Identical Spinless Particles Research Articles

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.

Quantum backflow for dissipative two-identical-particle systems

In this work, dissipative quantum backflow is studied for a superposition of two stretched Gaussian wave packets and two identical spinless particles within the Caldirola-Kanai framework. Backflow is mainly an interference process and dissipation is not able to suppress it in the first case. For two identical spinless particles, apart from interference terms, the symmetry of the wave function seems to be crucial in this dynamics. The combined properties make bosons display this effect, even for the dissipative regime but, for fermions, backflow is not exhibited in any regime, dissipative and nodissipative. The anti-symmetric character of the corresponding wave function seems to be strong enough to prevent it. For bosons, backflow is also analyzed in terms of fidelity of one-particle states which is a well-known property of two quantum states. At very small values of fidelity, this effect is not seen even for bosons.

Stability and Clustering for Lattice Many-Body Quantum Hamiltonians with Multiparticle Potentials

We analyze a quantum system of N identical spinless particles of mass m, in the lattice $$\mathbb {Z}^d$$ , given by a Hamiltonian $$H_N=T_N+V_N$$ , with kinetic energy $$T_N\ge 0$$ and potential $$V_N=V_{N,2}+V_{N,3}$$ composed of attractive pair and repulsive 3-body contact-potentials. This Hamiltonian is motivated by the desire to understand the stability of quantum field theories, with massive single particles and bound states in the energy-momentum spectrum, in terms of an approximate Hamiltonian for their N-particle sector. We determine the role of the potentials $$V_{N,2}$$ and $$V_{N,3}$$ on the physical stability of the system, such as to avoid a collapse of the N particles. Mathematically speaking, stability is associated with an N-linear lower bound for the infimum of the $$H_N$$ spectrum, $$\underline{\sigma }(H_N)\ge -cN$$ , for $$c>0$$ independent of N. For $$V_{N,3}=0$$ , $$H_N$$ is unstable, and the system collapses. If $$V_{N,3}\not =0$$ , $$H_N$$ is stable and, for strong enough repulsion, we obtain $$\underline{\sigma }(H_N)\ge -c' N$$ , where $$c'N$$ is the energy of (N/2) isolated bound pairs. This result is physically expected. A much less trivial result is that, as N varies, we show $$[\,\underline{\sigma }(V_N)/N\,]$$ has qualitatively the same behavior as the well-known curve for minus the nuclear binding energy per nucleon. Moreover, it turns out that there exists a saturation value $$N_s$$ of N at and above which the system presents a clustering: the N particles distributed in two fragments and, besides lattice translations of particle positions, there is an energy degeneracy of all two fragments with particle numbers $$N_r$$ and $$N_s-N_r$$ , with $$N_r=1,\ldots ,N_s-1$$ .

Acceleration of adiabatic transport of interacting particles and rapid manipulations of a dilute Bose gas in the ground state

We show a method to accelerate quantum adiabatic transport of identical spinless particles interacting with each other by developing the preceding fast-forward scaling theory formed for one-particle systems [Masuda and Nakamura, Proc. R. Soc. A 466, 1135 (2010)]. We derive a driving potential which accelerates adiabatic dynamics of quantum systems composed of identical particles in order to obtain the final adiabatic states in any desired short time. We also exhibit an ideal rapid manipulation of dilute Bose gas in the ground state without energy excitation by using the fast-forward scaling theory.

Techniques of amplitude analysis for two-pseudoscalar systems

The analytical tools needed for an amplitude analysis of two-pseudoscalar systems are described in some detail. Analyses involving two identical spinless particles require a new type of polynomial; the general form of such a polynomial is given for the first time.

Exact solution of the bound-state Faddeev-Yakubovsky equations for one-dimensional systems with a delta -function interaction.

The Faddeev-Yakubovsky equations for four identical spinless particles in one-dimensional space interacting via {delta}-function potentials are solved analytically for a bound state.

Exact solution of the four-body Faddeev-Yakubovsky equations for the harmonic oscillator.

The Faddeev-Yakubovsky equations for four identical spinless particles in three-dimensional space interacting via harmonic-oscillator two-body potentials are solved analytically for any excitation and for an arbitrary value of the total angular momentum.

Exact solution of the Faddeev equations for the harmonic oscillator.

The Faddeev equations for three identical spinless particles in three-dimensional space interacting via harmonic-oscillator two-body potentials are solved analytically for any excitation and for an arbitrary value of the total angular momentum.

Exotic statistics on surfaces

We investigate the allowed spectrum of statistics for n identical spinless particles on an arbitrary closed two-manifold M, by using a powerful topological approach to the study of quantum kinematics. On a surface of genus g⩾1 statistics other than Bose on Fermi can only be obtained by utilizing multi-component state transforming as an irreducible unitary representation of the fundamental group of the n-particle configuration space. These multi-component (or nonscalar) quantizations allow the possibility of fractional statistics, as well as other exotic, nonfractional statistics some of whose properties we discuss. On an orientable surface of genus g⩾0 only anyons with rational statistical parameter θ π = p q are allowed, and their number is restricted to be sq-g+1 (sϵ Z) . For nonorientable surfaces only θ=0, π are allowed. Finally, we briefly comment on systems of spinning particles and make a comparison with the results for solitons in the O(3)-invariant nonlinear sigma model with space manifold M.

Faddeev-Yakubovsky theory for four-body systems with three-body forces and its one-dimensional integral equations from the hyperspherical-harmonics expansion in momentum space

The generalized Faddeev-Yakubovsky equation is derived for the four-body system where three-body forces are included. There result twenty-two coupled equations which, in the case of four identical spinless particles, can be reduced to three. In addition, by using the hyperspherical-harmonics expansion in momentum space, as suggested by Dzhibuti and his collaborators, and the Raynal-Revai transformation, it is possible to write these as one-dimensional coupled integral equations. Numerical solutions are straightforward and, for sample potentials, suggest relatively fast convergence in the number of harmonics required. Results obtained so far offer fresh hope that this method may provide a means for quick and accurate computation of four-body scattering quantities.

SU(n) bundles over the configuration space of three identical particles moving on R3

The authors study the systems of three identical spinless particles moving on R3 and possessing an SU(n) gauge symmetry. Three such systems are possible, corresponding to the three non-isomorphic SU(n) bundles over C3(R3), the configuration space. The authors retract C3(R3) to a subcomplex which shows clearly how its homology arises. The three bundles can be realised as pull-backs of the universal bundle S7 to S4 using three non-homotopic maps C3(R3) to S4. The two non-trivial bundles admit no flat connection; they do not correspond to Bose, Fermi or parastatistics.

Three-particle dynamics in the case of finite-range forces

A reduction of the three-particle integral equations of motion, that may be done in the case of pair forces with the finite range, is described. By solving the reduced equations, the binding energy of a system of three identical spinless particles in the ground state and the binding energy of∑-hypertriton are calculated.

Exact solution of the Faddeev equations for the harmonic oscillator ground state

The Faddeev equations in $N$ space dimensions for three identical spinless particles in their ground state interacting via harmonic oscillator, two-body potentials is solved analytically. Unlike the well-known Schr\"odinger solution, the individual Faddeev amplitudes may be negative and contain a long-range component of arbitrary strength. Upon symmetrizing to obtain the full wave function, this component vanishes and the Schr\"odinger solution results. Three-dimensional plots of the various wave functions are presented.NUCLEAR STRUCTURE Faddeev calc., configuration space, harmonic oscillator.

Angular momentum reduction in the four-body problem

A complete angular momentum analysis of the integral equations of motion for four identical spinless particles is given. After separation of the variables associated with rotation, the equations of motion take the form of an infinite set of coupled integral equations in three continuous variables. In general, the four-particle kinematic factors in the integral kernels are expressed as bilinear combinations of factors of the three-particle kinematic factors type. For the case of a separable two-particle interaction the equations obtained are simplified so that they are reduced to coupled integral equations in two continuous variables.

Soluble model involving four identical particles

Using a nonrelativistic field theoretic formalism, a soluble model of scattering involving four identical spinless particles is developed and solved numerically. In addition to the elementary "nucleon" (the $n$), two composite particles meant to approximate the deuteron and triton are introduced with the couplings $d\ensuremath{\leftrightarrow}n+n$ and $t\ensuremath{\leftrightarrow}d+n$. By consistently excluding all particle-exchange contributions to the three-body sector, four-body integral equations are obtained for the two-to-two processes: $nt\ensuremath{\rightarrow}nt$, $nt\ensuremath{\rightarrow}dd$ as well as for $dd\ensuremath{\rightarrow}dd$ and $dd\ensuremath{\rightarrow}nt$. Numerical solutions of the equations are found to satisfy unitarity constraints above the two-, three-, and four-body thresholds. The positions of the four-body bound states are obtained and a complete phase shift calculation is performed. The sum of the total three- and four-body breakup cross sections predicted by the model are displayed as a function of energy and the angular distributions for all 2 \ensuremath{\rightarrow} 2 reactions are compared with the four-nucleon data.NUCLEAR REACTIONS Four-body problem. Spinless model of the four-nucleon system.

Efimov effect and higher bound states in a three-particle system

We study the J = 0 bound states for a system of three identical spinless particles interacting in pairs through delta-shell potentials. The Efimov states are identified, and their wave functions obtained. We have found a new family of bound states, which occur for higher values of the attractive coupling strength.

Integral equations for four identical particles with separable two-particle interactions

The problem of four identical spinless particles with separable two-particle interactions is considered. The integral equations of motion for four particles in the bound states and in the continuum states with two non-ihteracting subsystems in the initial channel are formulated. The angular-momentum reduction of the obtained equations is given.

A Sasakawa approach to three-particle scattering

An idea introduced by Sasakawa is used to convert the Faddeev equations into integral equations in which the inhomogeneous term contains the amplitudes for inelastic scattering and rearrangement collisions. The kernels are thereby modified and made nonsingular for energies below the three-particle break-up threshold. The equations are derived for a system of three nonidentical spinless particles interacting through central two-particle potentials. An iterative method for solving the modified equations is developed. It is shown how to symmetrize the equations for a system of three identical spinless particles. The equations are written out in detail for the angular-momentum-zero state of three identical spinless particles interacting through ans-wave two-particle force which produces only one two-particle bound state. The lowest-order approximation to the solution of the equations for this simple system is worked out. It is found that this approximation satisfies unitarity for energies below the three-particle break-up threshold.

Quasiparticle calculations with local potentials above the three-particle break-up threshold

The Alt-Grassberger-Sandhas equations are solved above the three-particle break-up threshold in the case of three identical spinless particles interacting via Yukawa potentials in the zeroth- and first-order quasi-Born approximation. The differences in the calculated phase shifts and total break-up cross sections between the two approximations indicate the importance of the non-separable part of the local potential, which is taken into account by the first-order approximation.

T-Matrix Perturbation Theory for the Faddeev Equations

A $T$-matrix perturbation theory for the Faddeev equations is developed in two different ways giving slightly differing results for the first-order correction to the ground-state energy of the three-body system, and both expressions differ from the result recently obtained by Fuda. The application of the theory to the system of three identical spinless particles is discussed.