442 research outputs found
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
Extrapolation-CAM Theory for Critical Exponents
By intentionally underestimating the rate of convergence of
exact-diagonalization values for the mass or energy gaps of finite systems, we
form families of sequences of gap estimates. The gap estimates cross zero with
generically nonzero linear terms in their Taylor expansions, so that
for each member of these sequences of estimates. Thus, the Coherent Anomaly
Method can be used to determine . Our freedom in deciding exactly how to
underestimate the convergence allows us to choose the sequence that displays
the clearest coherent anomaly. We demonstrate this approach on the
two-dimensional ferromagnetic Ising model, for which . We also use it
on the three-dimensional ferromagnetic Ising model, finding , in good agreement with other estimates.Comment: 21 pages, Submitted to Journal of Physics A; new section added
discussing rate of convergence and relation to Finite-Size Scalin
Factorizing the time evolution operator
There is a widespread belief in the quantum physical community, and in
textbooks used to teach Quantum Mechanics, that it is a difficult task to apply
the time evolution operator Exp{-itH/h} on an initial wave function. That is to
say, because the hamiltonian operator generally is the sum of two operators,
then it is a difficult task to apply the time evolution operator on an initial
wave function f(x,0), for it implies to apply terms operators like (a+b)^n. A
possible solution of this problem is to factorize the time evolution operator
and then apply successively the individual exponential operator on the initial
wave function. However, the exponential operator does not directly factorize,
i. e. Exp{a+b} is not equal to Exp{a}Exp{b}. In this work we present a useful
procedure for factorizing the time evolution operator when the argument of the
exponential is a sum of two operators, which obey specific commutation
relations. Then, we apply the exponential operator as an evolution operator for
the case of elementary unidimensional potentials, like the particle subject to
a constant force and the harmonic oscillator. Also, we argue about an apparent
paradox concerning the time evolution operator and non-spreading wave packets
addressed previously in the literature.Comment: 24 pages; added references; one figure change
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
Kinetics of Hydrogen Isotopes at Surfaces and Bulks of Plasma Facing Materials Based on Group 5 Metals
Hidden Order and Dimerization Transition in Chains
We study ground state properties of the quantum antiferromagnetic chain
with a bond alternation H = \sum_{j} [ 1 + \delta (-1)^j ] \mbox{\boldmath
$S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} by a Quantum Monte Carlo
calculation. We find that the hidden symmetry is broken for
while it is unbroken in the other regions. This confirms
the successive dimerization transitions first predicted by Affleck and Haldane.
Our result shows that these transitions can be understood in terms of the
hidden symmetry breaking, as was discussed using the
Valence-Bond-Solid states. Furthermore, we find that the behavior of the
generalized string correlation is qualitatively very similar to that in the
Valence-Bond-Solid states, including the location of zeroes as a function of
the angle parameter.Comment: 3 pages (LaTex with jpsj-style files
(ftp://ftp.u-tokyo.ac.jp/pub/SOCIETY/JPSJ)) and 1 Postscript figur
Recent Developments of World-Line Monte Carlo Methods
World-line quantum Monte Carlo methods are reviewed with an emphasis on
breakthroughs made in recent years. In particular, three algorithms -- the loop
algorithm, the worm algorithm, and the directed-loop algorithm -- for updating
world-line configurations are presented in a unified perspective. Detailed
descriptions of the algorithms in specific cases are also given.Comment: To appear in Journal of Physical Society of Japa
Accessing the dynamics of large many-particle systems using Stochastic Series Expansion
The Stochastic Series Expansion method (SSE) is a Quantum Monte Carlo (QMC)
technique working directly in the imaginary time continuum and thus avoiding
"Trotter discretization" errors. Using a non-local "operator-loop update" it
allows treating large quantum mechanical systems of many thousand sites. In
this paper we first give a comprehensive review on SSE and present benchmark
calculations of SSE's scaling behavior with system size and inverse
temperature, and compare it to the loop algorithm, whose scaling is known to be
one of the best of all QMC methods. Finally we introduce a new and efficient
algorithm to measure Green's functions and thus dynamical properties within
SSE.Comment: 11 RevTeX pages including 7 figures and 5 table
Kinetics of Hydrogen Isotopes at Surfaces and Bulks of Plasma Facing Materials Based on Group 5 Metals
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