25,815 research outputs found
The density matrix renormalization group for ab initio quantum chemistry
During the past 15 years, the density matrix renormalization group (DMRG) has
become increasingly important for ab initio quantum chemistry. Its underlying
wavefunction ansatz, the matrix product state (MPS), is a low-rank
decomposition of the full configuration interaction tensor. The virtual
dimension of the MPS, the rank of the decomposition, controls the size of the
corner of the many-body Hilbert space that can be reached with the ansatz. This
parameter can be systematically increased until numerical convergence is
reached. The MPS ansatz naturally captures exponentially decaying correlation
functions. Therefore DMRG works extremely well for noncritical one-dimensional
systems. The active orbital spaces in quantum chemistry are however often far
from one-dimensional, and relatively large virtual dimensions are required to
use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its
computational cost, and its properties are discussed. Two important aspects to
reduce the computational cost are given special attention: the orbital choice
and ordering, and the exploitation of the symmetry group of the Hamiltonian.
With these considerations, the QC-DMRG algorithm allows to find numerically
exact solutions in active spaces of up to 40 electrons in 40 orbitals.Comment: 24 pages; 10 figures; based on arXiv:1405.1225; invited review for
European Physical Journal
Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
We propose a Lie-algebraic duality approach to analyze non-equilibrium
evolution of closed dynamical systems and thermodynamics of interacting quantum
lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems).
The first part of the paper utilizes a geometric Hilbert-space-invariant
formulation of unitary time-evolution, where a quantum Hamiltonian is viewed as
a trajectory in an abstract Lie algebra, while the sought-after evolution
operator is a trajectory in a dynamic group, generated by the algebra via
exponentiation. The evolution operator is uniquely determined by the
time-dependent dual generators that satisfy a system of differential equations,
dubbed here dual Schrodinger-Bloch equations, which represent a viable
alternative to the conventional Schrodinger formulation. These dual
Schrodinger-Bloch equations are derived and analyzed on a number of specific
examples. It is shown that deterministic dynamics of a closed classical
dynamical system occurs as action of a symmetry group on a classical manifold
and is driven by the same dual generators as in the corresponding quantum
problem. This represents quantum-to-classical correspondence. In the second
part of the paper, we further extend the Lie algebraic approach to a wide class
of interacting many-particle lattice models. A generalized Hubbard-Stratonovich
transform is proposed and it is used to show that the thermodynamic partition
function of a generic many-body quantum lattice model can be expressed in terms
of traces of single-particle evolution operators governed by the dynamic
Hubbard-Stratonovich fields. Finally, we derive Hubbard-Stratonovich dynamical
systems for the Bose-Hubbard model and a quantum spin model and use the
Lie-algebraic approach to obtain new non-perturbative dual descriptions of
these theories.Comment: 25 pages, 1 figure; v2: citations adde
Spin facilitated Ising model with long range interaction
We study the dynamics of a spin facilitated Ising model with long range
kinetic constraints. To formulate those restrictions within an analytical
approach we introduce the size of a kinetic active environment of a given spin.
Based on a Master equation in second quantized form, the spin-autocorrelation
function is calculated. It exhibits a pronounced slow dynamics, manifested by a
logarithmic decay law of the spin-autocorrelation function. In case of an
infinite kinetic interaction the mean field solution yields an asymptotic exact
expression for the autocorrelation function which is in excellent agreement
with Monte Carlo Simulations for finite interaction lengths. With increasing
size of the active zone the cooperative processes, characterizing the
facilitated model with short range kinetic interaction, become irrelevant. We
demonstrate that the long range kinetic interaction dominates the actual spin
configurations of the whole system and the mean field solution is the exact
one.Comment: 18 pages, 5 figure
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