Abstract
We give a characterization of the class of gapped Hamiltonians introduced in Part I (Ogata, A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015). The Hamiltonians in this class are given as MPS (Matrix product state) Hamiltonians. In Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), we list up properties of ground state structures of Hamiltonians in this class. In this Part II, we show the converse. Namely, if a (not necessarily MPS) Hamiltonian H satisfies five of the listed properties, there is a Hamiltonian H′ from the class by Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), satisfying the following: The ground state spaces of the two Hamiltonians on the infinite interval coincide. The spectral projections onto the ground state space of H on each finite intervals are approximated by that of H′ exponentially well, with respect to the interval size. The latter property has an application to the classification problem with open boundary conditions.
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References
Arveson, W.B.: Continuous analogues of Fock space I. Mem. Am. Math. Soc. 80 (1969)
Bachmann S., Ogata Y.: C 1-Classification of gapped parent Hamiltonians of quantum spin chains. Commun. Math. Phys. 338, 1011–1042 (2015)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1986)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1996)
Bratteli O., Jorgensen P.E.T.: Endomorphisms of B(H) II. Finitely correlated states on O n . J. Funct. Anal. 145, 323–373 (1997)
Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)
Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated pure states. J. Funct. Anal. 144, 443–490 (1992)
Matsui, T.: A characterization of pure finitely correlated states. Infinite Dimensional Analysis, Quantum Probability and Related Topics, pp. 647–661 (1998)
Matsui, T.: Boundedness of entanglement entropy and split property of quantum spin chains. Rev. Math. Phys., 1350017 (2013)
Michalakis S., Pytel J.: Stability of frustration-free hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)
Ogata, Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I (2015). doi:10.1007/s00220-016-2696-6
Wolf, M.M.: Quantum channels and operations (2012, unpublished)
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Communicated by R. Seiringer
Supported in part by the Grants-in-Aid for Scientific Research, JSPS.
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Ogata, Y. A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization II. Commun. Math. Phys. 348, 897–957 (2016). https://doi.org/10.1007/s00220-016-2697-5
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DOI: https://doi.org/10.1007/s00220-016-2697-5