Abstract
The Luttinger model was introduced to illustrate the possibility of a perturbative treatment of the singularity at the Fermi surface, already known to be “anomalous” from the results of the theory of Tomonaga, via an exactly soluble model. It became soon the subject of great interest also on the part of Mathematical Physics and a key to the investigations of the mathematical properties of Condensed Matter Physics. This paper reviews aspects of the above developments relevant for renormalization group methods by illustrating the conceptual development of the renormalization group approach to the ground state theory of the 1-dimensional spinless Fermi gas at small coupling.
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REFERENCES
G. Benfatto and G. Gallavotti, Perturbation theory of the Fermi surface in a quantum liquid. A general quasi particle formalism and one dimensional systems, J. Stat. Phys. 59:541–660 (1990); see also [BGPS94].
G. Benfatto and G. Gallavotti, Renormalization Group (Princeton University Press, 1995), pp. 1–143.
G. Benfatto, G. Gallavotti, and V. Mastropietro, Renormalization group and the Fermi surface in the Luttinger model, Phys. Rev. B 45:5468–5480 (1992).
G. Benfatto, G. Gallavotti, A. Procacci, and B. Scoppola, Beta function and Schwinger functions for many fermion systems in one dimension. Anomaly of the Fermi surface, Commun. Math. Phys. 160:93–172 (1994).
G. Benfatto and V. Mastropietro, Renormalization group, hidden symmetries and approximate Ward identities in the XYZ model, I, II, in print on Communications in Mathematical Physics, preprint 2000, in http://ipparco.roma1.infn.it.
G. Benfatto and V. Mastropietro, private communication.
F. Bonetto and V. Mastropietro, 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).
J. Feldman and E. Trubowitz, Perturbation theory for many fermion systems, Helvetica Physica Acta 63:157–260 (1990). And: The flow of an electron-positron system to the superconducting state, Helv. Phys. Acta 64:213–357 (1991).
G. Gallavotti, Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods, Rev. Modern Phys. 57:471–562 (1985).
G. Gentile and V. Mastropietro, Renormalization Group for One-Dimensional Fermions: A Review on Mathematical Results, preprint, http://ipparco.roma1.infn.it, in print in Physics reports.
M. Gomes and J. H. Lowenstein, Asymptotic scale invariance in a massive Thirring model, Nucl. Phys. B 45:252–266 (1972).
G. Gentile and B. Scoppola, Renormalization group and the ultraviolet problem in the Luttinger model, Commun. Math. Phys. 154:135–179 (1993); and G. Gentile, Thesis, 1991, unpublished.
F. D. M. Haldane, “Luttinger liquid theory” of one-dimensional quantum fluids: I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas, J. Phys. C 14:2585–2609 (1981).
M. Jimbo, T. Miwa, Y. Mori, and M. Sato, Density matrix of an impenetrable gas and the fifth Painlevétranscendent, Physica D 1:80–158 (1980).
W. Kohn and J. M. Luttinger, Ground state energy of a many fermion system, Phys. Rev. 118:41–45 (1960).
M. Kac and J. M. Luttinger, Bose–Einstein condensation in the presence of impurities I and II, J. Math. Phys. 14:1626–1630 (1973); and 15:183–186 (1974).
T. Kennedy, E. Lieb, and B. S. Shastry, The XY model has long range order for all spins and all dimensions greater than one, Phys. Rev. Lett. 61:2582–2584 (1988).
A. Lesniewski, Effective actions for the Yukawa2 quantum field theory, Commun. Math. Phys. 108:437–467 (1987).
E. Lieb and W. Lininger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130:1606–1615 (1963), reprinted in [ML66]. See also E. Lieb, Exact analysis of an interacting Bose gas. I. The excitation spectrum, Phys. Rev. 130:1616–1624 (1963).
J. M. Luttinger, Fermi surface and some simple equilibrium properties of a system of interacting fermions, Phys. Rev. 119:1153–1163 (1960).
J. M. Luttinger, An exactly soluble model of a many fermion system, J. Math. Phys. 4:1154–1162 (1963).
J. M. Luttinger and J. Ward, Ground state energy of a many fermion system. II, Phys. Rev. 118:1417–1427 (1960).
E. Lieb and F. Y. Wu, Absence of Mott transition in an exact solution of the short-range, one-band model in one-dimension, Phys. Rev. Lett. 20:1445–1448 (1968).
E. Lieb and J. Yngvason, The Ground State Energy of a Dilute Two-dimensional Bose Gas, preprint 00-63 in mp_arc@ma.utexas.ed, 2000.
V. Mastropietro, private communication.
D. Mattis, Band theory of magnetism in metals in context of exactly soluble model, Physics 1:183–193 (1964).
V. Mastropietro, Interacting soluble Fermi systems in one dimension, Nuovo Cimento B 1:304–312 (1994).
V. Mastropietro, Anomalous BCS equation for a Luttinger superconductor, Modern Phys. Lett. 13:585–597 (1999). And Anomalous superconductivity for coupled Luttinger liquids, in print on Reviews in Mathematical Physics, preprint 1999, in http://ipparco.roma1.infn.it.
V. Mastropietro, A renormalization group computation of the XYZ correlation functions, Lett. Math. Phys. 47:339–352 (1999).
V. Mastropietro, private communication.
W. Metzner and C. Di Castro, Conservation laws and correlation functions in the Luttinger liquids, Phys. Rev. B 47:16107–16123 (1993).
D. Mattis and E. Lieb, Exact solution of a many fermion system and its associated boson field, J. Math. Phys. 6:304–312 (1965) (reprinted in: [ML66]).
D. Mattis and E. Lieb, Mathematical Physics in One Dimension (Academic Press, New York, 1966).
R. Baxter, Exactly Solved Models (Academic Press, London, 1982).
R. Shankar, Renormalization group approach to interacting fermions, Rev. Modern Phys. 66:129-192 (1994).
B. Simon, The P(φ) 2 Euclidean (Quantum) Field Theory (Princeton University Press, 1974).
J. Sólyom, The Fermi gas model of one-dimensional conductors, Adv. in Phys. 28:201–303 (1979).
T. Spencer, A Mathematical Approach to Universality in Two Dimensions, preprint, 1999.
S. Tomonaga, Remarks on Bloch's method of sound waves applied to many fermion problems, Progr. Theoret. Phys. 5:349–374 (1950), (reprinted in [ML66]).
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Gallavotti, G. The Luttinger Model: Its Role in the RG-Theory of One Dimensional Many Body Fermi Systems. Journal of Statistical Physics 103, 459–483 (2001). https://doi.org/10.1023/A:1010381030262
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DOI: https://doi.org/10.1023/A:1010381030262