Universal $p$-wave tetramers in low-dimensional fermionic systems with three-body interaction

Inspired by the narrow Feshbach resonance in systems with the two-body interaction, we propose the two-channel model of three-component fermions with the three-body interaction that takes into account the finite-range effects in low dimensions. Within this model, the $p$-wave Efimov-like effect in the four-body sector is predicted in fractional dimensions above 1D. The impact of the finite-range interaction on the formation of the four-body bound states in $d=1$ is also discussed in detail.


I. INTRODUCTION
A few-body quantum physics is known to be drastically different from its classical counterpart.The most famous example is the Efimov effect [1][2][3] realizing, as an infinite tower of three-body bound states, in the threeboson system at resonant two-body interaction.Similar behavior is found [4,5] in the three-body sector of twocomponent fermions.In the fermionic case, the Efimov effect occurs in the odd-wave (typically p-wave) channels on the arbitrary intervals of particle mass ratios.Being well-established in three dimensions, this effect, in general, has a limitation on spacial dimensionality [6].Even for non-equal particle masses the universal three-body bound states emerge in the non-integer dimensions between 2 < d < 4. Exactly in 2D, the super Efimov effect was found [7] in a system of the spin-polarized fermions with resonant p-wave interaction.Attempt to find universal Efimov-like behavior in lower dimensions necessarily requires the mixed-dimension geometries [8], or higher-order few-body interactions, namely, four-body [9] in 1D and three-body [10] in 2D.In the latter cases, the system parameters should be highly fine-tuned to provide the resonant highest-order interaction and vanishing all lower-order couplings.
In recent years, research in this field has shifted to fractional dimensions.Particularly, interesting analytical results for a model with zero-range two-body potential were obtained in Refs.[11][12][13].The existence of the Efimov-like bound states with characteristic scaling low in the four-body sector was reported in [14] for a bosonic system with the resonant three-body interaction in noninteger (1 < d < 2) dimensions.Independently of the spatial dimension, for particles interacting through the realistic two-body potential, the three-body forces typically appear [15][16][17] as the effective induced interaction.
The latter becomes dominant [18] when the two-body potential is tuned to a zero crossing.For the effective field theories above the upper critical dimension, the introduction of three-body terms in the Lagrangians is mandatory to provide the UV completeness of theories.In context of the Efimov physics, such a completion was proposed in seminal papers by Bedaque et al. [19,20], and more recently in Ref. [21] for a model of 1D spin-polarized fermions with the contact two-body p-wave interaction.A start for active investigation of 1D few-and many-body systems with three-body interaction should be identified with the publication of several articles [22][23][24][25], describing general properties of SU (3) fermions and (mostly) three-body states of bosons.Similarly to 2D systems of particles interacting with each other by a two-body δ-(pseudo)potential [26], the three-body contact interaction exhibits the quantum scale anomaly in 1D, which predetermines universal properties and thermodynamics of diluted bosons [27,28] and fermions [29,30].In the thermodynamic limit, the three-body interaction is responsible for the formation of the quantum droplet state [31,32] in the 1D system of bosons and for the crossover transition [33] from tightly bound vacuum trimers to the so-called Cooper triples.
The present work deals with the two-channel model of three-component fermionic particles with unequal masses that interact through the three-body forces in fractional dimension.The simplified version of the proposed setup in harmonic trapping potential was studied in [34] using the high-temperature series at finite densities of constituents.In particular, we are interested in finding regions in the parameter space for the emergence of the four-body bound states and revealing their peculiarities.Previously, important aspects of a few-body physics of SU (3) fermions with the contact three-body interaction in 1D were discussed by McKenney and Drut in Ref. [35].In this context, we fully complement the four-body sector by revealing the bound states for a system with unequal masses of different fermionic species and generalize these results to a case of finite interaction ranges and higher dimensions.

II. MODEL AND RENORMALIZATION
We consider a model of three-component Galileaninvariant fermions with different masses m σ (σ = 1, 2, 3) in d-dimensional space.Any two-body interactions between particles are assumed to be suppressed and only the three-body potential [15] is switched on between fermions of different sorts.In order to take into account the finite-range effects of the three-body interaction, we consider a two-channel model that is very similar to that of the narrow Feshbach resonance but in the three-body sector where ε σ (p) = p 2 2mσ , p = dp (2π) d and M = m 1 + m 2 + m 3 is the mass of composite fermion; g and δν Λ are the coupling and the detuning.The latter depends on the ultraviolet (UV) cutoff Λ that restricts the upper integration limit in the above momentum integrals.The anti-commutators of the field operators are fixed as follows ) and zero all other pairs.Before we proceed with the four-body bound and scattering states, let us first consider the simplest interacting system of three fermions.The latter solution sets up the correct renormalization of the cutoff-dependent detuning δν Λ and the relation of our model to the system with three-body δ-like interaction [14].The three-body state that describes particles with zero total momentum can be written down as follows: where |0⟩ is the vacuum state, and amplitudes A and A p1,p2,p3 are subject to the Schrödinger equation.Denoting energy of the system by E, we have The non-trivial solution to these coupled equations for the bound states ϵ g (negative Es) reads where for latter convenience we have introduced an auxiliary function here and below Π 23 First, let us discuss the limit of broad resonance.Keeping energy fixed while setting g → ∞, we can identify the three-body bare coupling constant g −1 3,Λ = −δν Λ /g 2 which absorbs the UV divergence of the integral in D(E).This procedure relates the observable three-body coupling g 3 to the three-body bound state energy ϵ ∞ at the broad resonance Restoring the finite magnitude of g and repeating the above calculation procedure, one obtains the transcendental equation for the composite fermion bound state energy at narrow resonance There is always a single solution to this equation for positive g 3 s.At large coupling g it recovers the broadresonant result ϵ ∞ , while in a case of small gs asymptotically behaves as ϵ g ≈ −g 2 /g 3 .The unitarity limit, ϵ g = 0, is reached at infinite g 3 for all dimensions above d = 1.The one-dimensional limit of Eq. 2.8 is also welldefined although for the three-body coupling (2.7) is not.From Eq. (2.9) it can be readily concluded that Finally, the Galilean invariance allows us to generalize the above results to a non-zero composite fermion momentum p: the only modification is the energy shift E → E − p 2 2M in Eq. (2.6).

III. FOUR-BODY PROBLEM
The wave function of an arbitrary four-body (composite fermion+f 1 -atom) state with zero center-of-mass momentum reads where 1 −p2 is anti-symmetric function of first two arguments.By acting with the Hamiltonian H on the ansatz (3.10) one obtains the system of two coupled equations for the wave-function with eigenvalue E. For negative energies E = ϵ 4 (four-body bound states) Eqs.(3.11), (3.12) can be readily transformed into a single integral equation for the function with the shorthand notation for function Because of the Fermi statistics, the four-body bound states can potentially occur only in the odd-wave channels.The one requiring the shallowest potential well is the p-wave with the wave function of the following form B p = (np/p)B p (here B p depends on modulus p with n being a unit vector).As it is expected for states with the orbital quantum number l = 1, this is not a single wave function, but d mutually-orthogonal states parameterized by different n i satisfying the condition n i n j = δ ij .Plugging the p-wave harmonics back into Eq.(3.13) and performing the d-dimensional hyperangle integration, we obtained the one-dimensional integral equation (see Appendix) for amplitude p .

A. d > 1
Before discussing numerical results, let us consider the limit of broad resonance g → ∞ and disappearing threebody bound state g 3 → ∞.In the case of bosons, this limit is characterized by the universal log-periodic wave function that signals an emergence [14] of the Efimovlike effect in the four-body sector.A very similar situation should be observed in our system.However, its fermionic nature (and, consequently, non-zero total angular momentum) demands the total mass of the second and third particles M 23 should be small enough in comparison to m 1 in order to provide the effective attracting potential for trapping of four particles.The scaling limit (g, g 3 → ∞) supposes the power-law behavior B p = 1/p 1−d/2±η of the wave function.Then, the integral equation transforms into the algebraic one on parameter η with Γ(z) and 2 F 1 (a, b; c; z) being the gamma and hypergeometric functions [36], respectively.Truly imaginary solutions η = iη 0 determine a region (see Fig. the four-body p-wave Efimov-like effect in the considered system, with the universal ratio of energy levels = e 2π/η0 , (n ≫ 1).
Note that η 0 depends only on the spatial dimension and mass ratio m 1 /M 23 .The asymptotic form of the appropriate wave functions is a linear combination of two partial solutions where Λ 0 is an arbitrary momentum scale related to the lowest Efimov state.An emergent discrete scale invariance of the asymptotic expression (3.16) for the wave function in the scaling region is the characteristic feature of the Efimov-like effects.To verify the above predictions about the behavior of our model in the four-particle sector, we have numerically solved the integral equation (3.13) in the p-wave channel by discretizing its kernel.The parameters of the system were specially chosen to reveal the Efimov behavior as simple as possible.In particular, we put d = 1.5 and mass ratios M 23 /m 1 = 1/50 small enough to provide a strong induced attractive potential between two f 1 -atoms.The three-body binding energy is sent to zero ϵ g → 0, and coupling g = m1m2m3 , and the numerical prefactors for the first five levels are gathered in table I.For the identification of the level, the wave functions were  also calculated (see Fig. 2).In the third column, we show the ratio of neighboring eigenenergies (3.15) which should tend to e 2π/η0 = 9.290926... at large n.It is seen, however, that already for the fourth excited state this quantity ϵ 4 deviates from the universal value by a tenth of a percent.

B. d = 1
The one-dimensional case is of particular interest due to potential realization in experiments and/or numerical simulations.According to the previous section, there is no four-body Efimov-like effect in the considered system in 1D, but a strong imbalance between masses of atoms may lead to the formation of the bound states.Projection onto p-wave states in the four-body Shrödinger equation (3.13) in d = 1 reduces to choosing the odd B p = −B −p solutions.An appropriate eigenvalues ϵ 4 at finite ranges (g < ∞) of the interaction potential, should be deeper than the three-body bound-states ϵ g .Therefore, it is natural to pick dimensionless units in a way that the four-body energies are measured in units of ϵ g , and the effective range is determined by closeness of the three-body bound state energy to its broad-resonance value γ = ln ϵ∞ ϵg .Then γ = 0 corresponds to zero range r 0 = 0 case, while γ > 0 refers to the model with narrow resonance.Introducing momentum scale p 0 such that m1+M 2m1M p 2 0 = |ϵ g |, one can readily rewrite Eq. 3.13 in dimensionless units in 1D with λ = ϵ 4 /ϵ g being dimensionless eigenvalue and B p0p = b p .Recall that due to the fermionic nature of the considered system, solutions for b p should be searched on a class of odd functions.The results of numerical diagonalization are presented in Fig. 3. Since, both m 2 and m 3 enters Eq. (3.17)only in combination M 23 , we can freely take them equal to each other m 2 = m 3 .Our calculations demonstrate that even at the broad resonance γ = 0, which is the most favorable limit for the existence of the four-body bound states, the first level emerges at m 1 /m 2 ≈ 2.5 (the second and the third ones at ≈ 13.0 and ≈ 31.5, respectively).With an increase of the effective range, these values shift towards larger m 1 /m 2 ratios.Particularly, when ϵ g = ϵ ∞ /e, the first three tetramers are found at m 1 /m 2 ≈ 5.5, m 1 /m 2 ≈ 32.5 and m 1 /m 2 ≈ 78.5.We have also obtained the fourbody ground-state wave functions in 1D (see Fig. 4) with m 1 /m 2 = 15.5 at various effective ranges.For comparison, the first two four-body eigenstates are depicted in Fig. 5 for zero γ = 0 and non-zero γ = 0.5 effective ranges of the three-body interaction.Figure 3 (especially zero-range case) reveals an intrinsic dependence of the eigenvalues at the large mass imbalance.Indeed, integral in Eq. 3.17 possesses a nonintegrable singularity at q = m1 M p when m 1 /M = 1.This fact together with parity properties of the wave function b p allows for the asymptotic (with logarithmic precision) determination of the four-body bound states Constant C γ should be determined for every four-body energy level separately by imposing the solution to Eq. (3.18) to be consistent with the large-m1 M23 tail of the eigenvalue behavior found numerically.From Eq. (3.18) one immediately recognizes the linear dependence of ϵ 4 on m 1 /m 2,3 (in the limit m 1 /m 2,3 ≫ 1) at broad resonance.Although the asymptotic solution (3.18) qualitatively explains the eigenvalue behavior at large mass imbalance, the logarithmic accuracy does not allow the quantitative description of the numerically calculated curves in Fig. 3 at intermediate mass ratios.

IV. SUMMARY
In conclusion, we have proposed for the first time the effective model of contact three-body interaction that includes the effects of finiteness of the potential range.Applying this effective description to the three-component Fermi system with the suppressed two-body interactions in the fractional dimension above d = 1, we have predicted the emergence of the Efimov-like physics in the p-wave channel of the four-body sector.An analytic estimations in the scaling limit are supported by numerically exact calculations for finite effective ranges at unitarity.The detailed analysis of the one-dimensional problem revealed the necessary conditions for the occurrence of negative eigenvalues in the four-body spectrum both in the case of broad and narrow resonances.Particularly, it is shown that depending on mass ratios of fermions with the three-body interaction, one can in principle observe an arbitrarily large number of the tetramer levels.The effect is suppressed for the non-zero effective ranges towards larger mass imbalance.case the above equation reduces, after passing to dimensionless variables, to Eq. (3.17) in the main text.
FIG.1: Region, where the p-wave Efimov-like effect in the four-body sector of the three-component Fermi system with the three-body interaction emerges.

3 = 0 .
scale r 0 which is related to the effective range.Actually, this is the only dimensionful parameter at unitary limit g −1 The four-body p-wave energies ϵ (n) 4 are measured in units of m1+M 2m1M r 2 0

(n) 4 (in units of m 1 +M 2m 1 M r 2 0 )FIG. 2 :
FIG.2:The ground state and the first few excited states wave functions B (n) p (unnormalized) of the four-body problem in the p-wave channel.