Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0502396 · AIP conference proceedings

Time evolution of one-dimensional Quantum Many Body Systems

Salvatore R. Manmana

Published Jan 1, 2005• 84 citations
arXivHTMLPDF

Abstract

The level of current understanding of the physics of time‐dependent strongly correlated quantum systems is far from complete, principally due to the lack of effective controlled approaches. Recently, there has been progress in the development of approaches for one‐dimensional systems. We describe recent developments in the construction of numerical schemes for general (one‐dimensional) Hamiltonians: in particular, schemes based on exact diagonalization techniques and on the density matrix renormalization group method (DMRG). We present preliminary results for spinless fermions with nearest‐neighbor‐interaction and investigate their accuracy by comparing with exact results.

DOI
10.1063/1.2080353

Related papers

Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems

R. M. Noack

In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.

arXiv:cond-mat/0510321•AIP conference proceedings•Jan 1, 2005•118 citations

Time-dependent Density-Matrix Renormalization-Group Methods

Ulrich Schollwöck

In this paper recent substantial progress in applying the density-matrix renormalization-group (DMRG) to the simulation of the time-evolution of strongly correlated quantum systems in one dimension is reviewed. Various approaches to generating a time-evolution numerically are considered. The key focus of this review is on current strategies to circumvent the limitations of the quite small subspace well approximated by DMRG, by either enlarging or changing it as time evolves. All these approaches can be extended to the simulation of mixed, i.e. finite-temperature states. While this paper is phrased almost entirely in standard DMRG language, I finish by considering the alternative formulation of time-evolutions in the language of matrix product states which is less well-known but conceptually more powerful.

Journal of the Physical Society of Japan•Jan 1, 2005•25 citations

Modelling and simulating strongly correlated fermions

Didier Poilblanc

I shall briefly review here some aspects of the Exact Diagonalization technique and put it in perspective with other numerical techniques such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo methods (partially) covered in this AIP Series by Reinhard Noack and Richard Scalettar respectively. I will then illustrate this lecture in discussing some simple applications, mainly for one dimensional systems.

AIP conference proceedings•Jan 1, 2007•0 citations

RECENT PROGRESS IN THE DENSITY-MATRIX RENORMALIZATION GROUP

Ulrich Schollwöck

Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.

International Journal of Modern Physics B•May 30, 2007•5 citations

Efficient Simulation of One-Dimensional Quantum Many-Body Systems

Guifré Vidal

We present a numerical method to simulate the time evolution, according to a generic Hamiltonian made of local interactions, of quantum spin chains and systems alike. The efficiency of the scheme depends on the amount of entanglement involved in the simulated evolution. Numerical analysis indicates that this method can be used, for instance, to efficiently compute time-dependent properties of low-energy dynamics in sufficiently regular but otherwise arbitrary one-dimensional quantum many-body systems. As by-products, we describe two alternatives to the density matrix renormalization group method.

arXiv:quant-ph/0310089•Physical Review Letters•Jul 19, 2004•1,771 citations

Current Trends in Density Matrix Renormalization Group Methods Ulrich Schollwo¨ck

Ulrich Schollwo¨ck Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Germany Since its invention in 1992 by White [1,2], the density matrix renormalization group (DMRG) has firmly established itself as the currently most powerful method to calculate static and more recently also dynamic [3-5] properties of low-lying eigenstates of strongly correlated Hamiltonians defined on onedimensional quantum lattices. At the same time, its extension to the analysis of two-dimensional classical [6] and one-dimensional quantum [7,8] transfer matrices has given access to highly precise thermodynamics in classical two-dimensional and quantum one-dimensional systems. It has even been extended to the numerically much more demanding study of non-Hermitian (pseudo-)Hamiltonians emerging in the analysis of classical steady states in one-dimensional systems [9-11]. However, it also turned out quite quickly that extending DMRG to the study of two-dimensional systems did not lead to similarly huge successes.

CRC Press eBooks•Nov 2, 2010•0 citations
The level of current understanding of the physics of time‐dependent strongly correlated quantum systems is far from complete, principally due to the lack of effective controlled approaches. Recently, there has been progress in the development of approaches for one‐dimensional systems. We describe recent developments in the construction of numerical schemes for general (one‐dimensional) Hamiltonians: in particular, schemes based on exact diagonalization techniques and on the density matrix renormalization group method (DMRG). We present preliminary results for spinless fermions with nearest‐neighbor‐interaction and investigate their accuracy by comparing with exact results.
In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.
In this paper recent substantial progress in applying the density-matrix renormalization-group (DMRG) to the simulation of the time-evolution of strongly correlated quantum systems in one dimension is reviewed. Various approaches to generating a time-evolution numerically are considered. The key focus of this review is on current strategies to circumvent the limitations of the quite small subspace well approximated by DMRG, by either enlarging or changing it as time evolves. All these approaches can be extended to the simulation of mixed, i.e. finite-temperature states. While this paper is phrased almost entirely in standard DMRG language, I finish by considering the alternative formulation of time-evolutions in the language of matrix product states which is less well-known but conceptually more powerful.
Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.
We present a numerical method to simulate the time evolution, according to a generic Hamiltonian made of local interactions, of quantum spin chains and systems alike. The efficiency of the scheme depends on the amount of entanglement involved in the simulated evolution. Numerical analysis indicates that this method can be used, for instance, to efficiently compute time-dependent properties of low-energy dynamics in sufficiently regular but otherwise arbitrary one-dimensional quantum many-body systems. As by-products, we describe two alternatives to the density matrix renormalization group method.
Ulrich Schollwo¨ck Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Germany Since its invention in 1992 by White [1,2], the density matrix renormalization group (DMRG) has firmly established itself as the currently most powerful method to calculate static and more recently also dynamic [3-5] properties of low-lying eigenstates of strongly correlated Hamiltonians defined on onedimensional quantum lattices. At the same time, its extension to the analysis of two-dimensional classical [6] and one-dimensional quantum [7,8] transfer matrices has given access to highly precise thermodynamics in classical two-dimensional and quantum one-dimensional systems. It has even been extended to the numerically much more demanding study of non-Hermitian (pseudo-)Hamiltonians emerging in the analysis of classical steady states in one-dimensional systems [9-11]. However, it also turned out quite quickly that extending DMRG to the study of two-dimensional systems did not lead to similarly huge successes.