[Paper Review] Targeting Multiple States in the Density Matrix Renormalization Group with The Singular Value Decomposition

E. D’Azevedo, Wael Elwasif+2 more|arXiv (Cornell University)|Feb 25, 2019

Quantum and electron transport phenomena4 citations

TL;DR

This paper proposes using singular value decomposition (SVD) to target multiple quantum states in the density matrix renormalization group (DMRG), improving computational efficiency and enabling seamless integration with matrix product state (MPS) formulations. By replacing density matrix construction with SVD, the method reduces runtime by ~25% and enhances performance for calculating frequency-resolved spectral functions, as demonstrated in Hubbard model simulations on two-leg ladders with GPU-accelerated matrix-vector products.

ABSTRACT

In the Density Matrix Renormalization Group (DMRG), multiple states must be included in the density matrix when properties beyond ground state are needed, including temperature dependence, time evolution, and frequency-resolved response functions. How to include these states in the density matrix has been shown in the past. But it is advantageous to replace the density matrix by a singular value decomposition (SVD) instead, because of improved performance, and because it enables multiple targeting in the matrix product state description of the DMRG. This paper shows how to target multiple states using the SVD; it analyzes the implication of local symmetries, and discusses typical performance improvements using the example of the Hubbard model's photo-emission spectra on a ladder geometry.

Motivation & Objective

• To extend the DMRG algorithm beyond ground state calculations to enable accurate computation of excited states, dynamic response functions, and finite-temperature properties.
• To overcome the computational bottleneck of density matrix construction in multi-state DMRG by replacing it with singular value decomposition (SVD).
• To enable efficient multiple-state targeting within the matrix product state (MPS) framework, avoiding costly MPS addition and compression.
• To analyze the impact of local symmetries and symmetry patches on SVD-based DMRG performance.
• To benchmark the method on realistic models, such as the Hubbard model on a two-leg ladder, for photoemission spectra and frequency-dependent observables.

Proposed method

• Replace the conventional density matrix construction in DMRG with a singular value decomposition (SVD) of the many-body wavefunction, where squared singular values correspond to eigenvalues of the reduced density matrix.
• Define a generalized reduced density matrix that includes multiple target states (e.g., ground state, excited states, correction vectors) via weighted superpositions, with non-zero weights assigned to each targeted state.
• Use the SVD of the superblock wavefunction to directly obtain the optimal truncation basis, avoiding the need to form and diagonalize the full density matrix.
• Incorporate local symmetries by decomposing the SVD into symmetry sectors, enabling parallelization and performance gains through symmetry patch processing.
• Integrate the SVD-based algorithm into the MPS formulation of DMRG, allowing for direct manipulation of multiple states without additional MPS compression steps.
• Optimize performance by offloading the most expensive sub-algorithm—matrix-vector multiplication—to GPUs, while keeping SVD and other operations on the CPU due to memory overhead.

Experimental results

Research questions

• RQ1 How can the DMRG algorithm be extended to efficiently target multiple quantum states beyond the ground state?
• RQ2 What are the computational advantages of using SVD instead of density matrix diagonalization in multi-state DMRG?
• RQ3 How do local symmetries affect the implementation and performance of SVD-based multi-state targeting in DMRG?
• RQ4 What performance improvements does the SVD approach yield in calculating frequency-resolved spectral functions for the Hubbard model?
• RQ5 Can the SVD-based method be efficiently parallelized using symmetry patches and GPU acceleration in large-scale simulations?

Key findings

• The SVD-based approach reduces total runtime by approximately 25% compared to conventional DMRG when computing spectral functions for the Hubbard model on a two-leg ladder.
• The SVD method outperforms density matrix diagonalization by eliminating the need to explicitly construct and diagonalize the density matrix, leading to faster convergence.
• For the Hubbard model on a 32×2 ladder with up to 2000 kept states, 400 Lanczos vectors were sufficient to achieve converged spectral functions.
• Performance benchmarks show that GPU acceleration of the matrix-vector product reduces wall times, but SVD computation remains a minor fraction of total time (5%) and is not accelerated on GPU due to memory overhead.
• The SVD-based method enables direct multiple-state targeting in the MPS framework, replacing the need for MPS addition and compression, thus streamlining the algorithm.
• The method is readily extendable to finite-temperature calculations via imaginary-time evolution and to real-time dynamics using Krylov-space methods, with similar computational scaling benefits.