We establish exact results for coupled spin-1/2 chains for special values of the four-spin interaction
V and dimerization parameter $\ensuremath{\delta}$. The first exact result is at $\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2$. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}4$ and gives evidence that the system has a spontaneously dimerized ground state. Numerical diagonalization and bosonization techniques indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\ensuremath{\le}V<\ensuremath{-}2$.
We study the locations of the gapless points which occur for quantum spin chains of finite length (with a twisted boundary condition) at particular values of the nearest-neighbor dimerization, as a function of the spin
S and the number of sites. For strong dimerization and large values of
S, a tunneling calculation reproduces the same results as those obtained from more involved field theoretic methods using the nonlinear $\ensuremath{\sigma}$-model approach. A different analytical calculation of the matrix element between the two N\'eel states gives a set of gapless points; for strong dimerization, these differ significantly from the tunneling values. Finally, the exact diagonalization method for a finite number of sites yields a set of gapless points which are in good agreement with the N\'eel state calculations for all values of the dimerization, but the agreement with the tunneling values is not very good even for large
S. This raises questions about possible corrections to the tunneling results.
The thermodynamic properties of a one-dimensional model describing spin dynamics in the presence of a twofold orbital degeneracy are studied numerically using the transfer-matrix renormalization group. The model contains an integrable SU(4)-symmetric point and a gapless phase which is SU(4) invariant up to a rescaling of the velocities for spin and orbital degrees of freedom which allows detailed comparison of the numerical results with conformal field theory. We pay special attention to the correlation lengths which show an intriguing evolution with temperature. We find that the model shows an intrinsic tendency towards dimerization at finite temperature even if the ground state is not dimerized.
We study a class of spin-1/2 quantum antiferromagnetic chains using DMRG technique. The exchange interaction in these models decreases linearly as a function of the separation between the spins,
Jij=R−∣i−j∣ for
∣i−j∣≤R. For the separations beyond
R, the interaction is zero. The range parameter
R takes positive integer values. The models corresponding to all the odd values of
R are known to have the same exact doubly degenerate dimer ground state as for the Majumdar-Ghosh (MG) model. In fact, R=3 is the MG model. For even
R, the exact ground state is not known in general, except for R=2 (the Bethe ansatz solvable Heisenberg chain) and in the asymptotic limit of
R where the two MG dimer states again emerge as the exact ground state. In the present work, we numerically investigate the even-
R models whose ground state is not known analytically. In particular, for R=4, 6 and 8, we have computed a number of ground state properties. We find that, unlike R=2, the higher even-
R models are spin-gapped, and show strong dimer-dimer correlations of the MG type. Moreover, the spin-spin correlations decay very rapidly, albeit showing weak periodic revivals.
We study a one-dimensional spin-orbital model using both analytical and numerical methods. Renormalization group calculations are performed in the vicinity of a special integrable point in the phase diagram with
SU(4) symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is
SU(4) invariant at low energies apart from the presence of different velocities for spin and orbital degrees of freedom. The phase transition into a gapped dimerized phase is in a generalized Kosterlitz-Thouless universality class. The phase diagram of this model is sketched using the density matrix renormalization group technique.
We examine the ground state and the excitations of an one-dimensional Heisenberg spin 1/2 antiferromagnet with alternating dimers and four-spin plaquettes (dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact-numerical and perturbational results are presented. We find that the system is gapped for all parameter values. For the latter one we find an effective S=1 Haldane like behaviour in the limit of dominating dimer bonds. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system
CaV4O9.