We introduce a Lagrangian formulation of the Density Matrix Renormalization Group (DMRG). We present Lagrangians which when minimised yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz, as well as within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.
A Hamiltonian renormalization group is presented. Such a formulation is relevant for chiralic systems and more appropriate than the Lagrangian formalism. An application to 1D system is presented.
We present the perturbation theory for lattice magnon fields of $D$-dimensional O(3) Heisenberg ferromagnet. The effective Hamiltonian for the lattice magnon fields is obtained starting from the effective Lagrangian, with two dominant contributions that describe magnon-magnon interactions identified as a usual gradient term for the unit vector field and a part originating in the Wess-Zumino-Witten term of effective Lagrangian. Feynman diagrams for lattice scalar fields with derivative couplings are introduced, on basis of which we investigate the influence of magnon-magnon interactions on magn…
In this paper we give an introduction to the numerical density matrix renormalization group (DMRG) algorithm, from the perspective of the more general matrix product state (MPS) formulation. We cover in detail the differences between the original DMRG formulation and the MPS approach, demonstrating the additional flexibility that arises from constructing both the wavefunction and the Hamiltonian in MPS form. We also show how to make use of global symmetries, for both the Abelian and non-Abelian cases.
We present an overview of the Density Matrix Renormalization Group and its connections to Quantum Groups, Matrix Products and Conformal Field Theory. We emphasize some common formal structures in all these theories. We also propose two-dimensional extensions of the variational matrix product ansatzs.
The generalization of Density Matrix Renormalization Group (DMRG) approach as implemented in quantum chemistry, to the case of non-orthogonal orbitals is carefully analyzed. This generalization is attractive from the physical point of view since it allows a better localization of the orbitals. The possible implementation difficulties and drawbacks are estimated. General formulae for hamiltonian matrix elements useful in DMRG calculations are given.
The purpose of this paper is (i) to present a generic and fully functional implementation of the density-matrix renormalization group (DMRG) algorithm, and (ii) to describe how to write additional strongly-correlated electron models and geometries by using templated classes. Besides considering general models and geometries, the code implements Hamiltonian symmetries in a generic way and parallelization over symmetry-related matrix blocks.
Several density-matrix renormalization group methods have been proposed to compute the momentum- and frequency-resolved dynamical correlation functions of low-dimensional strongly correlated systems. The most relevant approaches are discussed in this contribution. Their applications in various studies of quasi-one-dimensional strongly correlated systems (spin chains, itinerant electron systems, electron-phonon systems) are reviewed.
We describe here the extension of the density matrix renormalization group algorithm to the case where Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the one-dimensional Hubbard model as the symmetry group is enlarged from $U(1) \times U(1)$, up to $SU(2) \times SU(2)$…
We develop a correction to the density matrix used in density matrix renormalization group calculations to take into account the incompleteness of the environment block. The correction allows successful calculations using only a single site in the center of the system, rather than the standard two sites, improving typical computation times by a factor of two to four. In addition, in many cases where ordinary DMRG can get stuck in metastable configurations, the correction eliminates the sticking. We test the new method on the Heisenberg S=1 chain.
We propose a density matrix renormalization group approach to tackle a two-state system coupled to a bosonic bath with continuous spectrum. In this approach, the optimized phonon scheme is applied to several hundred phonon modes which are divided linearly among the spectrum. Although DMRG cannot resolve very small energy scales, the delocalized-localized transition points of the two-state system are extracted by the extrapolation of the flow diagram results. The phase diagram is compared with the numerical renormalization group results and shows good agreement in both Ohmic and sub-Ohmic cases…
We show that the N-patch functional renormalization group (pFRG), a theoretical method commonly applied for correlated electron systems, is unable to implement consistently the matrix element interference arising from strong momentum dependence in the Bloch state contents or the interaction vertices. We show that such a deficit could lead to results incompatible with checkable limits of weak and strong coupling. We propose that the pFRG could be improved by a better account of momentum conservation.
By using a different quantum-to-classical mapping from the Trotter-Suzuki decomposition, we identify the entanglement structure of the maximal eigenvectors for the associated quantum transfer matrix. This observation provides a deeper insight into the problem of linear growth of the entanglement entropy in time evolution using conventional methods. Based on this observation, we propose a general method for arbitrary temperatures using the biorthonormal transfer-matrix renormalization group. Our method exhibits a competitive accuracy with a much cheaper computational cost in comparison with two…
We propose a density matrix renormalization group (DMRG) technique at finite temperatures. As is the case of the ground state DMRG, we use a single-target state that is calculated by making use of a regulated polynomial expansion. Both static and dynamical quantities are obtained after a random-sampling and averaging procedure. We apply this technique to the one-dimensional Hubbard model at half filling and find that this gives excellent results at low temperatures.
We apply a generalization of the time-dependent DMRG to study finite temperature properties of several quantum spin chains, including the frustrated $J_1-J_2$ model. We discuss several practical issues with the method, including use of quantum numbers and finite size effects. We compare with transfer-matrix DMRG, finding that both methods produce excellent results.
We propose an analytic response theory for the density matrix renormalization group whereby response properties correspond to analytic derivatives of density matrix renormalization group observables with respect to the applied perturbations. Both static and frequency-dependent response theories are formulated and implemented. We evaluate our pilot implementation by calculating static and frequency dependent polarizabilities of short oligo-di-acetylenes. The analytic response theory is competitive with dynamical density matrix renormalization group methods and yields significantly improved accu…
Wilson's numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White's density matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model to a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an ``unfolde…
We extended the Density Matrix Renormalization Group method to study the real time dynamics of interacting one dimensional spinless Fermi systems by applying the full time evolution operator to an initial state. As an example we describe the propagation of a density excitation in an interacting clean system and the transport through an interacting nano structure.
We describe how density-matrix renormalization group (DMRG) can be used to solve the full-CI problem in quantum chemistry. As an illustration of the potential of this method, we apply it to a paramagnetic molecule. In particular, we show the effect of various basis set, the scaling as the fourth power of the size of the problem, and compare the DMRG with other methods.
A kind of least action principle is introduced for the discrete time evolution of one-dimensional quantum lattice models. Based on this principle, we obtain an optimal condition for the matrix product states on succeeding time slices generated by the real-time density matrix renormalization group method. This optimization can also be applied to classical simulations of quantum circuits. We discuss the time reversal symmetry in the fully optimized MPS.
The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model (SI…
We describe an extension to the density matrix renormalization group method incorporating real time evolution into the algorithm. Its application to transport problems in systems out of equilibrium and frequency dependent correlation functions is discussed and illustrated in several examples. We simulate a scattering process in a spin chain which generates a spatially non-local entangled wavefunction.
Renormalization of non-magnetic and magnetic impurities due to electron double occupancy prohibition is derived analytically by an improved Gutzwiller approximation. Non-magnetic impurities are effectively weakened by the same renormalization factor as that for the hopping amplitude, whereas magnetic impurities are strengthened by the square root of the spin-exchange renormalization factor, in contrast to results by the conventional Gutzwiller approximation. We demonstrate it by showing that transition matrix elements of number operators between assumed excited states and between an assumed gr…
We extend the active space decomposition method, recently developed by us, to more than two active sites using the density matrix renormalization group algorithm. The fragment wave functions are described by complete or restricted active-space wave functions. Numerical results are shown on a benzene pentamer and a perylene diimide trimer. It is found that the truncation errors in our method decrease almost exponentially with respect to the number of renormalization states M, allowing for numerically exact calculations (to a few microhartrees or less) with M = 128 in both cases. This rapid conv…
We present a novel numerical method for the evaluation of dynamical response functions at finite temperatures in one-dimensional strongly correlated systems. The approach is based on the density-matrix renormalization group method, combined with the finite-temperature Lanczos diagonalization. The feasibility of the method is tested on the example of dynamical spin correlations in the anisotropic Heisenberg chain, in particular it yields nontrivial results for the critical behavior in the isotropic case.
The particle-hole version of the density-matrix renormalization-group method (PH-DMRG) is utilized to calculate the ground-state energy of an interacting two-dimensional quantum dot. We show that a modification of the method, termed generation-based PH-DMRG, leads to significant improvement of the results, and discuss its feasibility for the treatment of large systems. As another application we calculate the addition spectrum.
It is conjectured that the Haldane phase of the S=1 antiferromagnetic Heisenberg chain and the $S=1/2$ ferromagnetic-antiferromagnetic alternating Heisenberg chain is stable against any strength of randomness, because of imposed breakdown of translational symmetry. This conjecture is confirmed by the density matrix renormalization group calculation of the string order parameter and the energy gap distribution.
Path integral techniques for the density matrix of a one-dimensional statistical system near a boundary previously employed in black-hole physics are applied to providing a new interpretation of the density matrix renormalization group: its efficacy is due to the concentration of quantum states near the boundary.
A distributed-memory parallelization strategy for the density matrix renormalization group is proposed for cases where correlation functions are required. This new strategy has substantial improvements with respect to previous works. A scalability analysis shows an overall serial fraction of 9.4% and an efficiency of around 60% considering up to eight nodes. Sources of possible parallel slowdown are pointed out and solutions to circumvent these issues are brought forward in order to achieve a better performance.
We employ the density matrix renormalization group (DMRG) and the wave function factorization method for the numerical solution of large scale nuclear structure problems. The DMRG exhibits an improved convergence for problems with realistic interactions due to the implementation of the finite algorithm. The wave function factorization of fpg-shell nuclei yields rapidly converging approximations that are at the present frontier for large-scale shell model calculations.
We study the ground state quantum phase transition by means of entanglement in the one-dimensional asymmetric Hubbard model with open boundary condition. The local entanglement between the middle two sites and the rest of the system, and the block entanglement between the left and right portions of the system, are calculated using the density-matrix renormalization group (DMRG) method. We find that the entanglement shows interesting scaling and singular behavior around the phase transition line.
We generalize the recently introduced Density-Matrix Renormalization Group (DMRG-X) [Khemani et al, PRL 2016] algorithm to obtain Floquet eigenstates of one-dimensional, periodically driven many-body localized systems. This generalization is made possible by the fact that the time-evolution operator for a period can be efficiently represented using a matrix-product operator. We first benchmark the method by comparing to exact diagonalization for small systems. We then obtain Floquet eigenstates for larger systems and show unambiguously that the characteristic area-law scaling remains robust.
This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.
We present a new hybrid multiconfigurational method based on the concept of range-separation that combines the density matrix renormalization group approach with density functional theory. This new method is designed for the simultaneous description of dynamical and static electron-correlation effects in multiconfigurational electronic structure problems.
In some cases the state of a quantum system with a large number of subsystems can be approximated efficiently by the density matrix renormalization group, which makes use of redundancies in the description of the state. Here we show that the achievable efficiency can be much better when performing density matrix renormalization group calculations in the Heisenberg picture, as only the observable of interest but not the entire state is considered. In some non-trivial cases, this approach can even be exact for finite bond dimensions.
We present the first implementation of the relativistic quantum chemical two- and four-component density matrix renormalization group algorithm that includes a variational description of scalar-relativistic effects and spin-orbit coupling. Numerical results based on the four-component Dirac-Coulomb Hamiltonian are presented for the standard reference molecule for correlated relativistic benchmarks: thallium hydride.
We present the first implementation of a density matrix renormalization group algorithm embedded in an environment described by density functional theory. The frozen density embedding scheme is used with a freeze-and-thaw strategy for a self-consistent polarization of the orbital-optimized wavefunction and the environmental densities with respect to each other.
We have proposed a density-matrix renormalization group (DMRG) scheme to optimize the one-electron basis states of molecules. It improves significantly the accuracy and efficiency of the DMRG in the study of quantum chemistry or other many-fermion system with nonlocal interactions. For a water molecule, we find that the ground state energy obtained by the DMRG with only 61 optimized orbitals already reaches the accuracy of best quantum Monte Carlo calculation with 92 orbitals.
We present an implementation of the relativistic quantum-chemical density matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin-orbit coupling from which we can calculate, for example, first-order electric and magnetic properties in a relativistic framework. While complementing our pilot implementation (S. Knecht et al., J. Chem. Phys., 140, 041101 (2014)) this work exploits all features provided by its underlying n…
