Abstract
We consider systems of weakly interacting fermions on a lattice. The corresponding free fermionic system is assumed to have a ground state separated by a gap from the rest of the spectrum. We prove that, if both the interaction and the free Hamiltonian are sums of sufficiently rapidly decaying terms, and if the interaction is sufficiently weak, then the interacting system has a spectral gap as well, uniformly in the lattice size. Our approach relies on convergent fermionic perturbation theory, thus providing an alternative method to the one used recently by Hastings (The stability of free Fermi hamiltonians, arXiv:1706.02270, 2017), and extending the result to include non-selfadjoint interaction terms.
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Aza, N.J.B., Bru, J.B., de Siqueira Pedra, W., Müssnichh, L.C.P.A.M.: Large deviations in weakly interacting fermions I—generating functions as Gaussian Berezin integrals and bounds on large Pfaffians. preprint pdfs.semanticscholar.org (2017)
Bachmann S., Michalakis S., Nachtergaele S., Sims R.: Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309, 835–871 (2012)
Bachmann S., Bols A., De Roeck W., Fraas M.: Quantization of conductance in gapped interacting systems. Ann. Henri Poincaré 19, 695 (2018)
Borgs C., Koteckỳ R., Ueltschi D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409–446 (1996)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States Models in Quantum Statistical Mechanics. Springer-Verlag, Heidelberg (1996)
Bravyi S., Hastings M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609–627 (2011)
Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)
Bru J.-B., Pedra W.: Universal bounds for large determinants fromnon-commutative Hölder inequalities in fermionic constructive quantum field theory. Math. Models Methods Appl. Sci. 27, 1963 (2017)
Brydges, D.C.: A short course on cluster expansions. In: Osterwalder, K., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories (Les Houches 1984) (1986)
Datta N., Fernández R., Fröhlich J.: Low-temperature phase diagrams of quantum lattice systems I Stability for quantum perturbations of classical systems with finitely-many ground states. J. Stat. Phys. 84, 455–534 (1996)
Gil, M.I.: Operator Functions and Localization of Spectra. Springer, (2003)
Giuliani A., Mastropietro V., Porta M.: Universality of the Hall conductivity in interacting electron systems. Commun. Math. Phys. 349, 1107–1161 (2016)
Hastings, M.B.: The stability of free Fermi hamiltonians (2017). arXiv preprint arXiv:1706.02270
Hastings M.B., Michalakis S.: Quantization of Hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015)
Hastings M.B., Wen X.-G.: Quasiadiabatic continuation of quantum states: the stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005)
Hofstadter D.R.: Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239 (1976)
Katsura H., Schuricht D., Takahashi M.: Exact ground states and topological order in interacting Kitaev chains. Phys. Rev. B 92, 115137 (2015)
Mariën M., Audenaert K.M.R., Van Acoleyen K., Verstraete F.: Entanglement rates and the stability of the area law for the entanglement entropy. Commun. Math. Phys. 346, 35–73 (2016)
Michalakis S., Zwolak J.P.: Stability of frustration-free hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)
Monaco, D., Teufel, S.: Adiabatic currents for interacting electrons on a lattice (2017). arXiv preprint arXiv:1707.01852
Nachtergaele B.: The spectral gap for some spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565–606 (1996)
Nachtergaele, B., Sims, R., Young, A.:Lieb––Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems (2017). arXiv preprint arXiv:1705.08553
PedraW.A. de S., Salmhofer M.: Determinant bounds and the Matsubara UV problem of many-fermion systems. Commun. Math. Phys. 282, 797–818 (2008)
Salmhofer M.: Renormalization: An Introduction. Springer Verlag, Heidelberg (1998)
Salmhofer M.: Clustering of fermionic truncated expectation values via functional integration. J. Stat. Phys. 134, 941–952 (2009)
Salmhofer M., Wieczerkowski C.: Positivity and convergence in fermionic quantum field theory. J. Stat. Phys. 99, 557–586 (2000)
Yarotsky D.A.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Commun. Math. Phys. 261, 799–819 (2006)
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Communicated by H.-T. Yau
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De Roeck, W., Salmhofer, M. Persistence of Exponential Decay and Spectral Gaps for Interacting Fermions. Commun. Math. Phys. 365, 773–796 (2019). https://doi.org/10.1007/s00220-018-3211-z
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DOI: https://doi.org/10.1007/s00220-018-3211-z