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We introduce a versatile and practical framework for applying matrix product state techniques to continuous quantum systems. We divide space into multiple segments and generate continuous basis functions for the many-body state in each segment. By combining this mapping with existing numerical Density-Matrix Renormalization Group routines, we show how one can accurately obtain the ground-state wave function, spatial correlations, and spatial entanglement entropy directly in the continuum. For a prototypical mesoscopic system of strongly-interacting bosons we demonstrate faster convergence than standard grid-based discretization. We illustrate the power of our approach by studying a superfluid-insulator transition in an external potential. We outline how one can directly apply or generalize this technique to a wide variety of experimentally relevant problems across condensed matter physics and quantum field theory.
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We report a systematic study of the fractional quantum Hall effect (FQHE) using the density-matrix renormalization group (DMRG) method on two different geometries: the sphere and the cylinder. We provide convergence benchmarks based on model Hamiltonians known to possess exact zero-energy ground states, as well as an analysis of the number of sweeps and basis elements that need to be kept in order to achieve the desired accuracy.The ground state energies of the Coulomb Hamiltonian at $ u=1/3$ and $ u=5/2$ filling are extracted and compared with the results obtained by previous DMRG implementations in the literature. A remarkably rapid convergence in the cylinder geometry is noted and suggests that this boundary condition is particularly suited for the application of the DMRG method to the FQHE.
A quantum dot coupled to ferromagnetically polarized one-dimensional leads is studied numerically using the density matrix renormalization group method. Several real space properties and the local density of states at the dot are computed. It is shown that this local density of states is suppressed by the parallel polarization of the leads. In this case we are able to estimate the length of the Kondo cloud, and to relate its behavior to that suppression. Another important result of our study is that the tunnel magnetoresistance as a function of the quantum dot on-site energy is minimum and negative at the symmetric point.
The numerical study of anyonic systems is known to be highly challenging due to their non-bosonic, non-fermionic particle exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behaviour as a result. Meanwhile, the density matrix renormalisation group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging both in terms of programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground state energy of the Golden Chain, which has become the benchmark system for the numerical study of anyons, and is shown to produce results comparable to those of the anyonic TEBD algorithm and superior to the variationally optimised anyonic MERA, at far lesser computational cost.
We improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems.The dynamical DMRG is used to compute the linear response of a finite system to an applied AC source-drain voltage, then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the DC conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theory such as the renormalization of the conductance in an homogeneous conductor, the universal effects of a single barrier, and the resonant tunneling through a double barrier.
We introduce the transcorrelated Density Matrix Renormalization Group (tcDMRG) theory for the efficient approximation of the energy for strongly correlated systems. tcDMRG encodes the wave function as a product of a fixed Jastrow or Gutzwiller correlator and a matrix product state. The latter is optimized by applying the imaginary-time variant of time-dependent (TD) DMRG to the non-Hermitian transcorrelated Hamiltonian. We demonstrate the efficiency of tcDMRG at the example of the two-dimensional Fermi-Hubbard Hamiltonian, a notoriously difficult target for the DMRG algorithm, for different sizes, occupation numbers, and interaction strengths. We demonstrate fast energy convergence of tcDMRG, which indicates that tcDMRG could increase the efficiency of standard DMRG beyond quasi-monodimensional systems and provides a generally powerful approach toward the dynamic correlation problem of DMRG.
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