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arXiv:0707.0159

Direct Extension of Density-Matrix Renormalization Group toward 2-Dimensional Quantum Lattice Systems: Studies for Parallel Algorithm, Accuracy, and Performance

Yamada, S., Masahiko Okumura, Masahiko Machida

Published Jul 2, 2007• 0 citations
arXivHTMLPDF

Abstract

We parallelize density-matrix renormalization group to directly extend it to 2-dimensional ($n$-leg) quantum lattice models. The parallelization is made mainly on the exact diagonalization for the superblock Hamiltonian since the part requires an enormous memory space as the leg number $n$ increases. The superblock Hamiltonian is divided into three parts, and the correspondent superblock vector is transformed into a matrix, whose elements are uniformly distributed into processors. The parallel efficiency shows a high rate as the number of the states kept $m$ increases, and the eigenvalue converges within only a few sweeps in contrast to the multichain algorithm.

DOI
10.48550/arxiv.0707.0159

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We parallelize density-matrix renormalization group to directly extend it to 2-dimensional (nnn-leg) quantum lattice models. The parallelization is made mainly on the exact diagonalization for the superblock Hamiltonian since the part requires an enormous memory space as the leg number nnn increases. The superblock Hamiltonian is divided into three parts, and the correspondent superblock vector is transformed into a matrix, whose elements are uniformly distributed into processors. The parallel efficiency shows a high rate as the number of the states kept mmm increases, and the eigenvalue converges within only a few sweeps in contrast to the multichain algorithm.
We parallelize the density-matrix renormalization group method to directly extend it to two-dimensional ( n -leg) quantum lattice models. The parallelization is performed mainly on the diagonalization for the superblock Hamiltonian since this step requires enormous memory space as the leg number n increases. The superblock Hamiltonian is divided into three parts, and the corresponding superblock vectors are transformed into matrices whose elements are uniformly distributed into processors. Parallel efficiency increases as the number of states kept, m , increases, and the obtained ground-state energy rapidly converges within a few sweeps. Furthermore, the present algorithm applied to doped 4-leg Hubbard ladders reaches their ground states, which satisfy the Lieb–Mattis theorem with m much smaller than those used in different indirect algorithms.
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.
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.
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.
In this study, we report a parallel algorithm for the infinite-size density matrix renormalization group (iDMRG) that is applicable to one-dimensional (1D) quantum systems with ℓ\ellℓ-site periods, where ℓ\ellℓ is an even number. It combines Hida's iDMRG applied to random 1D spin systems with a variant of McCulloch's wavefunction prediction. This allows us to apply ℓ/2\ell/2ℓ/2 times of computational power to accelerate the investigation of multi-leg frustrated quantum systems in the thermodynamic limit, which is a challenging simulation. We performed benchmark calculations for a spin-1/2 Heisenberg model on a Kagome cylinder YC8 using the parallel iDMRG, and found that the proposed iDMRG was efficiently parallelized for shared memory and distributed memory systems, and provided such bulk physical quantities as total energy, bond strength on nearest neighbor spins, and spin--spin correlation functions and their correlation lengths without finite-size effects. Moreover, the variant of the wavefunction prediction sped up Lanczos methods in the parallel iDMRG by approximately three times.
In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that are needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.