One of the most fascinating issues in modern condensed matter physics is to understand highly-correlated electronic structures and propose their novel device designs toward the reduced carbon-dioxide future. Among various developed numerical approaches for highly-correlated electrons, the density matrix renormalization group (DMRG) has been widely accepted as the most promising numerical scheme compared to Monte Carlo and exact diagonalization in terms of accuracy and accessible system size. In fact, DMRG almost perfectly resolves one-dimensional chain like long quantum systems. In this paper, we suggest its extended approach toward higher-dimensional systems by high-performanc computing techniques. The computing target in DMRG is a huge non-uniform sparse matrix diagonalization. In order to efficiently parallelize the part, we implement communication step doubling together with reuse of the mid-point data between the doubled two steps to avoid severe bottleneck of all-to-all communications essential for the diagonalization. The technique is successful even for clusters composed of more than 1000 cores and offers a trustworthy exploration way for two-dimensional highly-correlated systems.
One of the most fascinating issues in modern condensed matter physics is to understand highly-correlated electronic structures and propose their novel device designs toward the reduced carbon-dioxide future. Among various developed numerical approaches for highly-correlated electrons, the density matrix renormalization group (DMRG) has been widely accepted as the most promising numerical scheme compared to Monte Carlo and exact diagonalization in terms of accuracy and accessible system size. In fact, DMRG almost perfectly resolves one-dimensional chain like long quantum systems. In this paper, we suggest its extended approach toward higher-dimensional systems by high-performanc computing techniques. The computing target in DMRG is a huge non-uniform sparse matrix diagonalization. In order to efficiently parallelize the part, we implement communication step doubling together with reuse of the mid-point data between the doubled two steps to avoid severe bottleneck of all-to-all communications essential for the diagonalization. The technique is successful even for clusters composed of more than 1000 cores and offers a trustworthy exploration way for two-dimensional highly-correlated systems.
Density matrix renormalization group (DMRG), as a numerical method of solving the ground state of one-dimensional strongly-correlated lattice model with very high accuracy, requires expensive computational and memory cost when applied to two- and quasi-two-dimensional problems. The number of DMRG kept states is generally very large to achieve a reliable accuracy for these applications, which results in numerous matrix and vector operations and unbearably consuming time in the absence of the proper parallelization. However, due to its sequential nature, the parallelization of DMRG algorithm is usually not straightforward. In this work, we propose a new hybrid parallelization strategy for the DMRG method. It takes advantage of the computing capability of both central processing unit (CPU) and graphics processing unit (GPU) of the computer. In order to achieve as many as DMRG kept states within a limited GPU memory, we adopt the four-block formulation of the Hamiltonian rather than the two-block formulation. The later consumes much more memories, which has been used in another pioneer work on the hybrid parallelization of the DMRG algorithm, and only a small number of DMRG kept states are available. Our parallel strategy focuses on the diagonalization of the Hamiltonian, which is the most time-consuming part of the whole DMRG procedure. A hybrid parallelization strategy of diagonalization method is implemented, in which the required data for diagonalization are distributed on both the host and GPU memory, and the data exchange between them is negligible in our data partitioning scheme. The matrix operations are also shared on both CPU and GPU when the Hamiltonian acts on a wave function, while the distribution of these operations is determined by a load balancing strategy. Taking fermionic Hubbard model for example, we examine the running performance of the hybrid parallelization strategy with different DMRG kept states and provide corresponding performance benchmark. On a 4-leg ladder, we employ the conserved quantities with U(1) symmetry of the model and a good-quantum number based task scheduling to further reduce the GPU memory cost. We manage to obtain a moderate speedup of the hybrid parallelization for a wide range of DMRG kept states. In our example, the ground state energy with high accuracy is obtained by the extrapolation of the results, with different numbers of states kept, and we show charge stripes which are usually experimentally observed in high-temperature superconductors. In this case, we keep 104 DMRG states and the GPU memory cost is less than 12 Gigabytes.
Based on the contractor renormalization group (CORE) method and the density matrix renormalization group (DMRG) method, a new computational scheme, which is called the block density matrix renormalization group with effective interactions (BDMRG-EI), is proposed to deal with the numerical computation of quantum correlated systems. Different from the conventional CORE method in calculating the blocks and the fragments, where the DMRG method instead of the exact diagonalization is employed in BDMRG-EI, BDMRG-EI makes the calculations of larger blocks and fragments applicable. Integrating DMRG's advantage of high accuracy and CORE's advantage of low computational costs, BDMRG-EI can be widely used for the theoretical calculations of the ground state and low-lying excited states of large systems with simple or complicated connectivity. Test calculations on a 240 site one-dimensional chain and a double-layer polyacene oligomer containing 48 hexagons with the spin-1/2 Heisenberg Hamiltonian demonstrate the efficiency and potentiality of the method.
Numerical analysis of strongly correlated quantum lattice models has a great importance in quantum physics. The exponentially growing size of the Hilbert space makes these computations difficult, however sophisticated algorithms have been developed to balance the size of the effective Hilbert space and the accuracy of the simulation. One of these methods is the density matrix renormalization group (DMRG) algorithm which has become the leading numerical tool in the study of low dimensional lattice problems of current interest. In the algorithm a high computational problem can be translated to a list of dense matrix operations, which makes it an ideal application to fully utilize the computing power residing in both current multi-core processors and novel kilo-processor architectures.
Abstract The study of strongly correlated electron systems remains a fundamental challenge in condensed matter physics, particularly in two-dimensional (2D) systems hosting various exotic phases of matter including quantum spin liquids, unconventional superconductivity, and topological orders. Although Density Matrix Renormalization Group (DMRG) has established itself as a pillar for simulating one-dimensional quantum systems, its application to 2D systems has long been hindered by the notorious “local minimum” issues. Recent methodological breakthroughs have addressed this challenge by incorporating Gutzwiller-projected wave functions as initial states for DMRG simulations. This hybrid approach, referred to as DMRG guided by Gutzwiller-projected wave functions (or Gutzwiller-guided DMRG), has demonstrated remarkable improvements in accuracy, efficiency, and the ability to explore exotic quantum phases such as topological orders. This review examines the theoretical underpinnings of this approach, details key algorithmic developments, and showcases its applications in recent studies of 2D quantum systems.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
Tensor networks (TNs) and the breadth of algorithms acting on them have seen astounding success in simulating quantum many-body systems in the strongly interacting regime with both accuracy and efficiency. In the context of quantum chemistry, Steven White’s density matrix renormalization group (DMRG) continues to take center stage as the TN method of choice, seeing countless theoretical and computational breakthroughs in recent decades yet remaining fettered by a few persistent shortcomings, notably, a lack of size-extensivity and quasi-exactness . Here, we present a simple yet versatile framework for circumventing these issues: the bootstrap embedded density matrix renormalization group (BE-DMRG), and numerically validate its size-extensive and quasi-exact ground-state properties for a test bed of strongly correlated molecular systems (linear H - chains; quasi-linear E - polyacetylene; 2D and 3D H -lattices; 2D arene flakes ). Spanning a breadth of system sizes (10 to 200 orbitals) and entanglement topologies (linear to highly nonlinear), we demonstrate the robustness of the BE-DMRG for problems far beyond the reach of conventional DMRG implementations. Furthermore, by detailing BE-DMRG convergence behavior with respect to exact diagonalization, we find the rate of convergence with bond dimension to be significantly faster than, yet just as reliable as, that of conventional DMRG. Ultimately, we find that the embedded DMRG might serve as a natural extension of White’s original formulation to higher dimensions without the need for higher-order tensor networks. The coupling of tensor network theories to the framework of quantum embedding, more broadly, may become an incomparably powerful tool for the study of strongly correlated molecules and materials.