arXiv subjects

Johann Ostmeyer

Publications and source records attributed to Johann Ostmeyer.

At least 19 recordsLinked to original sources

arXiv2026

Stable and Efficient Algorithms for the Fermion Determinant

Some algorithms for the numerically exact treatment of fermion determinants are summarised. This is not supposed to be a review, rather a concise handbook. The audience is expected to have a basic understanding of how to put fermions on a computer. We primarily discuss different ways to work with the fermion matrix in the "sausage" (Green's function) formulation for quantum Monte Carlo (QMC). We emphasise the need for varied approaches in different space-time volume regimes. In particular, for small spatial volumes we describe a numerically stable method based on dense matrix operations. It is designed specifically to deal with very low temperature regimes. On the other hand, for (relatively) large volumes we describe a highly efficient and scalable sparse matrix approach.

physics.comp-ph↗

arXiv2026

Stable Determinant Monte Carlo Simulations at Large Inverse Temperature $β$

At low temperatures $T$ where $1/T=β\gg1$ the naïve implementation of determinant quantum Monte Carlo (DQMC) methods suffers from loss of precision and numerical instabilities when evaluating the fermion determinant. This instability propagates into the calculation of observables that rely on the evaluation of the inverse of the fermion matrix, or the Greens function. For DQMC methods that rely on the Hamiltonian Monte Carlo (HMC) algorithm, an additional complication comes from evaluating the force terms required for integrating Hamilton's equations of motion, since here loss of precision and numerical instabilities are also prevalent. We show how to address all these issues using various choices of matrix decompositions, allowing us to simulate at $β\gtrsim 90$, which corresponds to room temperature for graphene structures. Furthermore, our implementation has numerical costs that scale similarly to the naïve implementation, namely as $\mathcal{O}(N_x^3N_t)$, where $N_x$ ($N_t$) is the number of spatial (temporal) sites.

physics.comp-ph↗

arXiv2026

Lattice Field Theory Analysis of the Chiral Heisenberg Model

Motivated by ongoing interest in the universal behaviour of the Hubbard model of spinning electrons on honeycomb and $\pi$-flux lattices at the semi-metal -- Mott insulator phase transition, we formulate the 3D chiral Heisenberg model, a theory of relativistic fermions in three spacetime dimensions, as a lattice field theory using domain wall fermions. The contact interaction term preserves an SU(2) global symmetry. We perform numerical simulations using the Rational Hybrid Monte Carlo algorithm on system sizes $L^3\times L_s$ with $L\in\{8,\ldots,24\}$ and domain wall separation $L_s\in\{8,16,24\}$. We locate the phase transition corresponding to spontaneous SU(2)$\to$U(1) breaking, yielding critical exponent estimates $\nu^{-1}=0.63(3)$, $\eta_\Phi=1.42(8)$. These values are considerably removed from estimates obtained from simulations performed in (2+1)D, ie. with the time and spatial directions treated differently, but align more closely with analytic estimates obtained using 3D covariant field theory. We also present first results for the fermion correlator, ultimately needed for the determination of the exponent $\eta_\Psi$, highlighting the need to rotate the fermion source to a common reference direction in isospace in order to obtain a signal.

hep-lat↗

arXiv2026

Hamiltonian Monte Carlo enhanced by Exact Diagonalization

Strongly correlated fermionic systems are of great interest in condensed matter physics and numerical methods are indispensable tools for their study. However, existing approaches such as exact diagonalization (ED) and stochastic quantum Monte Carlo methods each suffer from fundamental limitations: ED is hindered by exponential scaling in system size, while Monte Carlo methods are plagued by sign problems and long autocorrelation times. These limitations restrict the accessible parameter space and developing algorithms that efficiently alleviate them remains a central challenge in computational physics. In this work, we propose a hybrid algorithm that combines ED and Hamiltonian Monte Carlo (HMC) to simulate 2D arrays of coupled quantum wires, modeled as interacting fermionic Hubbard chains. We demonstrate how our hybrid implementation of HMC, which we dub H$^2$MC, outperforms either method alone across several key simulation facets. When compared to pure ED, H$^2$MC has a much more favorable computational scaling, which allows us to push simulations to much larger 2D arrays. H$^2$MC also greatly alleviates the sign problem and reduces autocorrelation times when compared to pure HMC formulations utilizing either real or imaginary auxiliary fields. Our formalism demonstrates how complementary strengths of seemingly disparate methods can be leveraged to enable feasible simulations in an extended parameter space.

cond-mat.str-el↗

arXiv2026

Reducing the Gate Count with Efficient Trotter-Suzuki Schemes

Hamiltonian formulations of lattice field theories provide access to real-time dynamics, but their simulation is difficult to implement efficiently. Trotter-Suzuki decompositions are at the center of time evolution computation, either on quantum hardware or classically, for instance with the use of tensor networks. While low-order Trotterizations remain the standard choice due to their simplicity, higher-order schemes offer the potential for improved efficiency. In this work we outline a short guide to Trotter-Suzuki schemes and their implementations in general. To help with this, we highlight new efficient schemes found by our optimization framework, and demonstrate their performance on the Heisenberg model.

hep-lat↗

arXiv2026

Efficient Trotter-Suzuki Schemes for Long-time Quantum Dynamics

Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at $4^{\textrm{th}}$ and $6^{\textrm{th}}$ order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.

quant-ph↗

arXiv2025

A Comprehensive Stress Test of Truncated Hilbert Space Bases against Green's function Monte Carlo in U(1) Lattice Gauge Theory

A representation of Lattice Gauge Theories (LGT) suitable for simulations with tensor network state methods or with quantum computers requires a truncation of the Hilbert space to a finite dimensional approximation. In particular for U(1) LGTs, several such truncation schemes are known, which we compare with each other using tensor network states. We show that a functional basis obtained from single plaquette Hamiltonians -- which we call plaquette state basis -- outperforms the other schemes in two spatial dimensions for plaquette, ground state energy and mass gap, as it is delivering accurate results for a wide range of coupling strengths with a minimal number of basis states. We also show that this functional basis can be efficiently used in three spatial dimensions. Green's function Monte Carlo appears to be a highly useful tool to verify tensor network states results, which deserves further investigation in the future.

hep-lat↗

arXiv2025

The Truncated Hankel Correlator Method

We introduce a new method to approximate Euclidean correlation functions by exponential sums. The Truncated Hankel Correlator (THC) method builds a Hankel matrix from the full correlator data available and truncates the eigenspectrum of said Hankel matrix. It proceeds by applying the Prony generalised eigenvalue method to the thus obtained low-rank approximation. A large number of algebraic correlator analysis methods including (block) Prony (and equivalently (block) Lanczos) and the generalised eigenvalue problem (GEVP) can be reproduced as sub-optimal special cases of the THC method. Weights, for instance inverse square errors, can be included in the analysis, so that the result has a close to optimal $χ^2$-value. This makes the THC method very similar in spirit to a closed form solution to multi-state fits, naturally including the case of matrix-valued correlators. We show that, in general, finding approximations better than those provided by the THC method is exponentially hard in the number of exponentials. Moreover, the THC method is robust against noise and requires comparably little human oversight. Finally, when applied to symmetric data, the obtained energy spectrum is guaranteed to be symmetric up to machine precision.

hep-lat↗

arXiv2025

Fully ergodic simulations using radial updates

A sensible application of the Hybrid Monte Carlo (HMC) method is often hindered by the presence of large - or even infinite - potential barriers. These potential barriers separate the configuration space into distinct sectors and can lead to ergodicity violations that bias measurements. In this work, we address this problem by augmenting HMC with a multiplicative Metropolis-Hastings update in a so-called ''radial direction'' of the fields which enables crossing the potential barriers and ensures ergodicity of the sampling algorithm at comparably low computational cost. We demonstrate the algorithm on a simple toy model and show how it can be applied to the fermionic Hubbard model describing physics ranging from an exactly-solvable two-site system to the $C_{20}H_{12}$ perylene molecule. Our numerical results show that the radial updates successfully remove ergodicity violations, while simultaneously reducing autocorrelation times.

cond-mat.str-el↗

arXiv2025

Minimal Autocorrelation in Hybrid Monte Carlo simulations using Exact Fourier Acceleration

The hybrid Monte Carlo (HMC) algorithm is a ubiquitous method in computational physics with applications ranging from condensed matter to lattice QCD and beyond. However, HMC simulations often suffer from long autocorrelation times, severely reducing their efficiency. In this work two of the main sources of autocorrelations are identified and eliminated. The first source is the sampling of the canonical momenta from a sub-optimal normal distribution, the second is a badly chosen trajectory length. Analytic solutions to both problems are presented and implemented in the exact Fourier acceleration (EFA) method. It completely removes autocorrelations for near-harmonic potentials and consistently yields (close-to-) optimal results for numerical simulations of the Su-Schrieffer-Heeger and the Ising models as well as in lattice gauge theory, in some cases reducing the autocorrelation by multiple orders of magnitude. EFA is advantageous for and easily applicable to any HMC simulation of an action that includes a quadratic part.

hep-lat↗

arXiv2025

Exponential speed up in Monte Carlo sampling through Radial Updates

Recently, it has been shown that the hybrid Monte Carlo (HMC) algorithm is guaranteed to converge exponentially to a given target probability distribution $p(x)\propto e^{-V(x)}$ on non-compact spaces if augmented by an appropriate radial update. In this work we present a simple way to derive efficient radial updates meeting the necessary requirements for any potential $V$. We reduce the problem to finding a substitution for the radial direction $||x||=f(z)$ so that the effective potential $V(f(z))$ grows exponentially with $z\rightarrow\pm\infty$. Any additive update of $z$ then leads to the desired convergence. We show that choosing this update from a normal distribution with standard deviation $σ\approx 1/\sqrt{d}$ in $d$ dimensions yields very good results. We further generalise the previous results on radial updates to a wide class of Markov chain Monte Carlo (MCMC) algorithms beyond the HMC and we quantify the convergence behaviour of MCMC algorithms with badly chosen radial update. Finally, we apply the radial update to the sampling of heavy-tailed distributions and achieve a speed up of many orders of magnitude.

physics.comp-ph↗

arXiv2025

Dynamics in Hamiltonian Lattice Gauge Theory: Approaching the Continuum Limit with Partitionings of SU$(2)$

In this paper, we investigate a digitised SU$(2)$ lattice gauge theory in the Hamiltonian formalism. We use partitionings to digitise the gauge degrees of freedom and show how to define a penalty term based on finite element methods to project onto physical states of the system. Moreover, we show for a single plaquette system that in this framework the limit $g\to0$ can be approached at constant cost.

hep-lat↗

arXiv2025

The Physicist's Guide to the HMC

The hybrid Monte Carlo (HMC) algorithm is arguably the most efficient sampling method for general probability distributions of continuous variables. Together with exact Fourier acceleration (EFA) the HMC becomes equivalent to direct sampling for quadratic actions $S(x)=\frac12 x^\mathsf{T} M x$ (i.e. normal distributions $x\sim \mathrm{e}^{-S(x)}$), only perturbatively worse for perturbative deviations of the action from the quadratic case, and it remains viable for arbitrary actions. In this work the most recent improvements of the HMC including EFA and radial updates are collected into a numerical recipe.

hep-lat↗

arXiv2024

On the equivalence of Prony and Lanczos methods for Euclidean correlation functions

We investigate the oblique Lanczos method recently put forward in arXiv:2406.20009 for analysing Euclidean correlators in lattice field theories and show that it is analytically equivalent to the well known Prony Generalised Eigenvalue Method (PGEVM). Moreover, we discuss that the signal-to-noise problem is not aleviated by either of these two methods. Still, both methods show clear advantages when compared to the standard effective mass approach.

hep-lat↗

arXiv2024

High-precision Quantum Monte-Carlo study of charge transport in a lattice model of molecular organic semiconductors

We use first-principle Quantum Monte-Carlo (QMC) simulations and numerical exact diagonalization to analyze the low-frequency charge carrier mobility within a simple tight-binding model of molecular organic semiconductors on a two-dimensional triangular lattice. These compounds feature transient localization, an unusual charge transport mechanism driven by dynamical disorder. The challenges of studying the transient localization of charge carriers in the low-frequency/long-time limit from first principles are discussed. We demonstrate that a combination of high-precision QMC data with prior estimates of frequency-dependent charge carrier mobility based on the static disorder approximation for phonon fields allows for improved estimates of mobility in the low-frequency limit. We also point out that a simple relaxation time approximation for charge mobility in organic semiconductors is not consistent with the QMC data. Physical similarities with charge transport in quark-gluon plasma are highlighted.

cond-mat.mtrl-sci↗

arXiv2024

Overcoming Ergodicity Problems of the Hybrid Monte Carlo Method using Radial Updates

Despite its many advantages, the sensible application of the Hybrid Monte Carlo (HMC) method is often hindered by the presence of large - or even infinite - potential barriers. These potential barriers partition the configuration space into distinct sectors, which leads to ergodicity violations and biased measurements of observables. In this work, we address this problem by augmenting the HMC method with a multiplicative Metropolis-Hastings update in a so-called "radial direction" of the fields, which enables jumps over the aforementioned potential barriers at comparably low computational cost. The effectiveness of this approach is demonstrated for the Hubbard model, formulated in a non-compact space by means of a continuous Hubbard-Stratonovich transformation. Our numerical results show that the radial updates successfully resolve the ergodicity violation, while simultaneously reducing autocorrelations.

cond-mat.str-el↗

arXiv2024

First-principle quantum Monte-Carlo study of charge carrier mobility in organic molecular semiconductors

We present a first-principle numerical study of charge transport in a realistic two-dimensional tight-binding model of organic molecular semiconductors. We use the Hybrid Monte Carlo (HMC) algorithm to simulate the full quantum dynamics of phonons and either a single or multiple charge carriers without any tunable parameters. We introduce a number of algorithmic improvements, including efficient Metropolis updates for phonon fields based on analytic insights, which lead to negligible autocorrelation times and allow to reach sub-permille precisions at small computational cost of $O(1)$ CPU-hour. Our simulations produce charge mobility estimates that are in good agreement with experiment and that also justify the phenomenological Transient Localisation approach.

cond-mat.mtrl-sci↗

arXiv2024

Reducing the Sign Problem with simple Contour Deformation

We apply constant imaginary offsets to the path integral for a reduction of the sign problem in the Hubbard model. These simple transformations enhance the quality of results from HMC calculations without compromising the speed of the algorithm. This method enables us to efficiently calculate systems that are otherwise inaccessible due to a severe sign problem. To support this claim, we present observables of the C20 and C60 fullerenes. Furthermore, we demonstrate that at a certain offset, the sign problem is completely lifted in the limit of large chemical potential.

cond-mat.str-el↗