Publication: Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism
Loading...
Official URL
Full text at PDC
Publication Date
2012-12
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Physical Society
Citations
Exportar
Abstract
We study Hamiltonians which have Kitaev's toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of projected entangled pair states, and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians."
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] A. Kitaev, Ann. Phys. (Amsterdam) 303, 2 (2003).
[2] S. Bravyi, M. Hastings, and S. Michalakis, J. Math. Phys. (N.Y.) 51, 093512 (2010).
[3] S. Bravyi and M. B. Hastings, arXiv:1001.4363.
[4] F. Verstraete and J. I. Cirac, Phys. Rev. A 70, 060302 (2004); F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066.
[5] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006).
[6] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun.Math. Phys. 144, 443 (1992).
[7] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007).
[8] M. B. Hastings, Phys. Rev. B 76, 035114 (2007).
[9] C.V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. A 81, 052338 (2010).
[10] N. Schuch, I. Cirac, and D. Pérez-García, Ann. Phys. (Amsterdam) 325, 2153 (2010).
[11] M.A. Levin and X.-G.Wen, Phys. Rev. B 71, 045110 (2005).
[12] A. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988).
[13] S. Michalakis and J. Pytel, arXiv:1109.1588.
[14] X. Chen, B. Zeng, Z. C. Gu, I. L. Chuang, and X. G. Wen, Phys. Rev. B 82, 165119 (2010).
[15] M.Levin and X.-G.Wen, Phys.Rev.Lett. 96, 110405 (2006).
[16] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
[17] The parent Hamiltonian is usually obtained by considering local projectors on just two sites, but we need three sites for our construction in this specific case. A discussion on the need of either two or three sites and its relationship with bond and physical dimensions can be found in Ref. [18].
[18] C. Fernández-González, N. Schuch, M. M. Wolf, J. I. Cirac, and D. Pérez-García, arXiv:1210.6613.
[19] For periodic boundary conditions, domain walls come in pairs, i.e., for every │0 + 1> (‘‘up’’) domain wall there is a │1 + 0> (‘‘down’’) domain wall. Since both domain walls are in a momentum eigenstate, there is a nonzero probability that they are at adjacent sites, leading to a configuration │010> which is penalized by the Hamiltonian.
[20] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.109.260401 for a detailed proof of the structure of the kernel of the uncle Hamiltonian for the toric code (Part A), and the proof that the spectrum of the uncle Hamiltonian for the toric code is[0, ∞) (Part B).