Session V16: Quantum Manybody Systems: Theory and Computation II

Off-Diagonal Expansion Quantum Monte Carlo

I will discuss a novel, parameter-free Trotter error-free, quantum Monte Carlo technique designed to tackle a very broad range of physical models within a single unifying framework. The method builds on a series expansion of the canonical partition function around its classical component. The expansion further allows for a group-theoretical abstraction of the QMC algorithm. Some results showcasing the capabilities of the algorithm will be presented.   

FInite-temperature Auxiliary-Field Quantum Monte Carlo: Recent Developments and Applications

We present a highly accurate auxiliary-field quantum Monte Carlo (AFQMC) method to study finite-temperature properties of correlated fermion systems. This approach eliminates the minus sign problem by introducing constraints in auxiliary-field space. Building on earlier ideas [1] and incorporating the latest developments in zero-temperature methods, we introduce a self-consistent formalism [2] to improve the constraint in the finite-temperature framework. This with several other algorithmic advances leads to a more accurate, more efficient, and numerically more stable approach for finite-temperature calculations. We carry out systematic benchmark study in the 2D Hubbard model. Temperatures as low as T=1/80 (in units of hopping) are reached. The finite-temperature method is exact at high temperatures, and approaches the result of the zero-temperature constrained-path AFQMC as temperature is lowered. The benchmark shows that systematically accurate results are obtained for thermodynamic properties. The properties of the 2D doped Hubbard model as functions of temperature will be presented and discussed.

[1] Shiwei Zhang, Phys. Rev. Lett. 83, 2777 (1999).
[2] Mingpu Qin, Hao Shi, and Shiwei Zhang, Phys. Rev. B 94, 235119 (2016).   

Application of the linked cluster expansion to the many-particle path-integral

A diagrammatic expansion for the pair distribution can be derived by starting from the many-body path-integral and using the idea of cluster expansion. The expansion is written as a sum of nodal and non-nodal diagrams. The sum of all the nodal diagrams can be expressed in terms of the non-nodal diagrams using the hypernetted equation technique. The sum of the non-nodal diagrams, which are irreducible diagrams in momentum space, are written as a perturbation expansion in powers of the particle density. Our approach is analogous to the well-known many-body perturbation expansion of the n-body Green's function. The approach was tested on a system of distinguishable particles and our results agree very well with those obtained from the path-integral Monte Carlo.   

A multi-projective variational approach to the quantum lattice problem

We formulate a general method for constructing a variational expression of the total energy based on two projective ansatz from the weak and strong coupling limits. These two ansatz are then combined according to individual and mutual interactions of low and high energy degrees of freedom. We apply our approach to the single band Hubbar model, where the theory yields the double occupancy and the nonlocal single-particle density matrix. We compare to exact results for d=1 and d=infinity. For d=1, the insulating state is properly obtained at infinitesimal u; in addition to an accurate prediction of the ground state energy over a broad range of t/U. In infinite dimensions, we properly find a finite-U metal-insulator transition with reasonable quantitative accuraccy across all parameter space. Our approach has a negligible computational cost as compared to dynamical mean field theory and could be highly applicable in the context of total energies for strongly correlated materials and molecules.   

New practical methods for consistently closing truncated equations of motion

We discuss some new methods for truncating and closing the equations of motion for composite operators. Our formalism automatically generates a physical theory in the sense of having positive spectral weights. Another important advantage of our methods is that (for judiciously chosen set of operators) all averages necessary to close the formalism can be calculated self-consistently from within the theory itself. We discuss how the method can be applied to the Hubbard model in both fermionic and bosonic sectors.   

Semi-local observables from importance sampling diffusion Monte Carlo

An approach to evaluate a family of semi-local observables from the fixed-node ground state will be discussed. The celebrated importance-sampling diffusion Monte Carlo (DMC) method by Ceperley and Alder is a real space approach. Therefore, beyond the calculation of the total energy of the fixed-node ground state, DMC facilitates the evaluation of observables that are local in real space such as the density, albeit pure estimators or extrapolations are required. The evaluation of semi-local observables is expected to be more involved. It can be shown, however, that the DMC algorithm carries additional information of the fixed-node solution. DMC results for semi-local operators will be compared with exact results, in the hydrogen atom and the harmonic oscillator. The perspectives, computational cost, and potentials problems of this approach when applied to realistic many-body systems will be discussed.   

Accurate total energies of pseudopotentials

We present high accuracy total energies of pseudo-atoms, i.e, atoms with effective core potentials (ECP). The pseudo-atoms considered are the 1st, 2nd and 3rd-row elements with correlation consistent effective core potentials (ccECP) we constructed. For each element, we perform configuration interaction, coupled-cluster, and quantum Monte Carlo calculations with systematically eliminated/improved errors.
We carry out FCI/CCSDT(Q) calculations with aug-cc-pVnZ (n=3-6) basis set extrapolated to complete basis set limit to obtain highly accurate energies. For these methods, the achieved accuracy in the 1st row is typically 0.1-0.3 mHa, the 2nd row is 0.1-0.2 mHa, and 5-10 mHa in the 3rd row. Wherever possible, we included the kinetic energies obtained from FCI calculations to estimate the smoothness/curvature of the particle density. These accurate total energies will serve as a benchmark for independent calculations as well as the assessment of systematic and methodological errors.   

Compton Profile of Solid and Liquid Lithium from Quantum Monte Carlo

Compton scattering cross section is directly related to the electronic momentum distribution and provides an important check of many-body theory. We computed the Compton profile of solid and liquid lithium using quantum Monte Carlo (QMC) and compared with recent experiments obtaining excellent agreement. Importantly, we find it necessary to explicitly include the core electrons for quantitative accuracy. To account for lattice effects, we sampled finite-temperature configurations using molecular dynamics (MD), then performed pseudopotential diffusion Monte Carlo (DMC) simulations on each configuration. We used Slater-Jastrow wavefunction and twist-averaged boundary condition. QMC pseudopotential correction was derived from an all-electron DMC simulation of the perfect crystal. Our calculations provide the first all-electron QMC benchmark for the Compton profile of lithium under both liquid and solid conditions.   

Prediction of the singlet-triplet excitation energy for the spinel, MgTi2O4 via downfolding approach combined with first-principles Quantum Monte Carlo

The spinel, MgTi₂O₄ undergoes a transition into a dimerized state at low temperatures that is expected to be a spin singlet. However, no signature of a singlet-triplet transition as been discovered, in part due to the difficulty of predicting the energy of the transition from theory. Using high-accuracy first-principles quantum Monte Carlo combined with a novel model-fitting approach, we predict the singlet-triplet gap to be 350(50) meV, a higher energy than previous experimental observations have considered. Confirmation of our prediction would suggest that our approach should enable calculation of other excitation energies on the basis of first-principles quantum Monte Carlo combined with effective model calculations.   

An Efficient Calculation of the Total Energy Variation in DFT+DMFT Implemented using Orthogonal Basis sets

Calculations of the total energy variation in density functional theory plus dynamical mean field theory (DFT+DMFT) with respect to the structural change are important for atomic force and stress computations for strongly correlated materials. In this talk, we will show that the energy variation of DFT+DMFT implemented using orthogonal basis sets such as Wannier functions can be computed efficiently using analytic formula since the potential energy variation can be exactly cancelled out and the DMFT self-energy itself or its variation does not need to be explicitly accounted. We will use DMFT with the Continuous-Time Quantum Monte Carlo (CTQMC) impurity solver to compute its total energy variation of the two-dimensional one-band Hubbard model with respect to changes of hopping parameters and compare to the results of its analytic energy derivative formula to verify their quantitative agreement. We will also compare the potential energy contributions sampled directly from CTQMC and computed using the Migdal-Galitzki formula. The application of our formula to the calculations of forces and stress in DFT+DMFT will be also discussed.   

Systematic corrections to the Thomas-Fermi approximation without a gradient expansion

We improve on the Thomas–Fermi approximation for the single particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki–Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.   

Approximate States of the Hubbard Model from Heisenberg's Equation

Operators which add electrons with definite energies to states of interacting electrons are solutions of the time-independent Heisenberg equation. Expanding the operators in the number of electron-hole pairs generated to screen each added electron provides systematic approximations for the energies of these operators. This is illustrated for the simplest Hubbard model on a cubic lattice by relations between the energies and electron densities of ground states.   

Correlated states in magnetic quantum dots with multiple occupancy

The motivation to magnetically dope semiconductor quantum dots comes from the possibility for an enhanced control of magnetic ordering as compared to their bulklike counterparts. Unlike in the bulk structures, adding a single carrier in a magnetic QD can have important ramifications. An extra carrier can both strongly change the total carrier spin and the temperature of the onset of magnetization[1]. While QDs have been recognized for correlation effects, inherent in the analog of Wigner crystals referred to as Wigner molecules (WMs), the modification of underlying correlated states with magnetic doping is largely unexplored[2]. By focusing on Mn-doped II-VI QDs[3] we show correlated states, which can be viewed as a generalization of WMs, can be studied using exact diagonalization and conditional probability density. We explain how the formation of a shell structure in these Mn-doped QDs and magnetic frustrations are altered by the changes in strength of the Coulomb interaction.
[1] R. M. Abolfath, A. G. Petukhov, I. Zutic, PRL 101, 207202 (2008)
[2] J. M. Pientka et al., PRB 92, 155402 (2015)
[3] P. Zhang et al., arXiv:1801:05090
  

Resilience of the Mixed State Wigner Function

Solution of the quantum Vlasov equation shows the phase space dynamics of a mixed-state Wigner function is more resilient against changes in external electric and magnetic fields than a pure-state Wigner function. A particle in a mixed-state is harder to follow and localize, but the freedom to access a range of pure states increases its inertia against changes in external conditions. This could explain the electron's stability in the Shrodinger's atomic orbital model and perhaps provides a way to preserve the information of a qubit sufficiently long for quantum computing.