Imbalanced thee-component Fermi gas with attractive interactions: Multiple FFLO-pairing, Bose-Fermi and Fermi-Fermi mixtures versus collapse and phase separation

Imbalanced thee-component Fermi gas with attractive interactions: Multiple FFLO-pairing, Bose-Fermi and Fermi-Fermi mixtures versus collapse and phase separation Andreas L¨uscher1 and Andreas M. L¨auchli2 1Institut Romand de Recherche Num´erique en Physique des Mat´eriaux (IRRMA), EPFL, CH-1015 Lausanne, Switzerland 2Max Planck Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany (Dated: July 27, 2021) We present a detailed study of the population imbalanced three-component Hubbard chain with attractive interactions. Such a system can be realized experimentally with three different hyperfine states of ultra cold 6Li atoms in an optical lattice. We find that there are different phases that compete with each other in this system: A molecular superfluid phase in which the three fermion species pair up to form molecules (trions), a usual pairing phase involving two species with exactly opposite momenta, and a more exotic generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase consisting of three competing pairing tendencies with differ- ent non-zero center-of-mass momenta. At large attractive interactions the system exhibits strong tendencies towards collapse and phase separation. Employing the density-matrix-renormalization-group-method (DMRG) to determine the decay exponents of the various correlators we establish the phase diagram of this model for different fillings and interactions. We also discuss the experimentally relevant situation in a trap and report the existence of an additional region where two species are dynamically balanced. PACS numbers: 03.75.Ss, 03.75.Mn, 71.10.Fd, 71.10.Pm Introduction – Pair formation of fermions is an important phenomenon occurring in a variety of physical systems rang- ing from superconductors and superfluids to dense quark mat- ter. The pairing of two balanced species of fermions is well understood within the Bardeen-Cooper-Schrieffer (BCS) the- ory [1], but pairing and molecule formation involving three flavors are still intriguing, especially if the Fermi surfaces of the different flavors do not match. Recent advances in methods for trapping and controlling ultracold atoms have opened up the possibility of experimentally observing a three- component mixture of ultracold 6Li fermionic atoms with attractive interactions [2, 3]. These experimental prospects have fueled a variety of theoretical studies of systems with equal densities of flavors, in which the formation of three- body molecules is favored for strong attraction. In one dimen- sion, the molecular superfluid prevails for all attractive inter- actions [4], whereas in higher dimensions, an additional color superfluid phase has been found for smaller attractions [5]. In the more general, but still experimentally realizable case, where the flavour densities are different, it is an open question how the system maintains its binding tendencies and whether it favors trion or pair formation. In the two-flavor case – as exemplified by the imbalanced attractive Hubbard model – an extended Fulde-Ferrell-Ovchinnikov-Larkin (FFLO) phase with a well defined momentum Q = |kF ↑−kF ↓| of the Cooper pairs has been found [6]. In the three-flavor case there are three different possible FFLO wave vectors and it is un- clear which one will dominate the pairing properties. Moti- vated by these fundamental theoretical question and their rele- vance to upcoming experiments, we have investigated the case of generic filling in the three-component Hubbard chain with attractive interactions - the simplest possible model describing these binding tendencies. In this Letter, we present the pairing phase diagram for different imbalance and interactions. Model – We concern ourselves with a population imbal- free fermions (n1=n2=0, n3=n) BCS regime (n1=0, n2=n3) balanced (n1=n2=n3=n/3) FFLO regime (n1=0, n2 < n3) balanced- imbalanced (n1=n2, n3 > n1) (n2=n3, n1 < n2) central charge c=1 c=2 c=3 n3 n1 n2 n1+n2+n3=n n2=n3 n1=n2 n1=n3 n1 ≤ n2 ≤ n3 FIG. 1: (Color online) Density configurations for a given filling n form a triangle in n1-n2-n3 space. Because of symmetry, it is suffi- cient to consider only the subset n1 ≤n2 ≤n3, shown on the right hand side. Dots represent the systems studied in this work and the three encircled numbers refer to selected configurations presented in Figs. 2 and 4. anced three-component Fermi gas in a one-dimensional (1D) optical lattice, which can be described by a Hubbard Hamil- tonian H = −t X l,α  c† l,αcl+1,α + H.c.  + X l X β>α Uαβ nl,αnl,β . (1) Here c† l,α (cl,α) creates (destroys) a fermion with flavor α = 1, 2, 3 on site l and nl,α = c† l,αcl,α is the state selective oc- cupation number operator. The parameters t, which we set to t →1, and Uαβ ≡U < 0 characterize the nearest-neighbor hopping and the SU(3) invariant attractive on-site interac- tion, respectively. The flavor density configurations compati- ble with a given total density n := 1/L P α nα, where L is the length of the chain, form a triangle in n1-n2-n3-s

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