Density Matrix Renormalization Group Research Articles

Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet.

We propose a simple family of valence-bond crystals as potential ground states of the S=1/2 and S=1 Heisenberg antiferromagnet on the pyrochlore lattice. Exponentially numerous in the linear size of the system, these can be visualized as hard-hexagon coverings, with each hexagon representing a resonating valence-bond ring. This ensemble spontaneously breaks rotation, inversion, and translation symmetries. A simple, yet accurate, variational wave function allows a precise determination of the energy, confirmed by the density matrix renormalization group and numerical linked cluster expansion, and extended by an analysis of excited states. The identification of the origin of the stability indicates applicability to a broad class of frustrated lattices, which we demonstrate for the checkerboard and ruby lattices. Our work suggests a perspective on such quantum magnets, in which unfrustrated motifs are effectively uncoupled by the frustration of their interactions.

Combining the in-medium similarity renormalization group with the density matrix renormalization group: Shell structure and information entropy

We propose a novel many-body framework combining the density matrix renormalization group (DMRG) with the valence-space (VS) formulation of the in-medium similarity renormalization group. This hybrid scheme admits for favorable computational scaling in large-space calculations compared to direct diagonalization. The capacity of the VS-DMRG approach is highlighted in ab initio calculations of neutron-rich nickel isotopes based on chiral two- and three-nucleon interactions, and allows us to perform converged ab initio computations of ground and excited state energies. We also study orbital entanglement in the VS-DMRG, and investigate nuclear correlation effects in oxygen, neon, and magnesium isotopes. The explored entanglement measures reveal nuclear shell closures as well as pairing correlations.

Minimizing non-radiative decay in molecular aggregates through control of excitonic coupling

The widely known “Energy Gap Law” (EGL) predicts a monotonically exponential increase in the non-radiative decay rate (knr) as the energy gap narrows, which hinders the development of near-infrared (NIR) emissive molecular materials. Recently, several experiments proposed that the exciton delocalization in molecular aggregates could counteract EGL to facilitate NIR emission. In this work, the nearly exact time-dependent density matrix renormalization group (TD-DMRG) method is developed to evaluate the non-radiative decay rate for exciton-phonon coupled molecular aggregates. Systematical numerical simulations show, by increasing the excitonic coupling, knr will first decrease, then reach a minimum, and finally start to increase to follow EGL, which is an overall result of two opposite effects of a smaller energy gap and a smaller effective electron-phonon coupling. This anomalous non-monotonic behavior is found robust in a number of models, including dimer, one-dimensional chain, and two-dimensional square lattice. The optimal excitonic coupling strength that gives the minimum knr is about half of the monomer reorganization energy and is also influenced by system size, dimensionality, and temperature.

Magnetic-field-driven rise and fall of a bipartite entanglement in a spin-liquid phase of a spin-1/2 Heisenberg branched chain

The spin-1/2 Heisenberg branched chain with the unit cell composed of three spins in the main backbone and one spin at a side branching of one-dimensional chain is investigated with the help of density-matrix renormalization group (DMRG) and quantum Monte Carlo (QMC) methods. The DMRG simulations were employed to calculate zero-temperature magnetization curves and to construct the ground-state phase diagram, which is composed from four different ground states classified as gapped zero-plateau and one-half plateau phase, a gapless spin-liquid phase and a fully saturated phase. It is shown that the one-half magnetization plateau vanishes at the Kosterlitz–Thouless quantum critical point, which is wedged into a parameter space of the gapless quantum spin-liquid phase. The bipartite quantum entanglement between four distinct nearest-neighbor pairs of the spin-1/2 Heisenberg branched chain is quantified through the concurrence. It is found that the concurrence varies continuously within the quantum spin-liquid phase, where it may display a continuous rise, a continuous fall or eventually an intriguing rise-and-fall behavior. On the contrary, the concurrence is kept constant within two gapful zero- and one-half plateau phases. Temperature and magnetic-field dependencies of the magnetization and magnetic susceptibility computed within the QMC method uncover clear signatures of the quantum critical point at finite temperatures.

Quantum Circuit Matrix Product State Ansatz for Large-Scale Simulations of Molecules.

As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.

Equation-of-Motion Block-Correlated Coupled Cluster Method for Excited Electronic States of Strongly Correlated Systems.

An equation-of-motion block-correlated coupled cluster method based on the generalized valence bond wave function (EOM-GVB-BCCC) is proposed to describe low-lying excited states for strongly correlated systems. The EOM-GVB-BCCC2b method with up to two-pair correlation has been implemented and tested for a few strongly correlated systems. For a water hexamer with stretched O-H bonds, which is beyond the capability of the CASSCF method, EOM-GVB-BCCC2b provides very close results as the density matrix renormalization group (DMRG). For four conjugated diradical species with triplet ground states, we found that their vertical S-T gaps from EOM-GVB-BCCC2b are also quite consistent with the DMRG results. This new method is expected to be a promising theoretical tool for describing the low-lying excited states of strongly correlated systems with large active spaces.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments.

Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (101-2) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (102-3) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein-Nesbet perturbation theory (ENPT), without recourse to the expensive high order n-electron reduced density matrices (n-RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system-environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.

Fermion Disorder Operator at Gross-Neveu and Deconfined Quantum Criticalities.

The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang etal., arXiv:2209.07103]. Here we study, by means of large-scale quantum MonteCarlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.

Three Lagrangians for the complete-active space coupled-cluster method.

Three fully variational formulations of the complete-active space coupled-cluster method are derived. The formulations include the ability to approximate the model vectors by smooth manifolds, thereby opening up the possibility for overcoming the exponential wall of scaling for model spaces of complete-active space type. In particular, model vectors of matrix-product states are considered, and it is argued that the present variational formulation allows not only favorably scaling multireference coupled-cluster calculations but also systematic correction of tailored coupled-cluster calculations and of quantum chemical density-matrix renormalization group methods, which are fast and polynomial scaling but lack the ability to properly resolve dynamical correlation at chemical accuracy. The extension of the variational formulations to the time domain is also discussed, with derivations of abstract evolution equations.

Multiversality and Unnecessary Criticality in One Dimension.

We present microscopic models of spin ladders which exhibit continuous critical surfaces whose properties and existence, unusually, cannot be inferred from those of the flanking phases. These models exhibit either "multiversality"-the presence of different universality classes over finite regions of a critical surface separating two distinct phases-or its close cousin, "unnecessary criticality"-the presence of a stable critical surface within a single, possibly trivial, phase. We elucidate these properties using Abelian bosonization and density-matrix renormalization-group simulations, and attempt to distill the key ingredients required to generalize these considerations.

Observation of a magnetic phase transition in monolayer NiPS3

Monolayer magnet of $\mathit{XY}$ type composes a pivotal part of the two-dimensional magnetism. As an important suspected $\mathit{XY}$-type antiferromagnet, whether $1L \mathrm{NiP}{\mathrm{S}}_{3}$ exhibits magnetic order remains elusive. Herein by helicity-resolved Raman and ultrafast spectroscopy of $\mathrm{NiP}{\mathrm{S}}_{3}$ from bulk to monolayer, we find that $1L \mathrm{NiP}{\mathrm{S}}_{3}$ is magnetically ordered with a Berezinskii-Kosterlitz-Thouless (BKT) transition at ${T}_{\mathrm{BKT}}\ensuremath{\approx}140$ K. We have also performed large-scale density-matrix renormalization-group calculations to verify the ground state zigzag antiferromagnetic order and Monte Carlo simulation to confirm the BKT-transition temperature in $1L \mathrm{NiP}{\mathrm{S}}_{3}$. Our investigations establish $1L \mathrm{NiP}{\mathrm{S}}_{3}$ to be an $\mathit{XY}$ antiferromagnet with a relatively high-temperature BKT transition, providing an important platform for investigating complex couplings and topological excitations.

Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs

DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state $\mathrm{\ensuremath{\Psi}}$ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in $H\mathrm{\ensuremath{\Psi}}$ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.

Boundary quantum phase transitions in the spin- 12 Heisenberg chain with boundary magnetic fields

We consider the spin $\frac{1}{2}$ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin $\frac{1}{4}$, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground state phase transition, we find that the system may undergo an eigenstate phase transition (EPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground state properties by computing the magnetization profile $\langle S^z_j \rangle$ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.

Quantum Criticality and Universal Behavior in Molecular Dipolar Lattices of Endofullerenes

Fullerene cages allow the confinement of single molecules and the construction of molecular assemblies whose properties strongly differ from those of free species. In this work, we employ the density-matrix renormalization group method to show that chains of fullerenes filled with polar molecules (LiF, HF, and H2O) can form dipole-ordered quantum phases. In symmetry broken environments, these ordered phases are ferroelectric, a property that makes them promising candidates for quantum devices. We demonstrate that for a given guest molecule, the occurrence of these quantum phases can be enforced or influenced either by changing the effective electric dipole moment or by isotopic substitution. In the ordered phase, all systems under consideration are characterized by universal behavior that depends only on the ratio of the effective electric dipole and of the rotational constant. A phase diagram is derived, and further molecules are proposed as candidates for dipole-ordered endofullerene chains.

Pair density wave in the doped three-band Hubbard model on two-leg square cylinders

A pair density wave (PDW) is a superconducting (SC) state with a spatially modulated order parameter. Although much is known about the properties of the PDW state, its realization in microscopic models with divergent susceptibility has been challenging. Here we report a density-matrix renormalization group study of a three-band Hubbard model (also known as the Emery model) for cuprates on long two-leg square cylinders. Upon light doping, we find that the ground state of the system is consistent with that of a PDW state with mutually commensurate and power-law SC, charge (CDW), and spin (SDW) density wave correlations. The SC correlations are dominant between neighboring Cu sites with $d$-wave pairing symmetry. Interestingly, we find that the near-neighbor interactions, especially the near-neighbor attractive ${V}_{pd}$ interaction between neighboring Cu and oxygen sites, can notably enhance the SC correlations while simultaneously suppressing the CDW correlations. For a modestly strong attractive ${V}_{pd}$, the SC correlations become quasi-long-ranged with a divergent PDW susceptibility.

Superconductivity in doped triangular Mott insulators: The roles of parent spin backgrounds and charge kinetic energy

We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e., ${120}^{\ensuremath{\circ}}$ antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ of nearest-neighbor hopping and next-nearest-neighbor hopping $({t}_{1},{t}_{2})$. Based on density matrix renormalization group calculations, we find that, with finite ${t}_{2}$ and specific sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, these superconductivity phases in triangular lattices are commonly characterized by short-ranged spin correlations and two charges per stripe. In the robust superconductivity phase realized at larger ${t}_{2}/{t}_{1}$, flipping the signs ${\ensuremath{\tau}}_{2}$ and ${\ensuremath{\tau}}_{1}$ gives rise to the stripe phase without strong pairing and a pseudogaplike phase without Cooper-pair phase coherence, respectively. Interestingly, the roles of the two hopping signs are switched at smaller ${t}_{2}/{t}_{1}$. Moreover, different sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ would stabilize distinct phases including superconductivity, charge density waves, spin density waves, and pseudogaplike phases accordingly. Our findings suggest the important role of charge kinetic energy in realizing superconductivity in doped triangular-lattice Mott insulators.

Ground-state and spectral properties of the doped one-dimensional optical Hubbard-Su-Schrieffer-Heeger model

We present a density matrix renormalization group study of the doped one-dimensional (1D) Hubbard-Su-Schrieffer-Heeger (Hubbard-SSH) model, where the atomic displacements linearly modulate the nearest-neighbor hopping integrals. Focusing on an optical variant of the model in the strongly correlated limit relevant for cuprate spin chains, we examine how the SSH interaction modifies the model's ground- and excited-state properties. The SSH coupling weakly renormalizes the model's single- and two-particle response functions for electron-phonon $(e\text{\ensuremath{-}}\mathrm{ph})$ coupling strengths below a parameter-dependent critical value ${g}_{\mathrm{c}}$. For larger $e\text{\ensuremath{-}}\mathrm{ph}$ coupling, the sign of the effective hopping integrals changes for a subset of orbitals, which drives a lattice dimerization distinct from the standard nesting-driven picture in 1D. The spectral weight of the one- and two-particle dynamical response functions are dramatically rearranged across this transition, with significant changes in the ground-state correlations. We argue that this dimerization results from the breakdown of the linear approximation for the $e\text{\ensuremath{-}}\mathrm{ph}$ coupling and thus signals a fundamental limitation of the linear SSH interaction. Our results have consequences for our understanding of how SSH-like interactions can enter the physics of strongly correlated quantum materials, including the recently synthesized doped cuprate spin chains.

Non-Abelian anyons with Rydberg atoms

We study the emergence of topological matter in two-dimensional systems of neutral Rydberg atoms in Ruby lattices. While Abelian anyons have been predicted in such systems, non-Abelian anyons, which would form a substrate for fault-tolerant quantum computing, have not been generated. To generate anyons with non-Abelian braiding statistics, we consider systems with mixed-boundary punctures. We obtain the topologically distinct ground states of the system numerically using the infinite Density Matrix Renormalization Group technique. We discuss how these topological states can be created using ancilla atoms of a different type. We show that a system with $2N+2$ punctures and an equal number of ancilla atoms leads to $N$ logical qubits whose Hilbert space is determined by a set of stabilizing conditions on the ancilla atoms. Quantum gates can be implemented using a set of gates acting on the ancilla atoms that commute with the stabilizers and realize the braiding group of non-Abelian Ising anyons.

Triangular lattice Hubbard model physics at intermediate temperatures

Moire systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on $\text{WSe}_2/\text{WS}_2$ moire bilayers [Y. Tang et al., Nature 579, 353-358 (2020)], which realizes a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t \lesssim T < U$. Employing finite temperature Lanczos calculations, and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $\Theta$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $|\Theta|$ on decreasing the density below half filling and the sign change of $\Theta$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|\Theta|$ is also seen in the $t$-$J$ model (the low energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero temperature phase transition, which we explore with ground state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $\Theta$ seen in the experiment.

High-Performance Computing for Density Matrix Renormalization Group

Abstract: In the last decades, many algorithms have been developed to use high-performance computing (HPC) techniques to accelerate the density matrix renormalization group (DMRG) method, an effective method for solving large active space strong correlation problems. In this article, the previous DMRG parallelization algorithms at different levels of the parallelism are introduced. The heterogeneous computing acceleration methods and the mixed-precision implementation are also presented and discussed. This mini-review concludes with some summary and prospects for future works.