arXiv subjects

Markus Holzmann

Publications and source records attributed to Markus Holzmann.

At least 19 recordsLinked to original sources

arXiv2026

On the discrete spectrum of Dirac operators with Lorentz-scalar $\delta$-shell interactions supported on unbounded curves

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

math.SP↗

arXiv2026

Neural Wave Functions for High-Pressure Atomic Hydrogen

We leverage the power of neural quantum states to describe the ground state wave function of solid and liquid atomic hydrogen, including both electronic and protonic degrees of freedom. For static protons, the resulting Born-Oppenheimer energies are consistently comparable to or lower than all previous projector Monte Carlo results for systems containing up to $128$ hydrogen atoms. The same level of accuracy is preserved upon inclusion of nuclear quantum effects, thus going beyond the Born-Oppenheimer approximation. In addition, our description overcomes major limitations of current wave functions, notably by avoiding any explicit symmetry assumption on the expected quantum crystal, and sidestepping efficiency issues of imaginary time evolution with disparate mass scales. As a first application, we examine crystal formation in an extremely high-density region up to pressure-induced melting.

cond-mat.str-el↗

arXiv2026

Variance reduction for forces and pressure in variational Monte Carlo

We present simple and practical strategies to reduce the variance of Monte Carlo estimators. Our focus is on variational Monte Carlo calculations of atomic forces and pressure in electronic systems, although we show that the underlying ideas apply more broadly to other observables, like pair-correlation and angular-distribution functions, and other methods, including molecular dynamics. For Pulay-type contributions, we show that a minor modification based on the Metropolis acceptance ratio softens the power-law divergence of the variance to a logarithmic one, and that inexpensive regularizations can further suppress outliers at the price of a controlled small bias. For Hellmann-Feynman forces, we derive compact variance-reduced estimators for periodic systems that are straightforward to implement in standard Monte Carlo codes. The approach is illustrated for high-pressure metallic hydrogen with more than a hundred atoms described by neural quantum states, including an application to molecular dynamics driven by the improved forces.

cond-mat.str-el↗

arXiv2026

Nonperturbative computation of thermal conductivity based on Path Integral Monte Carlo methods

The calculation of thermal conductivity in insulating solids at temperatures below the Debye temperature is problematic, due to the breakdown of classical and semi-classical approaches. In this work, we present a fully non-perturbative quantum methodology to compute thermal conductivity based on Path Integral Monte Carlo (PIMC) simulations combined with the Green-Kubo linear response theory. The method is applied to rare gas solids modeled by a Lennard-Jones potential, paradigmatic systems where quantum effects strongly affect both thermodynamic and transport properties. From PIMC simulations, we obtain the temperature-dependent phonon frequencies, lifetimes, and specific heat. From the imaginary time correlations of the energy current, we extract the thermal transport coefficients based on a physically motivated prior. We show that the experimentally observed increase of the thermal conductivity of argon and neon at low temperatures cannot be explained within a Peierls-Boltzmann framework using phonon line-widths at equilibrium. In contrast, a distinct transport lifetime emerges from the analysis of heat-current correlations. Our results demonstrate that quantum Monte Carlo methods provide a robust, nonperturbative framework to investigate heat transport in insulating solids, beyond the limits of classical molecular dynamics without relying on perturbative or semi-classical approximations.

cond-mat.stat-mech↗

arXiv2026

Approximation of magnetic Schrödinger operators with $δ$-interactions supported on networks

This paper deals with the approximation of a magnetic Schrödinger operator with a singular $δ$-potential that is formally given by $(i \nabla + A)^2 + Q + αδ_Σ$ by Schrödinger operators with regular potentials in the norm resolvent sense. This is done for $Σ$ being the finite union of $C^2$-hypersurfaces, for coefficients $A$, $Q$, and $α$ under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and $Q$ and $α$ are allowed to be complex-valued functions. In particular, $Σ$ can be a graph in $\mathbb{R}^2$ or the boundary of a piecewise $C^2$-domain. Moreover, spectral implications of the mentioned convergence result are discussed.

math.SP↗

arXiv2025

Weak coupling for Schrödinger operators with complex potentials

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].

math.SP↗

arXiv2025

Bath-induced Zeno localization in driven many-body quantum systems

We study a quantum interacting spin system subject to an external drive and coupled to a thermal bath of spatially localized vibrational modes, serving as a model of Dynamic Nuclear Polarization. We show that even when the many-body eigenstates of the system are ergodic, a sufficiently strong coupling to the bath may effectively localize the spins due to many-body quantum Zeno effect, as manifested by the hole-burning shape of the electron paramagnetic resonance spectrum. Our results provide an explanation of the breakdown of the thermal mixing regime experimentally observed above 4 - 5 Kelvin.

cond-mat.stat-mech↗

arXiv2025

Approximation of Dirac operators with $\boldsymbolδ$-shell potentials in the norm resolvent sense, I. Qualitative results

In this paper the approximation of Dirac operators with general $δ$-shell potentials supported on $C^2$-curves in $\mathbb{R}^2$ or $C^2$-surfaces in $\mathbb{R}^3$, which may be bounded or unbounded, is studied. It is shown under suitable conditions on the weight of the $δ$-interaction that a family of Dirac operators with regular, squeezed potentials converges in the norm resolvent sense to the Dirac operator with the $δ$-shell interaction.

math.SP↗

arXiv2025

Approximation of Dirac operators with $\boldsymbolδ$-shell potentials in the norm resolvent sense, II. Quantitative results

This paper is devoted to the approximation of two and three-dimensional Dirac operators $H_{\widetilde{V} δ_Σ}$ with combinations of electrostatic and Lorentz scalar $δ$-shell interactions in the norm resolvent sense. Relying on results from \cite{BHS23} an explicit smallness condition on the coupling parameters is derived so that $H_{\widetilde{V} δ_Σ}$ is the limit of Dirac operators with scaled electrostatic and Lorentz scalar potentials. Via counterexamples it is shown that this condition is sharp. The approximation of $H_{\widetilde{V} δ_Σ}$ for larger coupling constants is achieved by adding an additional scaled magnetic term.

math.SP↗

arXiv2025

Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.

math.SP↗

arXiv2025

On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval

We describe all self-adjoint realizations of the restricted fractional Laplacian $(-Δ)^a$ with power $a \in (\frac{1}{2}, 1)$ on a bounded interval by imposing boundary conditions on the functions in the domain of a maximal realization; such conditions relate suitable weighted Dirichlet and Neumann traces. This is done in a systematic way by using the abstract concept of boundary triplets and their Weyl functions from extension and spectral theory of symmetric and self-adjoint operators in Hilbert spaces. Our treatment follows closely the well-known one for classical Laplacians on intervals and it shows that all self-adjoint realizations have purely discrete spectrum and are semibounded from below. To demonstrate the method, we focus on three self-adjoint realizations of the restricted fractional Laplacian: the Friedrichs extension, corresponding to Dirichlet-type boundary conditions, the Krein--von Neumann extension, and a Neumann-type realization. Notably, the Neumann-type realization exhibits a simple negative eigenvalue, thus it is not larger than the Krein--von Neumann extension.

math.SP↗

arXiv2025

Non-self-adjoint Dirac operators on graphs

In this paper we introduce and study generally non-self-adjoint realizations of the Dirac operator on an arbitrary finite metric graph. Employing the robust boundary triple framework, we derive, in particular, a variant of the Birman Schwinger principle for its eigenvalues, and with an example of a star shaped graph we show that the point spectrum may exhibit diverse behaviour. Subsequently, we find sufficient and necessary conditions on transmission conditions at the graph's vertices under which the Dirac operator on the graph is symmetric with respect to the parity, the time reversal, or the charge conjugation transformation.

math-ph↗

arXiv2025

Two-dimensional Schrödinger operators with non-local singular potentials

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $Σ$. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at $-\infty$. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic $δ$-shell interactions].

math.SP↗

arXiv2024

The liquid-liquid phase transition of hydrogen and its critical point: Analysis from ab initio simulation and a machine-learned potential

We simulate high-pressure hydrogen in its liquid phase close to molecular dissociation using a machine-learned interatomic potential. The model is trained with density functional theory (DFT) forces and energies, with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. We show that an accurate NequIP model, an E(3)-equivariant neural network potential, accurately reproduces the phase transition present in PBE. Moreover, the computational efficiency of this model allows for substantially longer molecular dynamics trajectories, enabling us to perform a finite-size scaling (FSS) analysis to distinguish between a crossover and a true first-order phase transition. We locate the critical point of this transition, the liquid-liquid phase transition (LLPT), at 1200-1300 K and 155-160 GPa, a temperature lower than most previous estimates and close to the melting transition.

cond-mat.stat-mech↗

arXiv2024

Phase diagram and crystal melting of helium-4 in two dimensions

We study the zero-temperature phase diagram of two-dimensional helium-4 using neural quantum states. Our variational description allows us to address liquid and solid phases using the same functional form as well as exploring possible melting scenarios, for instance via an intermediate hexatic phase. Notably, this is achieved by performing fixed pressure variational Monte Carlo calculations. Within the isobaric ensemble framework, we are able to clearly identify the first-order liquid-solid phase transition. However, in an intermediate region of nearly constant pressure, we find that simulations of $N=30$ atoms continuously transition from liquid to solid, with signatures of a hexatic order coexisting with a small condensate fraction. Calculations for larger systems follow the metastable liquid and solid branches in this transient region. We additionally compute the Rényi-2 entanglement entropy across the liquid-solid phase transition and find a sharp decrease upon freezing.

cond-mat.stat-mech↗

arXiv2024

High temperature melting of dense molecular hydrogen from machine-learning interatomic potentials trained on quantum Monte Carlo

We present results and discuss methods for computing the melting temperature of dense molecular hydrogen using a machine learned model trained on quantum Monte Carlo data. In this newly trained model, we emphasize the importance of accurate total energies in the training. We integrate a two phase method for estimating the melting temperature with estimates from the Clausius-Clapeyron relation to provide a more accurate melting curve from the model. We make detailed predictions of the melting temperature, solid and liquid volumes, latent heat and internal energy from 50 GPa to 180 GPa for both classical hydrogen and quantum hydrogen. At pressures of roughly 173 GPa and 1635K, we observe molecular dissociation in the liquid phase. We compare with previous simulations and experimental measurements.

physics.chem-ph↗

arXiv2024

Data Subsampling for Bayesian Neural Networks

Markov Chain Monte Carlo (MCMC) algorithms do not scale well for large datasets leading to difficulties in Neural Network posterior sampling. In this paper, we propose Penalty Bayesian Neural Networks - PBNNs, as a new algorithm that allows the evaluation of the likelihood using subsampled batch data (mini-batches) in a Bayesian inference context towards addressing scalability. PBNN avoids the biases inherent in other naive subsampling techniques by incorporating a penalty term as part of a generalization of the Metropolis Hastings algorithm. We show that it is straightforward to integrate PBNN with existing MCMC frameworks, as the variance of the loss function merely reduces the acceptance probability. By comparing with alternative sampling strategies on both synthetic data and the MNIST dataset, we demonstrate that PBNN achieves good predictive performance even for small mini-batch sizes of data. We show that PBNN provides a novel approach for calibrating the predictive distribution by varying the mini-batch size, significantly reducing predictive overconfidence.

stat.ML↗

arXiv2024

Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities

For a family of self-adjoint Dirac operators $-i c (α\cdot \nabla) + \frac{c^2}{2}$ subject to generalized MIT bag boundary conditions on domains in $\mathbb R^3$ it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large $c$.

math.SP↗