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arXiv:2501.02646 · Physical Review Letters

Diagrammatic Monte Carlo for Finite Systems at Zero Temperature

Stefano Brolli, C. Barbieri, E. Vigezzi

Published May 9, 2025• 1 citation
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Abstract

High-order virtual excitations play an important role in microscopic models of nuclear reactions at intermediate energies. However, the factorial growth of their complexity has prevented their consistent inclusion in ab initio many-body calculations. For infinite systems at finite temperature, such drawbacks can be overcome using diagrammatic Monte Carlo techniques to resum entire series of Feynman diagrams. We present a diagrammatic Monte Carlo algorithm that can be applied to self-bound systems with discrete energy levels at zero temperature, and demonstrate its potential for the Richardson model of nuclear pairing. We show that sampling the topological space of diagrams allows the inclusion of high-order excitations that are neglected in state-of-the-art approximations used in nuclear physics and quantum chemistry. We propose that sampling the diagrammatic space can overcome the long-standing gap between our microscopic understanding of structure and reactions in nuclear physics.

DOI
10.1103/physrevlett.134.182502

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High-order virtual excitations play an important role in microscopic models of nuclear reactions at intermediate energies. However, the factorial growth of their complexity has prevented their consistent inclusion in ab initio many-body calculations. For infinite systems at finite temperature, such drawbacks can be overcome using diagrammatic Monte Carlo techniques to resum entire series of Feynman diagrams. We present a diagrammatic Monte Carlo algorithm that can be applied to self-bound systems with discrete energy levels at zero temperature, and demonstrate its potential for the Richardson model of nuclear pairing. We show that sampling the topological space of diagrams allows the inclusion of high-order excitations that are neglected in state-of-the-art approximations used in nuclear physics and quantum chemistry. We propose that sampling the diagrammatic space can overcome the long-standing gap between our microscopic understanding of structure and reactions in nuclear physics.
Diagrammatic Monte Carlo provides a systematically improvable framework for stochastically resumming many-body expansions to high orders through direct sampling of diagram topologies. We advance our earlier work by introducing a novel time-ordered Diagrammatic Monte Carlo algorithm for the single-particle Green's function. The algorithm is tailored to finite nuclei, formulated in discrete model spaces and applicable to arbitrary two-body interactions. The new time-ordered diagrammatic Monte Carlo algorithm is based on the on-the-fly evaluation of time-ordered Goldstone diagrams, avoiding explicit diagram enumeration and expensive frequency integration. We show the algorithm by computing 16{}^{16}16O up to fifth order in a reduced model space using optimized reference state orbitals and including effective three-body forces. Benchmarking against established truncation schemes in ab initio nuclear theory demonstrates its potential to overcome the limitations of current many-body approaches.
This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
We present an ab initio auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the ab initio phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, ab initio finite temperature method. We demonstrate that the method produces internal energies within chemical accuracy of exact diagonalization results across a wide range of temperatures for H 2 O (STO-3G), C 2 (STO-6G), the one-dimensional hydrogen chain (STO-6G), and the multiorbital Hubbard model. Our method effectively controls the phase problem through importance sampling, often even without invoking the phaseless approximation, down to temperatures at which the systems studied approach their ground states and may therefore be viewed as exact over wide temperature ranges. This technique embodies a versatile tool for studying the finite temperature phase diagrams of a plethora of systems whose properties cannot be captured by a Hubbard U term alone. Our results moreover illustrate that the severity of the phase problem for model Hamiltonians far exceeds that for many molecules at all of the temperatures studied.
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We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynman's diagrammatic series for the single-particle self-energy we can study moderate values of the on-site repulsion (U/t∼4U/t \sim 4U/t∼4) and temperatures down to T/t=1/40T/t=1/40T/t=1/40. We compare our results with high temperature series expansion and with single-site and cluster dynamical mean-field theory.