Abstract
We present a review of different field theory techniques that have proved very useful in the study of quantum magnets in low dimensions We first review the application of the spin-wave analysis and non-linear σ-model techniques in one and two dimensional quantum antiferromagnets. We discuss in particular the emergence of Haldane’s conjecture for spin chains and ladders within this formalism. We also present a brief discussion on the non-linear σ-model description for the two-dimensional antiferromagnet in the square lattice. In a second part we review the method of abelian bosonization and its application to the study of the XXZ spin 1/2 chain and its generalizations, such as the dimerized chain. Non-abelian bosonization is used to describe both SU (2) symmetric chains with arbitrary spin S and 2 leg ladders, rederiving Haldane’s conjecture within this formalism. The inclusion of charge degrees of freedom leading to a Hubbard or a t—J model is also discussed. Finally, we apply the abelian bosonization approach to the study of N-leg ladders in amagnetic field, which leads to a further extension of Haldane’s conjecture.
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References
R. Shankar, Nuc. Phys. B330, 433 (1990).
E. Fradkin, Field Theories Of Condensed Matter Physics, Addison Wesley (1991).
M. Nakahara, Geometry, Topology and Physics, Bristol, UK: Hilger (1990).
I. Affleck, in Fields, Strings and Critical Phenomena, Les Houches, Session XLIX, eds. E. Brézin, J. Zinn-Justin (North-Holland, Amsterdam, 1988).
G. Sierra, Lectures Notes In Physics, Vol. 478, eds. G. Sierra and M.A. Martin-Delgado, Springer-Verlag (1997).
A.M. Polyakov. Gauge Fields and Strings, Harwood Academic Publishers (1987).
A.B. Zarnolodchikov, Al.B. Zamolodchikov, Nucl. Phys. B133, 525 (1978).
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996).
R. Shankar, N. Read, Nucl. Phys. B336, 457 (1990).
F.M.D. Haldane. Phys. Lett. A93, 464 (1983); Phys. Rev. Lett. 50. 1153 (1983); J Appl. Phys. 57, 3359 (1985).
T. Dombre, N. Read, Phys. Rev. B39, 6797 (1989)
F. Wilczek, A. Zee, Phys. Rev. Lett. 51, 2250(1983).
F.M.D. Haldane, Phys. Rev. Lett. 61, 1029 (1988).
A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, Bosonization and Strongly Correlated Electron Systems, Cambridge University Press, Cambridge (1998).
H.J. Schulz, Int. J. Mod. Phys. B5, 57 (1991); p. 533 in Proceedings of Les Houches Summer School LXI, eds. E. Akkermans, G. Montambaux, J. Pichard and J. Zinn-Justin (Elsevier. Amsterdam, 1995).
J. Voit, Rep. Prog. Phys. 58, 977 (1995).
D. Sénéchal, preprint cond-mat/9908262; S. Rao, D. Sen, preprint cond-mat/0005492.
D.C. Cabra, A. Honecker, P. Pujol, Phys. Rev. Lett. 79, 5126 (1997); Phys. Rev. B58, 6211 (1998).
See e.g. J. Von Dolft, H. Schoeller, Annalen der Physik 7, 225 (1998) and references therein.
M. Fowler, J. Phys. C: Solid St. Phys. 13, 1459 (1980).
S. Lukyanov, A. Zamolodchikov, Nucl. Phys. B493, 571 (1997).
N.M. Bogoliubov, V.G. Izergin, V.E. Korepin, Nucl. Phys. B275, 687 (1986): V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge (1993).
G.I. Dzhaparidze, A.A. Nersesvan, JETP Lett, 27, 334 (1978).
V.L. Pokrovsky, A.L. Talapov, Phys. Rev. Lett. 42, 65 (1979).
H.J. Schulz, Phys. Rev. B22, 5274 (1980).
J.M. Kosterlitz, J. Phys. C: Solid State Phys. 7, 1046 (1974).
S. Eggert, I. Affleck and, M. Takahashi, Phys. Rev. Lett. 73, 332 (1994).
E. Witten, Comm. Math. Phys. 92, 455 (1984).
V.G. Knizknik and A.B. Zamolodchikov, Nucl. Phys. B247, 83 (1984).
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer-Verlag New York (1997).
K. Totsuka, M. Suzuki, J. Phys.: Condensed Matter 7, 6079 (1995).
D.G. Shelton, A.A. Nersesyan, A.M. Tsvelik, Phys. Rev. B53, 8521 (1996).
E. Lieb and F.Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).
H. Frahm and V.E. Korepin, Phys. Rev. B42, 10553 (1990): Phys. Rev. B43, 5653 (1991).
F.H.L. Essler and H. Frahm, Phys. Rev. B60, 8540 (1999).
K. Penc and J. Sólyom Phys. Rev. B47, 6273 (1993).
D.C. Cabra, A. De Martino, A. Honecker, P. Pujol, P. Simon, Phys. Lett. A268, 418 (2000): Phys. Rev. B63, 094406 (2001).
F.H.L. Essler, V.E. Korepin, Phys. Rev. B46, 9147 (1992), and references therein.
I. Affleck, Nucl. Phys. B265 [FS15], 409 (1986).
C. Mudry and E. Fradkin, Phys. Rev. B50, 11409 (1994).
C. Itoi and H. Mukaida, J. Phys. A: Math. Gen. 27, 4695 (1994).
S.G. Naculich and H.J. Schnitzer, Nucl. Phys. B347, 687 (1990).
A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 62, 215 (1985).
D. Gepner, E. Witten, Nucl. Phys. 278, 493 (1986).
D. Gepner, Z. Qiu, Nucl. Phys. 285[FS19], 423 (1987).
V.A. Fateev, Int. Jour. Mod. Phys. A12, 2109 (1991).
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Cabra, D.C., Pujol, P. (2004). Field-theoretical methods in quantum magnetism. In: Schollwöck, U., Richter, J., Farnell, D.J.J., Bishop, R.F. (eds) Quantum Magnetism. Lecture Notes in Physics, vol 645. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0119596
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