Abstract
This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by Yarotsky that develops a perturbation theory for relatively bounded quantum perturbations of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltonian of a Matrix product state into a classical Hamiltonian plus some perturbation. We can thus extend Yarotsky’s results to provide a perturbation theory for parent Hamiltonians of Matrix product states and recover some of the findings of the independent contributions (Cirac et al in Phys Rev B 8(11):115108, 2013) and (Michalakis and Pytel in Comm Math Phys 322(2):277–302, 2013).
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Affleck, I., Kennedy, T., Lieb, E., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59(7), 799–802 (1987)
Albanese, C.: Unitary dressing transformations and exponential decay below the threshold for quantum spin systems. Comm. Math. Phys. 134(1–27), 237–272 (1990)
Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics. Springer, Berlin (1996)
Bravyi, S., Hastings, M., Michalakis, S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)
Cirac, J., Michalakis, S., Péres-García, D., Schuch, N.: Robustness in projected entangled pair states. Phys. Rev. B 8(11), 115108 (2013). arXiv:1306.4003
Fannes, M., Nachtergaele, B., Werner, R.: Finitely correlated states on quantum spin chains. Comm. Math. Phys. 144(3), 443–490 (1992)
Kennedy, T., Tasaki, H.: Hidden symmetry breaking in the Haldane phase S=1 quantum spin chains. Comm. Math. Phys. 147, 431–484 (1992)
Kennedy, T., Tasaki, H.: Hidden \(Z_2\times Z_2\) symmetry breaking in Haldane gap antiferromagnets. Phys. Rev. B 45, 304 (1992)
Kretschmann, D., Schlingemann, D., Wolf, M., Werner, R.: The information-disturbance tradeoff and the continuity of Stinespring’s theorem. IEEE Trans. Inf. Theory 54(4), 1708–1717 (2006)
Michalakis, S., Pytel, J.: Stability of frustration-free Hamiltonians. Comm. Math. Phys. 322(2), 277–302 (2013)
Nachtergaele, B.: The spectral gap for some quantum spin chains with discrete symmetry breaking. Comm. Math. Phys. 175, 565–606 (1996)
Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
Pérez-García, D., Verstaete, F., Wolf, M., Cirac, J.: Matrix product state representations. Quantum Inf. Comput. 7, 401–430 (2007)
Sanz, M., Péres-García, D., Wolf, M., Cirac, J.I: A quantum version of Wielandt’s inequality. IEEE Trans. on Inf. Theory, 56(9), 4668–4673 (2010)
Schuch, N., Péres-García, D., Cirac, J.I.: Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B, 84(16), 165139 (2011)
Spitzer, W., Starr, S.: Improved bounds on the spectral gap above frustration free ground states of quantum spin chains. Lett. Math. Phys. 63, 165–177 (2002)
Szehr, O., Reeb, D., Wolf, M.: Spectral convergence bounds for classical and quantum Markov processes. Comm. Math. Phys. 333(2), 565–595. doi:10.1007/s00220-014-2188-5
Szehr, O., Wolf, M.: Topology of irreducible maps and the classification of 1D quantum phases. Forthcoming 2015
Verstraete, F., Cirac, J.I., Latorre, J.I., Rico, E., Wolf, M.M.: Renormalization-group transformations on quantum states. Phys. Rev. Lett. 94, 140601 (2005)
Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)
Watrous, J.: Notes on Super-Operator Norms Induced by Schatten Norms (2004). arXiv:0411077v1
White, S.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992)
Wilson, K.: The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773–840 (1975)
Yarotsky, D.: Perturbations of ground states in weakly interacting quantum spin systems. J. Math. Phys. 45, 2134–2152 (2004)
Yarotsky, D.A.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Comm. Math. Phys. 261(3), 799–819 (2004)
Yarotsky, D.: Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems. J. Stat. Phys. 118, 119–144 (2005)
Acknowledgments
We acknowledge financial support from the QCCC programme of the Elite Network of Bavaria, the CHIST-ERA/BMBF Project CQC and the Alfried Krupp von Bohlen und Halbach–Stiftung.
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Szehr, O., Wolf, M.M. Perturbation Theory for Parent Hamiltonians of Matrix Product States. J Stat Phys 159, 752–771 (2015). https://doi.org/10.1007/s10955-015-1204-2
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DOI: https://doi.org/10.1007/s10955-015-1204-2