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Springer eBooks

Exact Results of the One-Dimensional 1/r 2 Supersymmetric t-J Model without Translational Invariance

D. F. Wang, C. Gruber

Published Jan 1, 1994• 0 citations
Publisher
DOI
10.1007/978-1-4615-0585-3_15

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Dynamical properties of the one-dimensional supersymmetric t-J model

Yasuhiro Saiga, Shinji Watanabe, Mitsuhiro Arikawa +5 more

•Jan 1, 1999•0 citations
In this work, we continue the study of the supersymmetric t-J model with 1/r2{\mathit{r}}^{2}r2 hopping and exchange without translational invariance. A set of Jastrow eigenfunctions are obtained for the system, with eigenenergies explicitly calculated. The ground state of the t-J model is included in this set of wave functions. The spectrum of this t-J model consists of equidistant energy levels which are highly degenerate.
We consider the one-dimensional SU(N)-invariant t-J model, which consists of electrons with N spin components on a lattice with nearest-neighbor hopping t constrained by the excluded multiple occupancy of the sites and spin-exchange J between neighboring lattice sites. The model is integrable and has been diagonalized in terms of nested Bethe ansatze at the supersymmetric point t=J. The low-T specific heat is proportional to T. The γ-coefficient is extracted from the thermodynamic Bethe-ansatz equations and is expressed in terms of the spin wave velocities and the group velocity of the charges for arbitrary N, band filling, and splitting of the levels (magnetic and crystalline fields). Our results contain the following special cases: (i) For N=2 the traditional spin-1/2 supersymmetric t-J model, (ii) for exactly one electron per site the SU(N)-Heisenberg chain, and (iii) for N=4 the two-band supersymmetric t-J model with crystalline field splitting.
We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.