Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

Hyperscaling inequalities for percolation

A set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved. One of the consequences of the inequalities is the lower boundd _C6 for the upper critical dimension. The proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as Ising ferromagnets.Supported by NSF Grant PHY-85-15288-A01     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

A rigorous control of logarithmic corrections in four-dimensional ϕ4 spin systems

Using Gawedzki and Kupiainen's rigorous block spin transformation method, we study critical phenomena in ⁴ spin systems in four dimensions. In Part I of this work we investigate in detail the renormalization group trajectory of the system not exactly at the critical point.     

Enzymatic Synthesis and Polymerization of Cyclic Trimethylene Carbonate Monomer with/without Methyl Substituent

The direct enzymatic synthesis of a cyclic trimethylene carbonate (1,3‐dioxane‐2‐one) monomer with/without a methyl substituent was carried out using dimethyl or diethyl carbonate and 1,3‐diol with the objective of producing aliphatic poly(trimethylene carbonate), a typical biodegradable synthetic plastic. The lipase‐catalyzed condensation of dimethyl or diethyl carbonate with aliphatic 1,3‐diols using immobilized Candida antarctica lipase (lipase CA) in an organic solvent at 70 °C afforded the corresponding methyl‐substituted and unsubstituted cyclic trimethylene carbonates. The cyclic trimethylene carbonates obtained by the reaction of dimethyl or diethyl carbonates with 1,3‐propanediol and 2‐methyl‐1,3‐propanediol were polymerized by lipase to produce the corresponding polycarbonates.

Total TMC yield as a function of the reaction time.     

Measurement theory in the Lax-Phillips formalism

It is shown that the application of the Lax-Phillips scattering theory to quantum mechanics provides a natural framework for the realization of the ideas of the Many-Hilbert-Space theory of Machida and Namiki to describe the development of decoherence in the process of measurement. We show that if the quantum mechanical evolution is pointwise in time, then decoherence occurs only if the Hamiltonian is time-dependent. If the evolution is not pointwise in time (as in Liouville space), then the decoherence may occur even for closed systems. These conclusions apply as well to the general problem of mixing of states.