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arXiv:1712.10209

Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions

Simon Becker, Alessandro Michelangeli, Andrea Ottolini

Published Dec 29, 2017• 3 citations
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Abstract

We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.

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Bound States of $$2+1$$ Fermionic Trimers on Lattice at Strong Couplings

J. I. Abdullaev, Ahmad Khalkhuzhaev, Shokhrukh Yu. Kholmatov

Abstract In this paper, we investigate the bound states of $$2+1$$ 2 + 1 fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator $$H_{\gamma ,\lambda }(K),$$ H γ , λ ( K ) , focusing on energies below the continuum and within its gap. Depending on the quasi-momentum K , we show that if the mass ratio $$\gamma >0$$ γ > 0 between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling $$\lambda >0.$$ λ > 0 . Conversely, if $$\gamma $$ γ exceeds this threshold, $$H_{\gamma ,\lambda }(K)$$ H γ , λ ( K ) admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as $$\lambda \rightarrow +\infty .$$ λ → + ∞ . For $$K=0,$$ K = 0 , the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.

arXiv:2502.01099•Annales Henri Poincaré•Jun 25, 2026•0 citations

Binding properties of the (2+1)-fermion system with zero-range interspecies interaction

Alessandro Michelangeli, Christian Schmidbauer

We study analytically and numerically the binding properties, in particular the ground state, of the so-called $(2+1)$-fermion system, which is a three-dimensional system of two identical fermions interacting with a third particle of different species through a zero-range interaction. We model the system with a specific self-adjoint point interaction Hamiltonian recently constructed in the mathematical literature. First we characterize the internal symmetry of the bound states in the attractive case. Then we show that the system confines in a precise regime of masses and that its ground-state energy stays finite as the mass becomes close to the critical point of the collapse of the system.

Physical Review A•May 6, 2013•17 citations

Critical mass ratio and phase transition in three-particle lattice systems: comparison of bosonic and fermionic cases

J. I. Abdullaev, A.I. Eshniyоzov, M. V. Dolgopolov

We study three-particle Schrödinger operators on the two-dimensional lattice Z 2 and show that a critical mass ratio γ c ≈ 2.75194 governs the existence of a bound trimer in the fermionic 2 + 1 configuration (two identical fermions and a third particle). For γ < γ c there is a topological prohibition (Pauli suppression) of a three-body bound state, whereas for γ > γ c a doubly degenerate eigenvalue emerges below the essential spectrum with the strong-coupling asymptotics z(γ, λ) = −λ+e 0 (γ)+O(λ −1 ). Within a unified framework based on the Birman–Schwinger principle and strong-coupling asymptotic analysis, we compare this behaviour with the bosonic case of three identical particles, where two bound states exist below the essential spectrum and the ground-state energy satisfies z 1 s (µ) = −3µ + C 2 + O(µ −1 ). The resulting second-order phase transition with respect to the mass ratio γ is relevant for the design of experiments on fermionic trimers in optical lattices and for modelling excitonic complexes and defect-bound states in two-dimensional nanomaterials, where the critical value γ c serves as a design guideline for the observability of three-body bound states. We also outline a modified three-particle lattice model with two competing interaction channels, for which the Birman–Schwinger analysis naturally leads to a Landau-type scenario of a first-order phase transition in the space of trimer bound states. In the bosonic case we prove a strong-coupling theorem describing the existence and asymptotics of trimer bound states, while in the fermionic 2+1 case we establish a spectral phase-transition theorem that identifies an explicit critical mass ratio γ c separating the trimer and non-trimer regimes.

Nanosystems Physics Chemistry Mathematics•Apr 30, 2026•0 citations

The point spectrum of the three-particle Schrödinger operator for a system comprising two identical bosons and one fermion on Z

Zahriddin Ishkobilovich Muminov, Актамова Васила Уктамовна

We consider the Hamiltonian of a system of three quantum particles (two identical bosons and a fermion) on the one-dimensional lattice interacting by means of zero-range attractive or repulsive potentials. We investigate the point spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ T which possesses infinitely many eigenvalues depending on repulsive or attractive interactions, under the assumption that the bosons in the system have infinite mass

Nanosystems Physics Chemistry Mathematics•Aug 30, 2024•3 citations

Absence of a four-body Efimov effect in the 2+2 fermionic problem

Shimpei Endo, Yvan Castin

In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant $s$-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this $2+2$ fermionic system can exhibit (as the $3+1$ fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where $s$ is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of $s$ are such that $M(s)$ has a zero eigenvalue; here the operator $M(s)$, that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of $M(s)$, analytical and for an arbitrary imaginary $s$ for the continuous spectrum and numerical and limited to $s=0$ and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of $M(0)$ crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of $M(s)$ has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the $2+2$ fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-$1/2$ Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary $s$ axis of $s$ times the logarithmic derivative of the determinant of $M(s)$ summed over all angular momenta. The conjectured value is in contradiction with the experimental results.

arXiv:1507.05580•Physical Review A•Nov 24, 2015•30 citations

Three two-component fermions with contact interactions: correct formulation and energy spectrum

O. I. Kartavtsev, A. V. Malykh

Properties of two identical particles of mass $m$ and a distinct particle of mass $m_1$ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum $L$ and parity $P$. For the unambiguous formulation of the problem in the interval $μ_r(L^P) < m/m_1 \le μ_c(L^P)$ ($μ_r(1^-) \approx 8.619$ and $μ_c(1^-) \approx 13.607$, $μ_r(2^+) \approx 32.948$ and $μ_c(2^+) \approx 38.630$,~etc.) in each $L^P$ sector an additional parameter $b$ determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on $m/m_1$ and $b$ in the sector of angular momentum and parity $L^P$ is calculated for $L \le 5$ and analysed with the aid of a simple model. A number of the bound states for each $L^P$ sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters $m/m_1$ and $b$.

arXiv:1904.04943•Apr 9, 2019•1 citation
We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.
Bound States of 2+12+12+1 Fermionic Trimers on Lattice at Strong Couplings
Abstract In this paper, we investigate the bound states of 2+12+12+1 2 + 1 fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator Hγ,λ(K),H_{\gamma ,\lambda }(K),Hγ,λ​(K), H γ , λ ( K ) , focusing on energies below the continuum and within its gap. Depending on the quasi-momentum K , we show that if the mass ratio γ>0\gamma >0γ>0 γ > 0 between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling λ>0.\lambda >0.λ>0. λ > 0 . Conversely, if γ\gamma γ γ exceeds this threshold, Hγ,λ(K)H_{\gamma ,\lambda }(K)Hγ,λ​(K) H γ , λ ( K ) admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as λ→+∞.\lambda \rightarrow +\infty .λ→+∞. λ → + ∞ . For K=0,K=0,K=0, K = 0 , the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.
We study analytically and numerically the binding properties, in particular the ground state, of the so-called (2+1)(2+1)(2+1)-fermion system, which is a three-dimensional system of two identical fermions interacting with a third particle of different species through a zero-range interaction. We model the system with a specific self-adjoint point interaction Hamiltonian recently constructed in the mathematical literature. First we characterize the internal symmetry of the bound states in the attractive case. Then we show that the system confines in a precise regime of masses and that its ground-state energy stays finite as the mass becomes close to the critical point of the collapse of the system.
We study three-particle Schrödinger operators on the two-dimensional lattice Z 2 and show that a critical mass ratio γ c ≈ 2.75194 governs the existence of a bound trimer in the fermionic 2 + 1 configuration (two identical fermions and a third particle). For γ < γ c there is a topological prohibition (Pauli suppression) of a three-body bound state, whereas for γ > γ c a doubly degenerate eigenvalue emerges below the essential spectrum with the strong-coupling asymptotics z(γ, λ) = −λ+e 0 (γ)+O(λ −1 ). Within a unified framework based on the Birman–Schwinger principle and strong-coupling asymptotic analysis, we compare this behaviour with the bosonic case of three identical particles, where two bound states exist below the essential spectrum and the ground-state energy satisfies z 1 s (µ) = −3µ + C 2 + O(µ −1 ). The resulting second-order phase transition with respect to the mass ratio γ is relevant for the design of experiments on fermionic trimers in optical lattices and for modelling excitonic complexes and defect-bound states in two-dimensional nanomaterials, where the critical value γ c serves as a design guideline for the observability of three-body bound states. We also outline a modified three-particle lattice model with two competing interaction channels, for which the Birman–Schwinger analysis naturally leads to a Landau-type scenario of a first-order phase transition in the space of trimer bound states. In the bosonic case we prove a strong-coupling theorem describing the existence and asymptotics of trimer bound states, while in the fermionic 2+1 case we establish a spectral phase-transition theorem that identifies an explicit critical mass ratio γ c separating the trimer and non-trimer regimes.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant sss-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this 2+22+22+2 fermionic system can exhibit (as the 3+13+13+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where sss is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of sss are such that M(s)M(s)M(s) has a zero eigenvalue; here the operator M(s)M(s)M(s), that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of M(s)M(s)M(s), analytical and for an arbitrary imaginary sss for the continuous spectrum and numerical and limited to s=0s=0s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of M(0)M(0)M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of M(s)M(s)M(s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the 2+22+22+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-1/21/21/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary sss axis of sss times the logarithmic derivative of the determinant of M(s)M(s)M(s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results.
Properties of two identical particles of mass mmm and a distinct particle of mass m1m_1m1​ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum LLL and parity PPP. For the unambiguous formulation of the problem in the interval μr(LP)<m/m1≤μc(LP)μ_r(L^P) < m/m_1 \le μ_c(L^P)μr​(LP)<m/m1​≤μc​(LP) (μr(1−)≈8.619μ_r(1^-) \approx 8.619μr​(1−)≈8.619 and μc(1−)≈13.607μ_c(1^-) \approx 13.607μc​(1−)≈13.607, μr(2+)≈32.948μ_r(2^+) \approx 32.948μr​(2+)≈32.948 and μc(2+)≈38.630μ_c(2^+) \approx 38.630μc​(2+)≈38.630,~etc.) in each LPL^PLP sector an additional parameter bbb determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on m/m1m/m_1m/m1​ and bbb in the sector of angular momentum and parity LPL^PLP is calculated for L≤5L \le 5L≤5 and analysed with the aid of a simple model. A number of the bound states for each LPL^PLP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1m/m_1m/m1​ and bbb.