Density Matrix Renormalization Group Approach Research Articles

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

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.

Plaquette valence bond state in the spin- 12 J1−J2 XY model on the square lattice

We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at ${J}_{2}/{J}_{1}<0.50$ and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.

Non-Hermitian topological quantum states in a reservoir-engineered transmon chain

Dissipation in open systems enriches the possible symmetries of the Hamiltonians beyond the Hermitian framework, allowing the possibility of novel non-Hermitian topological phases which exhibit long-living end states that are protected against disorder. So far, non-Hermitian topology has been explored in settings where probing genuine quantum effects has been challenging. We theoretically show that a non-Hermitian topological quantum phase can be realized in a reservoir-engineered transmon chain. The spatial modulation of dissipation is obtained by coupling each transmon to a quantum circuit refrigerator, allowing in situ tuning of dissipation strength in a wide range. By solving the many-body Lindblad master equation using a combination of the density matrix renormalization group and Prosen-Seligman third quantization approaches, we show that the topological end modes and the associated phase transition are visible in simple reflection measurements with experimentally realistic parameters. Finally, we demonstrate that genuine quantum effects are observable in this system via robust and slowly decaying long-range quantum entanglement of the topological end modes, which can be generated passively starting from a locally excited transmon.

Majorana correlations in quantum impurities coupled to a topological wire

In this paper, we investigate Majorana correlations in quantum impurities coupled to a topological superconducting wire. By employing a density matrix renormalization group approach, we have calculated the Majorana correlation functions in the ground state of the system for different positions of the impurities along the wire. We observe that the correlations decrease exponentially with the distance between the impurities and the ends of the wire. Moreover, different electron-electron interactions on the wire and on the impurities were also analyzed, and the results showed a stronger effect as the electron occupation in the wire is increased. In addition, while changing the occupation of the impurities by tuning their energy levels, we observe that, for specific values of $\ensuremath{\varepsilon}$, the absolute value of correlation is highly peaked when the chemical potential lies within the window of the topological regime. These peaks are associated with Coulomb blockade phenomena in the impurities, revealing very clearly its effect on the Majorana bound states in the topological phase of the system.

Ferroelectric water chains in carbon nanotubes: Creation and manipulation of ordered quantum phases.

Systems composed of molecular rotors are promising candidates as quantum devices. In this work, we employ our recently developed density matrix renormalization group approach to study such a rotor system, namely, linear chains of rotating para-water molecules encapsulated in a (6,5)-carbon nanotube. We show that the anisotropic environment provided by the nanotube breaks the inversion symmetry of the chain. This symmetry breaking lifts the degeneracy of the ground state and leads to a splitting between the left- and right-polarized states. In turn, a ferroelectric phase in nanoscopic systems is created, with a polarization that can be switched in a manner analogous to that of a supramolecular qubit. We present results for a few low-lying states and discuss the effect of external electric fields on the energy splitting and the occurrence of a quantum phase transition.

Full Configuration Interaction Excited-State Energies in Large Active Spaces from Subspace Iteration with Repeated Random Sparsification.

We present a stable and systematically improvable quantum Monte Carlo (QMC) approach to calculating excited-state energies, which we implement using our fast randomized iteration method for the full configuration interaction problem (FCI-FRI). Unlike previous excited-state quantum Monte Carlo methods, our approach, which is based on an asymmetric variant of subspace iteration, avoids the use of dot products of random vectors and instead relies upon trial vectors to maintain orthogonality and estimate eigenvalues. By leveraging recent advances, we apply our method to calculate ground- and excited-state energies of challenging molecular systems in large active spaces, including the carbon dimer with 8 electrons in 108 orbitals (8e,108o), an oxo-Mn(salen) transition metal complex (28e,28o), ozone (18e,87o), and butadiene (22e,82o). In the majority of these test cases, our approach yields total excited-state energies that agree with those from state-of-the-art methods─including heat-bath CI, the density matrix renormalization group approach, and FCIQMC─to within sub-milliHartree accuracy. In all cases, estimated excitation energies agree to within about 0.1 eV.

Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group.

In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate 2P and 2D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 1019 number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.

A theoretical simulation of the magnetic properties of nanocomposites based on graphene nanoflakes

AbstractUsing exact diagonalization, the density matrix renormalization group approach and the quantum Monte‐Carlo method, we studied the peculiarities of the magnetization profiles and the possibility of spin switching for the Heisenberg spin models of low‐dimensional composite magnets formed by graphene nanoclusters and complex transition‐metal compounds. For these models, we found the presence of intermediate plateaus in the low‐temperature magnetization profiles, which is in agreement with known experimental and theoretical studies of the magnetization profiles of nanostructured magnets with the ferrimagnetic spin ordering in the ground state. For spin‐chain model of composite magnets proposed, we also found a possibility of single and double spin switching with the macroscopic jump of the ground state spin results from frustrated spin interactions.

Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators.

In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron-electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.

Magnetoelectric coupling on fused azulene oligomers

The global magnetic phase diagram for fused azulene oligomers is obtained by using a fermionic Hubbard model, which is an intermediate between the molecular Pariser-Parr-Pople empiric Hamiltonian and the spin-$1/2$ antiferromagnetic Heisenberg model. We employ the density matrix renormalization group (DMRG) approach to explore the ground state properties of azulene-like molecules as a function of the electronic correlation and the oligomer size. It is shown that, depending on the length of the oligomer, fused azulene transitions from a singlet ($S=0$) to a higher-spin ($S=1,2$) ground state. Near the quantum magnetic phase transition the electric dipole moment, present on fused azulene molecules, couples with the magnetic moment leading to a divergent magnetoelectric susceptibility at the boundary lines of the magnetic phase diagram. These spontaneous electric and magnetic polarizations, together with the magnetoelectric coupling between them, indeed corroborate that these fused azulene oligomers can be viewed as a purely organic multiferroic material, being a magnetoelectric molecule.

Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation---indicated by a larger leading ground-state coefficient---is obtained when combining the symmetric group ${\mathcal{S}}_{n}$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group $\mathrm{U}(n)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

On the highest oxidation states of the actinoids in AnO4 molecules (An=Ac - Cm): A DMRG-CASSCF study.

Actinoid tetroxide molecules AnO4 (An=Ac - Cm) are investigated with the ab initio density matrix renormalization group (DMRG) approach. Natural orbital shapes are used to read out the oxidation state (OS) of the f-elements, and the atomic orbital energies and radii are used to explain the trends. The highest OSs reveal a "volcano"-type variation: For An=Ac - Np, the OSs are equal to the number of available valence electrons, that is, AcIII , ThIV , PaV , UVI , and NpVII . Starting with plutonium as the turning point, the highest OSs in the most stable AnO4 isomers then decrease as PuV , AmV , and CmIII , indicating that the 5f-electrons are hard to be fully oxidized off from Pu onward. The variations are related to the actinoid contraction and to the 5f-covalency characteristics. Combined with previous work on OSs, we review their general trends throughout the periodic table, providing fundamental understanding of OS-relevant phenomena.

Density matrix renormalization group approach to the low temperature thermodynamics of correlated 1D fermionic models

The low temperature thermodynamics of correlated 1D fermionic models with spin and charge degrees of freedom is obtained by exact diagonalization (ED) of small systems and followed by density matrix renormalization group (DMRG) calculations that target the lowest hundreds of states {E(N)} at system size N instead of the ground state. Progressively larger N reaches T<0.05t in correlated models with electron transfer t between first neighbors and bandwidth 4t. The size dependence of the many-fermion basis is explicitly included for arbitrary interactions by scaling the partition function. The remaining size dependence is then entirely due to the energy spectrum {E(N)} of the model. The ED/DMRG method is applied to Hubbard and extended Hubbard models, both gapped and gapless, with Ne=N or N/2 electrons and is validated against exact results for the magnetic susceptibility χ(T) and entropy S(T) per site. Some limitations of the method are noted. Special attention is given to the bond-order-wave phase of the extended Hubbard model with competing interactions and low T thermodynamics sensitive to small gaps.

Coexistence of spontaneous dimerization and magnetic order in a transverse-field Ising ladder with four-spin interactions

The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor exchange and plaquette four-spin interaction ${J}_{4}$ is studied analytically and numerically with the density matrix renormalization group approach. The quantum phase diagram in the transverse field $B$ versus ${J}_{4}$ plane has been obtained. There are three different phases: a paramagnetic (PM) phase for high values of the transverse field and, for low values of $B$, a ferromagnetic (FM) ordered phase for small ${J}_{4}$, and a dimerized-rung (DR) phase for large negative values of ${J}_{4}$. All phases are separated by quantum phase transition lines meeting at a multicritical point. The critical lines have been obtained by exploring the entanglement entropy. The results show that along the critical lines the central charge is $c=1/2$, while at the multicritical point one has $c=1$. The scaling dimension of the energy operator is ${X}_{\ensuremath{\epsilon}}=1$, in agreement with the universality class of the critical behavior of the quantum Ising chain. An effective field theory for the multicritical point is also discussed. The FM and the DR order parameters have also been computed and we found a region where the FM and the DR phases coexist.

Foreword to the 60th Sanibel Proceedings

Sanibel has supplied the forum for presenting, discussing, educating, criticising, and inspiring the development of the electronic structure and dynamics of atoms, molecules, and solids since 1961. Later, major developments presented at Sanibel included the density matrix renormalisation group approach (DMRG) and the introduction of explicit R12 and F12 interaction terms to correct for the failure of basis sets to better describe the critical correlation cusp in WFT calculations. These were the first hints that non-variational, linked diagram methods that guaranteed size-extensivity would eventually largely displace variational methods like CI in treating the correlation problem in WFT. [Extracted from the article] Copyright of Molecular Physics is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Analysis of a density matrix renormalization group approach for transport in open quantum systems

Understanding the intricate properties of one-dimensional quantum systems coupled to multiple reservoirs poses a challenge to both analytical approaches and simulation techniques. Fortunately, density matrix renormalization group-based tools, which have been widely used in the study of closed systems, have also been recently extended to the treatment of open systems. We present an implementation of such method based on state-of-the-art matrix product state (MPS) and tensor network methods, that produces accurate results for a variety of combinations of parameters. Unlike most approaches, which use the time-evolution to reach the steady-state, we focus on an algorithm that is time-independent and focuses on recasting the problem in exactly the same language as the standard Density Matrix Renormalization Group (DMRG) algorithm, initially put forward in [1]. Hence, it can be readily exported to any of the available DMRG platforms. We show that this implementation is suited for studying thermal transport in one-dimensional systems. As a case study, we focus on the XXZ quantum spin chain and benchmark our results by comparing the spin current and magnetization profiles with analytical results. We then explore beyond what can be computed analytically. Our code is freely available on github at [2].

Quantum Spin Torque Driven Transmutation of an Antiferromagnetic Mott Insulator.

The standard model of spin-transfer torque (STT) in antiferromagnetic spintronics considers the exchange of angular momentum between quantum spins of flowing electrons and noncollinear-to-them localized spins treated as classical vectors. These vectors are assumed to realize Néel order in equilibrium, ↑↓⋯↑↓, and their STT-driven dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation. However, many experimentally employed materials (such as archetypal NiO) are strongly electron-correlated antiferromagnetic Mott insulators (AFMIs) whose localized spins form a ground state quite different from the unentangled Néel state |↑↓⋯↑↓⟩. The true ground state is entangled by quantum spin fluctuations, leading to the expectation value of all localized spins being zero, so that LLG dynamics of classical vectors of fixed length rotating due to STT cannot even be initiated. Instead, a fully quantum treatment of both conduction electrons and localized spins is necessary to capture the exchange of spin angular momentum between them, denoted as quantum STT. We use a recently developed time-dependent density matrix renormalization group approach to quantum STT to predict how injection of a spin-polarized current pulse into a normal metal layer coupled to an AFMI overlayer via exchange interaction and possibly small interlayer hopping-mimicking, e.g., topological-insulator/NiO bilayer employed experimentally-will induce a nonzero expectation value of AFMI localized spins. This new nonequilibrium phase is a spatially inhomogeneous ferromagnet with a zigzag profile of localized spins. The total spin absorbed by AFMI increases with electron-electron repulsion in AFMIs, as well as when the two layers do not exchange any charge.

Topological and symmetry-enriched random quantum critical points

We study how symmetry can enrich strong-randomness quantum critical points and phases, and lead to robust topological edge modes coexisting with critical bulk fluctuations. These are the disordered analogues of gapless topological phases. Using real-space and density matrix renormalization group approaches, we analyze the boundary and bulk critical behavior of such symmetry-enriched random quantum spin chains. We uncover a new class of symmetry-enriched infinite randomness fixed points: while local bulk properties are indistinguishable from conventional random singlet phases, nonlocal observables and boundary critical behavior are controlled by a different renormalization group fixed point. We also illustrate how such new quantum critical points emerge naturally in Floquet systems.