Free Fermi Gas Research Articles

Effect of short-range correlation on the density dependence of nuclear symmetry energy at supra-saturation densities

Short-range correlation (SRC) induces a significant high-momentum component in the nucleon momentum distribution n(k) in both finite nuclei and nuclear matter. Recent experiments suggest that the amount of correlation in pure neutron matter (PNM) may be much less than that in symmetric nuclear matter (SNM). In this paper we investigate on a qualitative level how this affects the density dependence of the symmetry energy, especially at supra-saturation densities, in the framework of the well-established MDI energy density functional. In the MDI formalism, SRC affects both the kinetic energy and the potential energy through its effect on n(k). For the n(k) of SNM, we use a formula proposed based on an interesting relation between n(k) at high momentum k (i.e. k > k F ) and the average nucleon density of finite nuclei. For the n(k) of PNM, we use the prediction of a free Fermi gas model. The result is compared to the symmetry energy calculated assuming both the SNM and PNM to be Fermi-gas like. Our study shows that, to have a realistic estimation of the relative contributions to the symmetry energy from the kinetic part and the two-body and three-body potential part, one has to take the SRC effect into account. Furthermore, it is found that the density dependence of the symmetry energy at supra-saturation densities is sensitive to the competition between the SRC effect and the three-body force effect.

Relativistic dynamics compels a thermalized fermi gas to a unique intrinsic parity eigenstate

The Dirac equation describes the dynamics of a relativistic spin-1/2 particle regarding its spatial motion and intrinsic degrees of freedom. Here we adopt the point of view that the spinors describe the state of a massive particle carrying two qubits of information: helicity and intrinsic parity. We show that the density matrix for a gas of free fermions, in thermal equilibrium, correlates helicity and intrinsic parity. Our results introduce the basic elements for discussing the spin-parity correlation for a Fermi gas: (1) at the ultra-relativistic domains, when the temperature is quite high, , the fermions have no definite intrinsic parity (50% : 50%), which is maximally correlated with the helicity; (2) at very low temperature, K, a unique parity dominates (conventionally chosen positive), by to 1, while the helicity goes into a mixed state for spin up and down, and the quantum correlation decoheres. For the antifermions we get the opposite behavior. In the framework of quantum information, our result could be considered as a plausible explanation of why we accept, as a fact (consistent with the experimental observation), that fermions (and antifermions), in our present epoch of a cool universe, have a unique intrinsic parity. The framework for constructing spin-parity entangled states is established.

Universal quantum behavior of interacting fermions in one-dimensional traps: From few particles to the trap thermodynamic limit

We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e. in dilute regime, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size l keeping the particle number N fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-N behavior of systems at fixed N/l corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-l limit. When the trapped system is in the metallic phase, corrections at finite l are O(l^{-1}) and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as O(l^{-1/3}) only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.

Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization

We study determinantal random point processes on a compact complex manifold X associated to a Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a gas of free fermions on X subject to a U(1)-gauge field and when X is the Riemann sphere it specializes to various random matrix ensembles. Our general setup will also include the setting of weighted orthogonal polynomials in $${\mathbb{C}^{n}}$$ , as well as in $${\mathbb{R}^{n}}$$ . It is shown that, in the many particle limit, the empirical random measures on X converge exponentially towards the deterministic pluripotential equilibrium measure, defined in terms of the Monge–Ampère operator of complex pluripotential theory. More precisely, a large deviation principle (LDP) is established with a good rate functional which coincides with the (normalized) pluricomplex energy of a measure recently introduced in Berman et al. (Publ Math de l’IHÉS 117, 179–245, 2013). We also express the LDP in terms of the Ray–Singer analytic torsion. This can be seen as an effective bosonization formula, generalizing the previously known formula in the Riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson–fermion correspondence.

Determination of electron and hole effective masses in thermal oxide utilizing an n-channel silicon MOSFET

The gate tunnelling electron current and the substrate hole current obtained from carrier separation in an n-channel Si MOSFET are used to determine the parabolic electron and hole effective masses in the thermal oxide. The oxide voltages for electron and hole conduction in the n-MOSFET in inversion are formulated, and the carrier effective masses of 0.42m for electron and 0.58m for hole for a free Fermi gas model of carriers at the emitting electrode are determined using the Fowler-Nordheim tunnelling characteristics. These carrier masses are related to the band offsets in SiO2/Si MOS devices accurately through the MOSFET model for the first time. The carrier effective masses are predicted to be the same for all thickness of oxide. The technique can be extended to other insulating materials as well. Also, the 1/E model of the anode hole injection over the hole barrier of 4.6 eV completely explains the oxide breakdown in thin oxides of 5 to 10nm at high electric fields, and having a slope constant of 516 MV/cm.

Partial molar entropy of electrons in a jellium model: Implications for thermodynamics of ions in solution and electrons in metals

A universal relationship between the partial molar entropy of electrons in a conductor and the absolute thermoelectric power of the conductor was previously established using macroscopic thermodynamics. This relationship may depend on temperature but not on the type of material. Building on this, a recent comment published in this journal, as well as some earlier work, has argued that the partial molar entropy of electrons in a conductor is essentially equivalent to the absolute thermoelectric power of the metal. The argument was based on the thermodynamic and transport properties of a free electron Fermi gas. To further validate the relationship the present paper extends this approach to a jellium model of electronic structure. If the proposed equivalence between partial molar entropy and absolute thermoelectric power is valid it opens the way for an experimental thermodynamic method to measure quantities that have previously been considered un-measurable, such as partial molar entropies of ions in solution and electric fields in homogeneous conductors placed in a temperature gradient. It also relates to questions about the completeness of current thermodynamic theory and the possibility of a new principle or law of thermodynamics.

Landauer-Büttiker Formula and Schrödinger Conjecture

We study the entropy flux in the stationary state of a finite one-dimensional sample \({\mathcal{S}}\) connected at its left and right ends to two infinitely extended reservoirs \({\mathcal{R}_{l/r}}\) at distinct (inverse) temperatures \({\beta_{l/r}}\) and chemical potentials \({\mu_{l/r}}\) . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator \({h_{\mathcal{S}, L}= - \Delta + v}\) . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system \({\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}\) in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit \({L \to \infty}\) and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator \({h_\mathcal{S}}\) .

Thermodynamics of quantum gases for the entire range of temperature

We have analytically explored the thermodynamics of free Bose and Fermi gases for the entire range of temperature, and have extended the same for harmonically trapped cases. We have obtained approximate chemical potentials for the quantum gases in closed forms of temperature so that the thermodynamic properties of the quantum gases become plausible, especially in the intermediate regime between the classical and quantum limits.

Next-to-leading order non-Fermi-liquid corrections to the neutrino emissivity and cooling of the neutron star

In this work we derive the expressions of the neutrino mean free path (MFP) and emissivity with non-Fermi-liquid corrections up to next-to-leading order (NLO) in degenerate quark matter. The calculation has been performed for both the absorption and the scattering processes. Subsequently the role of these NLO corrections on the cooling of the neutron star has been demonstrated. The cooling curve shows moderate enhancement compared to the leading order non-Fermi-liquid result. Although the overall correction to the MFP and emissivity are larger compared to the free Fermi gas, the cooling behavior is not altered significantly.

The stability of a non-extensive relativistic Fermi system

Based on the statistical theory of non-extensive relativity, and using theoretical analysis and numerical simulation, the non-extensive mechanical stability of ultra-relativistic free Fermi gas is investigated. The expressions of the stability conditions under high and low temperatures are given, and the mechanisms of the influences of temperature, ultra-relativistic effect, and non-extensive parameter q on stability are analysed. Our results show that at high temperature and under the condition of q < 1, the stability of a non-extensive system is weaker than that of an extensive system, and the relativistic effect reduces system stability as compared with a non-relativistic system. However, under the condition of q > 1, the stability of the non-extensive system is stronger than that of the extensive system, and the relativistic effect strengthens the system stability as compared with the non-relativistic system. In addition, under the condition of low temperature, the variation of the stability of the non-extensive system with temperature has a turning point.

Spectral probes of the holographic Fermi ground state: Dialing between the electron star and AdS Dirac hair

We argue that the electron star and the anti--de Sitter (AdS) Dirac hair solution are two limits of the free charged Fermi gas in AdS. Spectral functions of holographic duals to probe fermions in the background of electron stars have a free parameter that quantifies the number of constituent fermions that make up the charge and energy density characterizing the electron star solution. The strict electron star limit takes this number to be infinite. The Dirac hair solution is the limit where this number is unity. This is evident in the behavior of the distribution of holographically dual Fermi surfaces. As we decrease the number of constituents in a fixed electron star background the number of Fermi surfaces also decreases. An improved holographic Fermi ground state should be a configuration that shares the qualitative properties of both limits.

Kähler-Einstein metrics emerging from free fermions and statistical mechanics

We propose a statistical mechanical derivation of Kahler-Einstein metrics, i.e. solutions to Einstein's vacuum field equations in Euclidean signature (with a cosmological constant) on a compact Kahler manifold X. The microscopic theory is given by a canonical free fermion gas on X whose one-particle states are pluricanonical holomorphic sections on X (coinciding with higher spin states in the case of a Riemann surface). A heuristic, but hopefully physically illuminating, argument for the convergence in the thermodynamical (large N) limit is given, based on a recent mathematically rigorous result about exponentially small fluctuations of Slater determinants. Relations to effective bosonization and the Yau-Tian-Donaldson program in Kahler geometry are pointed out. The precise mathematical details will be investigated elsewhere.

Fermi polaron in two dimensions: Importance of the two-body bound state

We investigate a single impurity interacting with a free two-dimensional atomic Fermi gas. The interaction between the impurity and the gas is characterized by an arbitrary attractive short-range potential, which, in two dimensions, always admits a two-particle bound state. We provide analytical expressions for the energy and the effective mass of the dressed impurity by including the two-body bound state, which is crucial for strong interactions, in the integral equation for the effective interaction. Using the same method we give also the results for the polaron parameters in one and three dimensions finding a pretty good agreement with previous known results. Thus our relations can be used as a simple way to estimate the polaron parameters once the two-body bound state of the interaction potential is known.

Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains

We rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions.

Renormalization group and the superconducting susceptibility of a Fermi liquid

A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does not. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.

A FUZZY BAG MODEL FOR NUCLEAR MATTER: A PRELIMINARY APPROACH

In this work we develop an effective formalism for nuclear matter based on the fuzzy bag model. The main objective of our study is to discuss the feasibility of using the fuzzy bag model to describe nuclear matter properties. The physical system is described in our approach by an internal energy function, which has a free term, corresponding to a free Fermi gas, and an interacting one. In the interacting part, pion exchange is taken into account via an effective potential. To avoid superposition of nucleons, we introduce an exclusion volume à la Van der Waals. The internal energy function depends on the nuclear matter density and also on a parameter which will determine the expected volume of a nucleon in matter. We then obtain results for the binding energy per nucleon for the symmetric nuclear matter and for neutron matter, as well as the equation of state within this model. We then determine the mass of neutron stars in hydrostatic equilibrium, using the TOV equations. In spite of utilizing a treatment that is still very preliminary, our results show the feasibility of using this treatment to describe nuclear matter properties.

A Special Case of a Conjecture by Widom with Implications to Fermionic Entanglement Entropy

We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space $L^2(\R^d)$. As already observed by Gioev and Klich, this implies that the bi-partite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.

Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms

Can a gas of spin-up and spin-down fermions become ferromagnetic because of repulsive interactions? We addressed this question, for which there is not yet a definitive theoretical answer, in an experiment with an ultracold two-component Fermi gas. The observation of nonmonotonic behavior of lifetime, kinetic energy, and size for increasing repulsive interactions provides strong evidence for a phase transition to a ferromagnetic state. Our observations imply that itinerant ferromagnetism of delocalized fermions is possible without lattice and band structure, and our data validate the most basic model for ferromagnetism introduced by Stoner.

Central Limit Theorem for Locally Interacting Fermi Gas

We consider a locally interacting Fermi gas in its natural non-equilibrium steady state and prove the Quantum Central Limit Theorem (QCLT) for a large class of observables. A special case of our results concerns finitely many free Fermi gas reservoirs coupled by local interactions. The QCLT for flux observables, together with the Green-Kubo formulas and the Onsager reciprocity relations previously established [JOP4], complete the proof of the Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models.

Quantum Monte Carlo simulations of the BCS-BEC crossover at finite temperature

The quantum Monte Carlo method for spin-$\frac{1}{2}$ fermions at finite temperature is formulated for dilute systems with an $s$-wave interaction. The motivation and the formalism are discussed along with descriptions of the algorithm and various numerical issues. We report on results for the energy, entropy, and chemical potential as a function of temperature. We give upper bounds on the critical temperature ${T}_{c}$ for the onset of superfluidity, obtained by studying the finite-size scaling of the condensate fraction. All of these quantities were computed for couplings around the unitary regime in the range $\ensuremath{-}0.5\ensuremath{\leqslant}{({k}_{F}a)}^{\ensuremath{-}1}\ensuremath{\leqslant}0.2$, where $a$ is the $s$-wave scattering length and ${k}_{F}$ is the Fermi momentum of a noninteracting gas at the same density. In all cases our data are consistent with normal Fermi gas behavior above a characteristic temperature ${T}_{0}g{T}_{c}$, which depends on the coupling and is obtained by studying the deviation of the caloric curve from that of a free Fermi gas. For ${T}_{c}lTl{T}_{0}$ we find deviations from normal Fermi gas behavior that can be attributed to pairing effects. Low-temperature results for the energy and the pairing gap are shown and compared with Green-function Monte Carlo results by other groups.