Free Fermi Gas Research Articles

Stability of the Enhanced Area Law of the Entanglement Entropy

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.

Relevance of the three-dimensional Thirring coupling at finite temperature and density

We studied the three dimensional Thirring model in the limit of infinite number of flavors at finite temperature and density. We calculated the number density as a function of temperature and the density at zero temperature serves as a relevant parameter. A three dimensional free fermion gas behavior as the density at zero temperature approaches zero smoothly crosses over to a two dimensional free fermion gas behavior as the density at zero temperature approaches infinity.

Superfluid condensate fraction and pairing wave function of the unitary Fermi gas

The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gases and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this work we present auxiliary-field lattice Monte Carlo simulations using a novel lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test we compute the ground state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy $E_{FG}$, we find $E_0/E_{FG}= 0.369(2), 0.372(2)$, using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\alpha = 0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be $\zeta_pk_F = 1.8(3) \hbar$, where $k_F$ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic P-wave and D-wave superfluids.

Tunneling conductance of long-range Coulomb interacting Luttinger liquid

The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.

Coincidence angle-resolved photoemission spectroscopy: Proposal for detection of two-particle correlations

The angle-resolved photoemission spectroscopy (ARPES) is one powerful experimental technique to study the electronic structure of materials. As many electron materials show unusual many-body correlations, the technique to detect directly these many-body correlations will play important roles in the study of their many-body physics. In this article, we propose a technique to detect directly the two-particle correlations, a coincidence ARPES (cARPES) where two incident photons excite two respective photoelectrons which are detected in coincidence. While the one-photon-absorption and one-photoelectron-emission ARPES provides the single-particle spectrum function, the proposed cARPES with two-photon absorption and two-photoelectron emission is relevant to a two-particle Bethe-Salpeter wave function. Examples of the coincidence detection probability of the cARPES for a free Fermi gas and a Bardeen-Cooper-Schrieffer (BCS) superconducting state are studied in detail. We also propose another two experimental techniques, a coincidence angle-resolved photoemission and inverse-photoemission spectroscopy (cARP/IPES) and a coincidence angle-resolved inverse-photoemission spectroscopy (cARIPES). As all of these proposed coincidence techniques can provide the two-particle frequency Bethe-Salpeter wave functions, they can show the momentum and energy dependent two-particle dynamical physics of the material electrons in the particle-particle or particle-hole channel. Thus, they can be introduced to study the Cooper-pair physics in the superconductor, the itinerant magnetism in the metallic ferromagnet/antiferromagnet, and the particle-hole pair physics in the metallic nematic state. Moreover, as the two-particle Bethe-Salpeter wave functions also involve the inner-pair dynamical physics, these proposed coincidence techniques can be used to study the inner-pair time-retarded physics.

Symmetry resolved entanglement: exact results in 1D and beyond

In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem’s reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher–Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement (‘flux-resolved entanglement’) for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.

Exotic quantum statistics and thermodynamics from a number-conserving theory of Majorana fermions

We propose a closed form for the statistical distribution of non-interacting Majorana fermions at low temperature. Majorana particles often appear in the contemporary many-body literature in the Kitaev, Fu–Kane, or Sachdev–Ye–Kitaev models, where the Majorana condition of self-conjugacy immediately results in non-conserved particle number, non-trivial braiding statistics, and the absence of a non-interacting limit. We deviate from this description and instead consider a gas of non-interacting, spin-1/2 Majorana fermions that obey the spin-statistics theorem via imposing a condensed matter analog of momentum conservation. This allows us to build a quantum statistical theory of the Majorana system in the low temperature, low density limit without the need to account for strong fluctuations in the particle number. A combinatorial analysis leads to a configurational entropy which deviates from the fermionic result with an increasing number of available microstates. A number-conserving Majorana distribution function is derived which shows signatures of a sharply-defined Fermi surface at finite temperatures. Such a distribution is then re-derived from a microscopic model in the form of a modified Kitaev chain with a bosonic pair interaction. The thermodynamics of this free Majorana system is found to be nearly identical to that of a free Fermi gas, except now distinguished by a two-fold ground state degeneracy and, subsequently, a residual entropy at zero temperature. Despite clear differences with the anyonic or Sachdev–Ye–Kitaev models, we nevertheless find surprising agreement between our theory and experimental signatures of Majorana excitations in several materials. Experimental realization of our exactly solvable model is also discussed in the realm of astrophysical and high-energy phenomena.

Constraining light fermionic dark matter with binary pulsars

A binary system embedded in a Dark Matter (DM) background may experience a change in its orbital period due to dynamical friction as the binary moves through a wind of DM particles. We compute such a perturbative effect on the binary evolution considering that DM is constituted of degenerate gas of free fermions. The analysis point out that the secular change of the orbital period is more sensitive, and likely measurable, to degenerate fermions with masses $\gtrsim50$ eV, depending slightly, but still being distinguishable, on the binary star configuration (e.g. NS-NS, NS-WD and WD-WD). Interestingly, we find that NS-NS binary systems with large orbital periods, $P_{b}\gtrsim100$ days, experience larger orbital period decays. We also show that this effect is clearly increased, under the former conditions, in binaries orbiting small DM halos, which correspond to extragalactic pulsars. This situation represents the best astrophysical scenario to test such effects of light fermionic DM. We use some available measurements of the orbital period time-derivative for long-period binaries in the Milky-Way to quantify more realistically this effect. For instance, measurements of the J1713+0747 pulsar set an upper bound on the fermion mass of $m_{f}\lesssim 1$ keV. This bound can be considerably improved by using pulsar timing observations of extragalactic pulsars. Under this perspective, high precision of timing pulsar observations will reveal whether DM dynamical friction effect may be tested with the upcoming generation of surveys leading to the possibility of constraining more strongly the properties of light fermionic DM.

Formulation of heat conduction and thermal conductivity of metals

Abstract The well-known low-pressure monatomic gas thermal conductivity expression is based on the Maxwell-Boltzmann velocity distribution and involves the mean particle velocity, the gas heat capacity at constant volume and the particle mean free path. The extension of the formula to a free electron Fermi gas, using the Fermi velocity along with the Sommerfeld electronic heat capacity, was demonstrated in the literature using the Boltzmann transport equation. A different formulation of heat conduction in sufficiently pure metals, yielding the same formula for the thermal conductivity, is provided in the present investigation using the free electron Fermi gas energy distribution with the thermal conductivity determined from the net heat transfer occurring due to random motions of the free electrons in the presence of temperature gradient. Potential applications of this approach include extension of the present kinetic model incorporating quantum effects to cases in which electron scattering occurs such as in nanowires and hollow nanowires.

Lower bound on the Hartree-Fock energy of the electron gas

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

Particle-number scaling of the quantum work statistics and Loschmidt echo in Fermi gases with time-dependent traps

We investigate the particle-number dependence of some features of the out-of-equilibrium dynamics of d-dimensional Fermi gases in the dilute regime. We consider protocols entailing the variation of the external potential which confines the particles within a limited spatial region, in particular sudden changes of the trap size. In order to characterize the dynamic behavior of the Fermi gas, we consider various global quantities such as the ground-state fidelity for different trap sizes, the quantum work statistics associated with the protocol considered, and the Loschmidt echo measuring the overlap of the out-of-equilibrium quantum states with the initial ground state. Their asymptotic particle-number dependences show power laws for noninteracting Fermi gases. We also discuss the effects of short-ranged interactions to the power laws of the average work and its square fluctuations, within the Hubbard model and its continuum limit, arguing that they do not generally change the particle-number power laws of the free Fermi gases, in any spatial dimensions.

Surface tension of hot and dense quark matter under strong magnetic fields

We study the surface tension of hot, highly magnetized three flavor quark matter droplets, focusing specifically on the thermodynamic conditions prevailing in neutron stars, hot lepton rich protoneutron stars and neutron star mergers. We explore the role of temperature, baryon number density, trapped neutrinos, droplet size and magnetic fields within the multiple reflection expansion formalism (MRE), assuming that astrophysical quark matter can be described as a mixture of free Fermi gases composed by quarks $u$, $d$, $s$, electrons and neutrinos, in chemical equilibrium under weak interactions. We find that the total surface tension is rather unaffected by the size of the drop, but is quite sensitive to the effect of baryon number density, temperature, trapped neutrinos and magnetic fields (specially above $eB \sim 5 \times 10^{-3} \mathrm{GeV}^2$). Surface tensions parallel and transverse to the magnetic field span values up to $\sim$ 25 MeV/fm$^2$. For $T \lesssim 100$ MeV the surface tension is a decreasing function of temperature but above 100 MeV it increases monotonically with $T$. Finally, we discuss some astrophysical consequences of our results.

Reinvestigation of the Bulk Modulus for fcc Al using a Helmholtz Energy Approach

While extensive studies on the bulk modulus of fcc Al have been conducted, there still exist controversies regarding to the experimental values. In the present work, we adopted a Helmholtz energy approach based on the Morse function, the free electron Fermi gas model, as well as a modified Debye–Gruneisen model ensuring intrinsic thermodynamic relationship satisfied. To identify consistent bulk modulus data, all the model parameters for fcc Al were evaluated by using comprehensively available experimental data on heat capacity, elastic modulus, thermal expansivity, molar volume, etc., over wide temperature and pressure ranges. Reasonable agreements have been achieved in this work without inconsistency among various thermodynamic and thermophysical properties. Our calculated bulk modulus of fcc Al agrees well with the data reported by Kamm and Alers, Gerlich and Fisher, Ho and Ruoff and the assessment done by Wang and Reeber. However, it is impossible to reconcile the parameters to fit the recent data from Raju et al., as well as the results from Tallon and Wolfenden due to the intrinsic thermodynamic constraints.

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling function characterising these entanglement measurements is the same as the one for homogeneous systems, but expressed in terms of a different variable. This new scaling variable is a non-trivial function of the subsystem length and system’s inhomogeneity that is easily written in terms of the curved metric. We test our predictions against exact numerical calculations in the free Fermi gas trapped by a harmonic potential, finding perfect agreement.

Background gravity correction to the limiting mass of white dwarfs

While computing the Fermi degeneracy pressure of electrons in a white dwarf star within the framework of hydrostatic equilibrium, we depart from the extant practice of treating the electrons as a free fermion gas, by including the effect of the background gravitational potential experienced by the electrons in the star, resulting from the mass of its constituent atoms (being the mass of all nucleons). Modifying the free particle Hamiltonian with this effective potential, we employ first order quantum mechanical perturbation theory to compute the degeneracy pressure, in order to study the effect of inclusion of this self-gravity of the star on the limiting mass. The final effect is found to be non-trivial, but perhaps a shade too small to alter any major observational result.

Quantum quench and thermalization of one-dimensional Fermi gas via phase-space hydrodynamics

By exploring a phase space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the Fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in an one dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion).

Ground-state properties of dilute Bose systems with synthetic dispersion laws

Experimental advances in synthesizing spin-orbit couplings in cold atomic Bose gases promise to create single-particle dispersion laws featuring energy minima that are degenerate on a ring or a sphere in momentum space. We show that for arbitrary space dimensionality the ground-state properties of a dilute system of spin-orbit coupled Bose particles with such dispersion and short-range repulsive interactions are universal: the chemical potential exhibits a quadratic dependence on the particle density as found in a one-dimensional free Fermi gas.

Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems

Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.

One-dimensional model of chiral fermions with Feshbach resonant interactions

We study a model of two species of 1D linearly dispersing fermions interacting via an s-wave Feshbach resonance at zero temperature. While this model is known to be integrable, it possesses novel features that have not previously been investigated. Here, we present an exact solution based on the coordinate Bethe Ansatz. In the limit of infinite resonance strength, which we term the strongly interacting limit, the two species of fermions behave as free Fermi gases. In the limit of infinitely weak resonance, or the weakly interacting limit, the gases can be in different phases depending on the detuning, the relative velocities of the particles, and the particle densities. When the molecule moves faster or slower than both species of atoms, the atomic velocities get renormalized and the atoms may even become non-chiral. On the other hand, when the molecular velocity is between that of the atoms, the system may behave like a weakly interacting Lieb–Liniger gas.

Probing the high-momentum component in the nucleon momentum distribution by nucleon emission from intermediate-energy nucleus-nucleus collisions

The free nucleon emission from intermediate-energy nucleus-nucleus collisions is proposed as a possible probe to study the high-momentum tail (HMT) in nucleon momentum distribution of nuclei. Within the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model, the effect of HMT on free nucleon emission from the reactions of a light system $^{12}\mathrm{C}+^{12}\mathrm{C}$ and heavy one $^{124}\mathrm{Sn}+^{124}\mathrm{Sn}$ at the beam energies of 50 and 140 MeV/nucleon was investigated. We show that the probability of free nucleon emission, especially high-energy nucleons, is sensitive to the high-momentum component in nucleon momentum distribution of nuclei. The momentum distribution with HMT can lead to an increase of both high-energy free proton and neutron emission as compared to the free Fermi gas case for both symmetric and asymmetric nuclear systems.