High-momentum Tail Research Articles

High Temperature Virial Expansion to Universal Quench Dynamics.

High temperature virial expansion is a powerful tool in equilibrium statistical mechanics. In this Letter we generalize the high temperature virial expansion approach to treat far-from-equilibrium quench dynamics. As an application of our framework, we study the dynamics of a Bose gas quenched from noninteracting to unitarity, and we compare our theoretical results with unexplained experimental results by the Cambridge group [Eigen et al., Nature 563, 221 (2018)]. We show that, during the quench dynamics, the momentum distribution decreases for the low-momentum part with k<k^{*}, and increases for high-momentum part with k>k^{*}, where k^{*} is a characteristic momentum scale separating the low- and the high-momentum regimes. We determine the universal value of k^{*}λ that agrees perfectly with the experiment, with λ being the thermal de Broglie wavelength. We also find a jump of the halfway relaxation time across k^{*}λ and the nonmonotonic behavior of energy distribution, both of which agree with the experiment. Finally, we address the issue whether the longtime steady state thermalizes or not, and we find that this state reaches a partial thermalization, namely, it thermalizes for the low-energy part with kλ≲1 but does not thermalize for the very high momentum tail with kλ≫1. Our framework can also be applied to quench dynamics in other systems.

Examination of an isospin-dependent single-nucleon momentum distribution for isospin-asymmetric nuclear matter in heavy-ion collisions

Within a transport model using nucleon momentum profiles as the input from a parameterized isospin-dependent single-nucleon momentum distribution, with a high momentum tail induced by short-range correlations, we employ \(^{197}\)Au + \(^{197}\)Au collisions at 400 MeV/nucleon to examine the effects of the short-range correlations on the pion and flow observables in probing the nuclear symmetry energy. We investigate how reliable this isospin-dependent single-nucleon momentum distribution is and determine the corresponding parameter settings. Apart from the significant effects of the short-range correlations on the pion and flow observables that are observed, we also find that the theoretical simulations of the \(^{197}\)Au + \(^{197}\)Au collisions with this momentum distribution using two sets of parameters, extracted from the experimental analysis and the self-consistent Green’s function prediction, can reproduce the neutron elliptic flows of the FOPI-LAND experiment and the \(\pi ^-/\pi ^+\) ratios of the FOPI experiment, respectively, under the symmetry energy setting in a particular range. Therefore, we conclude that this parameterized isospin-dependent single-nucleon momentum distribution is reliable for isospin-asymmetric nuclear matter. Correspondingly, two sets of parameters extracted from both the experimental analysis and the self-consistent Green’s function prediction cannot be excluded according to the available experimental information at present.

Renormalization Group and Scattering-equivalent Hamiltonians on a Coarse Momentum Grid

We consider the $\pi\pi$-scattering problem in the context of the Kadyshevsky equation. In this scheme, we introduce a momentum grid and provide an isospectral definition of the phase-shift based on the spectral shift of a Chebyshev angle. We address the problem of the unnatural high momentum tails present in the fitted interactions which reaches energies far beyond the maximal center-of-mass energy of $\sqrt{s}=1.4$ GeV. It turns out that these tails can be integrated out by using a block-diagonal generator of the SRG.

Proton-proton momentum correlation function as a probe of the high momentum tail of the nucleon-momentum distribution

Within an improved transport model, we examine effects of the high momentum tail of the nucleon momentum distribution induced by short-range correlations on the proton-proton momentum correlation function in $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon. It is found that the proton-proton momentum correlation function from preequilibrium emissions responds sensitively to the presence as well as fraction of nucleons in the high momentum tail of the nucleon momentum distribution, but is almost robustly insensitive to other factors including the symmetry energy and the uncertainty of cutoff value of nucleon effective high momentum. In terms of the sensitivity and clearness, we propose that the proton-proton momentum correlation function from preequilibrium emissions can be as an effective probe of the high momentum tail of the nucleon momentum distribution.

Neutron stars within a relativistic mean field theory compatible with nucleon-nucleon short-range correlations

Nucleon-nucleon short-range correlations (SRCs) result in the presence of high-momentum tail (HMT) above the Fermi surface in nucleon momentum distribution. In this work, we introduce HMT corrections in relativistic mean field (RMF) theory to make the theory compatible with the nucleon-nucleon SRCs. The properties of nucleonic stars and hyperon stars are studied within the RMF theory with HMT corrections. After introducing the HMT corrections, the equation of state (EoS) of the nucleonic star becomes softer and the maximum mass is smaller. The particle fractions of hyperon stars are significantly affected by the HMT corrections. The threshold densities of the presence of hyperons are changed dramatically. More importantly, with the introduction of HMT corrections, the mass-radius relations of nucleonic stars and hyperon stars can satisfy the stringent limits provided by the detection of gravitational wave.

Short-range correlations and the charge density

Sophisticated high-energy and large momentum-transfer scattering experiments combined with ab-initio calculations can reveal the short-distance behavior of nucleon pairs in nuclei. On an opposite energy and resolution scale, elastic electron scattering experiments are used to extract the charge density and charge radius of different nuclei. We show that even though the charge density has no obvious connection with nuclear short-range correlations, it can be used to extract properties of such correlations. This is accomplished by using the nuclear contact formalism to derive a relation between the charge density and the proton–proton nuclear contacts that describe the probability of two protons being at close proximity. With this relation, the values of the proton–proton contacts are extracted for various nuclei using only the nuclear charge density and a solution of the two-nucleon Schroedinger equation as inputs. For symmetric nuclei, the proton–neutron contacts can also be extracted from the charge density. Good agreement is obtained with previous extractions of the nuclear contacts. These results imply that one can predict (with reasonably good accuracy) the results of high-energy and large momentum-transfer electron-scattering experiments and ab-initio calculations of high momentum tails using only experimental data of elastic scattering experiments.

Universal relations and normal-state properties of a Fermi gas with laser-dressed mixed-partial-wave interactions

In a recent experiment [P. Peng, $et$ $al.$, Phys. Rev. A \textbf{97}, 012702 (2018)], it has been shown that the $p$-wave Feshbach resonance can be shifted toward the $s$-wave Feshbach resonance by a laser field. Based on this experiment, we study the universal relations and the normal-state properties in an ultracold Fermi gas with coexisting $s$- and $p$-wave interactions under optical control of a $p$-wave magnetic Feshbach resonance. Within the operator-product expansion, we derive the high-momentum tail of various observable quantities in terms of contacts. We find that the high-momentum tail becomes anisotropic. Adopting the quantum virial expansion, we calculate the normal-state contacts with and without a laser field for $^{40}$K atoms using typical experimental parameters. We show that the contacts are dependent on the laser dressing. We also reveal the interplay of laser dressing and different partial-wave interactions on various contacts. In particular, we demonstrate that the impact of the laser dressing in the $p$-wave channel can be probed by measuring the $s$-wave contacts, which is a direct manifestation of few-body effects on the many-body level. Our results can be readily checked experimentally.

Effects of the initialization of nucleon momentum in heavy-ion collisions at medium energies

Based on the Isospin-dependent transport model Boltzmann-Uehling-Uhlenbeck (IBUU), effects of the difference of the high momentum tails (HMTs) of nucleon momentum distribution in colliding nuclei on some isospin-sensitive observables are studied in the $^{197}\rm {Au}+^{197}\rm {Au}$ reactions at incident beam energy of 400 MeV/nucleon. It is found that the nucleon transverse and elliptic flows, the free neutron to proton ratio at low momenta are all less sensitive to the specific form of the HMT, while the free neutron to proton ratio at high momenta and the yields of $\pi^{-}$ and $\pi^{+}$ as well as the $\pi^{-}/\pi^{+}$ ratio around the Coulomb peak are sensitive to the specific form of the HMT. Combining the present studies with the experimental measurements at rare-isotope reaction facilities worldwide, one may get more insights into the nuclear short-range correlations in heavy nuclei or nuclear matter.

Spin-incoherent Luttinger liquid of one-dimensional SU( κ ) fermions

We theoretically investigate one-dimensional (1D) SU($\kappa$) fermions in the regime of spin-incoherent Luttinger liquid. We specifically focus on the Tonks-Girardeau gas limit where its density is sufficiently low that effective repulsions between atoms become infinite. In such case, spin exchange energy of 1D SU($\kappa$) fermions vanishes and all spin configurations are degenerate, which automatically puts them into spin-incoherent regime. In this limit, we are able to express the single-particle density matrices in terms of those of anyons. This allows us to numerically simulate the number of particles up to $N=32$. We numerically calculate single-particle density matrices in two cases: (1) equal populations for each spin components (balanced) and (2) all $S_z$ manifolds included. In contrast to noninteracting multi-component fermions, the momentum distributions are broadened due to strong interactions. As $\kappa$ increases, the momentum distributions are less broadened for fixed $N$, while they are more broadened for fixed number of particle per spin component. We then compare numerically calculated high momentum tails with analytical predictions which are proportional to $1/p^4$, in good agreement. Thus, our theoretical study provides a comparison with the experiments of repulsive multicomponent alkaline-earth fermions with a tunable SU($\kappa$) spin-symmetry in the spin-incoherent regime.

High-momentum tail and universal relations of a Fermi gas near a Raman-dressed Feshbach resonance

In a recent proposal [Jie and Zhang, Phys. Rev. A 95, 060701(R) (2017)], it has been shown that center-of-mass-momentum-dependent two-body interactions can be generated and tuned by Raman coupling the closed-channel bound states in a magnetic Feshbach resonance. Here we investigate the universal relations in a three-dimensional Fermi gas near such a laser modulated $s$-wave Feshbach resonance. Using the operator-product expansion approach, we find that, to fully describe the high-momentum tail of the density distribution up to $q^{-6}$ ($q$ is the relative momentum), four center-of-mass-momentum-dependent parameters are required, which we identify as contacts. These contacts appear in various universal relations connecting microscopic and thermodynamic properties. One contact is related to the variation of energy with respect to the inverse scattering length and determines the leading $q^{-4}$ tail of the high-momentum distribution. Another vector contact appears in the subleading $q^{-5}$ tail, which is related to the velocity of closed-channel molecules. The other two contacts emerge in the $q^{-6}$ tail and are respectively related to the variation of energy with respect to the range parameter and to the kinetic energy of closed-channel molecules. Particularly, we find that the $q^{-5}$ tail and part of the $q^{-6}$ tail of the momentum distribution show anisotropic features. We derive the universal relations and, as a concrete example, estimate the contacts for the zero-temperature superfluid ground state of the system using a mean-field approach.

Interplay of short-range correlations and nuclear symmetry energy in hard-photon production from heavy-ion reactions at Fermi energies

Within an isospin- and momentum-dependent transport model for nuclear reactions at intermediate energies, we investigate the interplay of the nucleon-nucleon short-range correlations (SRC) and nuclear symmetry energy $E_{sym}(\rho)$ on hard photon spectra in collisions of several Ca isotopes on $^{112}$Sn and $^{124}$Sn targets at a beam energy of 45 MeV/nucleon. It is found that over the whole spectra of hard photons studied, effects of the SRC overwhelm those due to the $E_{sym}(\rho)$. The energetic photons come mostly from the high-momentum tails (HMT) of single-nucleon momentum distributions in the target and projectile. Within the neutron-proton dominance model of SRC based on the consideration that the tensor force acts mostly in the isosinglet and spin-triplet nucleon-nucleon interaction channel, there are equal numbers of neutrons and protons, thus a zero isospin-asymmetry in the HMTs. Therefore, experimental measurements of the energetic photons from heavy-ion collisions at Fermi energies have the great potential to help us better understand the nature of SRC without any appreciable influence by the uncertain $E_{sym}(\rho)$. These measurements will be complementary to but also have some advantages over the ongoing and planned experiments using hadronic messengers from reactions induced by high-energy electrons or protons. Since the underlying physics of SRC and $E_{sym}(\rho)$ are closely correlated, a better understanding of the SRC will in turn help constrain the nuclear symmetry energy more precisely in a broad density range.

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.

Fermion superfluid with hybridized s- and p-wave pairings

Ever since the pioneering work of Bardeen, Cooper and Schrieffer in the 1950s, exploring novel pairing mechanisms for fermion superfluids has become one of the central tasks in modern physics. Here, we investigate a new type of fermion superfluid with hybridized $s$- and $p$-wave pairings in an ultracold spin-1/2 Fermi gas. Its occurrence is facilitated by the co-existence of comparable $s$- and $p$-wave interactions, which is realizable in a two-component $^{40}$K Fermi gas with close-by $s$- and $p$-wave Feshbach resonances. The hybridized superfluid state is stable over a considerable parameter region on the phase diagram, and can lead to intriguing patterns of spin densities and pairing fields in momentum space. In particular, it can induce a phase-locked $p$-wave pairing in the fermion species that has no $p$-wave interactions. The hybridized nature of this novel superfluid can also be confirmed by measuring the $s$-wave and $p$-wave contacts, which can be extracted from the high-momentum tail of the momentum distribution of each spin component. These results enrich our knowledge of pairing superfluidity in Fermi systems, and open the avenue for achieving novel fermion superfluids with multiple partial-wave scatterings in cold atomic gases.

Coarse-grained short-range correlations

We develop a scheme to take into account the effects of short-range nucleon-nucleon correlations in the nucleon-pair wave function by solving the Bethe-Goldstone equation for a coarse grained delta shell potential in S-wave configuration. The S-wave delta shell potential has been adjusted to reproduce the $^1$S$_0$ phase shifts of the AV18 potential for this partial wave up to 2 GeV in the laboratory kinetic energy. We show that a coarse grained potential can describe the high momentum tail of the back-to-back correlated pairs and the $G$-matrix in momentum space. We discuss the easiness and robustness of the calculation in coordinate space and the future improvements and utilities of this model. This work suggests the possibility of using perturbation theory for describing the short-range correlations, and related to this, to substitute the $G$-matrix by an appropriate coarse-grained potential.

Tan\u2019s contact of a harmonically trapped one-dimensional Bose gas: Strong-coupling expansion and conjectural approach at arbitrary interactions

We study Tan’s contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan’s contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan’s contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases.

Preequilibrium particle emissions and in-medium effects on the pion production in heavy-ion collisions

Within the framework of the Lanzhou quantum molecular dynamics (LQMD) transport model, pion dynamics in heavy-ion collisions near threshold energies and the emission of preequilibrium particles (nucleons and light complex fragments) have been investigated. A density, momentum and isospin dependent pion-nucleon potential based on the $\Delta$-hole model is implemented in the transport approach, which slightly leads to the increase of the $\pi^{-}/\pi^{+}$ ratio, but reduces the total pion yields. It is found that a bump structure of the $\pi^{-}/\pi^{+}$ ratio in the kinetic energy spectra appears at the pion energy close to the $\Delta$(1232) resonance region. The yield ratios of neutrons to protons from the squeeze-out particles perpendicular to the reaction plane are sensitive to the stiffness of nuclear symmetry energy, in particular at the high-momentum (kinetic energy) tails.

Effects of the high-momentum tail of nucleon momentum distribution on the isospin-sensitive observables

Based on the Isospin-dependent transport model Boltzmann-Uehling-Uhlenbeck (IBUU), effects of the high momentum tail (HMT) of nucleon momentum distribution in colliding nuclei on some isospin-sensitive observables are studied in the semi-central $^{197}\rm {Au}+^{197}\rm {Au}$ reactions at incident beam energy of 400 MeV/nucleon. It is found that the nucleon transverse flow, the difference of neutron and proton transverse flows, the nucleon elliptic flow and free neutron to proton ratio are all less sensitive to the HMT, while the isospin-sensitive nucleon elliptic flow difference is clearly affected by the HMT. Except at very high kinetic energies, the kinetic energy distributions of $\pi^{-}$, $\pi^{+}$ and charged pion ratio $\pi^{-}/\pi^{+}$ are all sensitive to the HMT.

Postquench dynamics and prethermalization in a resonant Bose gas

We explore the dynamics of a resonant Bose gas following its quench to a strongly interacting regime near a Feshbach resonance. For such deep quenches, we utilize a self-consistent dynamic field approximation and find that after an initial regime of many-body Rabi-like oscillations between the condensate and finite-momentum quasiparticle pairs, at long times, the gas reaches a pre-thermalized nonequilibrium steady state. We explore the resulting state through its broad stationary momentum distribution function, that exhibits a power-law high momentum tail. We study the dynamics and steady-state form of the associated enhanced depletion, quench-rate dependent excitation energy, Tan's contact, structure function and radio frequency spectroscopy. We find these predictions to be in a qualitative agreement with recent experiments.

Effects of the HMT on Nucleon Collective Flows within BUU Transport Model**Supported by the Fundamental Research Funds for the Central Universities under Grant No lzujbky-2014-170, the Research Fund for the Doctoral Program of Higher Education of China under Grant No 20120211120002, and the National Natural Science Foundation of China under Grant Nos 11205075 and 11375076.

Within the framework of a semiclassical Boltzmann–Uehling–Uhlenbeck (BUU) transport model, the high momentum tail (HMT) effects of nucleon momentum distribution in the nucleus on the nucleon collective flows are studied in semicentral Au+Au collisions. The HMT due to the isospin-dependent short-range correlations causes a smaller value of the collective flows. We find that the HMT effects on the nucleon collective flows are remarkable at beam energy of 300 MeV/nucleon and become weak as the incident beam energy increases. The results indicate that for the collective flow studies at intermediate energies, the HMT of nucleon momentum distribution in nucleus should be taken into account in transport models.

Efimov correlations in strongly interacting Bose gases

We compute the virial coefficients, the contact parameters, and the momentum distribution of a strongly interacting three-dimensional Bose gas by means of a virial expansion up to third order in the fugacity, which takes into account three-body correlations exactly. Our results characterize the nondegenerate regime of the interacting Bose gas, where the thermal wavelength is smaller than the interparticle spacing but the scattering length may be arbitrarily large. We observe a rapid variation of the third virial coefficient as the scattering length is tuned across the three-atom and the atom-dimer thresholds. The momentum distribution at unitarity displays a universal high-momentum tail with a log-periodic momentum dependence, which is a direct signature of Efimov physics. We provide a quantitative description of the momentum distribution at high momentum as measured by P. Makotyn et al. [Nat. Phys. 10, 116 (2014)], and our calculations indicate that the lowest trimer state might not be occupied in the experiment. Our results allow for a spectroscopy of Efimov states in the unitary limit.