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arXiv:2201.08313 · Physical Review C

Volume extrapolation via eigenvector continuation

Nuwan Yapa, S. König

Published Jul 18, 2022• 15 citations
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Abstract

We develop an extension of eigenvector continuation (EC) that makes it possible to extrapolate simulations of quantum systems in finite periodic boxes across large ranges of box sizes. The formal justification for this approach, which we call finite-volume eigenvector continuation (FVEC), is provided by matching periodic functions at different box sizes. As concrete FVEC implementation we use a discrete variable representation based on plane-wave states and present several applications calculated within this framework.

DOI
10.1103/physrevc.106.014309

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We develop an extension of eigenvector continuation (EC) that makes it possible to extrapolate simulations of quantum systems in finite periodic boxes across large ranges of box sizes. The formal justification for this approach, which we call finite-volume eigenvector continuation (FVEC), is provided by matching periodic functions at different box sizes. As concrete FVEC implementation we use a discrete variable representation based on plane-wave states and present several applications calculated within this framework.
Abstract Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.
Eigenvector continuation (EC) has recently attracted a lot attention in nuclear structure and reactions as a variational resummation tool for many-body expansions. While previous applications focused on ground-state energies, excited states can be accessed on equal footing. This work is dedicated to a detailed understanding of the emergence of excited states from the eigenvector continuation approach. For numerical applications the one-dimensional quartic anharmonic oscillator is investigated, which represents a strongly non-perturbative quantum system where the use of standard perturbation techniques break down. We discuss how different choices for the construction of the EC manifold affect the quality of the EC resummation and investigate in detail the results from EC for excited states compared to results from a full diagonalization as a function of the basis-space size.
First principles calculations of atomic nuclei based on microscopic nuclear forces derived from chiral effective field theory (EFT) have blossomed in the past years. A key element of such ab initio studies is the understanding and quantification of systematic and statistical errors arising from the omission of higher-order terms in the chiral expansion as well as the model calibration. While there has been significant progress in analyzing theoretical uncertainties for nucleon-nucleon scattering observables, the generalization to multi-nucleon systems has not been feasible yet due to the high computational cost of evaluating observables for a large set of low-energy couplings. In this Letter we show that a new method called eigenvector continuation (EC) can be used for constructing an efficient and accurate emulator for nuclear many-body observables, thereby enabling uncertainty quantification in multi-nucleon systems. We demonstrate the power of EC emulation with a proof-of-principle calculation that lays out all correlations between bulk ground-state observables in the few-nucleon sector. On the basis of ab initio calculations for the ground-state energy and radius in 4He, we demonstrate that EC is more accurate and efficient compared to established methods like Gaussian processes.
In this work, we discuss a new method for calculation of extremal eigenvectors and eigenvalues in systems or regions of parameter space where direct calculation is problematic. This technique relies on the analytic continuation of the power series expansion for the eigenvector around a point in the complex plane. We start this document by introducing the background material relevant to understand the basics of quantum mechanics and quantum field theories on the lattice, how we perform our numerical simulations, and how this relates to the nuclear physics we probe. We then move to the mathematical formalism of the eigenvector continuation, which is rooted in analytic function theory and linear algebra. We then discuss how these techniques are implemented numerically, with a discussion about the computational costs. Finally, we discuss applications of this method to full, quantum many-body systems. These include neutron matter, the Bose-Hubbard model, the Lipkin model, and the Coulomb interaction in light nuclei with LO chiral forces. These systems cover two categories of interest to the field: systems with a substantial sign problem, or systems that exhibit quantum phase transitions.
Repeated evaluation of the single-particle Green's function (GF) across parameter space is a recurring bottleneck in many-body methods, limiting the resolution at which phase boundaries may be probed and the frequency resolution of computed spectra. We introduce MOR+EC, a framework that constructs reusable reduced-order models for computing single-particle Green's functions by combining eigenvector continuation (EC) for parameter space exploration and model order reduction (MOR) for extrapolation in frequency space. Contrary to conventional parameterized MOR, we construct these reduced-order models with a parameter-independent resolvent, further reducing the number of required full-space evaluations. Benchmarking against exact diagonalization as the impurity solver for our example case of DMFT calculations for single- and two-band models, MOR+EC reproduces the DMFT-converged impurity GF to an average relative error below 10−310^{-3}10−3, with a median wall-time speedup of 9.59.59.5-92×92\times92×. This accuracy and efficiency together resolve a fine-grained scan of the impurity occupation as doping is tuned and produce a high-resolution phase diagram of an orbital-selective Mott transition. The reduced-order model also reproduces real- and imaginary-frequency spectra at no additional cost in full-space evaluations. This framework applies broadly to GF-based methods requiring repeated parametric evaluation and high resolution of the frequency axis.
Quantum subspace diagonalization (QSD) methods are quantum-classical hybrid methods, commonly used to find ground and excited state energies by projecting the Hamiltonian to a smaller subspace. In applying these, the choice of subspace basis is critical from the perspectives of basis completeness and efficiency of implementation on quantum computers. In this work, we present Eigenvector Continuation (EC) as a QSD method, where low-energy states of the Hamiltonian at different points in parameter space are chosen as the subspace basis. This unique choice enables rapid evaluation of low-energy spectra, including ground and nearby excited states, with minimal hardware effort. As a particular advantage, EC is able to capture the spectrum across ground state crossovers corresponding to different symmetry sectors of the problem. We demonstrate this method for interacting spin models and molecules.