)
Laflorencie, N.: Quantum entanglement in condensed matter systems. Phys. Rep. 643, 1–59 (2016)
Leschke, H., Sobolev, A., Spitzer, W.: Scaling of Ré nyi entanglement entropies of the free Fermi-gas ground state: a rigorous proof. Phys. Rev. Lett. 112, 160403 (2014)
Lifshitz, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the Theory of Disordered Systems. Wiley, New York (1989)
Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996)
Pastur, L., Figotin, A.: Spectra of Random and Almost Periodic Operators. Springer, Berlin (1992)
Pastur, L., Slavin, V.: Area law scaling for the entanglement entropy of disordered quasifree fermion. Phys. Rev. Lett. 113, 1504 (2004)
Peschel, I.: Calculation of reduced density matrices from correlation functions. J. Phys. A: Math. Gen. 36, L205 (2003)
Shiryaev, A.N.: Probability. Springer, Berlin (1995)
Teschl, G.: Jacobi Operators and Completely Integrable Systems. American Mathematical Society, Providence (1999)
Acknowledgements
The work is supported in part by the Grant 4/16-M of the National Academy of Sciences of Ukraine.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pastur, L., Slavin, V. The Absence of the Selfaveraging Property of the Entanglement Entropy of Disordered Free Fermions in One Dimension. J Stat Phys 170, 207–220 (2018). https://doi.org/10.1007/s10955-017-1929-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-017-1929-1