Abstract
This note provides a C *-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C *-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models.
Article PDF
Similar content being viewed by others
References
Araki H., Moriya H.: Equilibrium statistical mechanics of fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003)
Bratteli O., Robinson D.W.: Unbounded derivations of C *-algebras. Commun. Math. Phys. 42, 253–268 (1975)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1, 2nd edn. Springer, New York (1987)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 2, 2nd edn. Springer, New York (1997)
Buchholz D.: On the implementation of supersymmetry. Lect. Notes Phys. 539, 211–220 (2000)
Buchholz D., Grundling H.: Algebraic supersymmetry: a case study. Commun. Math. Phys. 272, 699–750 (2007)
Cooper F., Khare A., Sukhatme U.: Supersymmetry in Quantum Mechanics. World Scientific Pub Co Inc, River Edge (2002)
Fendley P., Schoutens K., de Boer J.: Lattice models with N = 2 supersymmetry. Phys. Rev. Lett. 90, 120402 (2003)
Fendley P., Halverson J., Huijse L., Schoutens K.: Charge frustration and quantum criticality for strongly correlated fermions. Phys. Rev. Lett. 101, 146406 (2008)
Haag R.: Local Quantum Physics, 2nd edn. Springer, Berlin (1996)
Haag R., Łopuszański J.T., Sohnius M.: All possible generators of supersymmetries of the S matrix. Nucl. Phys. B 88, 257–274 (1975)
Hagendorf C.: Spin chains with dynamical lattice supersymmetry. J. Stat. Phys. 150, 609–657 (2013)
Jaffe A., Lesniewski A., Osterwalder K.: Quantum K-theory. I. The Chern character. Commun. Math. Phys. 118, 1–14 (1988)
Jaffe A., Lesniewski A., Osterwalder K.: On super-KMS functionals and entire cyclic cohomology. K-Theory 2, 675–682 (1988)
Jaffe A., Lesniewski A., Wisniowski M.: Deformations of super-KMS functional. Commun. Math. Phys. 121, 527–540 (1989)
Junker G.: Supersymmetric Methods in Quantum and Statistical Physics. Springer, Berlin (1996)
Kastler D.: Cyclic cocycles from graded KMS functionals. Commun. Math. Phys. 121, 345–350 (1989)
Manin Y.I., Gelfand S.I.: Methods of Homological Algebra. Springer, New York (2003)
Moriya, H.: On quasi-free dynamics on the resolvent algebra. Preprint. arXiv:1405.4462
Nicolai H.: Supersymmetry and spin systems. J. Phys. A: Math. Gen. 9, 1497–1505 (1976)
Robinson D.W.: Statistical mechanics of quantum spin systems. II. Commun. Math. Phys. 7, 337–348 (1968)
Strocchi F.: Symmetry Breaking Lecture. Notes in Physics, vol. 732. Springer, New York (2008)
van Eerten H.: Extensive ground state entropy in supersymmetric lattice models. J. Math. Phys. 46(12), 123302 (2005)
Witten E.: Dynamical breaking of supersymmetry. Nucl. Phys. B185, 513–554 (1981)
Witten E.: Constraints of supersymmetry breaking. Nucl. Phys. B 202, 253–316 (1982)
Weinberg S.: The Quantum Theory of Fields III. Cambridge University Press, Cambridge (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vieri Mastropietro.
Rights and permissions
About this article
Cite this article
Moriya, H. On Supersymmetric Fermion Lattice Systems. Ann. Henri Poincaré 17, 2199–2236 (2016). https://doi.org/10.1007/s00023-016-0461-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0461-1