arXiv subjects

Thomas Chuna

Publications and source records attributed to Thomas Chuna.

13 recordsLinked to original sources

arXiv2026

Analysis of the Ill-Conditioning of the Discrete Inverse Laplace transform in Monte Carlo Simulations of Quantum Many-Body Systems

Analytic continuation of imaginary-time quantum Monte Carlo data to real-frequency spectra requires the inversion of a severely ill-posed two-sided Laplace transform and arises naturally in quantum many-body calculations of dynamic properties. In this work, we distinguish the intrinsic ill-posedness of the continuous inverse two-sided Laplace problem from the conditioning of its finite-dimensional discretization. For equidistant sampling and reconstruction grids, we express the discrete problem via a diagonally scaled monomial Vandermonde matrix with exponentially distributed nodes. Imposing the physical detailed-balance symmetry transforms the discretization into a diagonally scaled Chebyshev-Vandermonde system. Exploiting these structures, we derive explicit lower and upper bounds on the condition numbers in terms of the physical and discretization parameters. For the unconstrained discretization, our bounds reveal super-exponential growth of the condition number with the reconstruction dimension, which cannot be removed by increasing the number of imaginary-time samples alone. Detailed-balance substantially improves the conditioning, especially in the practically relevant pre-asymptotic regime, although the asymptotic super-exponential dependence remains. In both cases, our bounds identify a low-dimensional regime in which the super-exponential contribution is suppressed and stable reconstruction remains feasible. These results provide a mathematical explanation for the effectiveness of low-dimensional spectral representations and detailed-balance in analytic continuation. While they do not remove the fundamental ill-posedness, they substantially mitigate the ill-conditioning of its finite-dimensional discretization, delay the onset of its catastrophic super-exponential growth and thereby enlarge the range of numerically accessible reconstructions.

math.NA↗

arXiv2026

Discovering a well-conditioned analytic continuation problem via dictionary learning

Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time $\tau$ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic $\omega$-dependent quantities from ITCFs is a notoriously difficult task, known as analytic continuation (AC), that amounts to solving an exponentially ill-conditioned inverse problem. Within the AC literature, there are competing stochastic and regularized approaches, as well as an emerging collection of works using parameterized models like neural networks. Here we transcend the traditional divides between the communities, introducing the regularized stochastic optimization method (RSOM). This method reformulates AC as a dictionary learning problem, discovering a sparse dictionary to represent the solution. Our approach is motivated by the astounding results dictionary learning has produced in many scientific fields. Remarkably, RSOM discovers a sparse dictionary that maps an ill-conditioned inverse problem to a low-dimensional problem that is well-conditioned. We demonstrate that the method yields competitive results for common synthetic test problems as well as for authentic QMC data from the finite temperature electron gas. This work exposes that a dictionary exists within all stochastic and regularized methods and that dictionary learning provides a new angle of attack for future AC methods.

physics.comp-ph↗

arXiv2026

Reweighting Estimators for Density Response in Path Integral Monte Carlo: Applications to linear, nonlinear and cross-species density response

We present density response estimators for Monte Carlo simulations that are based on a reweighting procedure, where the samples of an unperturbed system are used to estimate the properties of a system perturbed by an external harmonic potential. This allows the linear and nonlinear static density response to be estimated purely from simulations of the unperturbed system. The method is demonstrated for the uniform electron gas under warm dense matter and strongly coupled conditions using ab initio path integral Monte Carlo simulations. The performance of the method with respect to the number of particles and the number of imaginary time slices is investigated. The scheme is generalised to consider multiple external perturbations, acting on different species and with different wavenumbers, giving one access to additional cross-species density response functions and the complete quadratic response function resolved for both wave number arguments through mode coupling. The flexibility of the methodology opens the possibility to investigate numerous new density response properties to further advance our understanding of interacting quantum many-body systems across a broad range of applications.

physics.chem-ph↗

arXiv2026

Temperature dependence of the dynamic structure factor of the electron liquid via analytic continuation

We present new analytic continuation results for the dynamic structure factor $S(\mathbf{q},\omega)$ of the uniform electron liquid based on quasi-exact \emph{ab initio} path integral Monte Carlo (PIMC) data for the imaginary-time density--density correlation function $F(\mathbf{q},\tau)$ across a broad range of temperatures. For this purpose, we employ both a traditional maximum entropy method solver, and a pre-optimized sparse Gaussian kernel representation as it has been implemented in the recent \texttt{PyLIT} package [Benedix Robles \textit{et al.}, \textit{Comp.~Phys.~Comm.}~\textbf{319}, 109904 (2026)], and we identify potential advantages and disadvantages in both. We expect our results to be interesting for a broad range of topics, including the interpretation of x-ray Thomson scattering experiments with extreme states of matter and the construction of improved exchange--correlation kernels for linear-response time-dependent density functional theory.

physics.comp-ph↗

arXiv2026

Mermin's dielectric function and the f-sum rule

Mermin's dielectric function [N.D. Mermin, Phys. Rev. B 1, 2362 (1970)] is widely assumed to satisfy the f-sum rule because he constrains his ansatz with the continuity equation. However, we identify a moment-closure problem in Mermin's use of the continuity equation. Further, we show that the Mermin's model can be derived without invoking continuity. We describe how other approaches such as the ``completed Mermin'' model of Chuna and Murillo [Phys. Rev. E 111, 035206 (2025)] remedy this closure issue. We then inspect the f-sum rule for both the original and completed Mermin models and find for the Mermin ansatz that collision frequencies scaling as $\omega$ must violate the f-sum rule, whereas constant, real, positive collision frequencies will satisfy it, with the caveat that, in practice, convergence with respect to the upper integration limit $\omega_{\max}$ is sufficiently slow that finite-domain numerical evaluations exhibit apparent violations, regardless of wavenumber $q$. We also find that collision frequencies with constant imaginary components cause f-sum rule violations. We conclude that if Mermin's model is fit to data via optimizing its collision frequency, then the f-sum rule is not inherently satisfied; constraints, though broad, are needed in order to assume the f-sum rule is satisfied. Further, if the f-sum rule is theoretically satisfied, but violations still appear, then these deviations ought to be included in the error estimates.

cond-mat.stat-mech↗

arXiv2025

The noiseless limit and improved-prior limit of the maximum entropy method and their implications for the analytic continuation problem

Quantum Monte Carlo (QMC) methods are uniquely capable of providing exact simulations of quantum many-body systems. Unfortunately, the applications of a QMC simulation are limited because extracting dynamic properties requires solving the analytic continuation (AC) problem. Across the many fields that use QMC methods, there is no universally accepted analytic continuation algorithm for extracting dynamic properties, but many publications compare to the maximum entropy method. We investigate when entropy maximization is an acceptable approach. We show that stochastic sampling algorithms reduce to entropy maximization when the Bayesian prior is near to the true solution. We investigate when is Bryan's controversial optimization algorithm [Bryan, Eur. Biophys. J. 18, 165-174 (1990)] for entropy maximization (sometimes known as the maximum entropy method) appropriate to use. We show that Bryan's algorithm is appropriate when the noise is near zero or when the Bayesian prior is near to the true solution. We also investigate the mean squared error, finding a better scaling when the Bayesian prior is near the true solution than when the noise is near zero. We point to examples of improved data-driven Bayesian priors that have already leveraged this advantage. We support these results by solving the double Gaussian problem using both Bryan's algorithm and the newly formulated dual approach to entropy maximization [Chuna et al., J. Phys. A: Math. Theor. 58, 335203 (2025)].

physics.comp-ph↗

arXiv2025

PyLIT: Reformulation and implementation of the analytic continuation problem using kernel representation methods

Path integral Monte Carlo (PIMC) simulations are a cornerstone for studying quantum many-body systems. The analytic continuation (AC) needed to estimate dynamic quantities from these simulations is an inverse Laplace transform, which is ill-conditioned. If this inversion were surmounted, then dynamical observables (e.g. dynamic structure factor (DSF) $S(q,\omega)$) could be extracted from the imaginary-time correlation functions estimates. Although of important, the AC problem remains challenging due to its ill-posedness. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints. We use least-squares optimization regularized with a Bayesian prior to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the Wasserstein distance and $L^2$-distance as well as techniques for setting the regularization weight. A key outcome is the open-source package PyLIT (\textbf{Py}thon \textbf{L}aplace \textbf{I}nverse \textbf{T}ransform), which leverages Numba and unifies the presented formulations. PyLIT's core functionality is kernel construction and optimization. In our applications, we find PyLIT's DSF estimates share qualitative features with other more established methods. We identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the Wasserstein distance, a previously unexplored regularizer, performs as good as the entropic regularizer while benefiting from its linear gradient. Thirdly, future work can meaningfully combine regularized and stochastic optimization. (text cut for char. limit)

physics.comp-ph↗

arXiv2025

Estimates of the dynamic structure factor for the finite temperature electron liquid via analytic continuation of path integral Monte Carlo data

Understanding the dynamic properties of the uniform electron gas (UEG) is important for numerous applications ranging from semiconductor physics to exotic warm dense matter. In this work, we apply the maximum entropy method (MEM), as implemented in Chuna \emph{et al.}~[arXiv:2501.01869], to \emph{ab initio} path integral Monte Carlo (PIMC) results for the imaginary-time correlation function $F(q,\tau)$ to estimate the dynamic structure factor $S(q,\omega)$ over an unprecedented range of densities at the electronic Fermi temperature. To conduct the MEM, we propose to construct the Bayesian prior $\mu$ from the PIMC data. Constructing the static approximation leads to a drastic improvement in $S(q,\omega)$ estimate over using the more simple random phase approximation (RPA) as the Bayesian prior. We find good agreement with existing results by Dornheim \emph{et al.}~[\textit{Phys.~Rev.~Lett.}~\textbf{121}, 255001 (2018)], where they are available. In addition, we present new results for the strongly coupled electron liquid regime with $r_s=50,...,200$, which reveal a pronounced roton-type feature and an incipient double peak structure in $S(q,\omega)$ at intermediate wavenumbers. We also find that our dynamic structure factors satisfy known sum rules, even though these sum rules are not enforced explicitly. An advantage of our set-up is that it is not specific to the UEG, thereby opening up new avenues to study the dynamics of real warm dense matter systems based on cutting-edge PIMC simulations in future works.

cond-mat.str-el↗

arXiv2025

HEART: A New X-Ray Tracing Code for Mosaic Crystal Spectrometers

We introduce a new open-source Python x-ray tracing code for modelling Bragg diffracting mosaic crystal spectrometers: High Energy Applications Ray Tracer (HEART). HEART's high modularity enables customizable workflows as well as efficient development of novel features. Utilizing Numba's just-in-time (JIT) compiler and the message-passing interface (MPI) allows running HEART in parallel leading to excellent performance. HEART is intended to be used for modelling x-ray spectra as they would be seen in experiments that measure x-ray spectroscopy with a mosaic crystal spectrometer. This enables the user to make predictions about what will be seen on a detector in experiment, perform optimizations on the design of the spectrometer setup, or to study the effect of the spectrometer on measured spectra. However, the code certainly has further uses beyond these example use cases. Here, we discuss the physical model used in the code, and explore a number of different mosaic distribution functions, intrinsic rocking curves, and sampling approaches which are available to the user. Finally, we demonstrate its strong predictive capability in comparison to spectroscopic data collected at the European XFEL in Germany.

physics.plasm-ph↗

arXiv2025

Dual formulation of the maximum entropy method applied to analytic continuation of quantum Monte Carlo data

Many fields of physics use quantum Monte Carlo techniques, but struggle to estimate dynamic spectra via the analytic continuation of imaginary-time quantum Monte Carlo data. One of the most ubiquitous approaches to analytic continuation is the maximum entropy method (MEM). We supply a dual Newton optimization algorithm to be used within the MEM and provide analytic bounds for the algorithm's error. The MEM is typically used with Bryan's controversial algorithm [Rothkopf, "Bryan's Maximum Entropy Method" Data 5.3 (2020)]. We present new theoretical issues that are not yet in the literature. Our algorithm has all the theoretical benefits of Bryan's algorithm without these theoretical issues. We compare the MEM with Bryan's optimization to the MEM with our dual Newton optimization on test problems from lattice quantum chromodynamics and plasma physics. These comparisons show that in the presence of noise the dual Newton algorithm produces better estimates and error bars; this indicates the limits of Bryan's algorithm's applicability. We use the MEM to investigate authentic quantum Monte Carlo data for the uniform electron gas at warm dense matter conditions and further substantiate the roton-type feature in the dispersion relation.

physics.comp-ph↗

arXiv2024

Conservative dielectric functions and electrical conductivities from the multicomponent Bhatnagar-Gross-Krook equation

A considerable number of semi-empirical and first-principles models have been created to describe the dynamic response of a collisionally damped charged-particle system. However, known challenges persist for established dynamic structure factors (DSF), dielectric functions, and conductivities. For instance, the semi-empirical Drude-Smith conductivity [N.M. Smith, Phys. Rev. B 64, 155106 (2001)] lacks interpretability, and the first-principles Mermin dielectric function [N.D. Mermin, Phys. Rev. B, 1, 2362 (1970)] does not satisfy the frequency sum rule [G.S. Atwal and N.W. Ashcroft, Phys. Rev. B 65, 115109 (2002)]. In this work, starting from the multicomponent Bhatnagar-Gross-Krook (BGK) kinetic equation, we produce a multi-species susceptibility that conserves number and momentum, which we refer to as the ``completed Mermin'' susceptibility, and we explore its properties and uses. We show that the completed Mermin susceptibility satisfies the frequency sum (f-sum) rule. We compute the associated DSF and find that momentum conservation qualitatively impacts the DSF's shape for a carbon-contaminated deuterium and tritium plasma under NIF hot-spot conditions. In the appendices, we provide numerical implementations of the completed Mermin susceptibility, for the reader's convenience. Further, we produce a new non-Drude conductivity model, by taking the single-species limit and introducing free parameters in the terms that enforce number and momentum conservation. To illustrate how number and momentum conservation impact the dynamical conductivity shape, we apply our conductivity model to dynamical gold conductivity measurements [Z. Chen, et al., Nature communications, 12.1, 1638, (2021)]. Finally, comparing our model to the Drude-Smith conductivity model, we conclude that Smith's phenomenological parameter violates local number conservation.

physics.plasm-ph↗

arXiv2021

Efficient integration of gradient flow in lattice gauge theory and properties of low-storage commutator-free Lie group methods

The smoothing procedure known as the gradient flow that suppresses ultraviolet fluctuations of gauge fields plays an important role in lattice gauge theory calculations. In particular, this procedure is often used for high-precision scale setting and renormalization of operators. The gradient flow equation is defined on the SU(3) manifold and therefore requires geometric, or structure-preserving, integration methods to obtain its numerical solutions. We examine the properties and origins of the three-stage third-order explicit Runge-Kutta Lie group integrator commonly used in the lattice gauge theory community, demonstrate its relation to 2N-storage classical Runge-Kutta methods and explore how its coefficients can be tuned for optimal performance in integrating the gradient flow. We also compare the performance of the tuned method with two third-order variable step size methods. Next, based on the recently established connection between low-storage Lie group integrators and classical 2N-storage Runge-Kutta methods, we study two fourth-order low-storage methods that provide a computationally efficient alternative to the commonly used third-order method while retaining the convenient iterative property of the latter. Finally, we demonstrate that almost no coding effort is needed to implement the low-storage Lie group methods into existing gradient flow codes.

hep-lat↗

arXiv2018

Honors Thesis: On the faithfulness of the Burau representation at roots of unity

We study the kernel of the evaluated Burau representation through the braid element $σ_i σ_{i+1} σ_i$. The element is significant as a part of the standard braid relation. We establish the form of this element's image raised to the $n^{th}$ power. Interestingly, the cyclotomic polynomials arise and can be used to define the expression. The main result of this paper is that the Burau representation of the braid group of $n$ strands for $n \geq 3$ is unfaithful at any primitive root of unity, excepting the first three.

math.GT↗