Finding Exponential Product Formulas of Higher Orders

In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.Comment: 22 pages, 9 figures. To be published in the conference proceedings ''Quantum Annealing and Other Optimization Methods," eds. B.K.Chakrabarti and A.Das (Springer, Heidelberg

Long-distance final-state interactions and J/psi decay

To understand the short-distance vs long-distance final-state interactions, we have performed a detailed amplitude analysis for the two-body decay, J/psi into vector and pseudoscalar mesons. The current data favor a large relative phase nearly 90 degrees between the three-gluon and one-photon decay amplitudes. The source of this phase is apparently in the long-distance final-state interaction. Nothing anomalous is found in the magnitudes of the three-gluon and one-photon amplitudes. We discuss implications of this large relative phase in the weak decay of heavy particles.Comment: 11 pages, RevTe

General Formulation of Quantum Analysis

A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the G\^ateaux derivative or commutators. This yields a unified formulation of quantum analysis, namely the invariance of quantum derivatives, which are expressed by multiple integrals of ordinary higher derivatives with hyperoperator variables. Multivariate quantum analysis is also formulated in the present unified scheme by introducing a partial inner derivation and a rearrangement formula. Operator Taylor expansion formulas are also given by introducing the two hyperoperators $\delta_{A \to B} \equiv -\delta_A^{-1} \delta_B$ and $d_{A \to B} \equiv \delta_{(-\delta_A^{-1}B) ; A}$ with the inner derivation $\delta_A : Q \mapsto [A,Q] \equiv AQ-QA$. Physically the present noncommutative derivatives express quantum fluctuations and responses.Comment: Latex file, 29 pages, no figur

Analytic study of disoriented chiral condensate

Evolution of disoriented chiral condenstates is studied with the classical sigma model in 3+1 dimensions. By smoothly connecting a chiral symmetric solution of the formation period to a solution of the decay period, we obtain a complete spacetime evolution of the pion field for a simple and physically interesting source. The formation process is discussed quantitatively from the viewpoint of the axial-vector isospin conservation.Comment: Latex with no figure, 18 pages. Full postscript available from http://theor1.lbl.gov/www/theorygroup/papers/38931.p

Final state interaction in heavy hadron decay

I present a critical account of the final-state interaction (FSI) in two-body B decays from viewpoint of the hadron picture. I emphasize that the phase and the magnitude of decay amplitude are related to each other by a dispersion relation. In a model phase of FSI motivated by experiment, I illustrate how much the magnitude of amplitude can deviate from its factorization value by the FSI.Comment: 8 pages in sprocl.tex with 4 eps figures. A talk presented at the Third International Conference on B Physics and CP Violation, (Taipei, December 1999

Aging dynamics of ferromagnetic and reentrant spin glass phases in stage-2 Cu0.80_{0.80}C0.20_{0.20}Cl2_{2} graphite intercalation compound

Aging dynamics of a reentrant ferromagnet stage-2 Cu$_{0.8}$Co$_{0.2}$Cl$_{2}$ graphite intercalation compound has been studied using DC magnetic susceptibility. This compound undergoes successive transitions at the transition temperatures $T_{c}$ ($\approx 8.7$ K) and $T_{RSG}$ ($\approx 3.3$ K). The relaxation rate $S_{ZFC}(t)$ exhibits a characteristic peak at $t_{cr}$ below $T_{c}$. The peak time $t_{cr}$ as a function of temperature $T$ shows a local maximum around 5.5 K, reflecting a frustrated nature of the ferromagnetic phase. It drastically increases with decreasing temperature below $T_{RSG}$. The spin configuration imprinted at the stop and wait process at a stop temperature $T_{s}$ ($<T_{c}$) during the field-cooled aging protocol, becomes frozen on further cooling. On reheating, the memory of the aging at $T_{s}$ is retrieved as an anomaly of the thermoremnant magnetization at $T_{s}$. These results indicate the occurrence of the aging phenomena in the ferromagnetic phase ($T_{RSG}<T<T_{c}$) as well as in the reentrant spin glass phase ($T<T_{RSG}$).Comment: 9 pages, 9 figures; submitted to Physical Review

Quantum phase transitions in the sub-ohmic spin-boson model: Failure of the quantum-classical mapping

The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that such a mapping fails for the sub-ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) ~ omega^s. Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to have an upper-critical dimension at s = 1/2, with mean-field transition behavior controlled by a non-interacting fixed point for 0 < s < 1/2. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.Comment: 4 pages, 3 figs; (v2) discussion extended; (v3) marginal changes, final version as published; (v4) added erratum pointing out that main conclusions were incorrect due to subtle failures of the NR

The Free Energy and the Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

A nonlinear susceptibilities (the third derivative of a magnetization $m_S$ by a magnetic field $h$ ) of the $S$=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures $T.$ In both chains the nonlinear susceptibilities diverge as $T^{-6}$ and a linear susceptibilities diverge as $T^{-2}.$ The arbitrary spin $S$ Heisenberg ferromagnet $[$ ${\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}$ $(J>0),$ $]$ has a scaling relation between $m_S,$ $h$ and $T:$ $m_S(T,h) = F( S^2 Jh/T^2).$ The scaling function $F(x)$=(2$x$/3)-(44$x^{3}$/135) + O($x^{5}$) is common to all values of spin $S.$Comment: 16 pages (revtex 2.0) + 6 PS figures upon reques