Abstract
Systems of interacting non-relativistic fermions in d =1, as well as spin chains or interacting two dimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non-universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed.
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Anderson, P.W.: The theory of superconductivity in high T c cuprates. Princeton University Press, 1997
Affleck, I.: Field theory methods and quantum critical phenomena. In: Proc. Les Houches summer school on Critical phenomena, Random Systems, Gauge theories. Amsterdam: North Holland, 1984
Benfatto, G., Gallavotti, G.: Perturbation Theory of the Fermi Surface in a Quantum Liquid. A General Quasiparticle Formalism and One-Dimensional Systems. J. Stat. Phys. 59, 541–664 (1990)
Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta Functions and Schwinger Functions for a Many Fermions System in One Dimension. Commun. Math. Phys. 160, 93–171 (1994)
Benfatto, G., Mastropietro, V.: Renormalization Group, hidden symmetries and approximate Ward identities in the XYZ model. Rev. Math. Phys. 13, 1323–1435 (2001)
Benfatto, G., Mastropietro, V.: On the density-density critical indices in interacting Fermi systems. Commun. Math. Phys. 231, 97–134 (2002)
Benfatto, G., Mastropietro, V.: Ward identities and vanishing of the Beta function for d=1 interacting Fermi systems. J. Stat. Phys. 115, 143–184 (2004)
Bonetto, F., Mastropietro, V.: Beta Function and Anomaly of the Fermi Surface for a d=1 System of Interacting Fermions in a Periodic Potential. Commun. Math. Phys. 172, 57–93 (1995)
Dzyaloshinky, I.E., Larkin, A.I.: Correlation functions for a one-dimensional Fermi system with long-range interaction (Tomonaga model). Soviet Phys. JETP 38, 202–208 (1974)
Giamarchi, T.: Quantum Physics in one dimension. International Series of Monographs on Physics. 121, Oxford: Clarendon Press, 2004
Gentile, G., Mastropietro, V.: Renormalization Group for fermions: a review of mathematical results. Phys. Rep. 352, 273–437 (2001)
Hurd, T.R.: Soft breaking of Gauge invariance in regularized Quantum Electrodynamics. Commun. Math. Phys. 125, 515–526 (1989)
Lesniewski, A.: Effective action for the Yukawa 2 quantum field Theory. Commun. Math. Phys. 108, 437–467 (1987)
Mastropietro, V.: Small denominators and anomalous behaviour in the incommensurate Hubbard-Holstein model. Commun. Math. Phys. 201, 81–115 (1999)
Mastropietro, V.: Coupled Ising models with quartic interaction at criticality. Commun. Math. Phys. 244, 595–642 (2004)
Metzner, W., Di Castro, C.: Conservation laws and correlation functions in the Luttinger liquid. Phys. Rev. B 47, 16107 (1993)
Mattis, D., Lieb, E.: Exact solution of a many fermion system and its associated boson field. J. Math. Phys. 6, 304–312 (1965)
Solyom, J.: The Fermi gas model of one-dimensional conductors. Adv. Phys. 28, 201–303 (1979)
Thirring, W.: a soluble relativistic field theory. Ann. Phys. 3, 91–112 (1958)
Zinn-Justin, J.: Quantum field theory and critical phenomena. Oxford: Oxford publications, 1989
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Communicated by G. Gallavotti
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Benfatto, G., Mastropietro, V. Ward Identities and Chiral Anomaly in the Luttinger Liquid. Commun. Math. Phys. 258, 609–655 (2005). https://doi.org/10.1007/s00220-005-1364-z
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DOI: https://doi.org/10.1007/s00220-005-1364-z