New trends in density matrix renormalization

Abstract

The Density Matrix Renormalization Group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamic and thermodynamic properties. Its field of applicability has now extended beyond Condensed Matter, and is successfully used in Quantum Chemistry, Statistical Mechanics, Quantum Information Theory, Nuclear and High Energy Physics as well. In this article, we briefly review the main aspects of the method and present some of the most relevant applications so as to give an overview on the scope and possibilities of DMRG. We focus on the most important extensions of the method such as the calculation of dynamical properties, the application to classical systems, finite temperature simulations, phonons and disorder, field theory, time-dependent properties and the ab initio calculation of electronic states in molecules. The recent quantum information interpretation, the development of highly accurate time-dependent algorithms and the possibility of using the DMRG as the impurity-solver of the Dynamical Mean Field Method (DMFT) give new insights into its present and potential uses. We review the numerous very recent applications of these techniques where the DMRG has shown to be one of the most reliable and versatile methods in modern computational physics.

Self-assembling tensor networks and holography in disordered spin chains

“…In contrast to traditional vMPS approaches to disordered systems, where the initial geometry of the MPS ignores the disorder and only takes it into account at the stage of variational sweeps [45], our approach incorporates the disorder into the fabric of its tensor network. We believe this strategy to be inherently more suited to disordered systems-the results presented here show that the accuracy of tSDRG is already comparable to vMPS without including any additional variational updates.…”

Scaling of entanglement support for matrix product states

“…There, the recent algorithms introduced by Vidal 1 based on the explicit use of Schmidt decompositions have been shown to deliver identical results to the very successful density matrix renormalization group ͑DMRG͒. [2][3][4] Actually, these two apparently wide-apart algorithms agree because they come down to represent the coefficients of a quantum state as a product of matrices, which is a matrix product state ͑MPS͒, 5 where s i labels a basis for the local degree of freedom ͑"spin"͒ of particle i, A i ͑s i ͒'s are matrices of some fixed finite size, , and N is the number of sites in the chain which will be taken to be infinite. 8 Under the assumption that the abovementioned algorithms do find a faithful description of the sought state, consistent with the MPS structure, we can forget about their details and describe their results as a consequence of the properties of MPS states.…”

Vector chiral and multipolar orders in the spin-12frustrated ferromagnetic chain in magnetic field

“…We fit the local spin polarization s z l and the longitudinal spin fluctuation correlation s z l s z l ′ − s z l s z l ′ to Eqs. (36) and (44) with p = 3, respectively, taking η and a as fitting parameters. We find that these correlators are fitted quite well by the formulas.…”

“…29,30,31,32 In this paper we concentrate on the ferromagnetic case (J 1 < 0) of the J 1 -J 2 spin chain (1) in magnetic field which partially polarizes spins to the +z direction. We show that the ground-state phase diagram in the case is a zoo of exotic quantum phases, using the numerical density-matrix renormalization group (DMRG) method, 33,34,35,36 exact-diagonalization method, and effective field theories. We find a phase with long-range vector chiral order and phases with various kinds of multipolar spin correlations, most of which have not been known to appear in this model.…”