Abstract
The familiar unrestricted Hartree-Fock variational principles is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn, provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.
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References
H. Araki, On quasifree states of CAR and Bogoliubov automorphisms,Publ. RIMS Kyoto 6: 385–442 (1970/71).
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity,Phys. Rev. 108:1175 (1957).
V. Bach, E. H. Lieb, M. Loss, and J. P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory,Phys. Rev. Lett. 72:2981–2983 (1994).
N. N. Bogoliubov, V. V. Tolmachev, and D. V. Shirkov,A New Method in the Theory of Superconductivity (Consultants Bureau, New York, 1959), Appendix 2.
J.-P. Blaizot and G. Ripka,Quantum Theory of Finite Systems (MIT Press, Cambridge, Massachusetts, 1986).
V. Bach, Error bound for the Hartree-Fock energy of atoms and molecules,Commun. Math. Phys. 147:527–548 (1992).
M. Cryot, Theory of Mott transition: Application to transient metal oxides,J. Phys. (Paris)33:125–134 (1972).
P. G. de Gennes,Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
E. Dagatto, Y. Fand, A. E. Ruckenstein, and S. Schmitt-Rink, Holes in the infiniteU Hubbard model. Instability of the Nagaoka state,Phys. Rev. B 40:7406–7409 (1989).
K. Dichtel, R. H. Jellito, and H. Koppe, The ground state of the neutral Hubbard model,Z. Physik 246:248–260 (1971); Thermodynamics of the Hubbard model,Z. Physik 251:173–184 (1972).
B. Doucot and X. G. Wen, Instability of the Nagaoka state with more than one hole,Phys. Rev. B 40:2719–2722 (1989).
E. Fradkin,Field Theories of Condensed Matter Systems (Addison-Wesley, Reading, Massachusetts, 1991).
M. Gaudin, Une démonstration simplifiée du théorème de Wick en méchanique statistique,Nucl. Phys. 15:89–91 (1960).
M. C. Gutzwiller, The effect of correlation on the ferromagnetism of transition metals,Phys. Rev. Lett. 10:159–162 (1963).
J. Hubbard, Electron correlations in narrow energy bands,Proc. R. Soc. Lond. A 276:238–257 (1963).
J. Kanamori, Electron correlation and ferromagnetism of transition metals,Prog. Theor. Phys. 30:275–289 (1963).
T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long range order,Physica 138A:320–358 (1986).
E. H. Lieb, Variational principle for many-fermion systems,Phys. Rev. Lett. 46:457–459 (1981); Errata47:69 (1981).
E. H. Lieb, Two theorems on the Hubbard model,Phys. Rev. Lett. 62:1201–1204 (1989).
E. H. Lieb, The Hubbard model: Some rigorous results and open problems, inAdvances in Dynamical Systems and Quantum Physics, V. Figariet al., eds. (World Scientific, Singapore, in press).
E. H. Lieb, Thomas-Fermi and Hartree-Fock theory, inProceedings International Congress Mathematicians (Canadian Mathematical Society, 1975), pp. 383–386.
E. H. Lieb and M. Loss, Fluxes, Laplacians and Kasteleyn's theorem,Duke Math. J. 71:337–363 (1993).
E. H. Lieb, M. Loss, and R. J. McCann, Uniform density theorem for the Hubbard model,J. Math. Phys. 34:891–898 (1993).
E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,Commun. Math. Phys. 53:185–194 (1977).
Y. Nagaoka, Ferromagnetism in a narrow, almost half-filled S-band,Phys. Rev. 147:392–405 (1966).
D.R. Penn, Stability theory of the magnetic phases for a simple model of the transition metals,Phys. Rev. 142:350–365 (1966).
A. Sütõ, Absence of highest spin ground states in the Hubbard model,Commun. Math. Phys. 140:43–62 (1991).
B. S. Shastry, H. R. Krishnamurthy, and P. W. Anderson, Instability of the Nagaoka ferromagnetic state of theU=∞ Hubbard model,Phys. Rev. B. 41:2375–2379 (1990).
B. Tóth, Failure of saturated ferromagnetism for the Hubbard model with two holes,Lett. Math. Phys. 22:321–333 (1991).
D. J. Thouless, Exchange in solid3He and the Heisenberg Hamiltonian,Proc. Phys. Soc. (London)86:893–904 (1965).
H. Tasaki, Extension of Nagaoka's theorem on the largeU Hubbard model,Phys. Rev. B 40:9192–9193 (1989).
W. Thirring,A Course in Mathematical Physics, Vol. 4 (Springer, Vienna, 1980), p. 48.
J. G. Valatin, Comments on the theory of superconductivity,Nuovo Cimento [X] 7:843–857 (1958).
C. N. Yang and S. C. Zhang,SO 4 symmetry in a Hubbard model,Mod. Phys. Lett. B 4:759–766 (1990).
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Dedicated to Philippe Choquard on his 65th birthday.
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Bach, V., Lieb, E.H. & Solovej, J.P. Generalized Hartree-Fock theory and the Hubbard model. J Stat Phys 76, 3–89 (1994). https://doi.org/10.1007/BF02188656
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DOI: https://doi.org/10.1007/BF02188656