Why is it called the density matrix renormalization group?

Photo of Thomas E. Baker
Thomas E. Baker
University of York, United Kingdom

Solving models in quantum physics is of paramount importance to understanding a variety of physical phenomena.  There are many methods that have been created to do this. Tensor networks are one strong candidate that have been extremely popular in solving models with local interactions [1,2].  These methods are motivated from a combination of information theory and renormalization group techniques.  They also naturally compute the entanglement of a given model, making them useful for studies of topological physics. In comparison with quantum Monte Carlo methods, they have no sign problem.  In comparison with exact diagonalization, the method circumvents the exponentially sized memory required and can be applied to hundreds or thousands of sites with a controlled error.

In this talk, I introduce a particular tensor network method known as the density matrix renormalization group [3]. This method works well in one dimension with local models, which apply to physically relevant models as provable with Kohn’s nearsightedness principle [4]. The method is applicable to many-body Hamiltonians which involve a quartic interaction term.  The method can be applied to ab initio systems, although the limitations of the method often prevent an efficient solution for general systems. I will review the background theory of this method and explain its context for quantum chemistry systems [5,6]. All concepts are explained through the matrix product state formalism [7].

[1] Thomas E. Baker, Samuel Desrosiers, Maxime Tremblay, Martin Thompson “Méthodes de calcul avec réseaux de tenseurs en physique” Canadian Journal of Physics 99, 4 (2021); “Basic tensor network computations in physics” arxiv: 1911.11566

[3] Steven R. White, “Density matrix formulation for quantum renormalization groups” Phys. Rev. Lett. 69, 19 (1992)

[4] M.B. Hastings, “Locality in quantum and Markov dynamics on lattices and networks” Phys. Rev. Lett. 93, 140402 (2004)

[5] S.R. White, R.L. Martin "Ab initio quantum chemistry using the density matrix renormalization group" 110, 9 (1999)

[6] G.K.L. Chan, M. Head-Gordon, “Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group" J. Chem. Phys. 116, 11 (2002)

[7] Stellan Östlund and Stefan Rommer, “Thermodynamic limit of density matrix renormalization,” Phys. Rev. Lett. 75, 3537 (1995)