Abstract
We use numerical methods to investigate the SU q(N) Perk–Schultz spin chain at the special quantum parameter value q = −e iπ/N. We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter q and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers n i of particles of different types (i = 0, 1, ..., N−1) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of q. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous SU q(N−1) vertex model that is involved in constructing the system of Bethe-ansatz equations of a “matrioshka” structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
REFERENCES
H. A. Bethe, Z. Phys., 71, 205–226 (1931).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, New York (1982); V. E. Korepin, I. G. Izergin, and N. M. Bogoliubov, Quantum Inverse Scattering Method, Correlation Functions, and Algebraic Bethe Ansatz, Cambridge Univ. Press, Cambridge (1993); F. H. L. Essler and V. E. Korepin, Exactly Solvable Models of Strongly Correlated Electrons, World Scientific, Singapore (1994); P. Schlottmann, Internat. J. Mod. Phys. B, 11, 355–667 (1997).
F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Ann. Phys., 182, 280–343 (1988); F. C. Alcaraz, M. N. Barber, M. T. Batchelor, R. J. Baxter, and G. R. W. Quispel, J. Phys. A, 20, 6397–6409 (1987).
R. J. Baxter, Adv. Stud. Pure. Math., 19, 95–116 (1989).
V. Fridkin, Yu. G. Stroganov, and D. Zagier, J. Phys. A, 33, L121–L125 (2000); V. Fridkin, Yu. G. Stroganov, and D. Zagier, J. Stat. Phys., 102, 781–794 (2001); nlin.SI/0010021 (2000); Yu. G. Stroganov, J. Phys. A, 34, L179–L186 (2001); cond-mat/0012035 (2000).
R. J. Baxter, Stud. Appl. Math., 50, 51–69 (1971).
A. V. Razumov and Yu. G. Stroganov, J. Phys. A, 34, 3185–3190 (2001); cond-mat/0012141 (2000); M. T. Batchelor, J. de Gier, and B. Nienhuis, J. Phys. A, 34, L265-L270 (2001); cond-mat/0101385 (2001); A. V. Razumov and Yu. G. Stroganov, J. Phys. A, 34, 5335–5340 (2001); cond-mat/0102247 (2001); “Combinatorial nature of ground state vector of O(1) loopm odel,” math.CO/0104216 (2001); P. A. Pearce, V. Rittenberg, and J. de Gier, “Critical Q=1 Potts model and Temperly–Lieb stochastic processes,” cond-mat/0108051 (2001); A. V. Razumov and Yu. G. Stroganov, “O(1) loopmo del with different boundary conditions and symmetry classes of alternating-sign matrices,” cond-mat/0108103 (2001); J. de Gier, M. T. Batchelor, B. Nienhuis, and S. Mitra, “The XXZ spin chain at ? = ?1/2: Bethe roots, symmetric functions, and determinants,” math-ph/0110011 (2001); Yu. G. Stroganov, Theor. Math. Phys., 129, 1596–1608 (2001); N. Kitanine, J. M. Maillet, N. A. Slavnov, and V. Terras, J. Phys. A, 35, L385–L391 (2002); hep-th/0201134 (2002); M. T. Batchelor, J. de Gier, and B. Nienhuis, Internat. J. Mod. Phys. B, 16, 1883–1890 (2002); math-ph/0204002 (2002); J. de Gier, B. Nienhuis, P. A. Pearce, and V. Rittenberg, Phys. Rev. E, 67, 016101–016104 (2003); cond-mat/0205467 (2002).
Yu. G. Stroganov, “The 8-vertex model with a special value of the crossing parameter and the related XYZ spin chain,” in: Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (S. Pakuliak and G. von Gehlen, eds.), Kluwer, Amsterdam (2001), p. 315–319.
F. C. Alcaraz and Yu. G. Stroganov, J. Phys. A, 35, 3805–3820 (2002); cond-mat/0201354 (2002); 35, 6767–6788 (2002); cond-mat/0205567 (2002); 36, 2381–2397 (2003); cond-mat/0212475 (2002).
J. H. H. Perk and C. L. Schultz, Phys. Lett. A, 84, 407–410 (1981); C. L. Schultz, Phys. A, 122, 71–88 (1983).
B. Sutherland, Phys. Rev. B, 12, 3795–3805 (1975).
N. Yu. Reshetikhin and P. B. Wiegmann, Phys. Lett. B, 189, 125–131 (1987); H. J. de Vega, Internat. J. Mod. Phys. A, 4, 2371–2463 (1989).
L. Mezincescu, R. I. Nepomechie, P. K. Towsend, and A. M. Tsvelick, Nucl. Phys. B, 406, 681–707 (1993).
H. J. de Vega and A. Gonzáles-Ruiz, Nucl. Phys. B, 417, 553–578 (1994).
Yu. G. Stroganov, “A new way to deal with Izergin–Korepin determinant at root of unity,” math-ph/0204042 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stroganov, Y.G. Quasifree States in Some One-Dimensional Quantum Spin Models. Theoretical and Mathematical Physics 139, 542–556 (2004). https://doi.org/10.1023/B:TAMP.0000022746.81620.65
Issue Date:
DOI: https://doi.org/10.1023/B:TAMP.0000022746.81620.65