Abstract
We investigate the conventional tight-binding model of L π electrons on a ring-shaped molecule of L atoms with nearest-neighbor hopping. The hopping amplitudes t(w) depend on the atomic spacings w with an associated distortion energy V(w). A Hubbard-type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when L =4k +2 the minimum energy E occurs either for equal spacing or for alternating spacings (dimerization); nothing more chaotic can occur. In particular, this statement is true for the Peierls-Hubbard Hamiltonian, which is the case of linear t(w) and quadratic V(w), i.e., t(w)=t o−αw and V(w)=k(w −a)2,but our results hold for any choice of couplings or functions t (w) and V(w). When L =4k we prove that more chaotic minima can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L→∞. Our analysis suggests three interesting conjectures about how dimerization stabilizes for large systems. We also treat the spin-Peierls problem and prove that nothing more chaotic than dimerization occurs for L =4k +2 and L =4k.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Hückel, Z. Phys. 70, 204 (1931); 72, 310 (1931); 76, 628 (1932).
L. Salem, The Molecular Orbital Theory of Conjugated Systems (Benjamin, New York, 1966 ).
F. London, J. Phys. Radium Sér. VII, Tome VIII, 397 (1937).
H. Jones, Proc. R. Soc. London Ser. A 147, 396 (1934).
Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963).
R. Pariser and R. G. Parr, J. Chem. Phys. 21, 466 (1953).
J. A. Pople, Trans. Faraday Soc. 49, 1375 (1953).
E. H. Lieb, in Advances in Dynamical Systems and Quantum Physics,Proceedings of 1993 conference in honor of G. F. Dell'Antonio, edited by R. Figari (World Scientific, Singapore, in press); and in The Physics and Mathematical Physics of the Hubbard Model,Proceedings of 1993 NATO ASW, edited by D. K. Campbell and F. Guinea (Plenum, New York, in press). An updated version appears in Proceedings of the XIth International Congress of Mathematical Physics, Paris, 1994,edited by D. Iagolnitzer (Diderot/International, 1994).
W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B 22, 2099 (1980).
S. Kivelson and D. E. Heim, Phys. Rev. B 26, 4278 (1982).
D. Baeriswyl, D. K. Campbell, and S. Mazumdar, in Conjugated Conducting Polymers,edited by H. Kiess, Springer Series in Solid State Sciences Vol. 102 (Springer, New York, 1992), pp. 7-133.
D. Baeriswyl and E. Jeckelmann (unpublished).
R. E. Peieris, Quantum Theory of Solids ( Clarendon, Oxford, 1955 ), p. 108.
H. Fröhlich, Proc. R. Soc. London Ser. A 223, 296 (1954).
H. C. Longuet-Higgins and L. Salem, Proc. R. Soc. London Ser. A 251, 172 (1959).
H. Labhart, J. Chem. Phys. 27, 957 (1957).
Y. Ooshika, J. Phys. Soc. Jpn. 12, 1238 (1957).
T. Kennedy and E. H. Lieb, Phys. Rev. Lett. 59, 1309 (1987). We, take the opportunity to note three minor technical misstatements in this paper. (A) The paragraph after Eq. (6) is not correct for N=2 because T,3=t,t2+t314 and not t,í2 in this case. Thus z has to be replaced by 2z, but the rest of the argument works. Alternatively, one can compute the eigen-values of T2 explicitly since it reduces to a 2 X 2 matrix. (B) The uniqueness proof for N =2 needs strengthening. T 2 =(T 2 ) implies only that t,=t3 or t 2 =t 4 However, this case can be analyzed explicitly and uniqueness holds. (C) The statement in case 1 that TV(z)mTr(2yr+ztl)'n is an even function of z is correct only when N is even (because the sub-lattices are then themselves bipartite). However, in all cases W(z) is certainly concave and dW(z)/dz = fTr(2y2 +z11)-1/211, which is zero at z =0. Thus W(z) is decreasing for z 003E 0 and increasing for z 003C0, which is what is needed in case 1 and case 2.
Y. Imry, in Directions in Condensed Matter Physics,edited by G. Grinstein and G. Mazenko (World Scientific, Singapore, 1986), pp. 101-163.
P. W. Wiegmann, Physica C 153, 102 (1988).
E. H. Lieb, Hely. Phys. Acta 65, 247 (1992).
E. H. Lieb and M. Loss, Duke Math. J. 71, 337 (1993).
E. H. Lieb, Phys. Rev. Lett. 73, 2158 (1994).
Y. Ooshika, J. Phys. Soc. Jpn. 12, 1246 (1957).
P. J. Garratt, Aromaticity (McGraw-Hill, London, 1971).
P. W. Anderson, Phys. Rev. 115, 2 (1959).
D. B. Chesnut, J. Chem. Phys. 45, 4677 (1966).
P. Pincus, Solid State Commun. 9, 1971 (1971).
G. Beni and P. Pincus, J. Chem. Phys. 57, 3531 (1972).
M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).
Z. G. Soos, S. Kuwajima, and J. E. Mihalick, Phys. Rev. B 32, 3124 (1985).
M. Aizenman and B. Nachtergaele, Commun. Math. Phys. 164, 17 (1994).
lnteracting Electrons in Reduced Dimensions Vol. 213 of NATO Advanced Study Institute, Series B: Physics,edited by D. Baeriswyl and D. K. Campbell (Plenum, New York, 1989). 345. N. Dixit and S. Mazumdar, Phys. Rev. B 29, 1824 (1984). 35S. Kivelson, W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 58, 1899 (1987).
N. Dixit and S. Mazumdar, Phys. Rev. B 29, 1824 (1984).
S. Kivelson, W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 58, 1899 (1987).
Y. Meir, Y. Gefen, and O. Entin-Wohlman, Phys. Rev. Lett. 63, 798 (1989).
O. Entin-Wohlman, Y. Gefen, Y. Meir, and Y. Oreg, Phys. Rev. B 45, 11 890 (1992).
S. Fujimoto and N. Kawakami, Phys. Rev. B 48,.17406 (1993).
D. C. Mattis and W. D. Langer, Phys. Rev. Lett. 25, 376 (1970).
J. V. Pulé, A. Verbeure, and V. A. Zagrebnov, J. Stat. Phys. 76, 155 (1994).
K. R. Subbaswamy and M. Grabowski, Phys. Rev. B 24, 2168 (1981).
G. W."Hayden and E. J. Mele, Phys. Rev. B 24, 5484 (1986).
E. J. Mele and M. Rice, Phys. Rev. B 23, 5397 (1981).
K. C. Ung, S. Mazumdar, and D. Toussaint (unpublished).
S. Mazumdar, Phys. Rev. B 36, 7190 (1987).
S. Tang and J. E. Hirsch, Phys. Rev. B 37, 9546 (1988).
T. Kennedy and E. H. Lieb, Physica A 138, 320 (1986).
U. Brandt and R. Schmidt, Z. Phys. B 63, 45 (1986).
J. L. Lebowitz and N. Maoris, J. Stat. Phys. 76, 91 (1994).
E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989).
F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18, 335 (1978).
I Fröhlich, R. Israel, E. H. Lieb, and B. Simon, Commun. Math. Phys. 62, 1 (1978).
T. Kennedy, E. H. Lieb, and B. S. Shastry, J. Stat. Phys. 53, 1019 (1988).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H., Nachtergaele, B. (2004). Stability of the Peierls instability for ring-shaped molecules. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-662-06390-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06093-9
Online ISBN: 978-3-662-06390-3
eBook Packages: Springer Book Archive