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Density matrix renormalization group, 30 years on

Nature Reviews Physics volume 5, pages 273–276 (2023)Cite this article

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The density matrix renormalization group (DMRG) algorithm pioneered by Steven White in 1992 is a variational optimization algorithm that physicists use to find the ground states of Hamiltonians of quantum many-body systems in low dimensions. But DMRG is more than a useful numerical method, it is a framework that brought together ideas from theoretical condensed matter physics and quantum information, enabling advances in other fields such as quantum chemistry and the study of dissipative systems. It also fostered the development and widespread use of tensor networks as mathematical representations of quantum many-body states, whose applications now go beyond quantum systems. Today, it is one of the most powerful and widely used methods for simulating strongly correlated quantum many-body systems. Six researchers discuss the early history of DMRG and the developments it spurred over the past three decades.

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References

  1. Fannes, M., Nachtergaele, B. & Werner, R. F. Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992).

    Article  ADS  Google Scholar 

  3. Östlund, S. & Rommer, S. Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537 (1995).

    Article  ADS  Google Scholar 

  4. Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004).

    Article  ADS  Google Scholar 

  5. White, S. R. & Feiguin, A. E. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett. 93, 076401 (2004).

    Article  ADS  Google Scholar 

  6. Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Wilson, K. G. The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773–840 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  8. White, S. R. & Martin, R. L. Ab initio quantum chemistry using the density matrix renormalization group. J. Chem. Phys. 110.9, 4127–4130 (1999).

    Article  ADS  Google Scholar 

  9. Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).

    Article  Google Scholar 

  10. White, S. R. Early days of DMRG. Nat. Rev. Phys. https://doi.org/10.1038/s42254-023-00571-6 (2023).

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. DAMTP, University of Cambridge, Cambridge, UK

    Frank Verstraete

  2. Department of Physics, Ghent University, Ghent, The Netherlands

    Frank Verstraete

  3. Department of Physics, Graduate School of Science, Kobe University, Kobe, Japan

    Tomotoshi Nishino

  4. Ludwig-Maximilians-Universität, Munich, Germany

    Ulrich Schollwöck

  5. Munich Center for Quantum Science and Technology (MCQST), Munich, Germany

    Ulrich Schollwöck & Mari Carmen Bañuls

  6. Max Planck Institute of Quantum Optics, Garching, Germany

    Mari Carmen Bañuls

  7. Division of Chemistry and Chemical Engineering, Caltech, Pasadena, CA, USA

    Garnet K. Chan

  8. Flatiron Institute, New York City, NY, USA

    Miles E. Stoudenmire

Authors
  1. Frank Verstraete
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  2. Tomotoshi Nishino
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  3. Ulrich Schollwöck
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  4. Mari Carmen Bañuls
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  5. Garnet K. Chan
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  6. Miles E. Stoudenmire
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Corresponding authors

Correspondence to Frank Verstraete, Tomotoshi Nishino, Ulrich Schollwöck, Mari Carmen Bañuls, Garnet K. Chan or Miles E. Stoudenmire.

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The contributors

Frank Verstraete is the Leigh Trapnell Professor of Quantum Physics at the University of Cambridge and Professor at Ghent University. His research is focused on the mathematical and computational aspects of tensor networks, and more generally on the quantum simulation problem.

Tomotoshi Nishino is Associate Professor of Physics at Kobe University. He received his PhD from Osaka University in 1993 and was Assistant Professor at Tohoku University. He has been listing tensor-network-related arXiv preprints on his home page since 1998 and works on statistical properties of lattice models and their critical behaviours.

Ulrich Schollwöck is Full Professor of Theoretical Physics at the University of Munich. He studied at Munich and Oxford, and received his PhD from the University of Paris-Saclay in 1995. He has authored two big reviews on DMRG and MPS and works on a range of topics in condensed matter physics, quantum many-body physics, computational physics and ultracold atomic gases.

Mari Carmen Bañuls is group leader at the Max Planck Institute of Quantum Optics. She received a PhD in physics from Valencia University in 2000 and in computer science from the Technical University of Valencia in 2006. She works on the development and application of tensor-network algorithms for a variety of problems.

Garnet Kin-Lic Chan is the Bren Professor of Chemistry at the California Institute of Technology. He is interested in problems that lie at the intersection of quantum chemistry, condensed matter physics and quantum information.

Miles E. Stoudenmire is a research scientist at the Flatiron Institute, Center for Computational Quantum Physics, where he leads the Tensor Networks project and the development of the ITensor software.

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Verstraete, F., Nishino, T., Schollwöck, U. et al. Density matrix renormalization group, 30 years on. Nat Rev Phys 5, 273–276 (2023). https://doi.org/10.1038/s42254-023-00572-5

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