New Perspectives for Time-Evolution with DMRG

Jutho Haegeman, Christian Lubich, Ivan Valer'evich Oseledets +2 more

We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.

Bassano Vacchini

Alberto Baiardi, Markus Reiher

In the past two decades, the density matrix renormalization group (DMRG) has emerged as an innovative new method in quantum chemistry relying on a theoretical framework very different from that of traditional electronic structure approaches. The development of the quantum chemical DMRG has been remarkably fast: it has already become one of the reference approaches for large-scale multiconfigurational calculations. This perspective discusses the major features of DMRG, highlighting its strengths and weaknesses also in comparison with other novel approaches. The method is presented following its historical development, starting from its original formulation up to its most recent applications. Possible routes to recover dynamical correlation are discussed in detail. Emerging new fields of applications of DMRG are explored, such as its time-dependent formulation and the application to vibrational spectroscopy.

Emilio d’Emilio, Luigi E. Picasso

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.

K. Tamvakis