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Markus Holzmann

Publications and source records attributed to Markus Holzmann.

At least 55 records · Page 3Linked to original sources
arXiv2021

A simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy

In 1963, a Simple Approach was developed to study the ground state energy of an interacting Bose gas. It consists in the derivation of an Equation, which is not based on perturbation theory, and which gives the exact expansion of the energy at low densities. This Equation is expressed directly in the thermodynamic limit, and only involves functions of $3$ variables, rather than $3N$. Here, we revisit this approach, and show that the Equation yields accurate predictions for various observables for all densities. Specifically, in addition to the ground state energy, we have shown that the Simple Approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the Equation with Quantum Monte Carlo calculations, and have found remarkable agreement. We thus show that the Simple Approach provides a new theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which little is known.

cond-mat.quant-gas↗
arXiv2021

A note on the three dimensional Dirac operator with zigzag type boundary conditions

In this note the three dimensional Dirac operator $A_m$ with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $A_m$ is self-adjoint in $L^2(Ω;\mathbb{C}^4)$ for any open set $Ω\subset \mathbb{R}^3$ and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $Ω$. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $A_m$ consists of discrete eigenvalues that accumulate at $\pm \infty$ and one additional eigenvalue of infinite multiplicity.

math.SP↗
arXiv2020

Quasi-condensation of bilayer excitons in a periodic potential

We study two-dimensional excitons confined in a lattice potential, for high fillings of the lattice sites. We show that a quasi-condensate is possibly formed for small values of the lattice depth, but for larger ones the critical phase-space density for quasi-condensation rapidly exceeds our experimental reach, due to the increase of the excitons effective mass. On the other hand, in the regime of a deep lattice potential where excitons are strongly localised at the lattice sites, we show that an array of phase-independent quasi-condensates, different from a Mott insulating phase, is realised.

cond-mat.quant-gas↗
arXiv2020

Electronic structure and optical properties of quantum crystals from first principles calculations in the Born-Oppenheimer approximation

We develop a formalism to accurately account for the renormalization of electronic structure due to quantum and thermal nuclear motions within the Born-Oppenheimer approximation. We focus on the fundamental energy gap obtained from electronic addition and removal energies from Quantum Monte Carlo calculations in either the canonical or grand canonical ensembles. The formalism applies as well to effective single electron theories such as those based on Density Functional Theory. We show that electronic (Bloch) crystal momentum can be restored by marginalizing the total electron-ion wave function with respect to the nuclear equilibrium distribution, and we describe an explicit procedure to establish the band structure of electronic excitations for quantum crystals within the Born-Oppenheimer approximation. Based on the Kubo-Greenwood equation, we discuss the effects of nuclear motion on optical conductivity. Our methodology applies to the low temperature regime where nuclear motion is quantized and in general differs from the semi-classical approximation. We apply our method to study the electronic structure of C2/c-24 crystalline hydrogen at 200K and 250 GPa and discuss the optical absorption profile of hydrogen crystal at 200K and carbon diamond at 297K.

cond-mat.mtrl-sci↗
arXiv2020

Electronic energy gap closure and metal-insulator transition in dense liquid hydrogen

Using Quantum Monte Carlo (QMC) calculations, we investigate the insulator-metal transition observed in liquid hydrogen at high pressure. Below the critical temperature of the transition from the molecular to the atomic liquid, the fundamental electronic gap closure occurs abruptly, with a small discontinuity reflecting the weak first-order transition in the thermodynamic equation of state. Above the critical temperature, molecular dissociation sets in while the gap is still open. When the gap closes, the decay of the off-diagonal reduced density matrix shows that the liquid enters a gapless, but localized phase: there is a cross-over between the insulating and the metallic liquids. Compared to different DFT functionals, our QMC calculations provide larger values for the fundamental gap and the electronic density of states close to the band edges, indicating that optical properties from DFT potentially benefit from error cancellations.

cond-mat.mtrl-sci↗
arXiv2020

Quantum Monte Carlo determination of the principal Hugoniot of deuterium

We present Coupled Electron-Ion Monte Carlo results for the principal Hugoniot of deuterium together with an accurate study of the initial reference state of shock wave experiments. We discuss the influence of nuclear quantum effects, thermal electronic excitations, and the convergence of the energy potential surface by wave function optimization within Variational Monte Carlo and Projection Quantum Monte Carlo methods. Compared to a previous study, the new calculations also include low pressure-temperature (P,T) conditions resulting in close agreement with experimental data, while our revised results at higher (P,T) conditions still predict a more compressible Hugoniot than experimentally observed.

physics.comp-ph↗
arXiv2020

Energy gap closure of crystalline molecular hydrogen with pressure

We study the gap closure with pressure of crystalline molecular hydrogen. The gaps are obtained from grand-canonical Quantum Monte Carlo methods properly extended to quantum and thermal crystals, simulated by Coupled Electron Ion Monte Carlo. Nuclear zero point effects cause a large reduction in the gap ($\sim 2eV$). \CP{Depending on the structure,} the fundamental indirect gap closes \CP{between 380GPa and} 530GPa for ideal crystals and 330-380GPa for quantum crystals. Beyond this pressure the system enters into a bad metal phase where the density of states at the Fermi level increases with pressure up to $\sim$450\CP{-500} GPa when the direct gap closes. Our work partially supports the interpretation of recent experiments in high pressure hydrogen.

cond-mat.mtrl-sci↗
arXiv2019

Quantum Monte Carlo Compton profiles of solid and liquid lithium

We computed the Compton profile of solid and liquid lithium using quantum Monte Carlo (QMC) and compared with recent experimental measurements obtaining good agreement. Importantly, we find it crucial to account for proper core-valence orthogonalization and to address density differences when comparing with experiment. To account for disorder effects, we sampled finite-temperature configurations using molecular dynamics (MD), then performed diffusion Monte Carlo (DMC) simulations on each configuration. We used Slater-Jastrow wavefunctions and grand-canonical twist-averaged boundary conditions. A QMC pseudopotential correction, derived from an all-electron DMC simulation of the perfect crystal was also used. Our calculations provide the first all-electron QMC benchmark for the Compton profile of lithium crystal and pseudopotential-corrected QMC Compton profiles for both the liquid and solid.

cond-mat.mtrl-sci↗
arXiv2019

Computing general observables in lattice models with complex actions

The study of QFTs at finite density is hindered by the presence of the so-called sign problem. The action definition of such systems is, in fact, complex-valued making standard importance sampling Monte Carlo methods ineffective. In this work, we shall review the generalized density of states method for complex action systems and the Linear Logarithmic Relaxation algorithm (LLR). We will focus on the recent developments regarding the bias control of the LLR method and the evaluation of general observables in the DoS+LLR framework. Recent results on the well-known relativistic Bose gas will be presented, proving that in our approach the phase factor can be consistently evaluated over hundreds of orders of magnitude. A first exploratory study on the Thirring model in the DoS formalism will be presented as well.

hep-lat↗
arXiv2019

Electronic band gaps from Quantum Monte Carlo methods

We develop a method for calculating the fundamental electronic gap of semiconductors and insulators using grand canonical Quantum Monte Carlo simulations. We discuss the origin of the bias introduced by supercell calculations of finite size and show how to correct the leading and subleading finite size errors either based on observables accessible in the finite-sized simulations or from DFT calculations. Our procedure is applied to solid molecular hydrogen and compared to experiment for carbon and silicon crystals.

cond-mat.mtrl-sci↗
arXiv2019

Self-adjoint Dirac operators on domains in $\mathbb{R}^3$

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Ω; \mathbb{C}^4)$, where $Ω\subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $Ω$ is unbounded, and corresponding trace formulas.

math.SP↗
arXiv2019

Free energy of the self-interacting relativistic lattice Bose gas at finite density

The density of state approach has recently been proposed as a potential route to circumvent the sign problem in systems at finite density. In this study, using the Linear Logarithmic Relaxation (LLR) algorithm, we extract the generalised density of states, which is defined in terms of the imaginary part of the action, for the self-interacting relativistic lattice Bose gas at finite density. After discussing the implementation and testing the reliability of our approach, we focus on the determination of the free energy difference between the full system and its phase-quenched counterpart. Using a set of lattices ranging from $4^4$ to $16^4$ , we show that in the low density phase, this overlap free energy can be reliably extrapolated to the thermodynamic limit. The numerical precision we obtain with the LLR method allows us to determine with sufficient accuracy the expectation value of the phase factor, which is used in the calculation of the overlap free energy, down to values of ${\cal O}(10^{-480})$. When phase factor measurements are extended to the dense phase, a change of behaviour of the overlap free energy is clearly visible as the chemical potential crosses a critical value. Using fits inspired by the approximate validity of mean-field theory, which is confirmed by our simulations, we extract the critical chemical potential as the non-analyticity point in the overlap free energy, obtaining a value that is in agreement with other determinations. Implications of our findings and potential improvements of our methodology are also discussed.

hep-lat↗
arXiv2019

Orbital-dependent backflow wave functions for real-space quantum Monte Carlo

We present and motivate an efficient way to include orbital dependent many--body correlations in trial wave function of real--space Quantum Monte Carlo methods for use in electronic structure calculations. We apply our new orbital--dependent backflow wave function to calculate ground state energies of the first row atoms using variational and diffusion Monte Carlo methods. The systematic overall gain of correlation energy with respect to single determinant Jastrow-Slater wave functions is competitive with the best single determinant trial wave functions currently available. The computational cost per Monte Carlo step is comparable to that of simple backflow calculations.

physics.comp-ph↗
arXiv2019

Two-dimensional Dirac operators with singular interactions supported on closed curves

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $δ$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.

math.AP↗
arXiv2019

Boundary integral formulations of eigenvalue problems for elliptic differential operators with singular interactions and their numerical approximation by boundary element methods

In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $δ$- and $δ'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.

math.SP↗
arXiv2019

Two-dimensional Mixture of Dipolar Fermions: Equation of State and Magnetic Phases

We study a two-component mixture of fermionic dipoles in two dimensions at zero temperature, interacting via a purely repulsive $1/r^3$ potential. This model can be realized with ultracold atoms or molecules, when their dipole moments are aligned in the confinement direction orthogonal to the plane. We characterize the unpolarized mixture by means of the Diffusion Monte Carlo technique. Computing the equation of state, we identify the regime of validity for a mean-field theory based on a low-density expansion and compare our results with the hard-disk model of repulsive fermions. At high density, we address the possibility of itinerant ferromagnetism, namely whether the ground state can be fully polarized in the fluid phase. Within the fixed-node approximation, we show that the accuracy of Jastrow-Slater trial wave functions, even with the typical two-body backflow correction, is not sufficient to resolve the relevant energy differences. By making use of the iterative-backflow improved trial wave functions, we observe no signature of a fully-polarized ground state up to the freezing density.

cond-mat.quant-gas↗
arXiv2019

On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $δ$-shell interactions

In this article Dirac operators $A_{η, τ}$ coupled with combinations of electrostatic and Lorentz scalar $δ$-shell interactions of constant strength $η$ and $τ$, respectively, supported on compact surfaces $Σ\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $δ$-potentials are modelled by coupling conditions at $Σ$. In the proof of the self-adjointness of $A_{η, τ}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{η, τ}$ is possible. In particular, the essential spectrum of $A_{η, τ}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{η, τ}$ is computed and it is discussed that for some special interaction strengths $A_{η, τ}$ is decoupled to two operators acting in the domains with the common boundary $Σ$.

math.SP↗