PHYSICAL REVIEW A 85, 033633 (2012) One-dimensional multicomponent fermions with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi gas Xi-Wen Guan,1 Zhong-Qi Ma,2 and Brendan Wilson1 1Department of Theoretical Physics, 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 spin 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 Bosons. 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 gases with higher spin symmetries. Fermionic for multicomponent Fermi gases with higher spin symmetry alkaline-earth-metal 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 electron 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 phase 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 boson 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<jN 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-canonical ensemble, 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) + + − | |+ − thermodynamic limit. 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 ground state. 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.