Three-body Contact Interaction Research Articles

Quantum Mechanics of One-Dimensional Three-Body Contact Interactions

The quantum mechanical description of topologically nontrivial three-body contact interactions in one dimension is not well understood. This study explores the Hamiltonian description of these interactions using the path-integral formalism.

Topologically nontrivial three-body contact interaction in one dimension

Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.

How to renormalize coupled cluster theory

Coupled cluster theory is an attractive tool to solve the quantum many-body problem because its singles and doubles (CCSD) approximation is computationally affordable and yields about 90% of the correlation energy. Capturing the remaining 10%, e.g., via including triples, is numerically expensive. Here, we assume that short-range three-body correlations dominate and---following Lepage (arXiv:nucl-th/9706029)---that their effects can be included within CCSD by renormalizing the three-body contact interaction. We renormalize this contact in $^{16}\mathrm{O}$ and obtain systematically improved CCSD results for $^{24}\mathrm{O}, ^{20--34}\mathrm{Ne}, ^{40,48}\mathrm{Ca}, ^{78}\mathrm{Ni}, ^{90}\mathrm{Zr}$, and $^{100}\mathrm{Sn}$.

Cooper triples in attractive three-component fermions: Implication for hadron-quark crossover

We investigate many-body properties of equally populated three-component fermions with attractive three-body contact interaction in one dimension. A diagrammatic approach suggests the possible occurrence of Cooper triples at low temperature, which are three-body counterparts of Cooper pairs with a two-body attraction. We develop a minimal framework that bridges the crossover from tightly-bound trimers to Cooper triples with increasing chemical potential and show how the formation of Cooper triples occurs in the grand-canonical phase diagram. Moreover, we argue that this non-trivial crossover is similar to the hadron-quark crossover proposed in dense matter. A coexistence of medium-induced triples and the underlying Fermi sea at positive chemical potential is analogous to quarkyonic matter consisting of baryonic excitations and the underlying quark Fermi sea. The comparison with the existing quantum Monte Carlo results implies that the emergence of these kinds of three-body states can be a microscopic origin of the peak of the sound velocity along the crossover.

Three-body and Coulomb interactions in a quasi-two-dimensional dipolar Bose-condensed gas

In this paper, we studied a dilute quasi-two-dimensional dipolar Bose-condensed with two- and three-body contact, and Coulomb interactions using the Hartree–Fock–Bogoliubov–Popov approximation. We analyze numerically the effects of three-body contact, and Coulomb interactions on the energy spectrum, the quantum and thermal noncondensate fraction of the system. We show that increasing the three-body contact and Coulomb interactions leads to the appearance of rotonization and condensate instability at stronger dipole–dipole interaction. Also we find that the temperature dependence of the thermal noncondensate fraction is linear at low temperature.

Investigation of $$\varXi ^- nn$$ ($$S=-2$$) hypernucleus in low-energy pionless halo effective theory

We study the \(\varXi ^- nn\) (\(S=-2,\,I=3/2,\,J^P={1/2}^+\)) three-body system using low-energy effective field theory (EFT). Due to the acute inadequacy of empirical information in this sector, there exists substantial degree of ambiguity in determining various few-body observables, some of which are expected to yield vital clues to resolving longstanding contentious issues in hypernuclear physics. Moreover, in astrophysical studies, a precise determination of neutron star equation of state (EoS) of putative hyperonic cores relies on essential input from the \(S=-2\) sector. In this obscure current scenario, a pionless EFT analysis provides a systematic model-independent framework for assessing the feasibility of light three-particle-stable bound states, utilizing low-energy universality. Here we take recourse to a simplistic speculation of the three-body system by eliminating the repulsive spin-singlet \(\varXi ^- n\) sub-system, while retaining the predominantly attractive (possibly bound) spin-triplet \(\varXi ^- n\) and the virtual bound spin-singlet nn sub-systems. In particular, a qualitative leading order EFT investigation by introducing a sharp momentum ultraviolet cut-off parameter \(\varLambda _{\mathrm{reg}}\) into the coupled integral equations indicates a discrete scaling behavior akin to a renormalization group limit cycle, thereby suggesting the formal existence of Efimov states in the unitary limit, as \(\varLambda _{\mathrm{reg}}\rightarrow \infty \). Our subsequent non-asymptotic analysis indicates that the three-body binding energy \(B_3\) is sensitively dependent on the cut-off without the inclusion of three-body contact interactions. Furthermore, our analysis reproduces several values of the binding energy \(B_3\sim 3{-}4\) MeV, predicted in context of existing potential models, with the regulator \(\varLambda _{\mathrm{reg}}\) in the range \(\sim 350{-}460\) MeV. Finally, based on these model inputs for \(B_3\), a ballpark estimate of the three-body scattering length in the range \(2.6{-}4.9\) fm, is naively constrained by our EFT analysis. Despite approximations, the resulting Phillips line is expected to yield a robust feature of the halo-bound \(\varXi ^- nn\) system. For pedagogical reasons, using a simple toy model interacting three-bosons system, we highlight in the appendices the typical universal features leading to emergence of RG limit cycle and Efimov states which are amenable to a low-energy EFT formalism.

Virial coefficients of trapped and untrapped three-component fermions with three-body forces in arbitrary spatial dimensions

Using a coarse temporal lattice approximation, we calculate the first few terms of the virial expansion of a three-species fermion system with a three-body contact interaction in $d$ spatial dimensions, both in homogeneous space as well as in a harmonic trapping potential of frequency $\omega$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $\Delta b_4$ and $\Delta b_5$ as functions of $\Delta b_3$. Additionally, we argue that in the $\omega \to 0$ limit the relationship $b_n^\text{T} = n^{-d/2} b_n$ holds between the trapped (T) and homogeneous coefficients for arbitrary temperature and coupling strength (not merely in scale-invariant regimes). Finally, we point out an exact, universal (coupling- and frequency-independent) relationship between $\Delta b_3^\text{T}$ in 1D with three-body forces and $\Delta b_2^\text{T}$ in 2D with two-body forces.

Fermi-Fermi crossover in the ground state of one-dimensional few-body systems with anomalous three-body interactions

In one spatial dimension, quantum systems with an attractive three-body contact interaction exhibit a scale anomaly. In this work, we examine the few-body sector for up to six particles. We study those systems with a self-consistent, non-perturbative, iterative method, in the subspace of zero total momentum. Exploiting the structure of the contact interaction, the method reduces the complexity of obtaining the wavefunction by three powers of the dimension of the Hilbert space. We present results on the energy, and momentum and spatial structure, as well as Tan's contact. We find a Fermi-Fermi crossover interpolating between large, weakly bound trimers and compact, deeply bound trimers: at weak coupling, the behavior is captured by degenerate perturbation theory; at strong coupling, the system is governed by an effective theory of heavy trimers (plus free particles in the case of asymmetric systems). Additionally, we find that there is no trimer-trimer attraction and therefore no six-body bound state.

Quantum Anomaly and Thermodynamics of One-Dimensional Fermions with Three-Body Interactions.

We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.

Dipolar Bose gas with three-body interactions at finite temperature

We investigate effects of three-body contact interactions on a trapped dipolar Bose gas at finite temperature using the Hartree–Fock–Bogoliubov approximation. We analyze numerically the behavior of the transition temperature and the condensed fraction. Effects of the three-body interactions, anomalous pair correlations and temperature on the collective modes are discussed.

Quantum dilute droplets of dipolar bosons at finite temperature

We systematically study the properties of dipolar Bose gases with two- and three-body contact interactions at finite temperature using the Hartree–Fock–Bogoliubov–Popov approximation. In uniform case, we obtain an exciting new extension of the seminal Lee–Huang–Yang corrected equation of state that depends explicitly on the thermal fluctuations and on the coupling constant of the three-body interaction. We investigate, on the other hand, the effects of thermal fluctuations on the occurrence and stability of a droplet state in a Bose–Einstein condensate with strong dipole–dipole interactions. We find that at finite temperature, the droplet phase appears as a narrow peak surrounded by a broader thermal halo. We show that the number of particles inside the droplet decays with increasing temperature.

Three-body scattering problem in the fixed center approximation: The case of attraction

We study the scattering of a light particle on a bound pair of heavy particles (e.g., the deuteron) within the fixed center approximation in the case of light-heavy attraction, solving the integral equation for the three-body Green's function both in the coordinate and in the momentum space. The results for the three-body scattering amplitude appear to be ambiguous -- they depend on a single real parameter. This parameter may be fixed by a three-body input, e.g., the three-body scattering length. We also solve the integral equation for the three-body Green function in the momentum space, introducing a finite cut-off. We show that all three approaches are equivalent. We also discuss how our approach to the problem matches with the introduction of three-body contact interaction as done by other authors.

Description of light nuclei in pionless effective field theory using the stochastic variational method

We construct a coordinate-space potential based on pionless effective field theory with a Gaussian regulator. Charge-symmetry breaking is included through the Coulomb potential and through two- and three-body contact interactions. Starting with the effective field theory potential, we apply the stochastic variational method to determine the ground states of nuclei with mass number $A\leq 4$. At next-to-next-to-leading order, two out of three independent three-body parameters can be fitted to the three-body binding energies. To fix the remaining one, we look for a simultaneous description of the binding energy of $^4$He and the charge radii of $^3$He and $^4$He. We show that at the order considered we can find an acceptable solution, within the uncertainty of the expansion. We find that the EFT expansion shows good agreement with empirical data within the estimated uncertainty, even for a system as dense as $^4$He.

Three-body systems in physics of cold atoms and halo nuclei

Few-body systems, such as cold atoms and halo nuclei, share universal features at low energies, which are insensitive to the underlying inter-particle interactions at short ranges. These low-energy properties can be investigated in the framework of effective field theory with two-body and three-body contact interactions. I review the effective-field-theory studies of universal physics in three-body systems, focusing on the application in cold atoms and halo nuclei.

ΛΛ6He in cluster effective field theory

The bound state of \nuclide[6][\Lambda\Lambda]{He} is studied as a three-body ($\Lambda\Lambda\alpha$) system in a cluster effective field theory at leading order (LO). We find that the system exhibits the limit cycle which is associated with the formation of bound states called the Efimov states. This implies that the three-body contact interaction should be promoted to LO. The relationship of the binding energy and the $\Lambda\Lambda$ scattering length is discussed as well as the role of the scale in this system.

Investigation of thennΛbound state in pionless effective theory

The possibility of an $nn\Lambda$ bound state is investigated in the framework of pionless effective field theory at leading order. A system of coupled integral equations are constructed in the spin-isospin basis, of which numerical solutions are investigated. In particular, we make use of the limit cycle behavior, i.e., cyclic singularities of coupled integral equations of the system, which would be associated with the formation of a three-body bound state, so-called the Efimov state, in the unitary limit. Furthermore, we find that, when the sharp momentum cutoff introduced in the integral equations is taken significantly larger than the hard scale of the effective theory, the coupling of a three-body contact interaction becomes cyclically singular indicating the onset of Efimov-like bound state formation. However, the paucity of empirical information to determine the parameters of the theory precludes a definitive conclusion on the existence of such a bound state. As a simple test of the feasibility of the $nn\Lambda$ bound system in nature, we explore the cutoff dependence of the theory, and uncertainties of the present study are discussed as well.

Giant resonances based on unitarily transformed two-nucleon plus phenomenological three-nucleon interactions

We investigate giant resonances of spherical nuclei on the basis of the Argonne V18 potential after unitary transformation within the similarity renormalization group or the unitary correlation operator method supplemented by a phenomenological three-body contact interaction. Such Hamiltonians can provide a good description of ground-state energies and radii within Hartree–Fock plus low-order many-body perturbation theory. The standard random phase approximation is applied here to calculate the isoscalar monopole, isovector dipole, and isoscalar quadrupole excitation modes of the , , and nuclei. Thanks to the inclusion of the three-nucleon interaction and despite the minimal optimization effort, a reasonable agreement with experimental centroid energies of all three modes has been achieved. The role and scope of the Hartree–Fock reference state in RPA methods are discussed.

HeΛΛ6in cluster effective field theory

The hypernucleus $_{\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}}^{6}\mathrm{He}$ is studied as a three-body ($\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}\ensuremath{\alpha}$) cluster system in cluster effective field theory at leading order. We find that the three-body contact interaction exhibits the limit cycle when the cutoff in the integral equations is sent to the asymptotic limit and thus it should be promoted to leading order. We also derive a determination equation of the limit cycle which reproduces the numerically obtained limit cycle. We then study the correlations between the double $\ensuremath{\Lambda}$ separation energy ${B}_{\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}}^{}$ of $_{\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}}^{6}\mathrm{He}$ and the scattering length ${a}_{\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}}^{}$ of the $S$-wave $\mathrm{\ensuremath{\Lambda}\ensuremath{\Lambda}}$ scattering. The role of the scale in this approach is also discussed.

The Born Series for S-wave Quartet nd Scattering at Small Cutoff Values

Perturbative expansions, the Born series, of the scattering length and the amplitude of $S$-wave neutron-deuteron scattering for spin quartet channel below deuteron breakup threshold are studied in pionless effective field theory at small cutoff values. A three-body contact interaction is introduced when the integral equation is solved using the small cutoffs. After renormalizing the three-body interaction by using the scattering length, we expand the integral equation as the ordinary and inverted Born series. We find that the scattering length and the phase shift are considerably well reproduced with a few terms of the inverted Born series at relatively large cutoff values, $\Lambda\simeq 100$ MeV.

ΛΛ4H in halo effective field theory

The ${}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{1.0pt}{0ex}}4}$H bound state and the $S$-wave hypertriton(${}_{\ensuremath{\Lambda}}^{\phantom{\rule{0.16em}{0ex}}3}$H)-$\ensuremath{\Lambda}$ scattering in spin singlet and triplet channels below the hypertriton breakup momentum scale are studied in halo/cluster effective field theory at leading order by treating the ${}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{1.0pt}{0ex}}4}$H system as a three-cluster ($\ensuremath{\Lambda}$-$\ensuremath{\Lambda}$-deuteron) system. In the spin singlet channel, the amplitude is insensitive to the cutoff parameter ${\ensuremath{\Lambda}}_{c}$ introduced in the integral equation, and we find that there is no bound state. In this case, the scattering length of the hypertriton-$\ensuremath{\Lambda}$ scattering is found to be ${a}_{0}^{}=16.0\ifmmode\pm\else\textpm\fi{}3.0$ fm. In the spin triplet channel, however, the amplitude obtained by the coupled integral equations is sensitive to ${\ensuremath{\Lambda}}_{c}$, and we introduce the three-body contact interaction ${g}_{1}^{}({\ensuremath{\Lambda}}_{c})$. After phenomenologically fixing ${g}_{1}^{}({\ensuremath{\Lambda}}_{c})$, we find that the correlation between the two-$\ensuremath{\Lambda}$ separation energy ${B}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}$ from the ${}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{1.0pt}{0ex}}4}$H bound state and the scattering length ${a}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{}$ of the $S$-wave $\ensuremath{\Lambda}$-$\ensuremath{\Lambda}$ scattering is significantly sensitive to the value of ${\ensuremath{\Lambda}}_{c}$.