Imaginary-time Integral Research Articles

Inchworm quasi Monte Carlo for quantum impurities

The inchworm expansion is a promising approach to solving strongly correlated quantum impurity models due to its reduction of the sign problem in real and imaginary time. However, inchworm Monte Carlo is computationally expensive, converging as 1/N where N is the number of samples. We show that the imaginary-time integration is amenable to quasi Monte Carlo, with parametrically better 1/N convergence, by mapping the Sobol low-discrepancy sequence from the hypercube to the simplex with the so-called Root transform. This extends the applicability of the inchworm method to, e.g., multiorbital Anderson impurity models with off-diagonal hybridization, relevant for materials simulation, where continuous-time hybridization expansion Monte Carlo has a severe sign problem. Published by the American Physical Society 2024

Third-order Epstein-Nesbet perturbative correction to the initiator approximation of configuration space quantum Monte Carlo.

We implement Epstein-Nesbet perturbative corrections in the third-order for the initiator approximation of the configuration space quantum Monte Carlo. An improved sampling algorithm is proposed to address the stochastic noise of the corrections. The stochastic error for the perturbative corrections is considerably larger than that for the reference energy, and it fails to provide reasonable results unless a very long imaginary time integration is performed. The new sampling algorithm accumulates rejected walkers from multiple independent steps to cover a larger portion of the secondary space. The performance of the perturbative corrections is demonstrated for small molecules.

Monte Carlo Second- and Third-Order Many-Body Green's Function Methods with Frequency-Dependent, Nondiagonal Self-Energy.

We fully develop the Monte Carlo many-body Green's function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O(n4) and O(n5) of molecular size (n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O(n5) and O(n6) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.

A Time-Dependent Formulation of Coupled-Cluster Theory for Many-Fermion Systems at Finite Temperature.

We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.

Quantum action for time-dependent Ginzburg-Landau equations

A gauge-invariant formula is derived for the quantum action of a dirty superconductor with strong pair breaking. The major complication is the coupling between the order parameter and the electro-chemical potential, which is most simply expressed as an imaginary time integral. The perturbative modes of excitation are identified.

Perturbation Expansion for the Anderson Hamiltonian. III

Thermodynamical quantities are expanded in perturbation series with respect to the Coulomb repulsion U for the Anderson Hamiltonian with electron-hole symmetry and 'their general terms of perturbation are investigated. By these processes the mutual. relations among these quantities are discussed. It is confirmed that the thermodynamical quantities such as the specific heat. and the scattering t-matrix are expressed in terms of two quantities; the even and odd parts of the susceptibility as far as low-lying excitations are concerned. These results are entirely consistent with the Nozieres phenomenological Fermi liquid theory based on the s-d exchange model. . §I. Introduction In the :first paper1h*> of' this series, we presented a perturbation theoretical approach to the Anderson Hamiltonian2> with electron-hole symmetry and showed that each term of the perturbation expansion for the thermodynamical quantities, in particular for the free energy, can be expressed by an imaginary time integral of the product of two antisymmetric determinants constructed by unperturbed local d-electron temperature Green's functions. In the second paper by Yam ada, B) each term in the perturbation expansions for the specific heat, the susceptibility, the resistivity and the density-of~state s for the localized d-state has been calculated up to fourth order. In the course of calculation it . has been found that some general relations hold between thermodynamical quantities, in particular, the T-linear specific heat· is proportional to the even part of the susceptibility. This finding immediately leads us to the result that the ratio of the susceptibility to the coefficient of the T-linear specific heat becomes twice as large in the s-d limit as the value in the case of no correlation. This paper deals with further development along this line: Here, discussion is mainly concentrated on the. general properties of the perturbation series for thermodynamical quantities in the Anderson model. The basic Hamiltonian in this paper is the Anderson Hamiltonian with electron-hole symmetry (Ed= -tU), which is divided into two parts, the unperturbed part H 0 and the perturbation H' as follows:

Perturbation Expansion for the Anderson Hamiltonian. II

Thermodynamical quantities are expanded in perturbation series with respect to the Coulomb repulsion U for the Anderson Hamiltonian with electron-hole symmetry and 'their general terms of perturbation are investigated. By these processes the mutual. relations among these quantities are discussed. It is confirmed that the thermodynamical quantities such as the specific heat. and the scattering t-matrix are expressed in terms of two quantities; the even and odd parts of the susceptibility as far as low-lying excitations are concerned. These results are entirely consistent with the Nozieres phenomenological Fermi liquid theory based on the s-d exchange model. . §I. Introduction In the :first paper1h*> of' this series, we presented a perturbation theoretical approach to the Anderson Hamiltonian2> with electron-hole symmetry and showed that each term of the perturbation expansion for the thermodynamical quantities, in particular for the free energy, can be expressed by an imaginary time integral of the product of two antisymmetric determinants constructed by unperturbed local d-electron temperature Green's functions. In the second paper by Yam ada, B) each term in the perturbation expansions for the specific heat, the susceptibility, the resistivity and the density-of~state s for the localized d-state has been calculated up to fourth order. In the course of calculation it . has been found that some general relations hold between thermodynamical quantities, in particular, the T-linear specific heat· is proportional to the even part of the susceptibility. This finding immediately leads us to the result that the ratio of the susceptibility to the coefficient of the T-linear specific heat becomes twice as large in the s-d limit as the value in the case of no correlation. This paper deals with further development along this line: Here, discussion is mainly concentrated on the. general properties of the perturbation series for thermodynamical quantities in the Anderson model. The basic Hamiltonian in this paper is the Anderson Hamiltonian with electron-hole symmetry (Ed= -tU), which is divided into two parts, the unperturbed part H 0 and the perturbation H' as follows:

Integration of the Schrödinger equation in imaginary time

Numerical techniques for the integration of the Schrödinger equation in imaginary time are investigated. Because the spatial dependence of the solution in the limit of large imaginary time is that of the ground state of the Hamiltonian, the method can be applied to bound states of quantum-mechanical three-body systems. Knowledge of the analytic dependence of the asymptotic form of the wavefunction on the trial eigenvalue is not required. In this paper, the first of a series, we have confined our attention to the problems arising from the imaginary time integration and reserve for a later date a discussion of the problems that are due to a multidimensional spatial grid.