PHYSICAL REVIEW A 85, 033633 (2012)

One-dimensional multicomponent with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi

Xi-Wen Guan,1 Zhong-Qi Ma,2 and Brendan Wilson1 1Department of Theoretical , Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China (Received 23 January 2012; published 26 March 2012) We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional κ- component fermions with repulsive and attractive δ-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the κ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You’s result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as κ →∞is the same as for spinless .

DOI: 10.1103/PhysRevA.85.033633 PACS number(s): 03.75.Ss, 03.75.Hh, 02.30.Ik, 34.10.+x

I. INTRODUCTION Although the model was solved long ago by Sutherland [4], solutions of the Fredholm equations for the model are far from The recently established experimental control over the being thoroughly investigated except for the two-component effective spin-spin interaction between atoms is opening up Fermi gas [16]. From a theoretical point of view, finding new avenues for studying spin effects in low-dimensional a general form of the solutions to the Fredholm equations atomic quantum with higher spin symmetries. Fermionic for multicomponent Fermi gases with higher spin symmetry alkaline-earth- atoms display an exact SU(κ) spin sym- imposes a number of challenges. metry with κ = 2I + 1 where I is the nuclear spin [1]. In the present paper, using the method which we recently For example, a recent experiment [2]for171Yb dramatically developed in the analytical study of the 1D two-component realised the model of fermionic atoms with SU(2) ⊗ SU(6) Fermi gas [16], we approximately solve the Fredholm equa- symmetry where spin decouples from its nuclear spin tions of the 1D κ-component fermions with polarization I = 5/2. Such fermionic systems with enlarged SU(κ)spin for the (A) strongly repulsive regime, (B) weakly repulsive symmetry are expected to display a remarkable diversity of regime, (C) weakly attractive regime, and (D) strongly at- new quantum phases and quantum critical phenomena due to tractive regime. We thus obtain the ground-state energy of the rich linear and nonlinear Zeeman effects. The study of the κ-component Fermi gas with polarization that provides low-dimensional cold atomic Fermi gases with higher pseu- a fundamental understanding of the ground-state properties, dospin symmetries has become a new frontier in cold-atom such as diagram, magnetism, and quantum statistical physics. effects. In contrast to the two-component Fermi gas [16], the One-dimensional (1D) quantum Fermi gases with δ- two sets of Fredholm equations for the multicomponent Fermi function interaction are important exactly solvable quantum gas with weakly repulsive and attractive interactions cannot many-body systems and have had tremendous impact in be unified by the density mapping as the interacting strength quantum statistical mechanics. The spin-1/2 Fermi gas with vanishes. The multiple spin degrees of freedom impose subtle arbitrary polarization was solved long ago by Yang [3]using intricacies of quantum statistics in the ground-state properties the Bethe ansatz (BA) hypothesis. Sutherland [4] generalized of the systems. We further study the local pair-correlation the result of the spin-1/2 Fermi gas to 1D multicomponent functions for the κ-component interacting fermions. We find Fermi gas in 1968. The study of multicomponent attractive that the local pair correlation as κ →∞is the same as for Fermi gases was initiated by Yang [5] and by Takahashi spinless bosons. This result is consistent with Yang and You’s [6]. Using Yang and Yang’s method [7] for the case, finding [17] that the energy per particle as κ →∞is the same Takahashi [8] and Lai [9] derived the thermodynamic Bethe as for spinless Bosons. ansatz (TBA) equations for spin-1/2 fermions. In the same fashion, Schlottmann [10] derived the TBA equations for SU(κ) fermions with repulsive and attractive interactions, II. FREDHOLM EQUATIONS respectively. Recently, the thermodynamics of the multicom- ponent Fermi gas was obtained by solving the TBA equations The Hamiltonian for the 1D N-body problem is [3,4] in Ref. [11]. These models with enlarged spin symmetries have received a renewed interest in cold-atom physics [12–15]. 2 N 2  In this paper, we consider 1D κ-component fermions h¯ ∂ H =− + g1D δ(xi − xj ). (1) 2m ∂x2 with repulsive and with attractive δ-function interactions. i=1 i 1i

1050-2947/2012/85(3)/033633(10)033633-1 ©2012 American Physical Society XI-WEN GUAN, ZHONG-QI MA, AND BRENDAN WILSON PHYSICAL REVIEW A 85, 033633 (2012)

It describes N fermions of the same mass m confined to a here c>0 for repulsive interaction and c<0 for attractive 1D system of length L interacting via a δ-function potential. interaction. Following the method used for the two-component There are κ possible hyperfine states |1,|2,...,|κ that Fermi gas [16], we rewrite the Fredholm equations (4) as the fermions can occupy. For an irreducible representation m−1  [κNκ ,(κ − 1)Nκ−1 ,...,2N2 ,1N1 ], the Young diagram has κ = − − − i i+1 rm(k) β0 Km s (k λ)rs(λ)dλ columns with the quantum numbers N = N − N , here | | i s=0 λ >Bs the N i is the numbers of fermions at the ith hyperfine levels  Bm+1 such that N 1  N 2  ··· N κ . This system has SU(κ)spin + K1(k − λ)rm+1(λ)dλ, (6) symmetry and U(1) charge symmetry. The coupling constant −Bm+1 g1D can be expressed in terms of the interaction strength   − 2 where 0 m κ 1. The associating integration boundaries c =−2/a1D as g1D = h¯ c/m where a1D is the effective 1D Bm are determined by the conditions scattering length. Here c>0 for repulsive fermions, and c<0  B for attractive fermions. M m ≡ = r (k)dk, 0    κ − 1, (7) The energy eigenspectrum is given in terms of the quasimo-    L −B { } = N 2 menta ki of the fermions via E = k , which in terms  j 1 j where M = N is the total number of fermions, N = M − − of the function e (x) = (x + ibc/2)/(x − ibc/2) satisfy the 0  1 b M is the number of fermions in the th hyperfine state. Here BA equations [4–6]  Mκ = 0. The ground-state energy E per unit length is given M1   by (1) exp(iki L) = e1 ki − λ ,  α B0 α=1 2 E = k r0(k)dk. (8) − M−1   B0 () − (−1) e1 λα λβ The model has SU(κ) symmetry in the spin sector and β=1 U(1) symmetry in the charge sector. Therefore the quantum M   M+1   numbers of each spin states are conserved. Thus the system () () () (+1) { } =− e2 λ − λ e−1 λ − λ , (2) has κ chemical potentials μ in regard to these conserved α η α δ  η=1 δ=1 numbers N . The ground-state energy (8) is a smooth function of the densities of n = N /L with  = 1,2,...,κ for the = = { ()} where i 1,...,N, α 1,...,M and the parameters λα unbalanced case. In the grand-, we can = − with  1,2,...,κ 1 are the spin rapidities while we denote also get the chemical potentials μ via μ = ∂E/∂n. (0) = (κ) =  =   λα kα, λα 0 and c c/2. The BA quantum numbers = κ−1 − + = Mi j=i (j i 1)Nj+1 with Mκ 0intheaboveBA B. Attractive regime equations (2). In the attractive regime, it is found that complex string solutions of k also satisfy the BA equations [5,6,18]. The A. Repulsive regime j quasimomenta kj may appear as bound states of the m atom The fundamental physics of the model are determined up to length 2,...,κ. A bound state in quasimomentum space by the set of transcendental BA equations (2) which can of length m takes on the form be transformed to generalized Fredholm equations in the m,j = (m−1) + + − | |+ − . The Fredholm equations for repulsive kα λα i(m 1 2j) c O( exp( δL)), (9) and attractive regimes are significantly different. From the BA where j = 1,...,m. The number of bound states with length {k } {λ()}   (m−1) equations (2), the quasimomenta i are real, but all α 1 m κ is denoted as Nm. Its real part is λα .Akα bound (1) are real only for the . For the ground state, the state of the m atom will be accompanied by a λα string of c> − (2) − generalized Fredholm equations for 0aregivenby[4–6] length m 1, a λα string of length m 2, and so on until a  (m−1) B1 λα string of length 1. Each accompanying string state in    (1) (2) (m−1) r0(k) = β0 + K1(k − k )r1(k )dk , (3) λ space, λ space, ...,λ space will share the same real −  −B1 (m 1) Bm−1 part λα ;see[5,6,18] and the second reference in Ref. [11]. rm(k) = K1(k − λ)rm−1(λ)dλ The unpaired atoms have real quasimomenta ki ’s. − Bm−1 From these quasimomentum bound states, the Fredholm Bm equations for the model with an attractive interaction are given − − K2(k λ)rm(λ)dλ by [5,6] −Bm  −  Bm+1 m1 κ Qs + − K1(k λ)rm+1(λ)dλ, (4) ρm(λ) = mβ0 + Ks+m−2r (λ − )ρs ( )d −B + − m 1 r=1 s=r Qs   − =  where 1 m κ 1 and β0 1/(2π), r0(k) is the particle κ Qs quasimomentum distribution function whereas rm(k) with m  + Ks−m(λ − )ρs ( )d , (10) − 1 are the distribution functions for the κ − 1 spin rapidities. s=m+1 Qs The kernel function K(k) is defined as where ρ1(k) is the density distribution function of single 1 c fermions, whereas ρ (k) is the density distribution function K (k) = , (5) m  2π (c/2)2 + k2 for the bound state of the m atom with 1

033633-2 ONE-DIMENSIONAL MULTICOMPONENT FERMIONS WITH ... PHYSICAL REVIEW A 85, 033633 (2012)  = κ total number of fermions is given by N m=1 mNm.The III. ASYMPTOTIC SOLUTIONS OF THE FREDHOLM integration boundaries Qm, characterizing the Fermi points in EQUATIONS each Fermi sea, are determined by A. Strong repulsion  Qm − = ≡ Nm = For the balanced case, i.e., Mm Mm+1 N/κ, the follow- nm ρm(k)dk. (11) B →∞ L −Q ing theorem shows that all integration boundaries m m except for m = 0, namely, there are no finite Fermi points The ground-state energy per unit length is given by (without chemical biases between different spin states).

κ   Theorem. For the κ-component system with the  Qm 2 − 2 m(m 1) 2 repulsive δ-function interaction, Bm →∞for m = 1,...,κ − E = mk − c ρm(k)dk. (12) − 12 1 if the relation m=1 Qm Mm−1 − Mm = Mm − Mm+1 (17) In the attractive regime, for convenience, we can define the holds, and vice versa. effective chemical potentials for the cluster bound sates μm = + + 2 − 2 = Proof. Integrating both sides of the Fredholm equations (4) μ Hm/m (m 1)c /12 with m 1,...,κ, where Hm is = the effective magnetic field (or Zeeman splitting parameter) for with infinite boundary (where m 0), we obtain  ∞ the bound state of the m atom, see [11]. The Zeeman energy = Mm−1 − Mm + Mm+1 = N−1 rm(k)dk . (18) per unit length can be written as Ez m=1 HmNm/L.In −∞ L L L this regime, the effective chemical potentials for the bound If the condition (17) holds, thus we have states of different sizes can be derived from the energies of the  ∞ ground state, i.e., Mm rm(k)dk = , (19)  −∞ L κ Qs 1 ∂ 2 =∞ μm = sk ρs (k)dk. (13) where the integration boundary Bm is inferred. Con- m ∂nm − s=1 Qs versely, if Bm with m  1 tend to infinity, from Eqs. (4) and (7) we have Following [11], the full phase diagrams of the model can be  ∞ determined by the field-energy transfer relations Mm Mm−1 Mm Mm+1 = rm(k)dk = − + .  L −∞ L L L = 1 2 − 2 2 + − Hm 12 m(κ m )c m(μm μκ ) (14) For the balanced case, the integration boundaries Bm  with m = 1,2,...,κ. In the next section, we shall derive with m 1 are infinitely large. The Fermi B0 the explicit form of the ground-state energy that can give is always finite. For convenience, we introduce the notation r (k) = r (k) + r (k) where highly accurate effective chemical potentials μm to determine 0 0in 0out magnetism and full phase diagrams of the model. r0(k)for|k|  B0, Similarly, the Fredholm equations can be rewritten as r0in(k) =  0for|k| >B0,

= − − ρm(λ) β0 K1(λ )ρm−1( )d 0for|k|  B0, | | >Qm−1 r0 (k) =  out r (k)for|k| >B. κ Qs 0 0 + − Ks−m(λ )ρs ( )d . (15) Taking the Fourier transformation with the Fredholm equations −Q s=m+1 s (4) for the balanced case, we obtain the following relations:

We see that there is a particular mapping between the two sets Fm+1(ω) F2(ω) r˜ (ω) = r˜ − (ω), r˜ (ω) = r˜ (ω), of Fredholm equations (6) and (15), i.e., m F (ω) m 1 1 F (ω) 0in m 1 (20) κ−m ρ → r − ,Q→ B − ,  m κ m m κ m −(κ−m+2s)c|ω|/2    (16) Fm(ω) = e , Q ∞ B − m κ m = → − ,c→−c. s 0 − −∞ − Qm Bκ−m where 2  m  κ − 1 and Fκ = 1. We denote the Fourier transform Fˆ[r (k)] = r˜ (ω). From the Fredholm equation (3), This connection is useful in the analysis of the symmetric m m we see that a closed form of the distribution function r (k)is structure between the two sides. However, we find that the 1 essential for the calculation of the ground-state energy. After a Fredholm equation (6) for the repulsive regime and Eq. (15) for straightforward calculation with the relations (20), we obtain the attractive regime do not preserve the mapping which exists the closed form between the two sets of the Fredholm equations for the two- 1 − | | component Fermi gas [16]. This leads to particular intricacies r˜0in (ω)sinh 2 (κ m) ω c r˜m(ω) = (21) in the analytical behavior of the ground-state energy as the sinh 1 κ|ω|c interaction strength vanishes. In the present paper, we mainly 2 that gives the distribution function concentrate on the solutions of the two sets of the Fredholm  equations for multicomponent Fermi gas with attractive and 1 ∞ = −iωλ repulsive interactions. rm(λ) r˜m(ω)e dω. (22) 2π −∞

033633-3 XI-WEN GUAN, ZHONG-QI MA, AND BRENDAN WILSON PHYSICAL REVIEW A 85, 033633 (2012)

E function B oson s 1 NY0(k) EY2(k) −4 3 r0(k) = + − + O(c ), (25) 10 2π 2πL 4π 2 where the function 4  ∞ iωk −c|ω|/2 α e e ω F2(ω) 1 2 Yα(k) ≈ dω. (26) −∞ F1(ω) γ After some algebra, we obtain the two functions used in 5 10 15 20 25 30 1 Eq. (25) 2Z 2Z k2 4Z Y (k) = 1 − 3 + O(c−4),Y(k) = 3 + O(c−4), 2 0 c c3 2 c3 with 3     1 1 1 Z =− ψ + C ,Z= κ−3 ζ 3, − ζ (3) . FIG. 1. (Color online) The ground-state energy per unit length 1 κ κ 3 κ = = = vs γ cL/N in natural units of 2m h¯ 1 for the balanced (27) multicomponent Fermi gas: κ = 2,4,10 and for spinless in the whole interacting regime. For the repulsive regime, the Here ζ (z,q) and ζ (z) are the Riemann ζ functions, ψ(p) discrepancy between the energies of the κ = 10 Fermi gas and the denotes the Euler ψ function, C denotes the Euler constant. →∞ spinless Bose gas is negligible. As κ , the ground-state energy When κ = 2wehaveZ1 = ln 2,Z3 = (3/4)ζ (3) that are of the balanced repulsive Fermi gas fully coincides with that of the consistent with the result given in Ref. [16]. spinless Bose gas [17]. Substituting Eq. (25) into Eqs. (7) and (8) and solving by iteration, the Fermi boundary B0 and the ground-state energy of balanced κ-component Fermi gas with a strong repulsion − In the above equations r˜ (ω) can be determined from the are given explicitly [up to order O(c 3)], 0in   following relation: 2Z 4Z2 8Z3 4Z π 2 B ≈ nπ 1 − 1 + 1 − 1 + 3 , (28)   0 2 3 3 B0 ∞ γ γ γ 3γ I(k) = K (k − λ)r (λ)dλ = K (k − λ)r (λ)dλ,    1 0 1 0in 3 2 2 2 −B −∞ n π 4Z 12Z 32 Z π 0 E ≈ 1 − 1 + 1 − Z3 − 3 , (29) (23) 3 γ γ 2 γ 3 1 15 where the dimensionless interaction strength γ = c/n.The where their Fourier transform reads I˜(ω) = e−c|ω|/2r˜ (ω). 0in ground-state energy (29) with κ = 2 reduces to the result The ground-state energy per unit length (8) can be calculated either numerically or asymptotically. In Fig. 1,weplotthe ground-state energy of 1D balanced multicomponent Fermi gas for κ = 2, 4, 10 and for spinless Bose gas [19]. We see E E that as κ →∞, the ground-state energy of the balanced κ-component gas with repulsion coincides with that of the 2 2 1D spinless Bose gas [17]. In order to capture the statistical γ γ 5 5 10 15 5 5 10 15 nature for this connection, we shall calculate the first few terms 2 2 of the expansion of the ground-state energy. The strong repulsion condition, i.e., cL/N 1, naturally 4 (a) 2 4 (b) 4 gives c B0, where B0 is proportional to the Fermi velocity E E = kF nπ. Following the method used in Ref. [16], we take 3.0 a Taylor expansion with respect to λ for the kernel function 2 2.5 2.0 K1(k − λ)inEq.(23). Using the relations (7), (8), (22), and γ 5 5 10 15 1.5 (23), we find 1.0 2 (d)  0.5 Bosons 2 2 4 (c) 10 γ ≈ − Eω ≈ F2(ω) − Eω 0 5 10 15 20 r˜0in (ω) n , r˜1(ω) n . (24) 2 F1(ω) 2 FIG. 2. (Color online) The ground-state energy per unit length In the above equations, we used the following formulas: vs γ = cL/N in natural units of 2m = h¯ = 1 for the balanced multicomponent Fermi gas: κ = 2,4,10 and for spinless Bose gas. Fˆ[(c2/4 + k2)−2] = 2πc−3e−c|ω|/2[c|ω|+2], The lines are the numerical solution obtained from the two sets of Fredholm equations. The dashed lines are plotted directly 2 2 −3 −5 −c|ω|/2 2 2 Fˆ[(c /4 + k ) ] = πc e [c |ω| + 6c|ω|+12]. from the asymptotic expansion ground-state energies (29) and (44) for a repulsive interaction and Eqs. (55) and (59) for an attractive Substituting Eq. (24) into the Fredholm equation (3), interaction. High precision of the asymptotic expansion ground-state we obtain an asymptotic form of the density distribution energy is seen throughout strongly and weakly coupled regimes.

033633-4 ONE-DIMENSIONAL MULTICOMPONENT FERMIONS WITH ... PHYSICAL REVIEW A 85, 033633 (2012) given in Ref. [16]. In view of the quasimomentum distribution (6) become r0(k) (25), the system can be taken as an with a  ∞ less exclusive fractional statistics than the Fermi statistics. r (k) = β + K (k − λ)r (λ)dλ, It is interesting to see that the ground-state energy (29) 0 0 1 1 −∞ reduces to the energy of the spinless Bose gas [see Eq. (10) ∞ →∞ = − − in Ref. [20]] as κ ;alsosee[17]. Here we find that rm(k) β0 Km(k λ)r0out (λ)dλ −∞ →∞ = →∞ = limκ Z1 limκ Z3 1.  ∞ A significant interpolation between fermions and bosons + K1(k − λ)rm+1(λ)dλ, (32) can be conceived from the parameters Z1 and Z2 in Eq. (29) −∞ that encodes the quantum statistical and dynamical effects. In Fig. 2, we see a good agreement between the asymptotic where 1  m  κ − 1. By iteration with r1, r2,...,rκ−1,the expansion result (29) and numerical result obtained from the first equation of Eq. (32) becomes Fredholm equation (3) with the density distribution (22). κ−1  In general, the boundaries Bm with m>1arehardtobe  ∞ estimated for the imbalanced case in a strong repulsive regime. r0(k) = κβ0 − K2s (k − λ)r0out(λ)dλ. (33) −∞ However, for strong repulsion, the spin sector is dominated by s=1 the density-density interaction. Therefore the spin polarization is not essential. We consider the high polarization case Using a Fourier transformation, we easily prove where Mm−1 − Mm N. In this case, the condition c Bm  always holds. Thus it is straightforward to calculate density B0 r (k) = β − R(k − λ)r (λ)dλ, distributions from Eqs. (3) and (4) with a proper Taylor 0out 0 0in −B0 −3 expansion [up to order O(c )], i.e.,  − (34)  ∞ κ−1 1  1 ikω −cs|ω| 2 R(k) = e e dω. 1 2M1 4k 2π −∞ r (k) ≈ + 1 − , s=0 0 2π cπL c2 = ≈ 1 − + Here we see that R(k) O(c). Substituting the leading term r1(k) [2N M1 2M2] = cπL of r0out (k) β0 into Eq. (33), we further obtain 2 k 8E   − [8N − M + 8M ] − , κ−1 3 1 2 3 κ 1 cs c πL c π r (k) = − arctan 1 0 2π 2π 2 B − k ≈ − + s=1 0 rm(k) [2Mm−1 Mm 2Mm+1]   cπL cs 2 + + O c2 k arctan + ( ) (35) − [8M − − M + 8M + ], (30) B0 k c3πL m 1 m m 1 | |  = for the region k B0. The energy can be calculated from where m 2,...,κ. It is also straightforward to calculate the Eqs. (7) and (8) with the distribution function (35), Fermi boundary B0 and E from Eqs. (7) and (8), i.e.,   2 3 2 3 2 2 π n 2 2 4m1 16m 64m 16π m1n E = + c(κ − 1)n /κ + O(c ). (36) B = nπ 1 − + 1 − 1 + 3κ2 0 c c2 c3 3c3 + O(c−4), When κ = 2 the energy (36) reduces to the result given in   [16]. This result clearly indicates a mean-field effect among n3π 2 8m 48m2 256m3 32π 2m n2 E = 1 − 1 + 1 − 1 + 1 the balanced κ-component weakly interacting fermions. The 2 3 3 3 c c c 5c kinetic energy part in Eq. (36) vanishes as κ →∞. The energy + O(c−4), (31) (36) as κ →∞is the same as for spinless Bosons with a weak  repulsion. κ−1 j+1 where m1 = M1/L with M1 = = N . For the highly For the imbalanced case with weak repulsion, we assume j 1   − polarized case, M1 N, the ground-state energy E per unit Bm−1 >Bm with 1 m κ 1. The calculation of the length, up to O(c−3), solely depends on the ground-state energy is very complicated because the κ integral M1. The spin states are not essential in this strongly repulsive equations are coupled with each other. We have to separate regime due to the freezing of spin transportation. This result the integration intervals case by case. Using the Fredholm agrees with the energy of the two-component Fermi gas with equations (6), we calculate the following integral in different Bm a strong repulsion; see [16]. It clearly indicates that strong regions. We denote the integral Im = rm(λ)K(k − λ)dλ. −Bm repulsion suppresses the spin effect. For the region |k| >Bm, we find

κ−1 cBm cBs 2 B. Weak repulsion Im =   +   + O(c ). 2π 2 k2 − B2 2π 2 k2 − B2 For the balanced case with weak repulsion, i.e., cL/N 1, m s=m+1 s all Bm with m  1 tend to infinity, so the Fredholm equations (37)

033633-5 XI-WEN GUAN, ZHONG-QI MA, AND BRENDAN WILSON PHYSICAL REVIEW A 85, 033633 (2012)

After a lengthy calculation, we calculate the integral for the Substituting Eq. (41) into Eq. (43), we finally obtain the region |k|  Bm, ground-state energy of the κ-component Fermi gas with weak repulsion, m−1 cBm ( − m + s + 1)cBs I = β −   −   κ−1 m 0 2 2 2 2 2 2  2π B − k 2π B − k 1 3 2 m s=0 s E = (m − − m ) π  3 i 1 i Bm+1 i=1 2 + + + − + rm 1(λ)K 1(k λ)dλ O(c ). (38) κ−1 κ −   Bm+1 2 + 2c (m − − m )(m − − m ) + O(c ), (44) | |  i 1 i j 1 j For Bp < k Bp−1 with p>m, we further calculate i=1 j=i+1 − m1 − + − + with m = n and m = 0. Here the linear density n = N/L (p m)(  s m  1)cBs 0 κ Im = (p − m)β0 − = 2π 2 B2 − k2 and the quantum numbers mi Mi /L. If we introduce s=0 s i i polarization pi = N /L with i = 1,2,...,κ − 1, where N p−1 − + − + − is the number of fermions in the ith level. Thus we have a − [(p s)( s m 1) 1]cBs 2 2 − 2 simple form of the ground-state energy per length s=m 2π Bs k κ κ−1 κ + − κ−1 1 ( p m)cBp cBs 2 = 3 2 + + 2 +   +   + O(c ). E pi π 2c pi pj O(c ). (45) 2π 2 k2 − B2 2π 2 k2 − B2 3 = = = + p s=p+1 s i 1 i 1 j i 1 (39) The first part is the kinetic energy of the κ-component With the help of these formulas, we are able to evaluate the fermions whereas the second part is the interaction energy. order of r (k) in the Fredholm equations (6), This result is valid for arbitrary spin imbalance in the weakly m repulsive regime. This result is in a good agreement with the rm(k) = O(c)for|k| >Bm−1, numerical calculation; see Fig. 2. For the balanced case, i.e., M − M + = N/κ, (44) reduces to the energy (36).Itis r (k) = β + O(c)forB + < |k|  B − , m m 1 m 0 m 1 m 1 interesting to note that the ground-state energy (45) presents a = − + | |  rm(k) (p m)β0 O(c)forBp < k Bp−1, mean-field theory of the two-body s-wave scattering physics. p>m+ 1. (40) From Eq. (7), we obtain the integration boundaries via C. Weak attraction

−   In the weakly attractive coupling regime, the Fredholm m1  −  Mm Bm Mm+1 c Bs Bm  = + + ln   equations give the distribution functions of clusters of different L π L 2π 2 B + B | | s=0 s m sizes. For weak attraction, i.e., c L/N 1, the two sets of   the Fredholm equations for repulsive and attractive regimes κ−1  −  Bm Bs  2 preserve the symmetry (16). Therefore the calculation of the + ln   + O(c ), (41) B + B ground-state energy E per unit length for weak attraction is s=m+1 m s similar to that for weak repulsion. For the balanced case with a   − where 0 m κ 1. In order to calculate the ground-state weak attraction, Nm = 0form = 1,...,κ − 1 and Nκ = N/κ. energy, we also need a lengthy calculation of the integral Thus we have the condition Qm = 0 for all Fermi points except  Q . In this case, the ground state is a spin singlet state. The Bm κ 2 λ rm(λ)dλ Fredholm equations (10) for the spin neutral bound state of a − Bm   κ-atom becomes m−1 κ−1 3 κ−1  Bm cBm  Qκ = + Bs + Bs 2 ρκ (λ) = κβ0 + K2s (λ − )ρκ ( )d . (46) 3π π = = + s 0 s m 1 −Q     s=1 κ m−1  −  κ−1  −  c 2 Bs Bm  2 Bm Bs  + B ln   + B ln   Under the mapping (16), the Fredhlom equations (46) for weak 2π 2 s B + B s B + B s=0 s m s=m+1 m s attraction map to Eq. (33) for weak repulsion. In order to  Bm+1 estimate the contribution of density distribution ρκin ,wetake 2 2 + λ rm+1(λ)dλ + O(c ). (42) the Fourier transformation of Eq. (46) and then prove that − Bm+1  Qκ Using Eqs. (8) and (42), the ground-state energy E per unit = − − ρκin (λ) β0 T (λ )ρκout ( )d , − length is given by Qκ  −  ∞ κ−1 1 κ−1 3 κ−2 κ−1 1 B c = iωλ −s|cω| E = m + T (λ) e e dω, 2 2π −∞ = 3π 2π = = + s=0 m 0 m 0 r m 1     B − B  = | |  = × + 2 + 2  m r  + 2 where T (λ) O(c). For λ Qκ , we thus have ρκ (λ) 4BmBr Bm Br ln   O(c ). (43) = + Bm + Br ρκ in(λ) β0 O(c). Substituting this leading order into

033633-6 ONE-DIMENSIONAL MULTICOMPONENT FERMIONS WITH ... PHYSICAL REVIEW A 85, 033633 (2012)

Eq. (46),wehave Eq. (11), we obtain the integration boundaries

 −   Qκ − | | κ | | m1  −  n Qκ (κ 1) c Nm Qm Ns c Qs Qm  = ρκ (λ)dλ = + = − − ln 2 2   κ −Q π 2π L π L 2π Q + Q κ s=m+1 s=1 s m κ−1     | | 2| |2 + 2 κ s c s c 4Qκ 2 Q − Q  + ln + O(c ). (47) +  m s  + 2 2κπ2 s2|c|2 ln   O(c ). (53) s= Q + Q 1 s=m+1 m s Similarly, from Eq. (12), we calculate the ground-state energy From Eq. (12), the ground-state energy E per unit length is of the balanced gas, given by

π 2n3 |c|(κ − 1)n2 κ 3 | | κ−1 κ E = − + O(c2). (48) Qm 2 c 2 E = − Q Q 3κ κ 3π π 2 s r m=1 s=1 r=s+1 We see that the energy of the balanced gas with weak attraction   | | κ−1 κ    −  (48) and with weak repulsion (36) continuously connect at c 2 2 Qs Qr  2 − Q + Q ln   + O(c ). c → 0. It is also seen that Eq. (48) with κ = 2 is consistent 2π 2 r s Q + Q s=1 r=s+1 s r with the result given in Ref. [16]. For the imbalanced case, the ground state has cluster states (54) of different sizes. Therefore we assume an ansatz in order to Substituting the integration boundaries Qm from Eq. (53) into calculate the integration boundaries for each of the Fermi seas, Eq. (54), we thus obtain the ground-state energy per unit length   − i.e., Qm >Qm+1 with 1 m  κ 1. Similarly, we should of the κ-component Fermi gas with a weak attraction, | | = 1 Qm  c rm(k)dk calculate the integrals Im 2π −Q (c/2)2+(λ−k)2 for different − m 1 κ κ 1 κ regions. After some algebra, we find E = p3π 2 − 2|c| p p , (55) 3 i i j | | κ | | i=1 i=1 j=i+1  c Qm c Qs 2 Im =   −   + O(c ) (49) i 2 2 − 2 2 2 − 2 where we introduced polarization pi = N /L with i = 2π λ Qm 2π λ Qs s=m+1 1,2,...,κ. The first part is the kinetic energy of the κ- for the region |λ| >Qm and component fermions whereas the second part is the interaction energy. This result is in a good agreement with numerical |c|Q |c|Q − i = I = β −  m  +  m 1  calculation; see Fig. 2. For the balanced case, i.e., N N/κ, m 0 2 2 − 2 2 2 − 2 2π Qm λ 2π Qm−1 λ Eq. (55) reduces to the energy (48). From the energies (45) and (55), we see that the ground-state energy of the κ-component m−2 | | + c Qs  gas with arbitrary polarization continuously connects at c = 0; 2π 2 Q2 − λ2 precisely speaking, the ground-state energy per unit length and s=1 s  its first derivative is continuously connected at c = 0. Again, κ Qs 2 the ground-state energy (55) presents a mean-field theory of − K+s−m(λ − )ρs ( )d + O(c ) (50) − the two-body s-wave scattering physics in weak interacting s=m+1 Qs regimes. for the region |λ|  Qm. In the region Qm+1 < |λ|  Qm,we further obtain D. Strong attraction m−2 = |c|Qs |c|Qm−1 Universal low- behaviors of isospin S Im = β0 +   +   − 2π 2 Q2 − λ2 2π 2 Q2 − λ2 1/2,1,3/2,...,(κ 1)/2 interacting fermions with an attrac- s=1 s m−1 tive interaction in one dimension shed light on the nature of |c|Q κ ( + 1)|c|Q trions, pairing, and quantum phase transitions. The existence −  m  −  s + O(c2). 2π 2 Q2 − λ2 2π 2 λ2 − Q2 of these internal degrees of freedom gives rise to some m s=m+1 s exotic superfluid phases. These models exhibit new quantum (51) phases of matter, which are characterized by bound states of different sizes underlying the symmetries. For example, three- | |  For λ Qm+1, we find component ultracold fermions give rise to a + | | + | | from a state of trions into the BCS pairing state under external = (  1) c Qm+1 − (2  1) c Qm Im fields. Recently, considerable interest has been paid to the low 2π 2 Q2 − λ2 2π 2 Q2 − λ2 m+1 m dimensional strongly interacting fermionic atoms with high ( − 1)|c|Q − +  m 1 + O(c2). (52) spin symmetries. 2 2 − 2 | | 2π Qm−1 λ For a strong attraction, i.e., c L/N 1, the bound states of different sizes form tightly bound molecules of different () = 2 − 2 From these equations (49)–(52), we are able to evaluate sizes with binding energies εb ( 1)c /12, where the leading-order contributions in the density distribution  = 2,3,...,κ. In this regime, all the Fermi momenta of functions, i.e., ρm(λ) = (m − p + 1)β0 + O(c)forQp+1 < the molecules are finite, i.e., |c| Qm with m = 1,...,κ. |λ|  Qp with p  m and ρm(λ) = O(c)for|λ| >Qm.From Here Q1 characterizes the Fermi momentum of the single

033633-7 XI-WEN GUAN, ZHONG-QI MA, AND BRENDAN WILSON PHYSICAL REVIEW A 85, 033633 (2012) spin-aligned atoms. In the canonical ensemble, external fields Here Ni , with i = 1,2 ...,are the numbers of the cluster state or nonlinear Zeeman splittings trigger rich quantum phases of the i atom. and magnetism [11]. A closed form of the ground-state energy After a lengthy calculation, we obtain explicit forms of the with polarization is essential to work out phase diagrams Fermi momenta and the energies of the molecules of different and magnetism at zero temperature. In view of the strong sizes for a strong attraction, attraction condition |c| Q and following the method [16],  m N π 4 16 proper Taylor expansions can be carried out for the three Q ≈ m 1 + F + F 2 m mL mL|c| m m2L2|c|2 m supplementary formulas   16 Q 4Q 16Q(Q2 + 3λ2) + 12F 3 − G π 2 , (57) K¯ (λ − )d = − 3 3| |3 m m  3 3  3m L c −Q π|c| 3π |c| 2 3 π Nm 8 48 2 256 3 4 2 2 4 Em ≈ 1 + Fm + F + F 64Q(Q + 10Q λ + 5λ ) − 3 | | 2 2| |2 m 3 3| |3 m + + O(c 7), 3mL mL c m L c m L c 5π5|c|5   16π 2 Q + − G + G /15 . (58) m3L3|c|3 m m (λ − )K¯ (λ − )d −Q Thus the ground-state energy of the gas with arbitrary 2 2 4Qλ 16Qλ(Q + λ ) − polarization per unit length in strong attractive regime is given =− + + O(c 5), π|c| π3|c|3 by  Q κ 2    K¯ (λ − )d () E = E − nε , (59) −Q b =1 3 3 2 2 4Q 16Q (3Q + 5λ ) − = − + O(c 5). where the binding energy of the molecule state of  atom is 3π|c| 15π3|c|3 () = 2 − 2 = given by εb ( 1)c /12. For κ 2, it covers the result In the above calculation, we denote the kernel function obtained for the two-component Fermi gas given in Ref. [16]. ¯ = 1 |c| K(k) 2π (c/2)2+k2 . The following calculations are valid for From the discrete BA equations, one of the authors and co- arbitrary polarization. For a strong attraction, all terms in the workers [11] derived the ground-state energy of κ-component Taylor expansion of the Fredholm equations are converged strongly attractive Fermi gas for up to the order O(1/c2). well; see the method proposed in Ref. [16]. From the Here we obtained a more accurate ground-state energy (58) Fredholm equations (10) and the boundary conditions (11), of the κ-component gas with arbitrary polarization from the it is straightforward to obtain the following expression [up to analytical study of the Fredholm equations. We noticed that − the order O(c 3)] the ground state of strongly attractive κ-component Fermi   2 gases can be effectively described by a super-Tonks-Girardeau Nm mQm 4 16π ≈ 1 − F + G , gaslike state via a proper mapping [21]. The explicit form of the L π mL|c| m 3m3L3|c|3 m  (56) ground-state energy can be used to study magnetism and full Qm 2 phase diagrams of the model from the relations (13) and (14) ρm( )d − in a straightforward way. The effective chemical potentials Qm   = κ 3 2 may be calculated from μm ∂( =1 E)/(m∂nm), where mQm 4 16π n = N /L. From these effective fields, the phase diagrams ≈ 1 − Fm + Gm , m m 3π mL|c| 15m3L3|c|3 can be determined by the energy-field transfer relations (14); where we denoted the functions see recent study of the three-component attractive Fermi gas − − [13,22]. m1 s N m1 N F = s + m m 2r − s + m − 2 2r s=1 r=1 r=1 IV. LOCAL PAIR CORRELATIONS κ m N There has been a considerable interest in studying universal + s , 2r + s − m − 2 nature of interacting fermions. Remarkably, Tan [23]showed s=m+1 r=1 that the momentum distribution exhibits universal C/k4 decay m−1 s 2 2 + 2 3 m−1 3 as the momentum tends to infinity. Here the constant C is s NmNs m Ns 2Nm Gm = + called universal contact that measures the probability of two s2(2r − s + m − 2)3 (2r)3 s=1 r=1 r=1 fermions with opposite spin at the same position. For 1D two- κ m 2 2 2 3 component Fermi gas [24], the universal contact is obtained by s N Ns + m N + m s , calculating the change of the interacting energy with respect to s2(2r + s − m − 2)3 s=m+1 r=1 interaction strength by Hellman-Feynman theorem, i.e., C = (2) (2) m−1 s m−1 4   2 2 + 2 3  3 2 n↑n↓g↑,↓(0). The local pair correlation g↑,↓ are accessible 9s NmNs 5m Ns 14Nm a1D G = + (2) m 2 3 3 = s (2r − s + m − 2) (2r) via exact Bethe ansatz solution through the relation g↑,↓(0) s=1 r=1 r=1 1 ∂E/∂c.HereE is the ground-state energy per unit length. κ m 9s2N 2 N + 5m2N 3 2n↑n↓ + m s s . In a similar way, for 1D κ-component Fermi gas, there exists s2(2r + s − m − 2)3 s=m+1 r=1 a 1D analog of the Tan adiabatic theorem where the universal

033633-8 ONE-DIMENSIONAL MULTICOMPONENT FERMIONS WITH ... PHYSICAL REVIEW A 85, 033633 (2012) contact is given by the local pair correlations for two fermions 3 3 with different spin states. The two-body local pair-correlation (a) 2 (b) 4 0 0 ' function is similar to the calculation of the expectation value 2 ' 2 σσ σσ 2 of the four-operator term in the second quantized Hamiltonian. 1 2 1 g For a homogenous and balanced κ-component Fermi gas, the g 0 0 local pair-correlation function is given by 3 0 5 10 15 3 0 5 10 15 γ γ (2) = κ ∂E gσ,σ  (0) (60) 3 3 (κ − 1)n2 ∂c (c) 10 (d) Bosons 0 0 ' ' 2 2 σσ σσ with κ>1. Where E is the ground-state energy per unit length. 2 1 2 1 From the asymptotic expansion result of the ground-state g g 0 0 energy of the balanced Fermi gas obtained in the above section, 3 0 5 10 15 3 0 5 10 15 (2) → | |→ we easily find gσ,σ  (0) 1as c 0. From the ground-state γ γ energy (29) of the balanced gas with a strong repulsion, we have the local pair correlation (2) FIG. 4. (Color online) The local pair correlation gσ,σ  (0) vs =   γ cL/N in natural units for the balanced multicomponent Fermi 2 2 gas: κ = 2,4,10 and for spinless Bose gas. The solid lines are (2) 4κπ 6Z1 24 3 Z3π g  (0) = Z − + Z − . σ,σ 3(κ − 1)γ 2 1 γ γ 2 1 15 the numerical solutions obtained from the two sets of Fredholm equations. The dashed lines are the asymptotic expansion result (61) obtained from Eqs. (61) and (63).

This local pair correlation reduces to the one for the spinless regime, Bose gas [20,25]asκ →∞. κ(κ + 1)|γ | 4π 2 For strong attractive interaction, we obtain the ground-state g(2) = + 6 3κ5(κ − 1)γ 2 energy of the balanced gas from the result (58),   6A2 24 B  × A + κ + A3 − κ + O(1/γ 4). π 2n3 4A 12A2 32A3 32B κ κ2|γ | κ4γ 2 κ 15 E = 1 + κ + κ + κ − κ 3κ4 κ2|γ | κ4γ 2 κ6|γ |3 15κ6γ 3 (63) − (κ) + 4 nκ εb O(1/γ ), (62) We see that the local pair correlation becomes divergent as κ →∞, i.e., goes to the limit for the attractive bosons. In   = κ−1 = κ−1 3 Fig. 3, we present the numerical solution of the local pair where Aκ r=1 1/r and Bκ r=1 1/r . From the re- lation (60), we obtained the local pair correlation for two correlation for two fermions with different spin states in the fermions with different spin states in a strong attractive multicomponent Fermi gas. In the whole interacting regime, the local pair correlations for the balanced κ-component Fermi gas tends to the limit value for the spinless bosons as κ → 3.0 ∞. High precision of these asymptotic expansion local pair correlations is seen in strong and weak interaction regions; see 2.5 Bosons Fig. 4.

10 2.0 V. CONCLUSION

0 4 ' 1.5 We have analytically studied the Fredholm equations for 2

g 2 the 1D κ-component fermions with repulsive and attractive δ- 1.0 function interactions in four regimes: (A) strong repulsion, (B) weak repulsion, (C) weak attraction, and (D) strong attraction. 0.5 Solving two sets of the Fredholm equations (4) and (10),we have obtained the first few terms of the asymptotic expansion 0.0 for the density distribution functions and the ground-state 2 0 2 4 energy E per unit length for both balanced and unbalanced cases in these regimes. We summarize our main result as (2) follows. FIG. 3. (Color online) The local pair correlation gσ,σ  (0) vs γ = cL/N in natural units for the balanced multicomponent Fermi (A) For the strong repulsive regime, the ground-state energy gas: κ = 2,4,10 and for spinless Bose gas. The solid lines are of the balanced κ-component gas has been given in Eq. (29) up 3 the numerical solutions obtained from the two sets of Fredholm to the order of 1/c . It presents a universal structure in terms equations. The dashed lines are the asymptotic limits obtained from of the parameters Z1 and Z3 (27) that characterize quantum the first term in Eq. (63). All local correlations pass the point statistical and dynamical effects of the model. It also confirms (2) = gσ,σ  (0) 1 at vanishing interaction strength. However, the local pair Yang and You’s result [17] that the energy per particle as correlation diverges for the attractive Bose gas. κ →∞is the same as for spinless bosons. In this regime,

033633-9 XI-WEN GUAN, ZHONG-QI MA, AND BRENDAN WILSON PHYSICAL REVIEW A 85, 033633 (2012) we have also obtained the ground-state energy for the highly of hard-core bosons. From the result of ground-state energy of imbalanced case (31). It clearly indicates that the internal spin the strongly attractive fermions, we can study magnetism and effect is strongly suppressed in this regime, i.e., the energy identify full phase diagrams of the gas with SU(κ) symmetry only depends on the quantum number M1. driven by the external fields. (B) For the weak repulsive regime, we have calculated Furthermore, for the balanced case, we have obtained the the ground-state energy E (36) for the balanced case and first few terms of the asymptotic expansion for the local pair Eq. (44) for the imbalanced case. The ground-state energy (44) correlations of two fermion with different spin states in the four presents a simple mean-field theory of the two-body s-wave regimes. A numerical solution of the local pair-correlation scattering physics. We see that the energy (36) for the balanced function has been obtained in the whole interacting regime. κ-component Fermi gas is the same as that for the spinless Bose These results we obtained provide insight into understanding gas as κ →∞, consistent with the result in Ref. [17]. quantum statistical and dynamical effects in 1D interacting (C) For the weak attractive regime, we have also calculated fermions with high spin symmetries. The explicit forms of the ground-state energy E (48) for the balanced case and the ground-state energy of the multicomponent Fermi gas Eq. (55) for the imbalanced case. We have found that the two with polarization give further applications to the study of sets of the Fredholm equations for the model with repulsive quantum phase transitions, magnetism, and phase diagrams, and attractive interactions preserves a particular mapping (16). which provides useful guides for future experiments with But they do not preserve the density mapping found for the ultracold atomic Fermi gases with high symmetries. two-component Fermi gas [16]. Our result shows that the ground-state energy of the model with arbitrary spin state is ACKNOWLEDGMENTS continuous at c → 0, as is the first derivative. (D) For the strong attractive regime, the highly accurate The authors thank Professor Chen Ning Yang for initiating ground-state energy [up to O(c−3)] of the κ component with this topic and for helpful discussions and suggestions. This arbitrary spin polarization has been given in Eq. (58).The work is partly supported by the Australian Research Council ground state of the largest cluster state can be effectively and by Natural Science Foundation of China under Grants No. described by the gaslike state in the super-Tonks-Girardeau gas 11075014 and No. 11174099.

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