SO(N) Singlet‐Projection Model on the Pyrochlore Lattice

N-(sulfoethyl) iminodiacetic acid-based lanthanide coordination polymers: Synthesis, magnetism and quantum Monte Carlo studies

Recent quantum Monte Carlo (QMC) studies of electronic structure have considered various trial function enhancements directed at improved fixed-node energies. In this study we investigate complete active space self-consistent field (CASSCF) trial functions in the diffusion Monte Carlo (DMC) method. We study longer CASSCF expansions than typically used in QMC studies and optimize correlation function parameters, basis function coefficients, and configuration state function mixing coefficients. To perform a stable, efficient wavefunction optimization, sample points are analytically obtained from an integrable probability density function or a Monte Carlo walk guided by a positive definite function. The approach is applied to acetylene and its dissociation fragments (C, CH, C2, C2H, C2H2). For these systems 70%–90% of the correlation energy is recovered with variational MC and 91%–98% with DMC.

The single-impurity Anderson model1 was invented nearly thirty years ago to describe dilute magnetic impurities in metallic hosts, including the formation and properties of itinerant local moments in alloys. The model is a prerequisite for the understanding of mixed valent and heavy-fermion phenomena.2 A limit of the model is the famous Kondo model3 which describes resistivity minima and saturation. The model is simple to state and its properties are readily measureable in the laboratory.4 However, the solution of the model is a difficult many-body problem. In recent years considerable progress in understanding the static properties of the model has been achieved using non-perturbative methods such as the Bethe ansatz,5 the renormalization group,6 and quantum Monte Carlo (QMC).7 Nevertheless, the understanding of the dynamical properties has remained elusive. These include the spectral density of the impurity state, the transport coeffecients, and the dynamical magnetic susceptibility. The spectral density has been calculated reliably only for large orbital degeneracy8 or for expansion parameters below the range of most physical interest.9 It has been obtained only at zero temperature by the renormalization group method,10 and it is extremely difficult to calculate reliably from QMC.11-15

Single point calculations of the ground state electronic structure of peroxynitrite anion have been performed at the optimized cis geometry using the restricted Hartree–Fock (RHF), Møller Plesset second order perturbation theory (MP2), generalized gradient approximation density functional theory (GGA DFT) in the B3LYP form and two quantum Monte Carlo (QMC) methods, variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). These calculations reveal differences in atomization energies estimated by B3LYP (287.03 kcal/mol), MP2 (290.84 kcal/mol), and DMC, 307.4(1.9) kcal/mol, as compared to experiment, 313(1) kcal/mol. The error associated with MP2 and B3LYP methods is attributed largely to differential recovery of correlation energies for neutral nitrogen and oxygen atoms relative to the oxygen and peroxynitrite anions. In addition, basis set studies were carried out to determine potential sources of error in MP2 and B3LYP valence energy values. Our studies indicate that MP2 and B3LYP valence energies are strongly dependent on the presence of diffuse functions for the negative ions O− and ONOO−.

A unique type of frustrated lattice is found in two A-site ordered spinel oxides, LiGaCr(4)O(8) and LiInCr(4)O(8). Because of the large size mismatch between Li(+) and Ga(3+)/In(3+) ions at the A site, the pyrochlore lattice, made up of Cr(3+) ions carrying spin 3/2, becomes an alternating array of small and large tetrahedra, i.e., a "breathing" pyrochlore lattice. We introduce a parameter, the breathing factor B(f), which quantifies the degree of frustration in the pyrochlore lattice: B(f) is defined as J'/J, where J' and J are nearest-neighbor magnetic interactions in the large and small tetrahedra, respectively. LiGaCr(4)O(8) with B(f)~0.6 shows magnetic susceptibility similar to that of conventional Cr spinel oxides such as ZnCr(2)O(4). In contrast, LiInCr(4)O(8) with a small B(f)~0.1 exhibits a spin-gap behavior in its magnetic susceptibility, suggesting a proximity to an exotic singlet ground state. Magnetic long-range order occurs at 13.8 and 15.9 K for LiGaCr(4)O(8) and LiInCr(4)O(8), respectively, in both cases likely owing to the coupling to structural distortions.

This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.

Accuratemodeling of heterogeneous catalysis requires the availabilityof highly accurate potential energy surfaces. Within density functionaltheory, these can—unfortunately—depend heavily on theexchange-correlation functional. High-level ab initio calculations, on the other hand, are challenging due to the systemsize and the metallic character of the metal slab. Here, we presenta quantum Monte Carlo (QMC) study for the benchmark system H2 + Cu(111), focusing on the dissociative chemisorption barrier height.These computationally extremely challenging ab initio calculations agree to within 1.6 ± 1.0 kcal/mol with a chemicallyaccurate semiempirical value. Remaining errors, such as time-steperrors and locality errors, are analyzed in detail in order to assessthe reliability of the results. The benchmark studies presented hereare at the cutting edge of what is computationally feasible at thepresent time. Illustrating not only the achievable accuracy but alsothe challenges arising within QMC in such a calculation, our studypresents a clear picture of where we stand at the moment and whichapproaches might allow for even more accurate results in the future.

First-principles calculations are used to investigate the effects of magnetic ordering on the minimum-energy structure and on the full phonon dispersion relation of CdCr${}_{2}$O${}_{4}$, focusing on the changes through the coupled magnetic/structural transition which shows relief of the geometric frustration of the antiferromagnetic ordering on the pyrochlore lattice. We computed the full phonon dispersion relations for the ferromagnetic and antiferromagnetic orderings in cubic and tetragonal structures of CdCr${}_{2}$O${}_{4}$. We extracted the phonon dispersion for the cubic paramagnetic phase and found that it compares well with the experimental results. The AFM ordering is seen to lower the symmetry and induce a lattice distortion comparable in magnitude to that observed in the transition. While the spin-phonon couplings are large for modes which involve displacement of the Cr atoms, there are no unstable modes at any point in the Brillouin zone for either of the magnetic orderings considered, and thus we conclude that the phase transition is driven not by spin-phonon coupling, but by the atomic forces and stresses induced by the magnetic order. Finally, by comparison of the phonon frequencies for structures with different magnetic orderings and structural distortions, we find that the spin-phonon coupling, rather than the coupling of the phonons to the structural change, is the dominant factor in the observed changes of phonon frequencies through the phase transition.

In order to investigate the validity of a conjecture that correlated electrons in ladders having even-numbered legs exhibit superconductivity accompanied by a spin gap while odd ones do not, the pairing correlation is studied for the three-leg ladder in the Hubbard model. Both the renormalization group for weak interactions and the quantum Monte Carlo (QMC) method for strong interactions are employed. The weak-coupling study predicts that the superconducting correlation dominates, which refutes the naive even-odd conjecture. A crucial point is that a spin gap for only some out of multiple spin modes can suffice to make the ladder superconduct with a pairing symmetry (d-like here) compatible with the gapped mode. A QMC study endorses the enhanced pairing correlation for stronger Hubbard repulsions. Effects of Umklapp processes are also elucidated. A small energy scale relevant here requires a care in the QMC related to the discreteness of one-electron levels. This argument is reinforced from the result for an exactly solvable, one-dimensional attractive Hubbard model.

The total nonrelativistic energies of the fluorine atom and its negative ion are calculated using the fixed-node quantum Monte Carlo (QMC) method. Over 90% of the correlation energy is obtained for both the neutral and the anion. Subtracting these energies yields an electron affinity of 3.45±0.11 eV, in excellent agreement with the recommended experimental value of 3.40 eV. The observed dependence of our Monte Carlo energies on the time step is discussed within the short-time QMC formalism. As in previous QMC studies in this series, only a single determinant, constructed with a small (double-zeta) basis set, multiplied by simple functions of electron–electron and electron–nuclear separation, is required as an importance function.

Large-scale quantum Monte Carlo (QMC) calculations of ground and excited singlet states of both conformers of azobenzene are presented. Remarkable accuracy is achieved by combining medium accuracy quantum chemistry methods with QMC. The results not only reproduce measured values with chemical accuracy but the accuracy is sufficient to identify part of experimental results which appear to be biased. Novel analysis of nodal surface structure yields new insights and control over their convergence, providing boost to the chemical accuracy electronic structure methods of large molecular systems.

The interplay of the magnetic field and interlayer correlations was experimentally investigated in a quasi-two-dimensional quantum magnet $\mathrm{Cu}(en){\mathrm{Cl}}_{2}$. For this purpose, extensive quantum Monte Carlo (QMC) studies of the finite-temperature properties of the spin-1/2 Heisenberg antiferromagnet (HAF) on the rectangular and the spatially anisotropic zig-zag square lattices were performed. The QMC studies revealed the equivalency of both models for any values of magnetic fields and spatial anisotropies $R={J}_{\mathrm{interchain}}/{J}_{\mathrm{intrachain}}$. The analysis based on the decomposition of both lattices into local Hamiltonians confirmed the equivalence. Despite the large influence of the interlayer coupling and rather complicated distribution of the intralayer exchange pathways in $\mathrm{Cu}(en){\mathrm{Cl}}_{2}$, considering several constraints in the data analysis enabled the extraction of the main features of the low-dimensional magnetic subsystem from the specific heat and magnetization. It was found that $\mathrm{Cu}(en){\mathrm{Cl}}_{2}$ can be treated as the realization of the spin-1/2 HAF on the rectangular/zig-zag square lattice with the intralayer spatial anisotropy $R\ensuremath{\approx}0.2$, the intrachain exchange coupling 2.20 \ifmmode\pm\else\textpm\fi{} 0.15 K, the saturation field 3.7 \ifmmode\pm\else\textpm\fi{} 0.1 T, the spin-flop field about 0.1 T, and the spin anisotropy of the orthorhombic symmetry. The possibility to investigate magnon instabilities in the strong-field regime of $\mathrm{Cu}(en){\mathrm{Cl}}_{2}$ is discussed.

Exchange interaction tends to favor collinear or coplanar magnetic orders in rotationally invariant spin systems. Indeed, such magnetic structures are usually selected by thermal or quantum fluctuations in highly frustrated magnets. Here we show that a complex noncoplanar magnetic order with a quadrupled unit cell is stabilized by itinerant electrons on the pyrochlore lattice. Specifically, we consider a Kondo-lattice model with classical localized moments at quarter filling. The electron Fermi "surface" at this filling factor is topologically equivalent to three intersecting Fermi circles. Perfect nesting of the Fermi lines leads to magnetic ordering with multiple wave vectors and a definite handedness. The chiral order might persist without magnetic order in a chiral spin liquid at finite temperatures.

There is a growing interest in the $U(1)$ Coulomb liquid in both quantum materials in pyrochlore ice and cluster Mott insulators and cold atom systems. We explore a paired hardcore boson model on a pyrochlore lattice. This model is equivalent to the XYZ spin model that was proposed for rare-earth pyrochlores with "dipole-octupole" doublets. Since this model has no sign problem for quantum Monte Carlo (QMC) simulations in a large parameter regime, we carry out both analytical and QMC calculations. We find that the $U(1)$ Coulomb liquid is quite stable and spans a rather large portion of the phase diagram with boson pairing. Moreover, we numerically find thermodynamic evidence that the boson pairing could induce a possible $\mathbb{Z}_2$ liquid in the vicinity of the phase boundary between Coulomb liquid and $\mathbb{Z}_2$ symmetry-broken phase. Besides the materials' relevance with quantum spin ice, we point to quantum simulation with cold atoms on optical lattices.

We investigate the ground state and critical temperature (Tc) phase diagrams of the classical and quantum S=12 pyrochlore lattice with nearest-neighbor Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider ferromagnetic and antiferromagnetic Heisenberg exchange interaction as well as direct and indirect DMI. At the classical level, three ground states are found: all-in/all-out, ferromagnetic, and a locally ordered XY phase, known as Γ5, which displays an accidental classical U(1) degeneracy at the mean-field level. Quantum zero-point energy fluctuations computed to order 1/S are found to lift the classical ground-state degeneracy and select the so-called ψ3 state out of the degenerate manifold in most parts of the Γ5 regime. Likewise, thermal fluctuations treated classically at the Gaussian level entropically select the ψ3 state at T=0+. In contrast to this low-temperature state-selection behavior, classical Monte Carlo simulations find that the system orders at Tc in the noncoplanar ψ2 state of Γ5 for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from ψ2 to ψ3 at a temperature TΓ5