Performance of the rigorous renormalization group for first order phase transitions and topological phases

Abstract

Expanding and improving the repertoire of numerical methods for studying quantum lattice models is an ongoing focus in many-body physics. While the density matrix renormalization group (DMRG) has been established as a practically useful algorithm for finding the ground state in 1D systems, a provably efficient and accurate algorithm remained elusive until the introduction of the rigorous renormalization group (RRG) by Landau et al. [Nature Physics 11, 566 (2015)]. In this paper, we study the accuracy and performance of a numerical implementation of RRG at first order phase transitions and in symmetry protected topological phases. Our study is motived by the question of when RRG might provide a useful complement to the more established DMRG technique. In particular, despite its general utility, DMRG can give unreliable results near first order phase transitions and in topological phases, since its local update procedure can fail to adequately explore (near) degenerate manifolds. The rigorous theoretical underpinnings of RRG, meanwhile, suggest that it should not suffer from the same difficulties. We show this optimism is justified, and that RRG indeed determines accurate, well-ordered energies even when DMRG does not. Moreover, our performance analysis indicates that in certain circumstances seeding DMRG with states determined by coarse runs of RRG may provide an advantage over simply performing DMRG.

arXiv comment8+4 pages, 2+5 figures

References in corpus (29)

• The density-matrix renormalization group in the age of matrix product states
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Entanglement spectrum of a topological phase in one dimension
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Identifying Topological Order by Entanglement Entropy
• DMRG and periodic boundary conditions: a quantum information perspective
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• The density matrix renormalization group for ab initio quantum chemistry
• Density matrix renormalization group algorithms with a single center site
• Topological characterization of fractional quantum Hall ground states from microscopic Hamiltonians
• Characterizing topological order by studying the ground states of an infinite cylinder
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Chiral spin liquid phase of the triangular lattice Hubbard model: a density matrix renormalization group study
• Spin Gaps in a Frustrated Heisenberg model for CaV${}_4$O${}_9$
• A Strictly Single-Site DMRG Algorithm with Subspace Expansion
• Real-Space Parallel Density Matrix Renormalization Group
• Efficient Matrix Product State Method for periodic boundary conditions
• An improved 1D area law for frustration-free systems
• Density matrix renormalization group on a cylinder in mixed real and momentum space
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Phase transitions and adiabatic preparation of a fractional Chern insulator in a boson cold atom model
• Multigrid Algorithms for Tensor Network States
• Obtaining topological degenerate ground states by the density matrix renormalization group
• Implementation of rigorous renormalization group method for ground space and low-energy states of local Hamiltonians
• Distributed-Memory DMRG via Sparse and Dense Parallel Tensor Contractions
• Density matrix renormalization group for a highly degenerate quantum system: Sliding environment block approach

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