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Small Clusters of $^4$He Atoms in Finite-Cutoff Effective Field Theory
Authors:
Betzalel Bazak
Abstract:
Small clusters of $^4$He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted t…
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Small clusters of $^4$He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted to the trimer and tetramer ground-state energies. We distinguish total binding energies from the one-atom separation energies $S_A^*=B_A^*-B_{A-1}$ of the shallow excited states. Ground-state energies through $A=8$ are reproduced at the few-percent level, but threshold-sensitive observables are less accurate. In particular, the calculated tetramer separation energy is $S_4^*=2.85$~mK, compared with $0.92$--$0.96$~mK in converged LM2M2 calculations, and the atom--trimer scattering length differs substantially from modern LM2M2 benchmarks. For $A\ge5$, the available excited-state benchmarks are sparse and strongly method dependent. Comparison with another soft-core interaction suggests that the discrepancies primarily reflect short-distance physics and higher-order operators omitted from the finite-cutoff Hamiltonian rather than numerical convergence of the stochastic variational calculation. Finite-cutoff EFT therefore provides an efficient description of ground-state systematics, whereas shallow excited states and atom--cluster scattering require a controlled higher-order and cutoff-dependence analysis.
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Submitted 20 August, 2026; v1 submitted 16 November, 2025;
originally announced November 2025.
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Few-body bound states of two-dimensional bosons
Authors:
G. Guijarro,
G. E. Astrakharchik,
J. Boronat,
B. Bazak,
D. S. Petrov
Abstract:
We study clusters of the type A$_N$B$_M$ with $N\leq M\leq 3$ in a two-dimensional mixture of A and B bosons, with attractive AB and equally repulsive AA and BB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered cluste…
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We study clusters of the type A$_N$B$_M$ with $N\leq M\leq 3$ in a two-dimensional mixture of A and B bosons, with attractive AB and equally repulsive AA and BB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered clusters are bound and that their energies are universal functions of the scattering lengths $a_{AB}$ and $a_{AA}=a_{BB}$, for sufficiently large attraction-to-repulsion ratios $a_{AB}/a_{BB}$. When $a_{AB}/a_{BB}$ decreases below $\approx 10$, the dimer-dimer interaction changes from attractive to repulsive and the population-balanced AABB and AAABBB clusters break into AB dimers. Calculating the AAABBB hexamer energy just below this threshold, we find an effective three-dimer repulsion which may have important implications for the many-body problem, particularly for observing liquid and supersolid states of dipolar dimers in the bilayer geometry. The population-imbalanced ABB trimer, ABBB tetramer, and AABBB pentamer remain bound beyond the dimer-dimer threshold. In the dipolar model, they break up at $a_{AB}\approx 2 a_{BB}$ where the atom-dimer interaction switches to repulsion.
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Submitted 7 April, 2020; v1 submitted 5 November, 2019;
originally announced November 2019.
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Universal Short Range Correlations in Bosonic Helium Clusters
Authors:
Betzalel Bazak,
Manuel Valiente,
Nir Barnea
Abstract:
Short-range correlations in bosonic Helium clusters, composed of $^4$He atoms, are studied utilizing the generalized contact formalism. The emergence of universal $n$-body short range correlations is formulated and demonstrated numerically via Monte Carlo simulations. The values of the $n$-particle contacts are evaluated for $n\le5$. In the thermodynamic limit, the two-body contact is extracted fr…
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Short-range correlations in bosonic Helium clusters, composed of $^4$He atoms, are studied utilizing the generalized contact formalism. The emergence of universal $n$-body short range correlations is formulated and demonstrated numerically via Monte Carlo simulations. The values of the $n$-particle contacts are evaluated for $n\le5$. In the thermodynamic limit, the two-body contact is extracted from available experimental measurements of the static structure factor of liquid $^4$He at high momenta, and found in a good agreement with the value extracted from our calculations.
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Submitted 23 January, 2020; v1 submitted 31 January, 2019;
originally announced January 2019.
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Efimov physics beyond three particles
Authors:
Betzalel Bazak
Abstract:
Efimov physics originally refers to a system of three particles. Here we review recent theoretical progress seeking for manifestations of Efimov physics in systems composed of more than three particles. Clusters of more than three bosons are tied to each Efimov trimer, but no independent Efimov physics exists there beyond three bosons. The case of a few heavy fermions interacting with a lighter at…
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Efimov physics originally refers to a system of three particles. Here we review recent theoretical progress seeking for manifestations of Efimov physics in systems composed of more than three particles. Clusters of more than three bosons are tied to each Efimov trimer, but no independent Efimov physics exists there beyond three bosons. The case of a few heavy fermions interacting with a lighter atom is also considered, where the mass ratio of the constituent particles plays a significant role. Following Efimov's study of the (2+1) system, the (3+1) system was shown to have its own critical mass ratio to become Efimovian. We show that the (4+1) system becomes Efimovian at a mass ratio which is smaller than its sub-systems thresholds, giving a pure five-body Efimov effect. The (5+1) and (6+1) systems are also discussed, and we show the absence of 6- and 7-body Efimov physics there.
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Submitted 8 January, 2020; v1 submitted 30 December, 2018;
originally announced December 2018.
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Four-Body Scale in Universal Few-Boson Systems
Authors:
Betzalel Bazak,
Johannes Kirscher,
Sebastian König,
Manuel Pavón Valderrama,
Nir Barnea,
Ubirajara van Kolck
Abstract:
The role of an intrinsic four-body scale in universal few-boson systems is the subject of active debate. We study these systems within the framework of effective field theory. For systems of up to six bosons we establish that no four-body scale appears at leading order (LO). However, we find that at next-to-leading (NLO) order a four-body force is needed to obtain renormalized results for binding…
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The role of an intrinsic four-body scale in universal few-boson systems is the subject of active debate. We study these systems within the framework of effective field theory. For systems of up to six bosons we establish that no four-body scale appears at leading order (LO). However, we find that at next-to-leading (NLO) order a four-body force is needed to obtain renormalized results for binding energies. With the associated parameter fixed to the binding energy of the four-boson system, this force is shown to renormalize the five- and six-body systems as well. We present an original ansatz for the short-distance limit of the bosonic $A$-body wave function from which we conjecture that new $A$-body scales appear at N$^{A-3}$LO. As a specific example, calculations are presented for clusters of helium atoms. Our results apply more generally to other few-body systems governed by a large scattering length, such as light nuclei and halo states, the low-energy properties of which are independent of the detailed internal structure of the constituents.
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Submitted 11 April, 2019; v1 submitted 2 December, 2018;
originally announced December 2018.
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Stable p-wave resonant two-dimensional Fermi-Bose dimers
Authors:
Betzalel Bazak,
Dmitry S. Petrov
Abstract:
We consider two-dimensional weakly-bound heterospecies molecules formed in a Fermi-Bose mixture with attractive Fermi-Bose and repulsive Bose-Bose interactions. Bosonic exchanges lead to an intermolecular attraction, which can be controlled and tuned to a p-wave resonance. Such attractive fermionic molecules can be realized in quasi-two-dimensional ultracold isotopic or heteronuclear mixtures. We…
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We consider two-dimensional weakly-bound heterospecies molecules formed in a Fermi-Bose mixture with attractive Fermi-Bose and repulsive Bose-Bose interactions. Bosonic exchanges lead to an intermolecular attraction, which can be controlled and tuned to a p-wave resonance. Such attractive fermionic molecules can be realized in quasi-two-dimensional ultracold isotopic or heteronuclear mixtures. We show that they are stable with respect to the recombination to deeply-bound molecular states and with respect to the formation of higher-order clusters (trimers, tetramers, etc.)
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Submitted 29 December, 2018; v1 submitted 18 July, 2018;
originally announced July 2018.
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Energy of N two-dimensional bosons with zero-range interactions
Authors:
Betzalel Bazak,
Dmitry S. Petrov
Abstract:
We derive an integral equation describing $N$ two-dimensional bosons with zero-range interactions and solve it for the ground state energy $B_N$ by applying a stochastic diffusion Monte Carlo scheme for up to 26 particles. We confirm and go beyond the scaling $B_N\propto 8.567^N$ predicted by Hammer and Son [Phys. Rev. Lett. {\bf 93}, 250408 (2004)] in the large-$N$ limit.
We derive an integral equation describing $N$ two-dimensional bosons with zero-range interactions and solve it for the ground state energy $B_N$ by applying a stochastic diffusion Monte Carlo scheme for up to 26 particles. We confirm and go beyond the scaling $B_N\propto 8.567^N$ predicted by Hammer and Son [Phys. Rev. Lett. {\bf 93}, 250408 (2004)] in the large-$N$ limit.
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Submitted 20 February, 2018; v1 submitted 7 November, 2017;
originally announced November 2017.
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Mass-imbalanced fermionic mixture in a harmonic trap
Authors:
Betzalel Bazak
Abstract:
The mass-imbalanced fermionic mixture is studied, where $N\le5$ identical fermions interact resonantly with an impurity, a distinguishable atom. The shell structure is explored, and the physics of a dynamic light-impurity is shown to be different from that of the static heavy-impurity case. The energies in a harmonic trap at unitarity are calculated and extrapolated to the zero-range limit. In doi…
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The mass-imbalanced fermionic mixture is studied, where $N\le5$ identical fermions interact resonantly with an impurity, a distinguishable atom. The shell structure is explored, and the physics of a dynamic light-impurity is shown to be different from that of the static heavy-impurity case. The energies in a harmonic trap at unitarity are calculated and extrapolated to the zero-range limit. In doing so, the scaling factor of the ground state, as well as of a few excited states, is calculated. In the $2 \le N \le 4$ systems, pure $(N+1)$ Efimov states exist for large enough mass ratio. However, no sign for a six-body Efimov state in the $(5+1)$ system is found in the mass ratio explored, $M/m \le 12$.
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Submitted 17 August, 2017; v1 submitted 18 June, 2017;
originally announced June 2017.
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Five-Body Efimov Effect and Universal Pentamer in Fermionic Mixtures
Authors:
Betzalel Bazak,
Dmitry S. Petrov
Abstract:
We show that four heavy fermions interacting resonantly with a lighter atom (4+1 system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding 2+1 and 3+1 thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non- Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal…
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We show that four heavy fermions interacting resonantly with a lighter atom (4+1 system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding 2+1 and 3+1 thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non- Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal 4+1 pentamer state, which continues the series of the 2+1 trimer predicted by Kartavtsev and Malykh and 3+1 tetramer discovered by Blume. We also show that the effective-range correction for the light-heavy interaction has a strong effect on all these states and larger effective ranges increase their tendency to bind.
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Submitted 26 February, 2017; v1 submitted 29 December, 2016;
originally announced December 2016.
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Effective Field Theory for Few-Boson Systems
Authors:
Betzalel Bazak,
Moti Eliyahu,
Ubirajara van Kolck
Abstract:
We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of th…
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We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for 4He atomic systems, and show that the results are in surprisingly good agreement with those of sophisticated 4He-4He potentials. Potential implications for the convergence of the EFT expansion are discussed.
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Submitted 3 November, 2016; v1 submitted 6 July, 2016;
originally announced July 2016.
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Generalized nuclear contacts and the nucleon's momentum distributions
Authors:
Ronen Weiss,
Betzalel Bazak,
Nir Barnea
Abstract:
The general nuclear contact matrices are defined, taking into consideration all partial waves and finite-range interactions, extending Tan's work for the zero range model. The properties of these matrices are discussed and the relations between the contacts and the one-nucleon and two-nucleon momentum distributions are derived. Using these relations, a new asymptotic connection between the one-nuc…
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The general nuclear contact matrices are defined, taking into consideration all partial waves and finite-range interactions, extending Tan's work for the zero range model. The properties of these matrices are discussed and the relations between the contacts and the one-nucleon and two-nucleon momentum distributions are derived. Using these relations, a new asymptotic connection between the one-nucleon and two-nucleon momentum distributions, describing the two-body short-range correlations in nuclei, is obtained. Using available numerical data, we extract few connections between the different contacts and verify their relations to the momentum distributions. The numerical data also allows us to identify the main nucleon momentum range affected by two-body short-range correlations. Utilizing these relations and the numerical data, we also verify a previous independent prediction connecting between the Levinger constant and the contacts. This work provides an important indication for the relevance of the contact formalism to nuclear systems, and should open the path for revealing more useful relations between the contacts and interesting quantities of nuclei and nuclear matter.
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Submitted 24 March, 2015;
originally announced March 2015.
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Photoassociation of Universal Efimov Trimers
Authors:
Betzalel Bazak,
Nir Barnea
Abstract:
In view of recent experiments in ultracold atomic systems, the photoassociation of Efimov trimers, composed of three identical bosons, is studied utilizing the multipole expansion. We study both the normal hierarchy case, where one-body current is dominant, and the strong hierarchy case, relevant for photoassociation in ultracold atoms, where two-body current is dominant. For identical particles i…
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In view of recent experiments in ultracold atomic systems, the photoassociation of Efimov trimers, composed of three identical bosons, is studied utilizing the multipole expansion. We study both the normal hierarchy case, where one-body current is dominant, and the strong hierarchy case, relevant for photoassociation in ultracold atoms, where two-body current is dominant. For identical particles in the normal hierarchy case, the leading contribution comes from the r^2 s-mode operator and from the quadrupole d-mode operator. The s-mode reaction is found to be dominant at low temperature, while as the temperature increases the d-mode becomes as significant. For the strong hierarchy case, the leading contribution comes from a 2-body s-wave δ operator. In both cases log periodic oscillations are found in the cross section. For large but finite scattering length the amplitude of the oscillations becomes larger in comparison to infinite scattering length case. We apply our theory to photoassociation of 7Li ultracold atoms and show a good fit to the available experimental results.
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Submitted 20 July, 2015; v1 submitted 25 February, 2015;
originally announced February 2015.
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Experimental evaluation of the nuclear neutron-proton contact
Authors:
Ronen Weiss,
Betzalel Bazak,
Nir Barnea
Abstract:
The nuclear neutron-proton contact is introduced, generalizing Tan's work, and evaluated from medium energy nuclear photodisintegration experiments. To this end we reformulate the quasi-deuteron model of nuclear photodisintegration and establish the bridge between the Levinger constant and the contact. Using experimental evaluations of Levinger's constant we extract the value of the neutron-proton…
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The nuclear neutron-proton contact is introduced, generalizing Tan's work, and evaluated from medium energy nuclear photodisintegration experiments. To this end we reformulate the quasi-deuteron model of nuclear photodisintegration and establish the bridge between the Levinger constant and the contact. Using experimental evaluations of Levinger's constant we extract the value of the neutron-proton contact in finite nuclei and in symmetric nuclear matter. Assuming isospin symmetry we propose to evaluate the neutron-neutron contact through measurement of photonuclear spin correlated neutron-proton pairs.
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Submitted 30 November, 2014; v1 submitted 12 May, 2014;
originally announced May 2014.
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Log-Periodic Oscillations in the Photo Response of Efimov Trimers
Authors:
Betzalel Bazak,
Nir Barnea
Abstract:
The photoassociation of Efimov trimer, composed of three identical bosons, is studied utilizing the multipole expansion. For identical particles the leading contribution comes from the r^2 s-mode operator and from the quadrupole d-mode operator. Log-periodic oscillations are found in the photoassociation response function, both near the energy threshold for the leading s-wave reaction, and in the…
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The photoassociation of Efimov trimer, composed of three identical bosons, is studied utilizing the multipole expansion. For identical particles the leading contribution comes from the r^2 s-mode operator and from the quadrupole d-mode operator. Log-periodic oscillations are found in the photoassociation response function, both near the energy threshold for the leading s-wave reaction, and in the high frequency tail for all partial waves.
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Submitted 31 October, 2013;
originally announced October 2013.
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Photo Reactions with Universal Trimers
Authors:
Betzalel Bazak,
Nir Barnea
Abstract:
Considering one-body and two-body currents, we study the photoassociation and photodissociation of universal bosonic trimers. Analyzing the relative importance of these currents we identify two physical scenarios (i) Normal hierarchy, where naive power counting holds and the one-body current dominates, and (ii) Strong hierarchy, where the one-body current is suppressed. For both scenarios we obser…
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Considering one-body and two-body currents, we study the photoassociation and photodissociation of universal bosonic trimers. Analyzing the relative importance of these currents we identify two physical scenarios (i) Normal hierarchy, where naive power counting holds and the one-body current dominates, and (ii) Strong hierarchy, where the one-body current is suppressed. For both scenarios we observe that at the high frequency tail, the response function exhibits log periodic oscillations in transition to or from any continuum state regardless of the reaction partial wave channel. In contrast, near threshold log periodic oscillations appear only in the leading $s$-wave components. These oscillations are the fingerprints of universal Efimov physics. We discuss the relevance of this effect to contemporary experiments in ultracold atoms.
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Submitted 26 February, 2014; v1 submitted 19 May, 2013;
originally announced May 2013.
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Multipole Analysis of Radio-Frequency Reactions in Ultra-Cold Atoms
Authors:
Betzalel Bazak,
Evgeny Liverts,
Nir Barnea
Abstract:
Using the multipole expansion we analyze photo induced reactions in an ultra-cold atomic gas composed of identical neutral bosons. While the Frank-Condon factor dominates the photo induced spin-flip reactions, we have found that for frozen-spin process where the atomic spins are conserved the reaction rate is governed by the monopole $r^2$ and the quadrupole terms. Consequently, the dependence of…
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Using the multipole expansion we analyze photo induced reactions in an ultra-cold atomic gas composed of identical neutral bosons. While the Frank-Condon factor dominates the photo induced spin-flip reactions, we have found that for frozen-spin process where the atomic spins are conserved the reaction rate is governed by the monopole $r^2$ and the quadrupole terms. Consequently, the dependence of the frozen-spin reaction rate on the photon wave number $k$ acquires an extra $k^4$ factor in comparison to the spin-flip process. Comparing the relative strength of the $r^2$ and quadrupole modes in dimer photoassociation we predict that the mutual importance of these two modes changes with temperature and scattering length.
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Submitted 20 September, 2012; v1 submitted 25 July, 2012;
originally announced July 2012.
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Gapless Superfluid in $SU(2n+1)$ Fermions Systems
Authors:
Betzalel Bazak,
Nir Barnea
Abstract:
We investigate the generalized Hubbard model of $(2n+1)$ Fermion species interacting via a symmetric contact attraction potential. We prove that the ground state of such system is a gapless superfluid, where a full Fermi surface coexists with a superfluid. Moreover, doing so we prove the existence of a free mode in a strongly interacting system, regardless of the potential strength. This proof hol…
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We investigate the generalized Hubbard model of $(2n+1)$ Fermion species interacting via a symmetric contact attraction potential. We prove that the ground state of such system is a gapless superfluid, where a full Fermi surface coexists with a superfluid. Moreover, doing so we prove the existence of a free mode in a strongly interacting system, regardless of the potential strength. This proof holds at the mean field level. A Grassmannian Gaussian integration technique is used to deal with the problem. Our predictions may be relevant to future high-spin cold atoms experiments.
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Submitted 8 April, 2011;
originally announced April 2011.