Abstract
In many body physics, we are usually faced with the intractable problem of finding the ground state wavefunction of a thermo dynamic ally large system of interacting particles. Typically this problem is solved perturbatively. That is, we start with some simple noninteracting Hamiltonian and its ground state (e.g. the free Fermi gas) and perturb in the interactions (e.g. the Coulomb potential):
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References
We are indebted to C. L. Kane for suggesting the clever title of this article.
R. P. Feynman, Statistical Mechanics, (Benjamin, New York, 1972).
G. D. Mahan, Many Particle Physics, second edition (Plenum, New York, 1990) and references therein.
D. P. Arovas, A. Auerbach, and F. D. M. Haldane, Phys. Rev. Lett. 60, 531 (1988).
S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986); see also S. M. Girvin in Chapter 9 of Ref[28].
To fix the sing of, we take.
By “macroscopically distinct”, we mean that the overlap between the two states decreases exponentially with the size of the system.
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 1388 and 1399 (1969); J. Phys. C 3, 911 (1970); P. M. van den Broek, Phys. Lett. 77A, 261 (1980).
W. J. Caspers, K. M. Emmett, and W. Magnus, J. Phys. A 17, 2697 (1984).
H. Bethe, Z. Phys. 71, 205 (1931).
F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 (1988).
N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1144 (1966).
F. D. M. Haldane, Phys. Rev. B 25, 4295 (1982).
S. S. Shastry and B. Sutherland, Phys. Rev. Lett. 47, 964 (1981). See also Ref[9].
A handful of exact excited states are known. See C. K. Majumdar, K. Krishnan, and V. Mubayi, J. Phys. C 5, 2896 (1972) and also W. J. Caspers and W. Magnus, Phys. Lett. 88A, 103 (1982).
P. Kulish and E. Sklyanin, in Lecture Notes in Physics (Springer-Verlag, Berlin, 1982), Vol. 151, p. 61; P. Kulish, N. Yu. Reshetikhin and E. Sklyanin, Lett. Math. Phys. 5, 393 (1981); L. Takhtajan, Phys. Lett. 90A, 479 (1982); J. Babudjian, Nucl. Phys. B125, 317 (1983).
W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
One can say that creates a state transforming by the adjoint representation of SU(2).
F. D. M. Haldane, unpublished.
Ian Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987); Comm. Math. Phys. 115, 477 (1988).
D. P. Arovas, Phys. Lett. 137A, 431 (1989).
F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. Rev. Lett. 50, 1153 (1983).
J. Solyom, Phys. Rev. B 36, 8642 (1987).
T. Kennedy, J. Phys.: Cond. Matter 2, 5737 (1990).
R. Jastrow, Phys. Rev. 98, 1479 (1955).
R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) and also in Chapter 7 of Ref[28].
E. J. Neves and J. F. Perez, Phys. Lett. 114A, 331 (1986).
The Quantum Hall Effect 2nd ed., edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1990).
J. M. Caillol, D. Levesque, J. J. Weis, and J. P. Hansen, J. Stat. Phys. 28, 2 (1985); M. Baus and J. P. Hansen, Phys. Rep. 59, 1 (1980); B. Jancovici, Phys. Rev. Lett. 46, 386 (1981).
F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983) and also in Chapter 8 of Ref[28].
See also S. A. Trugman and S. Kivelson, Phys. Rev. B 31, 5280 (1985).
W. Kohn, Phys. Rev. 123, 1242 (1961).
E. H. Rezayi, private communications.
I. Affleck, Phys. Rev. Lett. 54, 966 (1985); I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988).
D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988).
N. Read and S. Sachdev, Nucl. Phys. B316, 609 (1989); Phys. Rev. Lett. 62, 1694 (1989); Phys. Rev. B 42, 4568 (1990).
N. Read and D. M. Newns, J. Phys. C 16, 3273 (1983); N. E. Bickers, p59, 845, 1987.
I. Affleck, D. P. Arovas, J. B. Marston, and D. A. Rabson, “SU(2n) Quantum Antifer-romagnets with Exact C-Breaking Ground States” (accepted for publication in Nucl. Phys. B).
B. Sutherland, J. Math. Phys. 12, 246 (1971); J. Math. Phys. 12, 251 (1971); Phys. Rev. A 5, 1372 (1972).
C. L. Kane, S. Kivelson, D. H. Lee, and S. C. Zhang, Phys. Rev. B 43, 3255 (1991).
D. H. Lee and M. P. A. Fisher, Phys. Rev. Lett. 63, 903 (1989); M. P. A. Fisher and D. H. Lee, Phys. Rev. B 39, 2756 (1989); X. G. Wen and A. Zee, Int. J. Mod. Phys. B 4, 437 (1990).
Min-Chul Cha, Matthew P. A. Fisher, S. M. Girvin, Mats Wallin, and A. Peter Young, Phys. Rev. B (October 1, 1991).
Nandini Trivedi and D. M. Ceperley, Phys. Rev. B 40, 2737 (1989); Phys. Rev. B 41, 4552 (1990).
L. Reatto and G. V. Chester, Phys. Rev. 155, 88 (1967).
C. L. Kane, private communication.
Phillipe Nozières and David Pines, The Theory of Quantum Liquids, II (Addison-Wesley, Reading, MA, 1989).
Petter Minnhagen, Rev. Mod. Phys. 59, 1001 (1987).
S. M. Girvin, in Chapter 10 of Ref[28].
S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. 58, 1252 (1987).
N. Read, Phys. Rev. Lett. 62, 86 (1989).
S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).
J. M. Kosterlitz and D. J. Thouless, J. Phys. C 5, L124 (1972).
V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 61, 1144 (1971) [Sov. Phys. JETP 34, 610 (1972)].
J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961); C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
A. H. MacDonald and S. M. Girvin, Phys. Rev. B 38, 6295 (1988).
B. I. Halperin, Helv. Phys. Acta 56, 75 (1983).
P. W. Anderson (private communication).
J. M. Leinaas and J. Myerheim, II Nuovo Cimento 37B, 1 (1977).
F. Wilczek, Phys. Rev. Lett. 48, 11144 (1982); ibid. 49, 957 (1982).
G. S. Canright and S. M. Girvin, Science 247, 1197 (1990).
D. P. Arovas, J. R. Schrieffer, F. Wilczek, and A. Zee, Nucl. Phys. B251, 117 (1985).
R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988); Science 242, 525 (1988).
S. M. Girvin, in Appendix I of Ref[28].
E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 61, 1985 (1988).
J. K. Jain, Phys. Rev. B 41, 7653 (1990).
K. Rommelse and M. den Nijs, Phys. Rev. Lett. 59, 2578 (1987); M. Den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).
S. M. Girvin and D. P. Arovas, Physica Scripta T27, 156 (1989).
Y. Hatsugai and M. Kohmoto, “Numerical Study of the Hidden Antiferromagnetic Order in the Haldane Phase”, ISSP preprint, 1991.
I. Affleck, Nucl. Phys. B265[FS15], 409 (1986).
B. Sutherland, Phys. Rev. B 12, 3795 (1975).
N. Read (private communication).
Shoucheng Zhang and Daniel P. Arovas, Phys. Rev. B 40, 2708 (1989).
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Arovas, D.P., Girvin, S.M. (1992). Exact Questions to Some Interesting Answers in Many Body Physics. In: Ainsworth, T.L., Campbell, C.E., Clements, B.E., Krotscheck, E. (eds) Recent Progress in Many-Body Theories. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3466-2_21
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DOI: https://doi.org/10.1007/978-1-4615-3466-2_21
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