Improved hybrid parallel strategy for density matrix renormalization group method

We propose a new heterogeneous parallel strategy for the density matrix renormalization group(DMRG)method in the hybrid architecture with both central processing unit(CPU)and graphics processing unit(GPU).Focusing on the two most time-consuming sections in the finite DMRG sweeps,i.e.,the diagonalization of superblock and the truncation of subblock,we optimize our previous hybrid algorithm to achieve better performance.For the former,we adopt OpenMP application programming interface on CPU and use our own subroutines with higher bandwidth on GPU.For the later,we use GPU to accelerate matrix and vector operations involving the reduced density matrix.Applying the parallel scheme to the Hubbard model with next-nearest hopping on the 4-leg ladder,we compute the ground state of the system and obtain the charge stripe pattern which is usually observed in high temperature superconductors.Based on simulations with different numbers of DMRG kept states,we show significant performance improvement and computational time reduction with the optimized parallel algorithm.Our hybrid parallel strategy with superiority in solving the ground state of quasi-two dimensional lattices is also expected to be useful for other DMRG applications with large numbers of kept states,e.g.,the time dependent DMRG algorithms.

Real-space parallel density matrix renormalization group with adaptive boundaries

We propose an improved real-space parallel strategy for the density matrix renormalization group(DMRG)method,where boundaries of separate regions are adaptively distributed during DMRG sweeps.Our scheme greatly improves the parallel efficiency with shorter waiting time between two adjacent tasks,compared with the original real-space parallel DMRG with fixed boundaries.We implement our new strategy based on the message passing interface(MPI),and dynamically control the number of kept states according to the truncation error in each DMRG step.We study the performance of the new parallel strategy by calculating the ground state of a spin-cluster chain and a quantum chemical Hamiltonian of the water molecule.The maximum parallel efficiencies for these two models are 91%and 76%in 4 nodes,which are much higher than the real-space parallel DMRG with fixed boundaries.

Time-Dependent Density Matrix Renormalization Group Coupled with n-Mode Representation Potentials for the Excited State Radiationless Decay Rate:Formalism and Application to Azulene

We propose a method for calculating the nonradiative decay rates for polyatomic molecules including anharmonic effects of the potential energy surface(PES)in the Franck-Condon region.The method combines the n-mode repre-sentation method to construct the ab initio PES and the nearly exact time-dependent density matrix renormalization group method(TD-DMRG)to simulate quantum dynamics.In addition,in the framework of TD-DMRG,we further develop an algorithm to calculate the final-state-resolved rate coefficient which is very useful to analyze the contribution from each vibrational mode to the transition process.We use this method to study the internal conversion(IC)process of azulene after taking into account the anharmonicity of the ground state PES.The results show that even for this semi-rigid molecule,the intramode anharmonicity enhances the IC rate significantly,and after considering the two-mode coupling effect,the rate increases even further.The reason is that the anharmonicity enables the C-H vibrations to receive electronic energy while C-H vibrations do not contribute on the harmonic PES as the Huang-Rhys factor is close to 0.

Generation of Complex Matrices Belonging to Matrix Groups with Controlled Condition Number

The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical problems and complex control problems. In this work, the authors try to generate matrices such that： （1） the condition number of these types of matrices is controlled and （2） The algorithm used to generate these matrices preserves their structure.

A Survey of Mathematical Tools in Topology and Performance Integrated Modeling and Design of Robotic Mechanism

Topology and performance are the two main topics dealt in the development of robotic mechanisms.However,it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame.As the properties associated with topology and performance,finite motion and instantaneous motion of the robot play key roles in the procedure.On the purpose of providing a fundamental preparation for integrated modeling and design,this paper carries out a review on the existing unified mathematic frameworks for motion description and computation,involving matrix Lie group and Lie algebra,dual quaternion and pure dual quaternion,finite screw and instantaneous screw.Besides the application in robotics,the review of the work from these mathematicians concentrates on the description,composition and intersection operations of the finite and instantaneous motions,especially on the exponential-differential maps which connect the two sides.Furthermore,an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development.The presented review offers insightful investigation to the motion description and computation,and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots.

Exponential Set-Point Stabilization of Underactuated Vehicles Moving in Three-Dimensional Space

This paper investigates the stabilization of underactuated vehicles moving in a three-dimensional vector space.The vehicle’s model is established on the matrix Lie group SE(3),which describes the configuration of rigid bodies globally and uniquely.We focus on the kinematic model of the underactuated vehicle,which features an underactuation form that has no sway and heave velocity.To compensate for the lack of these two velocities,we construct additional rotation matrices to generate a motion of rotation coupled with translation.Then,the state feedback is designed with the help of the logarithmic map,and we prove that the proposed control law can exponentially stabilize the underactuated vehicle to the identity group element with an almost global domain of attraction.Later,the presented control strategy is extended to set-point stabilization in the sense that the underactuated vehicle can be stabilized to an arbitrary desired configuration specified in advance.Finally,simulation examples are provided to verify the effectiveness of the stabilization controller.

Ground-state phase diagram of the dimerizedspin-1/2 two-leg ladder

Dimerized spin-1/2 ladders exhibit a variety of phase structures,which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder.Using the density matrix renormalization group(DMRG)algorithm,we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J_(×).Two types of spin systems,staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder,are investigated.To clarify the phase transition behaviors,we simultaneously analyze the string order parameter(SOP),the twisted order parameter(TOP),as well as a measurement of the quantum information analysis.Based on measuring this different observables,we establish the phase diagram accurately and give the fitting functions of the phase boundaries.In addition,the phase transition of cross-coupled spin ladder(in the absence of intrinsic dimerization)is also discussed.

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).

Off-site trimer superfluid on a one-dimensional optical lattice

The Bose-Hubbard model with an effective off-site three-body tunneling,characterized by jumps towards one another,between one atom on a site and a pair atoms on the neighborhood site,is studied systematically on a one-dimensional（1D） lattice,by using the density matrix renormalization group method.The off-site trimer superfluid,condensing at momentum k = 0,emerges in the softcore Bose-Hubbard model but it disappears in the hardcore Bose-Hubbard model.Our results numerically verify that the off-site trimer superfluid phase derived in the momentum space from[Phys.Rev.A81,011601（R）（2010）]is stable in the thermodynamic limit.The off-site trimer superfluid phase,the partially off-site trimer superfluid phase and the Mott insulator phase are found,as well as interesting phase transitions,such as the continuous or first-order phase transition from the trimer superfluid phase to the Mott insulator phase.Our results are helpful in realizing this novel off-site trimer superfluid phase by cold atom experiments.

The Automorphism Group of U（4,Z）

In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.

Cryptographic Protocols Based on Nielsen Transformations

We introduce in this paper cryptographic protocols which use combinatorial group theory. Based on a combinatorial distribution of shares we present secret sharing schemes and cryptosystems using Nielsen transformations. Nielsen transformations are a linear technique to study free groups and general infinite groups. In addition the group of all automorphisms of a free group F, denoted by AUT (F), is generated by a regular Nielsen transformation between two basis of F, and each regular Nielsen transformation between two basis of F defines an automorphism of F.

Equivariant filtering framework for inertial-integrated navigation

This paper proposes an Equivariant Filtering(EqF)framework for the inertial-integrated state estimation.As the kin-ematic system of the inertial-integrated navigation can be naturally modeled on the matrix Lie group SE_(2)(3),the sym-metry of the Lie group can be exploited to design an equivariant filter which extends the invariant extended Kalman filtering on the group-affine system and overcomes the inconsitency issue of the traditional extend Kalman filter.We firstly formulate the inertial-integrated dynamics as the group-affine systems.Then,we prove the left equivariant properties of the inertial-integrated dynamics.Finally,we design an equivariant filtering framework on the earth-centered earth-fixed frame and the local geodetic navigation frame.The experiments show the superiority of the proposed filters when confronting large misalignment angles in Global Navigation Satellite Navigation(GNSS)/Inertial Navigation System(INS)loosely integrated navigation experiments.

The L^(*) Partial Order on the Set of Group Matrices

In this paper,we use the Lowner partial order and the star partial order to introduce a new partial order(denoted by"L^(*)")on the set of group matrices,and get some characteristics and properties of the new partial order.In particular,we prove that the L*partial order is a special kind of the core partial order and it is equivalent to the star partial order under some conditions.We also illustrate its difference from other partial orders with examples and find out under what conditions it is equivalent to other partial orders.

Bijective maps on standard Borel subgroup of symplectic group preserving commutators

Abstract Let F be a field, and let G be the standard Borel subgroup of the symplectie group Sp（2m, F）. In this paper, we characterize the bijective maps φ： G -- G satisfying φ[x, y] = [φ（x）, φ（y）].

Analytical and Numerical Studies of Quantum Plateau State in One Alternating Heisenberg Chain

By using the coupled duster method and the numerical density matrix renormalization group method, we investigate the properties of the quantum plateau state in an alternating Heisenberg spin chain. In the absence of a magnetic field, the results obtained from the coupled cluster method and density matrix renormalization group method both show that the ground state of the aiternating chain is a gapped dimerized state when the parameter a exceeds a critical point ac. The value of the critical points can be determined precisely by a detailed investigation of the behavior of the spin gap. The system therefore possesses an m = 0 plateau state in the presence of a magnetic field When a 〉 ac. In addition to the m = 0 plateau state, the results of density matrix renormaiization group indicate that there is an m = 1/4 plateau state that occurs between two critical fields in the alternating chain if a 〉 1. The mechanism for the m = 1/4 plateau state and the critical behavior of the magnetization as one approaches this plateau state are also discussed.

Zero Temperature Numerical Studies of Multiband Lattice Models of Strongly Correlated Electrons

Relative to single-band models,multiband models of strongly interacting electron systems are of growing interest because of their wider range of novel phenomena and their closer match to the electronic structure of real materials.In this brief review we discuss the physics of three multiband models(the three-band Hubbard,the periodic Anderson,and the Falicov-Kimball models)that was obtained by numerical simulations at zero temperature.We first give heuristic descriptions of the three principal numerical methods(the Lanczos,the density matrix renormalization group,and the constrainedpath Monte Carlo methods).We then present generalized versions of the models and discuss the measurables most often associated with them.Finally,we summarize the results of their ground state numerical studies.While each model was developed to study specific phenomena,unexpected phenomena,usually of a subtle quantum mechanical nature,are often exhibited.Just as often,the predictions of the numerical simulations differ from those of mean-field theories.