Two-body Contact Interactions Research Articles

Discrete Scale Invariance and U(2) Family of Two-Body Contact Interactions in One Dimension

Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U(2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U(2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.

Observation of 1/k^{4}-Tails after Expansion of Bose-Einstein Condensates with Impurities.

We measure the momentum density in a Bose-Einstein condensate (BEC) with dilute spin impurities after an expansion in the presence of interactions. We observe tails decaying as 1/k^{4} at large momentum k in the condensate and in the impurity cloud. These algebraic tails originate from the impurity-BEC interaction, but their amplitudes greatly exceed those expected from two-body contact interactions at equilibrium in the trap. Furthermore, in the absence of impurities, such algebraic tails are not found in the BEC density measured after the interaction-driven expansion. These results highlight the key role played by impurities when present, a possibility that had not been considered in our previous work [Chang etal., Phys. Rev. Lett. 117, 235303 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.235303]. Our measurements suggest that these unexpected algebraic tails originate from the nontrivial dynamics of the expansion in the presence of impurity-bath interactions.

Thermal conductivity of a weakly interacting Bose gas in quasi-one-dimension.

Transport coefficients are typically divergent for quantum integrable systems in one dimension, such as a Bose gas with a two-body contact interaction. However, when a one-dimensional system is realized by confining bosons into a tight matter waveguide, an effective three-body interaction inevitably arises as leading perturbation to break the integrability. This fact motivates us to study the thermal conductivity of a Bose gas in one dimension with both two-body and three-body interactions. In particular, we evaluate the Kubo formula exactly to the lowest order in perturbation by summing up all contributions that are naively higher orders in perturbation but become comparable in the zero-frequency limit due to the pinch singularity. Consequently, a self-consistent equationfor a vertex function is derived, showing that the thermal conductivity in quasi-one-dimension is dominated by the three-body interaction rather than the two-body interaction. Furthermore, the resulting thermal conductivity in the weak-coupling limit proves to be identical to that computed based on the quantum Boltzmann equation and its temperature dependence is numerically determined.

Effect of quartic–quintic beyond-mean-field interactions on a self-bound dipolar droplet

We study the effect of beyond-mean-field quantum-fluctuation (QF) Lee–Huang–Yang (LHY) and three-body interactions, with quartic and quintic nonlinearities, respectively, on the formation of a stable self-repulsive (positive scattering length a) and a self-attractive (negative a) self-bound dipolar Bose–Einstein condensate (BEC) droplet in free space under the action of two-body contact and dipolar interactions. Previous studies of dipolar droplets considered either the LHY interaction or the three-body interaction, as either of these interactions could stabilize a dipolar BEC droplet against collapse. We find that the effect of three-body recombination on the formation of a dipolar droplet could be quite large and for a complete description of the problem both the QF LHY and three-body interactions should be considered simultaneously, where appropriate. In the self-repulsive case for small a and in the self-attractive case, no appropriate LHY interaction is known and only three-body interaction should be used, otherwise both beyond-mean-field interactions should be used. We consider a numerical solution of a highly-nonlinear beyond-mean-field model as well as a variational approximation to it in this investigation and present results for size, shape and energy of a dipolar droplet of polarized 164Dy atoms. The shape is filament-like, along the polarization direction, and could be long, for a large number of atoms N, short for small N, thin for negative a and small positive a, and fat for large positive a.

Discrete scale-invariant boson-fermion duality in one dimension

We introduce models of one-dimensional $n(\geq3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"{o}dinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body S-matrix elements for arbitrary $n\geq3$, all of which enjoy discrete scale invariance or log-periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of $n=3$; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.

From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems

Quantum systems composed of N distinct particles in mathbb {R}^2 with two-body contact interactions of TMS type are shown to arise as limits—in the norm resolvent sense—of Schrödinger operators with suitably rescaled pair potentials.

Emergent Quasicrystalline Symmetry in Light-Induced Quantum Phase Transitions

The discovery of quasicrystals with crystallographically forbidden rotational symmetries has changed the notion of the ordering in materials, yet little is known about the dynamical emergence of such exotic forms of order. Here we theoretically study a nonequilibrium cavity-QED setup realizing a zero-temperature quantum phase transition from a homogeneous Bose-Einstein condensate to a quasicrystalline phase via collective superradiant light scattering. Across the superradiant phase transition, collective light scattering creates a dynamical, quasicrystalline optical potential for the atoms. Remarkably, the quasicrystalline potential is "emergent" as its eightfold rotational symmetry is not present in the Hamiltonian of the system, rather appears solely in the low-energy states. For sufficiently strong two-body contact interactions between atoms, a quasicrystalline order is stabilized in the system, while for weakly interacting atoms the condensate is localized in one or few of the deepest minima of the quasicrystalline potential.

Renormalization of a finite-range inverse-cube potential

We study the regularization and renormalization of a finite range inverse cube potential in the two- and three-body sectors. Specifically, we compare and contrast three different regulation schemes frequently used to study few-body systems as well as the associated renormalization group flows. We also calculate bound state and scattering observables over a wide range of cutoffs, demonstrating the sufficiency of a two-body contact interaction to renormalize two- and three-body observables. We supplement these plots with quantified analyses of the observables' residual cutoff dependence.

Spectral Theory of the Fermi Polaron

The Fermi polaron refers to a system of free fermions interacting with an impurity particle by means of two-body contact forces. Motivated by the physicists' approach to this system, the present article develops a general mathematical framework for defining many-body Hamiltonians with two-body contact interactions by means of a renormalization procedure. In the case of the Fermi polaron the well-known TMS Hamiltonians are shown to emerge. For the Fermi polaron in a two-dimensional box a novel variational principle, established within the general framework, links the low-lying eigenvalues of the system to the zero-modes of a Birman-Schwinger type operator. It allows us to show, e.g., that the \emph{polaron}- and \emph{molecule} energies, computed in the physical literature, are indeed upper bounds to the ground state energy of the system.

Stabilization of trapless Bose-Einstein condensates without any management

In higher dimensions, so far, both attractive and repulsive trapless Bose-Einstein condensates (BECs) have been stabilized with the help of temporal or spatial management of the two-body contact interaction. However, in the absence of nonlinear management, up to now, there is no model to stabilize the repulsive, trapless BECs at higher dimensions. Hence, in the present study, we tried to stabilize the same and achieved with help of the interplay between three-body and higher-order interactions. In addition, we show that there is an enhancement of the stability of attractive, trapless BECs due to the inclusion of the higher-order interaction along with the two- and three-body interactions. Further, our analytical predictions are corroborated with 4th order Runge-Kutta and split-step Crank-Nicholson numerical results.

Self-trapped quantum balls in binary Bose–Einstein condensates

We study the formation of a stable self-trapped spherical quantum ball in a binary Bose–Einstein condensate (BEC) with two-body inter-species attraction and intra-species repulsion employing the beyond-mean-field Lee–Huang–Yang and the three-body interactions. We find that either of these interactions or a combination of them can stabilize the binary BEC quantum ball with very similar stationary results, and for a complete description of the problem both the terms should be considered. These interactions lead to higher-order nonlinearities, e.g. quartic and quintic, respectively, in a nonlinear dynamical equation compared to the cubic nonlinearity of the two-body contact interaction in the mean-field Gross–Pitaevskii equation. The higher-order nonlinearity makes the energy infinitely large at the center of the binary ball and thus avoids its collapse. In addition to the formation of stationary binary balls, we also study a collision between two such balls. At large velocities, the collision is found to be elastic, which turns out to be inelastic as the velocity is lowered. We consider the numerical solution of a beyond-mean-field model for the binary ball as well as a single-mode variational approximation to it in this study.

Tan's Contact for Trapped Lieb-Liniger Bosons at Finite Temperature.

The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, local-density approximation, and exact quantum MonteCarlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.

Information Entropy for a Two-Dimensional Rotating Bose–Einstein Condensate

We study the information entropy, order, disorder, and complexity for the two-dimensional (2D) rotating and nonrotating Bose-Einstein condensates. The choice of our system is a complete theoretical laboratory where the complexity is controlled by the two-body contact interaction strength and the rotation frequency ($\Omega$) of the harmonic trap. The 2D nonrotating condensate shows the complexity of the category I where the disorder-order transition is triggered by the interaction strength. In the rotating condensates, $\Omega$ is chosen as the disorder parameter when the interaction strength is fixed. With respect to $\Omega$, the complexity shifts between the maximum and minimum confirm the existence of category II complexity in the rotating condensate. Also, We consider the interaction strength as the disorder parameter when $\Omega$ is unchanged and complexity as a function of interaction strength exhibits category III complexity. The present work also includes the calculation of upper bound and lower bound of entropy for 2D quantum systems.

Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

We study the ground state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained variational (LOCV) method, we are able to semi-quantitatively attain Bose-Einstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle $E_0/N \approx 2.5 N^{1/3} \hbar \omega$ and two-body contact $C_2/N \approx 16 N^{1/6}\sqrt{m\omega/\hbar}$ for a Bose gas on resonance, where $\omega$ is the trap frequency.

Renormalized contact interaction in degenerate unitary Bose gases

We renormalize the two-body contact interaction based on the exact solution of two interacting particles in a harmonic trap. This renormalization extends the validity of the contact interaction to large scattering lengths. We apply this renormalized interaction to a degenerate unitary Bose gas to study its stationary properties and elementary excitations using the mean-field theory and the hyperspherical method. Since the scattering length is no longer a relevant length scale at unitarity, universal properties are obtained that depend only on the average particle density. Our treatment shows that the universal relations for the total energy and for the two-body contact are $E/N=12.67\hbar^2\langle n^{2/3}\rangle/2m$ and $C_{2}/N=11.8\langle n^{1/3}\rangle$ respectively.

Thermodynamics and renormalized quasiparticles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wavefunction renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature $T$ and chemical potential $\mu$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization group calculations for thermodynamic observables at $\mu =0$ to finite values of $\mu$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, the susceptibility, and the correlation length of the two-dimensional spin-$1/2$ quantum $XY$ model with coupling $J$ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\mu \leq 0$ rather accurately up to relatively high temperatures $T \lesssim 0.1 J$.

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.

Dynamical stability of dipolar Bose-Einstein condensates with temporal modulation of the s-wave scattering length.

We study the stabilization properties of dipolar Bose-Einstein condensate by temporal modulation of short-range two-body interaction. Through both analytical and numerical methods, we analyze the mean-field Gross-Pitaevskii equation with short-range two-body and long-range, nonlocal, dipolar interaction terms. We derive the equation of motion and effective potential of the dipolar condensate by variational method. We show that there is an enhancement of the condensate stability due to the inclusion of dipolar interaction in addition to the two-body contact interaction. We also show that the stability of the dipolar condensate increases in the presence of time varying two-body contact interaction; the temporal modification of the contact interaction prevents the collapse of dipolar Bose-Einstein condensate. Finally we confirm the semi-analytical prediction through the direct numerical simulations of the governing equation.

Stable Dilute Supersolid of Two-Dimensional Dipolar Bosons.

We consider two-dimensional bosonic dipoles oriented perpendicularly to the plane. On top of the usual two-body contact and long-range dipolar interactions we add a contact three-body repulsion as expected, in particular, for dipoles in the bilayer geometry with tunneling. The three-body repulsion is crucial for stabilizing the system, and we show that our model allows for stable continuous space supersolid states in the dilute regime and calculate the zero-temperature phase diagram.

Creating and manipulating non-Abelian anyons in cold atom systems using auxiliary bosons

The possibility of realizing bosonic fractional quantum Hall effect in ultra-cold atomic systems suggests a new route to producing and manipulating anyons, by introducing auxiliary bosons of a different species that capture quasiholes and thus inherit their non-trivial braiding properties. States with localized quasiholes at any desired locations can be obtained by annihilating the auxiliary bosons at those locations. We explore how this method can be used to generate non-Abelian quasiholes of the Moore-Read Pfaffian state for bosons at filling factor $\nu=1$. We show that a Hamiltonian with an appropriate three-body interaction can produce two-quasihole states in two distinct fusion channels of the topological "qubit." Characteristics of these states that are related to the non-Abelian nature can be probed and verified by a measurement of the effective relative angular momentum of the auxiliary bosons, which is directly related to their pair distribution function. Moore-Read states of more than two quasiholes can also be produced in a similar fashion. We investigate some issues related to the experimental feasibility of this approach, in particular, how large the systems should be for a realization of this physics and to what extent this physics carries over to systems with the more standard two-body contact interaction.