Density-matrix Algorithm Research Articles

The tensor hypercontracted parametric reduced density matrix algorithm: Coupled-cluster accuracy with O(r4) scaling

Tensor hypercontraction is a method that allows the representation of a high-rank tensor as a product of lower-rank tensors. In this paper, we show how tensor hypercontraction can be applied to both the electron repulsion integral tensor and the two-particle excitation amplitudes used in the parametric 2-electron reduced density matrix (p2RDM) algorithm. Because only O(r) auxiliary functions are needed in both of these approximations, our overall algorithm can be shown to scale as O(r4), where r is the number of single-particle basis functions. We apply our algorithm to several small molecules, hydrogen chains, and alkanes to demonstrate its low formal scaling and practical utility. Provided we use enough auxiliary functions, we obtain accuracy similar to that of the standard p2RDM algorithm, somewhere between that of CCSD and CCSD(T).

Weakly Supervised Training of a Sign Language Recognition System Using Multiple Instance Learning Density Matrices

A system for automatically training and spotting signs from continuous sign language sentences is presented. We propose a novel multiple instance learning density matrix algorithm which automatically extracts isolated signs from full sentences using the weak and noisy supervision of text translations. The automatically extracted isolated samples are then utilized to train our spatiotemporal gesture and hand posture classifiers. The experiments were carried out to evaluate the performance of the automatic sign extraction, hand posture classification, and spatiotemporal gesture spotting systems. We then carry out a full evaluation of our overall sign spotting system which was automatically trained on 30 different signs.

Thermodynamics of multiferroic spin chains

The minimal model to describe many spin chain materials with ferroelectric properties is the Heisenberg model with ferromagnetic nearest neighbor coupling J1 and antiferromagnetic next-nearest neighbor coupling J2. Here we study the thermodynamics of this model using a density-matrix algorithm applied to transfer matrices. We find that the incommensurate spin-spin correlations - crucial for the ferroelectric properties and the analogue of the classical spiral pitch angle - depend not only on the ratio J2/|J1| but also strongly on temperature. We study small easy-plane anisotropies which can stabilize a vector chiral order as well as the finite-temperature signatures of multipolar phases, stable at finite magnetic field. Furthermore, we fit the susceptibilities of LiCuVO4, LiCu2O2, and Li2ZrCuO4. Contrary to the literature, we find that for LiCuVO4 the best fit is obtained with J2 ~ 90 K and J2/|J1| ~ 0.5 and show that these values are consistent with the observed spin incommensurability. Finally, we discuss our findings concerning the incommensurate spin-spin correlations and multipolar orders in relation to future experiments on these compounds.

Tensor Product State Formulation for the Spin 1/2 Antiferromagnetic XXZ Model on the Checkerboard Lattice

Vertical density matrix algorithm (VDMA), a tensor product state formulation of the ``higher-dimensional'' density matrix renormalization group, is applied to the spin 1/2 antiferromagnetic XXZ model on the checkerboard lattice. The VDMA was, in the preceding study, applied to the transverse field Ising model on the square lattice and the three-dimensional classical Ising model. In the present paper, its implementation procedure is modified in order to apply the VDMA to the XXZ model. Numerical accuracy of the VDMA is investigated for the XXZ model on the square lattice, which shows that the method gives reliable results for the ground state energy. In the frustrated region, VDMA results are compared with a simple calculation based on a magnetically disordered state. It is found that the weakly frustrated region is in the N\'{e}el ordered phase, while in the strongly frustrated region the realized phase cannot be identified clearly by the obtained results.

Self-consistent density matrix algorithm for electronic structure and excitations of molecules and aggregates

An ab initio density matrix algorithm for electronic structure computations of many-electron systems is proposed. The reduced single-electron density matrices are derived by mapping the density functional theory nonlinear optical response functions onto an effective multilevel system. These density matrices are then used as a zeroth order iteration into self-consistent equations whose solution should yield the exact energies and the complete set of (transition and diagonal) single-electron density matrices. Higher order (n electron) density matrices are not computed explicitly. The linear and nonlinear optical response functions may be obtained at a low computational cost. Application is made to constructing an exciton Hamiltonian for molecular aggregates using density matrices of isolated molecules, avoiding electronic structure calculations of the entire aggregate.

Vicinal surface with Langmuir adsorption: A decorated restricted solid-on-solid model

We study the vicinal surface of the restricted solid-on-solid model coupled with the Langmuir adsorbates which we regard as two-dimensional lattice gas without lateral interaction. The effect of the vapor pressure of the adsorbates in the environmental phase is taken into consideration through the chemical potential. We calculate the surface free energy $f$, the adsorption coverage $\Theta$, the step tension $\gamma$, and the step stiffness $\tilde{\gamma}$ by the transfer matrix method combined with the density-matrix algorithm. Detailed step-density-dependence of $f$ and $\Theta$ is obtained. We draw the roughening transition curve in the plane of the temperature and the chemical potential of adsorbates. We find the multi-reentrant roughening transition accompanying the inverse roughening phenomena. We also find quasi-reentrant behavior in the step tension.

Vertical density matrix algorithm: a higher-dimensional numerical renormalization scheme based on the tensor product state ansatz.

We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors that are iteratively optimized by the use of the "vertical density matrix" formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to 2D quantum systems, which we demonstrate for the 2D transverse field Ising model.

Stiffening Transition in Vicinal Surfaces with Adsorption

We study the vicinal surface with adsorption below the roughening temperature, using a solid-on-solid model coupled with the Ising model. We calculate the step tension γ and the step-interaction coefficient B using the density matrix algorithm. We find a temperature Ts where B vanishes and that the surface free energy has the form f (p) � f (0) = γp + (const) p 5 +···, where p is the surface gradient.

Trimer-Monomer Mixture Problem on (111) 1 x 1 Surface of Diamond Structure

We consider a system of trimers and monomers on the triangular lattice, which describes the adsorption problem on (111) $1 \times 1$ surface of diamond structure. We give a mapping to a 3-state vertex model on the square lattice. We treat the problem by the transfer-matrix method combined with the density-matrix algorithm, to obtain thermodynamic quantities.

121 O(N) 法による Tight-binding 分子動力学計算の高速化

Tight-binding method (TB) lies between the very accurate but expensive ab-initio calculations and the fast but limited empirical methods for description of atomic-level materials behaviors. As the size of system involved becomes larger, the more inevitable scaling a solved hamiltonian matrix down is. The number of operations required to diagonalize a hamiltonian matrix is proportional to the cube of the system size. Therefore, several algorithms of O (N) scaling have been developed. In the present paper, we demonstrate the improvement of tight-binding molecular dynamics calculation by the density matrix algorithm, compared with standard diagonalization method.

Ground-state density-matrix algorithm for excited-state adiabatic surfaces: application to polyenes

A low-cost method for computing excited-state adiabatic surfaces, which totally avoids the calculation of the excited-state many-electron wavefunctions, and scales favorably with molecular size is proposed. The technique combines standard ground-state calculations with a time-dependent density-matrix calculation of vertical optical excitations, using the ground-state single-electron density matrix as an input. Several surfaces are generated simultaneously. The structure and vibrational frequencies of the 1 B u state of hexatriene and octatetraene are calculated using this method. The computed sign of the excited-state equilibrium displacements facilitates rapid analysis of resonance Raman measurements.

A Density Matrix Algorithm for 3D Classical Models

We generalize the corner transfer matrix renormalization group, which consists of White's density matrix algorithm and Baxter's method of the corner transfer matrix, to three dimensional (3D) classical models. The renormalization group transformation is obtained through the diagonalization of density matrices for a cubic cluster. A trial application for 3D Ising model with m=2 is shown as the simplest case.

Corner Transfer Matrix Algorithm for Classical Renormalization Group

We report a real-space renormalization group (RSRG) algorithm, which is formulated through Baxter's corner transfer matrix (CTM), for two-dimensional ( d = 2) classical lattice models. The new method performs the renormalization group transformation according to White's density matrix algorithm, so that variational free energies are minimized within a restricted degree of freedom. As a consequence of the renormalization, spin variables on each corner of CTM are replaced by a m -state block spin variable. It is shown that the thermodynamic functions and critical exponents of the q = 2, 3 Potts models can be precisely evaluated by use of the renormalization group method.

Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems.

I extend the scope of the density-matrix renormalization-group technique developed by White to the calculation of dynamical correlation functions using the continuous fraction expansion of the Green's function. As an application and performance evaluation I calculate the spin dynamics of the one-dimensional S=1/2 Heisenberg chain.