Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems
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It has previously been suggested that small subsystems of closed quantum systems thermalize under some assumptions; however, this has been rigorously shown so far only for systems with very weak interaction between subsystems. In this work, we give rigorous analytic results on thermalization for translation-invariant quantum lattice systems with finite-range interaction of arbitrary strength, in all cases where there is a unique equilibrium state at the corresponding temperature. We clarify the physical picture by showing that subsystems relax towards the reduction of the global Gibbs state, not the local Gibbs state, if the initial state has close to maximal population entropy and certain non-degeneracy conditions on the spectrumare satisfied.Moreover,we showthat almost all pure states with support on a small energy window are locally thermal in the sense of canonical typicality. We derive our results from a statement on equivalence of ensembles, generalizing earlier results by Lima, and give numerical and analytic finite size bounds, relating the Ising model to the finite de Finetti theorem. Furthermore, we prove that global energy eigenstates are locally close to diagonal in the local energy eigenbasis, which constitutes a part of the eigenstate thermalization hypothesis that is valid regardless of the integrability of the model.
KeywordsInverse Temperature Boundary Condition Gibbs State Energy Eigenstates Periodic Boundary Condition
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- 14.Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics I and II. Springer, New York (2002)Google Scholar
- 18.Kliesch, M., Gogolin, C., Kastoryano, M.J., Riera, A., Eisert, J.: Locality of temperature. Phys. Rev. X 4, 031019 (2014). arXiv:1309.0816
- 20.Bhatia, R.:Perturbation bounds for matrix eigenvalues. SIAM Class. Appl. Math. (2007)Google Scholar
- 23.Deserno, M.: Microcanonical and canonical two-dimensional Ising model: an example. http://www.cmu.edu/biolphys/deserno/pdf/microcan
- 25.Brandão, F.G.S.L., Harrow, A.W., Horodecki, M.: Local random quantum circuits are approximate polynomial-designs. arXiv:1208.0692
- 26.Low, R.A.: Large deviation bounds for k-designs. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 465, pp. 3289–3308. The Royal Society, London (2009)Google Scholar
- 27.Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 813-822 (2008)Google Scholar