Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1309.1927 · Physical Review A

Universality and scaling in the N -body sector of Efimov physics

M. Gattobigio, Alejandro Kievsky

Published Jul 2, 2014• 49 citations
arXivHTMLPDF

Abstract

Universal behavior has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.

DOI
10.1103/physreva.90.012502

Related papers

Exact renormalization relation and binding energies for three identical bosons

Langxuan Chen, Pengfei Zhang

In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.

arXiv:2506.12531•Physical Review A•Aug 29, 2025•2 citations

Efimov Physics and Connections to Nuclear Physics

Alejandro Kievsky, M. Gattobigio, Luca Girlanda +1 more

Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by continuous scale invariance, which is broken into discrete scale invariance when three or more particles come into play. This symmetry induces a universal behavior for different systems that is independent of the details of the underlying interaction and rooted in the smallness of the ratio ℓ/ a B ≪ 1, where the length a B is associated with the binding energy of the two-body system [Formula: see text], and ℓ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the onset of system-specific nonuniversal effects. In this review, we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which a B → ∞, and we show that nuclear systems belong to this class of universality.

arXiv:2102.13504•Annual Review of Nuclear and Particle Science•Jul 6, 2021•44 citations

Universality and scaling limit of weakly-bound tetramers

M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio +2 more

The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.

AIP conference proceedings•Jan 1, 2012•6 citations

Some aspects of universality in Efimov physics

M. Gattobigio, Alejandro Kievsky

The Efimov effect is the appearance of an infinite tower of bound states, E n 3 , that accumulate in a geometrical way at zero energy in systems of three-identical bosons having an infinite large two-body scattering length a . In the zero-range limit the geometrical factor is E n +1 3 / E n 3 = e −2π/ s 0 with s 0 ≈ 1.00624 an universal number. The Efimov effect is the signature of the discrete scale invariance (DSI) present in these systems close to the unitary limit (1/ a = 0). The DSI constrains also the spectrum in N > 3 systems. In this contribution the two-level tree structure observed in bosons systems with N ≥ 4 will be discussed using potential models.

Journal of Physics Conference Series•Jul 7, 2014•1 citation

Efimov States in Nuclear and Particle Physics

H.‐W. Hammer, Lucas Platter

Particles with resonant short-range interactions have universal properties that do not depend on the details of their structure or their interactions at short distances. In the three-body system, these properties include the existence of a geometric spectrum of three-body Efimov states and a discrete scaling symmetry, which leads to log-periodic dependence of observables on the scattering length. Similar universal properties appear in the four-body system and possibly higher-body systems as well. For example, universal four-body states have recently been predicted and observed in experiments. These phenomena are often referred to as Efimov physics. We review their theoretical description and discuss applications in different areas of physics with a special emphasis on nuclear and particle physics.

arXiv:1001.1981•Annual Review of Nuclear and Particle Science•Oct 22, 2010•123 citations

Universal range corrections to Efimov trimers for a class of paths to the unitary limit

Alejandro Kievsky, M. Gattobigio

Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific $n$ level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length $a$ and the energy-scattering length ${a}_{B}$, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact ${C}_{2}$ at the unitary limit.

arXiv:1507.03402•Physical Review A•Dec 29, 2015•21 citations
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of N\ensuremath≤6N\ensuremath{\le}6N\ensuremath≤6 systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by continuous scale invariance, which is broken into discrete scale invariance when three or more particles come into play. This symmetry induces a universal behavior for different systems that is independent of the details of the underlying interaction and rooted in the smallness of the ratio ℓ/ a B ≪ 1, where the length a B is associated with the binding energy of the two-body system [Formula: see text], and ℓ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the onset of system-specific nonuniversal effects. In this review, we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which a B → ∞, and we show that nuclear systems belong to this class of universality.
The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.
The Efimov effect is the appearance of an infinite tower of bound states, E n 3 , that accumulate in a geometrical way at zero energy in systems of three-identical bosons having an infinite large two-body scattering length a . In the zero-range limit the geometrical factor is E n +1 3 / E n 3 = e −2π/ s 0 with s 0 ≈ 1.00624 an universal number. The Efimov effect is the signature of the discrete scale invariance (DSI) present in these systems close to the unitary limit (1/ a = 0). The DSI constrains also the spectrum in N > 3 systems. In this contribution the two-level tree structure observed in bosons systems with N ≥ 4 will be discussed using potential models.
Particles with resonant short-range interactions have universal properties that do not depend on the details of their structure or their interactions at short distances. In the three-body system, these properties include the existence of a geometric spectrum of three-body Efimov states and a discrete scaling symmetry, which leads to log-periodic dependence of observables on the scattering length. Similar universal properties appear in the four-body system and possibly higher-body systems as well. For example, universal four-body states have recently been predicted and observed in experiments. These phenomena are often referred to as Efimov physics. We review their theoretical description and discuss applications in different areas of physics with a special emphasis on nuclear and particle physics.
Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific nnn level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length aaa and the energy-scattering length aB{a}_{B}aB​, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact C2{C}_{2}C2​ at the unitary limit.