Quantum phases of one-dimensional Hubbard models with three- and four-body couplings

Fabrizio Dolcini and Arianna Montorsi

Phys. Rev. B 88, 115115 – Published 10 September 2013

Abstract

The experimental advances in cold atomic and molecular gases stimulate the investigation of lattice correlated systems beyond the conventional on-site Hubbard approximation, by possibly including multiparticle processes. We study fermionic extended Hubbard models in a one-dimensional lattice with different types of particle couplings, including also three- and four-body interactions up to nearest neighboring sites. By using the bosonization technique, we investigate the low-energy regime and determine the conditions for the appearance of ordered phases, for arbitrary particle filling. We find that three- and four-body couplings may significantly modify the phase diagram. In particular, diagonal three-body terms that directly couple the local particle densities have qualitatively different effects from off-diagonal three-body couplings originating from correlated hopping, and favor the appearance of a Luther-Emery phase even when two-body terms are repulsive. Furthermore, the four-body coupling gives rise to a rich phase diagram and may lead to the realization of the Haldane insulator phase at half-filling.

Received 27 July 2013

DOI:https://doi.org/10.1103/PhysRevB.88.115115

Authors & Affiliations

Fabrizio Dolcini^1,2 and Arianna Montorsi¹

¹Dipartimento di Scienza Applicata e Tecnologia del Politecnico di Torino corso Duca degli Abruzzi 24, 10129 Torino, Italy
²CNR-SPIN, I-80126 Napoli, Italy

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Vol. 88, Iss. 11 — 15 September 2013

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Images

Figure 1
Schematic representation of various phases with a gap in either the spin or the charge sector, at half-filling. (a) The LE liquid phase exhibits a $C_{P}^{(s)}$ order, where pairs of singly occupied sites with spin- $↑$ and spin- $↓$ fermions are correlated and localized close to each other along the 1D lattice. (b) The MI phase exhibits a $C_{P}^{(c)}$ order, where correlated doubly occupied sites (doublons) and empty site (holons) are localized near to each other. (c) The HI exhibits a nonvanishing $C_{S}^{(c)}$ correlation, which implies that doublons and holons appear in alternate order in the sublattice of nonsingly occupied sites. See Ref. 28 for details.Reuse & Permissions
Figure 2
The phase diagram of the extended Hubbard model for $U = 3 V = t / 4$ as a function of the filling factor $ρ$ and the three-body diagonal coupling $P$ . Transitions from LL to LE phases are possible both for positive and negative values of $P$ , depending on the filling factor. At half-filling $ρ = 1$ , the MI transition occurs for both repulsive and moderately attractive three-body coupling $P$ .Reuse & Permissions
Figure 3
Ground state phase diagram of the extended Hubbard model (1) as a function of the two-body nearest neighbors interaction parameter $V$ and of the diagonal three-body coupling $P$ , for $U = t / 2$ at half-filling $ρ = 1$ . For $V < U / 2$ , the $P$ coupling drives a transition from a LL to a MI phase, whereas for $V > U / 2$ the $P$ coupling drives a transition from a fully gapped CDW phase to a LE phase, characterized by a gapless charge sector and a gapped spin sector. Differently from the $U - V$ extended Hubbard model, the $P$ term makes the LE phase arise also for repulsive $U, V > 0$ . This phase diagram has to be compared with that of the off-diagonal three-body coupling $\tilde{X}$ originating from correlated hopping in Eq. (1), shown in Fig. 4.Reuse & Permissions
Figure 4
Ground state phase diagram of the extended Hubbard model (1) as a function of the two-body nearest neighbors interaction parameter $V$ and of the off-diagonal three-body coupling $\tilde{X}$ , for $U = t / 2$ at half-filling $ρ = 1$ . Differently from Fig. 3, no LE phase is present. Instead, when $V > 0$ , a HI phase, characterized by a gapless spin sector and a gapped charge sector, emerges. Furthermore, for $V > U / 2$ , the fully gapped CDW phase, already present in the $U - V$ model, persists.Reuse & Permissions
Figure 5
The phase diagram of the extended Hubbard model for $U = 2 V = t / 4$ as a function of the filling factor $ρ$ and the four-body diagonal coupling $Q$ . Transitions from LL to LE phases are possible both for positive and negative values of $Q$ , depending on the filling factor. At half-filling $ρ = 1$ , two transitions occur from a LE to a fully gapped BOW phase, and from the BOW phase to a MI phase, with varying the four-body coupling $Q$ from attractive to repulsive regime.Reuse & Permissions
Figure 6
Ground state phase diagram of the model in the $V - Q$ space for $U = t / 4$ , at half-filling $ρ = 1$ . All possible phases described in Table appear when varying the two-body coupling $V$ and the four-body coupling $Q$ . In particular, one observes the presence of a Haldane insulator phase for repulsive $U, V, Q > 0$ .Reuse & Permissions

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