Fermi-Dirac Research Articles

Generalization of Balian–Brezin decomposition for exponentials with linear fermionic part

Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick’s theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian–Brezin decomposition to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick’s theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the so(N) Lie algebra.

Exact Results for Bose–Einstein and Fermi–Dirac Statistics for Finite Generic Systems

Exact Results for Bose–Einstein and Fermi–Dirac Statistics for Finite Generic Systems

Computation of the Spatial Distribution of Charge-Carrier Density in Disordered Media.

The space- and temperature-dependent electron distribution n(r,T) determines optoelectronic properties of disordered semiconductors. It is a challenging task to get access to n(r,T) in random potentials, while avoiding the time-consuming numerical solution of the Schrödinger equation. We present several numerical techniques targeted to fulfill this task. For a degenerate system with Fermi statistics, a numerical approach based on a matrix inversion and one based on a system of linear equations are developed. For a non-degenerate system with Boltzmann statistics, a numerical technique based on a universal low-pass filter and one based on random wave functions are introduced. The high accuracy of the approximate calculations are checked by comparison with the exact quantum-mechanical solutions.

Determination of Nuclear Matter Radii by Means of Microscopic Optical Potentials: The Case of 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr

In this work we use microscopic Nucleon–Nucleus Optical Potentials (OP) to analyze elastic scattering data for the differential cross section of the 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr (p,p) 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr reaction, with the goal of extracting the matter radius and estimating the neutron skin, quantities that are both needed to determine the slope parameter L of the nuclear symmetry energy. Our analysis is performed with the factorized version of the microscopic OP obtained in a previous series of papers within the Watson multiple scattering theory at the first order of the spectator expansion, which is based on the underlying nucleon–nucleon dynamics and is free from phenomenological inputs. Differently from our previous applications, the proton and neutron densities are described with a two-parameter Fermi (2pF) distribution, which makes the extraction of the matter radius easier and allows us to make a meaningful comparison with the original analysis, that was performed with the Glauber model. With standard minimization techniques we performed data analysis and extracted the matter radius and the neutron skin. Our analysis produces a matter radius of Rm(rms)=4.12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_m^{\\mathrm{(rms)}} = 4.12$$\\end{document} fm, in good agreement with previous matter radii extracted from 76\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{76}$$\\end{document}Kr and 80\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{80}$$\\end{document}Kr, and a neutron skin of ΔRnp≃-0.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta R_{np} \\simeq - 0.1$$\\end{document} fm, compatible with a previous analysis. Our factorized microscopic OP, supplied with 2pF densities, is a valuable tool to perform the analysis of the experimental differential cross section and extract information such as matter radius and neutron skin. Without any free parameters it provides a reasonably good description of the experimental differential cross section for scattering angles up to ≈\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\approx $$\\end{document} 40 degrees. Compared to the Glauber model our OP can be applied to a wider range of scattering angles and allows one to probe the nuclear systems in a more internal region.

Numerically Exact Simulation of Photodoped Mott Insulators.

A description of long-lived photodoped states in Mott insulators is challenging, as it needs to address exponentially separated timescales. We demonstrate how properties of such states can be computed using numerically exact steady state techniques, in particular, the quantum Monte Carlo algorithm, by using a time-local ansatz for the distribution function with separate Fermi functions for the electron and hole quasiparticles. The simulations show that the Mott gap remains robust to large photodoping, and the photodoped state has hole and electron quasiparticles with strongly renormalized properties.

Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design.

Efficient ways to prepare Fermionic ground states on quantum computers are in high demand, and different techniques have been developed over the past few years. Despite having a vast set of methods, it is still unclear which method performs well for which system. In this work, we combine interpretable circuit designs with an effective basis approach in order to optimize a multiconfigurational valence bond wave function. Based on selected model systems, we show how this leads to an explainable performance. We demonstrate that the developed methodology outperforms related methods in terms of the size of the effective basis, as well as individual quantum resources for the involved circuits.

Creation of bound half-fermion pairs by solitons

In the presence of topologically nontrivial bosonic field configurations, the fermion number operator may take on fractional eigenvalues, because of the existence of zero-energy fermion modes. The simplest examples of this occur in 1+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+1$$\\end{document} dimensions, with zero modes attached to kink-type solitons. In the presence of a kink-antikink pair, the two associated zero modes bifurcate into positive and negative energy levels with energies ±ge-gΔ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm ge^{-g\\Delta }$$\\end{document}, in terms of the Yukawa coupling g≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ll 1$$\\end{document} and the distance Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} between the kink and antikink centers. When the kink and antikink are moving, it seems that there could be Landau–Zener-like transitions between these two fermionic modes, which would be interpretable as the creation or annihilation of fermion-antifermion pairs; however, with only two solitons in relative motion, this does not occur. If a third solitary wave is introduced farther away to perturb the kink-antikink system, a movement of the faraway kink can induce transitions between the discrete fermion modes bound to the solitons. These state changes can be interpreted globally as creation or destruction of a novel type of pair: a half-fermion and a half-antifermion. The production of the half-integral pairs will dominate over other particle production channels as long as the solitary waves remain well separated, so that there is a manifold of discrete fermion states whose energies are either zero or exponentially close to zero.

Pandemic Equation and COVID-19 Evolution

The Pandemic Equation describes multiple pandemic waves and has been applied to describe the COVID-19 pandemic. Using the generalized approaches of solid-state physics, we derive the Pandemic Equation, which accounts for the effects of pandemic mitigation measures and multiple pandemic waves. The Pandemic Equation uses slow and fast time scales for “curve flattening” and describing vaccination and mitigation measures and the Scaled Fermi–Dirac distribution functions for describing transitions between pandemic waves. The Pandemic Equation parameters extracted from the pandemic curves can be used for comparing different scenarios of the pandemic evolution and for extrapolating the pandemic evolution curves for the periods of time on the order of the instantaneous Pandemic Equation characteristic time constant. The parameter extraction for multiple locations could also allow for uncertainty quantification for such pandemic evolution predictions.

Beyond Conventional Charge Density Wave for Strongly Enhanced 2D Superconductivity in 1H-TaS2 Superlattices.

Noncentrosymmetric transition metal dichalcogenide (TMD) monolayers offer a fertile platform for exploring unconventional Ising superconductivity (SC) and charge density waves (CDWs). However, the vulnerability of isolated monolayers to structural disorder and environmental oxidation often degrade their electronic coherence. Herein, an alternative approach is reported for fabricating stable and intrinsic monolayers of 1H-TaS2 sandwiched between SnS blocks in a (SnS)1.15TaS2 van der Waals (vdW) superlattice. The SnS block layers not only decouple individual 1H-TaS2 sublayers to endow them with monolayer-like electronic characteristics, but also protect the 1H-TaS2 layers from electronic degradation. The results reveal the characteristic 3 × 3 CDW order in 1H-TaS2 sublayers associated with electronic rearrangement in the low-lying sulfur p band, which uncovers a previously undiscovered CDW mechanism rather than the conventional Fermi surface-related framework. Additionally, the (SnS)1.15TaS2 superlattice exhibits a strongly enhanced Ising-like SC with a layer-independent Tc of ≈3.0 K, comparable to that of the isolated monolayer 1H-TaS2 sample, presumably attributed to their monolayer-like characteristics and retained Fermi states. These results provide new insights into the long-debated CDW order and enhanced SC of monolayer 1H-TaS2, establishing bulk vdW superlattices as promising platforms for investigating exotic collective quantum phases in the 2D limit.

Charge Critical Phenomena in a Field Heterostructure with Two-Dimensional Crystal

The charge properties and regularities of mutual influence of the electro-physical parameters in a metal (M)/insulator (I)/two-dimensional crystal heterostructure were studied. In one case, the transition metal dichalcogenide (TMD) MoS2 was considered as a two-dimensional crystal, and in another the Weyl semi-metal (WSM) ZrTe5, representative of a quasi-two-dimensional crystal was chosen for this purpose. By self-consistently solving the electrostatic equations of the heterostructures under consideration and the Fermi–Dirac distribution, the relationship between such parameters as the concentration of charge carriers, chemical potential, and quantum capacitance of the TMD (WSM), as well as the capacitance of the I layer and the interface capacitance I–TMD (WSM), and their dependence on the field electrode potential, have been derived. The conditions for the emergence of charge instability and the critical phenomena caused by it are also determined.

Thermodynamic geometry of a system with unified quantum statistics.

We investigate the thermodynamic characteristics of unified quantum statistics, a framework exhibiting a crossover between Bose-Einstein and Fermi-Dirac statistics by varying a generalization parameter δ. An intrinsic statistical interaction becomes attractive for δ≤0.5, maintaining positive thermodynamic curvature across the entire physical range. In the range 0.5<δ<1, the system predominantly displays Fermi-like behavior at high temperatures. Conversely, at low temperatures, the thermodynamic curvature is positive, resembling bosonic behavior. Further temperature reduction induces a transition into the condensate phase. We introduce a critical fugacity (z=Z^{*}) at which the thermodynamic curvature changes sign. Below (z<Z^{*}) and above (z>Z^{*}) this critical point, the statistical behavior mimics fermions and bosons, respectively. We explore the system's statistical behavior for various δ values with respect to temperature, determining the critical fugacity and temperature-dependent condensation. Finally, we analyze specific heat as a function of temperature and condensation phase transition temperature for different δ values in various dimensions.

Random Green’s Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green’s function (rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H2O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H2O clusters at T = 0 K can reach an error of ∼ 1 meV per H2O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.

Theoretical Advances in Beta and Double-Beta Decay

Weak interaction processes continue to be hot topics in fundamental physics research. In this paper, we briefly review some recent advances in the theoretical study of beta and double-beta decays that include both the nuclear and atomic part of these processes. On the nuclear side, we present a statistical approach for the computation of the nuclear matrix elements (NME) for neutrinoless double-beta (0νββ). A range of NME values, the most probable value for NME, and the associated theoretical uncertainty are given. Correlations with other related observables are shown as well. On the atomic side, we first briefly review the methods used to obtain the electrons’ wave functions. Further, we use them for the computation of some relevant kinematic quantities such as Fermi functions, electron spectra, and angular correlation between the emitted electrons. Then, we present applications of these calculations to the experimental data analysis related to the search of the Lorentz invariance violation in two-neutrino double-beta (2νββ) decay and description of the decay rates and decay rate ratios for allowed and unique forbidden electron capture (EC) processes.

Exact asymptotics of long-range quantum correlations in a non-equilibrium steady state

Out-of-equilibrium states of many-body systems tend to evade a description by standard statistical mechanics, and their uniqueness is epitomized by the possibility of certain long-range correlations that cannot occur in equilibrium. In quantum many-body systems, coherent correlations of this sort may lead to the emergence of remarkable entanglement structures. In this work, we analytically study the asymptotic scaling of quantum correlation measures—the mutual information (MI) and the fermionic negativity—within the zero-temperature steady state of voltage-biased free fermions on a one-dimensional lattice containing a non-interacting impurity. Previously, we have shown that two subsystems on opposite sides of the impurity exhibit volume-law entanglement, which is independent of the absolute distances of the subsystems from the impurity. Here, we go beyond that result and derive the exact form of the subleading logarithmic corrections to the extensive terms of correlation measures, in excellent agreement with numerical calculations. In particular, the logarithmic term of the MI asymptotics can be encapsulated in a concise formula, depending only on simple four-point ratios of subsystem length scales and on the impurity scattering probabilities at the Fermi energies. This echoes the case of equilibrium states, where such logarithmic terms may convey universal information about the physical system. To compute these exact results, we devise a hybrid method that relies on Toeplitz determinant asymptotics for correlation matrices in both real space and momentum space, successfully circumventing the inhomogeneity of the system. This method could potentially find wider use for analytical calculations of entanglement measures in similar scenarios.

Extracting Kinetic Freeze-Out Properties in High-Energy Collisions Using a Multisource Thermal Model

We study the transverse momentum (pT) spectra of neutral pions and identified charged hadrons produced in proton–proton (pp), deuteron–gold (d–Au), and gold–gold (Au–Au) collisions at the center of mass energy sNN=200 GeV. The study is made in the framework of a multisource thermal model used in the partonic level. It is assumed that the contribution to the pT-value of any hadron comes from two or three partons with an isotropic distribution of the azimuthal angle. The contribution of each parton to the pT-value of a given hadron is assumed to obey any one of the standard (Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein) distributions with the kinetic freeze-out temperature and average transverse flow velocity. The pT spectra of the final-state hadrons can be fitted by the superposition of two or three components. The results obtained from our Monte Carlo method are used to fit the experimental results of the PHENIX and STAR Collaborations. The results of the present work serve as a suitable reference baseline for other experiments and simulation studies.

Variational Tensor Wave Functions for the Interacting Quantum Spin Hall Phase.

The quantum spin hall (QSH) phase, also known as the 2D topological insulator, is characterized by protected helical edge modes arising from time reversal symmetry. While initially proposed as band insulators, this phase can also manifest in strongly correlated systems where conventional band theory fails. To overcome the challenge of simulating this phase in realistic correlated models, we propose a novel framework utilizing fermionic tensor network states. Our approach involves constructing a tensor representation of the fixed-point wave function based on an exact solvable model, enabling us to derive a set of tensor equations governing the transformation rules of local tensors under symmetry operations. These tensor equations lead to the anomalous edge theory, which provides a comprehensive description of the QSH phase. By solving these tensor equations, we obtain variational ansatz for the QSH phase, which we subsequently verify its topological properties through numerical calculations. This method serves as an initial step toward employing tensor algorithms to simulate the QSH phase in strongly correlated systems, opening new avenues for investigating and understanding topological phenomena in complex materials.

Krylov Complexity of Fermionic and Bosonic Gaussian States

AbstractThe concept of complexity has become pivotal in multiple disciplines, including quantum information, where it serves as an alternative metric for gauging the chaotic evolution of a quantum state. This paper focuses on Krylov complexity, a specialized form of quantum complexity that offers an unambiguous and intrinsically meaningful assessment of the spread of a quantum state over all possible orthogonal bases. This study is situated in the context of Gaussian quantum states, which are fundamental to both Bosonic and Fermionic systems and can be fully described by a covariance matrix. While the covariance matrix is essential, it is insufficient alone for calculating Krylov complexity due to its lack of relative phase information is shown. The relative covariance matrix can provide an upper bound for Krylov complexity for Gaussian quantum states is suggested. The implications of Krylov complexity for theories proposing complexity as a candidate for holographic duality by computing Krylov complexity for the thermofield double States (TFD) and Dirac field are also explored.

Extracting many-body quantum resources within one-body reduced density matrix functional theory

Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matrix terms can universally be determined by the one-body reduced density matrix (1-RDM), thus avoiding the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, thus becoming the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information. Published by the American Physical Society 2024

All orders factorization and the Coulomb problem

In the limit of large nuclear charge, Z≫1, or small lepton velocity, β≪1, Coulomb corrections to nuclear beta decay and related processes are enhanced as Zα/β and become large or even nonperturbative (with α the QED fine structure constant). We provide a constructive demonstration of factorization to all orders in perturbation theory for these processes and compute the all-orders hard and soft functions appearing in the factorization formula. We clarify the relationship between effective field theory amplitudes and historical treatments of beta decay in terms of a Fermi function. Published by the American Physical Society 2024

Ab initio calculation of nonequilibrium quasiparticle-phonon dynamics in superconductors

Phonon-induced Cooper pair breaking, inciting nonequilibrium quasiparticle (QP) bursts, is known to deteriorate the performance of superconducting devices. However, a detailed understanding of QP-phonon dynamics is lacking due to the absence of a well-established theoretical framework. This paper presents a fully ab initio scheme of calculating nonequilibrium, polarization-dependent QP-phonon dynamics in superconductors. The authors find that with only an 8% deviation from the equilibrium phonon Bose–Einstein distribution, the resulting nonequilibrium QP population is 83 times larger than the equilibrium Fermi–Dirac distribution, and the longitudinal acoustic (LA) phonon polarization is most responsible for QP generation. The authors demonstrated that the qubit transition rate in Josephson junction-based transmon qubits is increased by orders of magnitude when taking these nonequilibrium distributions into account, compared to equilibrium distributions. This framework allows an in-depth exploration of nonequilibrium QP-phonon dynamics in various Josephson-junction-based superconducting devices. It paves the way for formulating advanced phonon shielding strategies to target the LA polarization, potentially leading to enhanced device performance, such as increased coherence time of transmon qubits or reduced thermal noise in cryogenics.