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Published January 2015 | Submitted
Journal Article Open

Frustration Free Gapless Hamiltonians for Matrix Product States

Abstract

For every matrix product state (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state space (the so-called non-injective case), we construct another 'uncle' Hamiltonian which is also local and frustration free, has the same ground state space, but is gapless, and its spectrum is R^+ . The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the parent Hamiltonian has a unique ground state (the so-called injective case) we also build such uncle Hamiltonian with the same properties in the thermodynamic limit.

Additional Information

© 2014 Springer-Verlag Berlin Heidelberg. Received: 13 November 2012. Accepted: 5 August 2014. Published online: 6 November 2014. Communicated by A. Winter. We specially thank Bruno Nachtergaele for his help. This work has been partially funded by the Spanish Grants MTM2011-26912 and QUITEMAD, the European projects QUEVADIS and CHISTERA CQC, the Alexander von Humboldt Foundation, the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information, the NSF Grant No. PHY-0803371, and the ARO Grant No. W911NF-09-1-0442. We also acknowledge the hospitality of the Centro de Ciencias de Benasque Pedro Pascual and of the Perimeter Institute, where part of this work was carried out.

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August 20, 2023
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