Abstract
We prove that translationally invariant Hamiltonians of a chain of n qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit \({n \rightarrow \infty}\) we show that any translationally invariant Hamiltonian of a chain of n qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e., Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly, in the limit \({n \rightarrow \infty}\) we show that any nearest-neighbour Hamiltonian of a chain of n qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.
Similar content being viewed by others
References
Mattis D.C.: The Theory of Magnetism 1, Statistics and Dynamics. Springer, Berlin (1988)
Sachdev S.: Quantum Phase Transitions, 2nd edn. Cambridge University Press, Cambridge (2011)
Bose S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003)
Osborne T.J., Linden N.: Propagation of quantum information through a spin system. Phys. Rev. A 69, 052315 (2004)
Vidal G., Latorre J.I., Rico E., Kitaev A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)
Latorre J.I., Rico E., Vidal G.: Ground state entanglement in quantum spin chains. Quantum Inform. Comput. 4, 48–92 (2004)
Jin B.Q., Korepin V.E.: Quantum spin chain, toeplitz determinants and the fisher-hartwig conjecture. J. Stat. Phys. 116, 79–95 (2004)
Its A.R., Jin B.Q., Korepin V.E.: Entanglement in the xy spin chain. J. Phys. A Math. Gen. 38, 2975–2990 (2005)
Korepin V.E.: Universality of entropy scaling in one dimensional gapless models. Phys. Rev. Lett. 92, 096402 (2004)
Keating J.P., Mezzadri F.: Random matrix theory and entanglement in quantum spin chains. Commun. Math. Phys. 252, 543–579 (2004)
Keating J.P., Mezzadri F.: Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory. Phys. Rev. Lett. 94, 050501 (2005)
Masanes L.: Area law for the entropy of low-energy states. Phys. Rev. A 80, 052104 (2009)
Arnesen M.C., Bose S., Vedral V.: Natural thermal and magnetic entanglement in the 1d heisenberg model. Phys. Rev. Lett. 87, 017901 (2001)
Gunlycke D., Kendon V.M., Vedral V.: Thermal concurrence mixing in a one-dimensional ising model. Phys. Rev. A 64, 042302 (2001)
Calabrese P., Cardy J.: Entanglement entropy and quantum field theory. J. Stat. Mech. Theory E 2004, 06002 (2004)
Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)
Linden N., Popescu S., Short A.J., Winter A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)
Goldstein S., Lebowitz J.L., Mastrodonato C., Tumulka R., Zanghi N.: Approach to thermal equilibrium of macroscopic quantum systems. Phys. Rev. E 81, 011109 (2010)
Deutsch J.M.: Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991)
Srednicki M.: Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994)
Rigol M., Dunjko V., Olshanii M.: Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008)
Atas, Y.Y., Bogomolny, E.: Spectral density of the quantum ising model in two fields: Gaussian and multi-gaussian approximations. ArXiv e-prints arXiv:1402.6858v1 (2014)
Keating, J.P., Linden, N., Wells, H.J.: Random matrices and quantum spin chains. Markov Processes and Related Fields (accepted) arXiv:1403.1114 (2014)
Hartmann M., Mahler G., Hess O.: Gaussian quantum fluctuations in interacting many particle systems. Lett. Math. Phys. 68, 103–112 (2004)
Hartmann M., Mahler G., Hess O.: Spectral densities and partition functions of modular quantum systems as derived from a central limit theorem. J. Stat. Phys. 119, 1139–1151 (2005)
Lenstra, H.W.: Vanishing sums of roots of unity. In: Proceedings Bicentennial Congress Wiskundig Genootschap, Math. Centre Tracts 100/101, pp. 249–268. Mathematisch Centrum, Amsterdam (1979)
Billingsley P.: Probability and Measure, 3rd edn. Wiley Interscience, New York (1995)
Nielsen, M.A.: The fermionic canonical commutation relations and the Jordan–Wigner transform. School of Physical Sciences The University of Queensland (2005)
Bernstein D.S.: Matrix Mathematics, 2nd edn, pp. 388–390. Princeton University Press, Oxfordshire (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Wolf
Rights and permissions
About this article
Cite this article
Keating, J.P., Linden, N. & Wells, H.J. Spectra and Eigenstates of Spin Chain Hamiltonians. Commun. Math. Phys. 338, 81–102 (2015). https://doi.org/10.1007/s00220-015-2366-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2366-0