Quantum spin models with exact dimer ground states

Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground state configuration of all the members of the family on a periodic chain. The ground state is two-fold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh hamiltonian with two-fold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.Comment: 10 pages, 8 figures, revtex, to appear in Phys. Rev.

Apoptosis and proliferation in the trigeminal placode

The neurogenic trigeminal placode develops from the crescent-shaped panplacodal primordium which delineates the neural plate anteriorly. We show that, in Tupaia belangeri, the trigeminal placode is represented by a field of focal ectodermal thickenings which over time changes positions from as far rostral as the level of the forebrain to as far caudal as opposite rhombomere 3. Delamination proceeds rostrocaudally from the ectoderm adjacent to the rostral midbrain, and contributes neurons to the trigeminal ganglion as well as to the ciliary ganglion/oculomotor complex. Proliferative events are centered on the field prior to the peak of delamination. They are preceded, paralleled and, finally, outnumbered by apoptotic events which proceed rostrocaudally from non-delaminating to delaminating parts of the field. Apoptosis persists upon regression of the placode, thereby exhibiting a massive “wedge” of apoptotic cells which includes the postulated position of the “ventrolateral postoptic placode” (Lee et al. in Dev Biol 263:176–190, 2003), merges with groups of lens-associated apoptotic cells, and disappears upon lens detachment. In conjunction with earlier work (Washausen et al. in Dev Biol 278:86–102, 2005) our findings suggest that apoptosis contributes repeatedly to the disintegration of the panplacodal primordium, to the elimination of subsets of premigratory placodal neuroblasts, and to the regression of placodes

Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title; corrected minor error

Exact ground states for a class of one-dimensional frustrated quantum spin models

We have found the exact ground state for two frustrated quantum spin-1/2 models on a linear chain. The first model describes ferromagnet- antiferromagnet transition point. The singlet state at this point has double-spiral ordering. The second model is equivalent to special case of the spin-1/2 ladder. It has non-degenerate singlet ground state with exponentially decaying spin correlations and there is an energy gap. The exact ground state wave function of these models is presented in a special recurrent form and recurrence technics of expectation value calculations is developed.Comment: 16 pages, 3 figures, RevTe

An extended massless phase and the Haldane phase in a spin-1 isotropic antiferromagnetic chain

We study the phase transition of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland model by using the SU(3)_1 WZW model with certain marginal perturbations. The unstable RG trajectory by a marginally relevant perturbation generates a mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at ULS point becomes the BKT type. Our results support recent numerical studies by F\'ath and S\'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU(\nu)-symmetric and asymmetric models.Comment: 19 pages, RevTe

Scalar chiral ground states of spin ladders with four-spin exchanges

We show that scalar chiral order can be induced by four-spin exchanges in the two-leg spin ladder, using the spin-chirality duality transformation and matrix-product ansatz. Scalar-chiral-ordered states are found to be exact ground states in a family of spin ladder models. In this scalar chiral phase, there is a finite energy gap above the doubly degenerate ground states and a $Z_2 \times Z_2 \times Z_2$ symmetry is fully broken. It is also shown that the SU(4)-symmetric model, which is self-dual under the duality transformation, is on a multicritical point surrounded by the staggered dimer phase, the staggered scalar chiral phase, and the gapless phase.Comment: 8 pages, 2 figures, to appear in Phys. Rev.

The spectral gap for some spin chains with discrete symmetry breaking

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples.Comment: 48 pages, Plain TeX, BN26/Oct/9