The density-matrix renormalization group in the age of matrix product states
arXiv:1008.3477 · doi:10.1016/j.aop.2010.09.012
Abstract
The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.
118 pages, 63 figures; minor changes in this final version: typos eliminated and a few references added
References in corpus (43)
- Many-Body Physics with Ultracold Gases
- Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
- The numerical renormalization group method for quantum impurity systems
- Real time evolution using the density matrix renormalization group
- Matrix Product Density Operators: Simulation of finite-T and dissipative systems
- Classical simulation of infinite-size quantum lattice systems in one spatial dimension
- Matrix product states represent ground states faithfully
- DMRG and periodic boundary conditions: a quantum information perspective
- Entanglement entropy of fermions in any dimension and the Widom conjecture
- The iTEBD algorithm beyond unitary evolution
- From density-matrix renormalization group to matrix product states
- Entropy scaling and simulability by Matrix Product States
- Sum-rule Conserving Spectral Functions from the Numerical Renormalization Group
- New Trends in Density Matrix Renormalization
- Scaling of entanglement support for Matrix Product States
- Minimally Entangled Typical Thermal State Algorithms
- Continuous Matrix Product States for Quantum Fields
- A Numerical Renormalization Group approach to Green's Functions for Quantum Impurity Models
- Atomic three-body loss as a dynamical three-body interaction
- Matrix Product States for dynamical simulation of infinite chains
- Spectral functions in one-dimensional quantum systems at T>0
- Time-step targetting methods for real-time dynamics using DMRG
- Entanglement scaling in critical two-dimensional fermionic and bosonic systems
- Applying matrix product operators to model systems with long-range interactions
- Spectral Function for the S=1 Heisenberg Antiferromagetic Chain
- The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
- Finite automata for caching in matrix product algorithms
- Exploring local quantum many-body relaxation by atoms in optical superlattices
- Exact edge singularities and dynamical correlations in spin-1/2 chains
- Spin-charge separation in cold Fermi-gases: a real time analysis
- A real-time study of diffusive and ballistic transport in spin-1/2 chains using the adaptive time-dependent density matrix renormalization group method
- Spectroscopy of ultracold atoms by periodic lattice modulations
- d-wave resonating valence bond states of fermionic atoms in optical lattices
- Density Matrix Renormalization Group in the Heisenberg Picture
- Tensor operators: constructions and applications for long-range interaction systems
- Broken time-reversal symmetry in strongly correlated ladder structures
- Real-time dynamics at finite temperature by DMRG: A path-integral approach
- Matrix product state comparison of the numerical renormalization group and the variational formulation of the density matrix renormalization group
- Vector chiral order in frustrated spin chains
- High Energy Dynamics of the Single Impurity Anderson Model
- Spectral Densities from Dynamic Density-Matrix Renormalization
- Magnetism, coherent many-particle dynamics, and relaxation with ultracold bosons in optical superlattices
- Observations Outside the Light-Cone: Algorithms for Non-Equilibrium and Thermal States
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- Linearized Tensor Renormalization Group Algorithm for Thermodynamics of Quantum Lattice Models
- Adaptive Lanczos-vector method for dynamic properties within the density-matrix renormalization group
- Landau-Zener sweeps and sudden quenches in coupled Bose-Hubbard chains
- Quantum Spinon Oscillations