Phase equilibrium of binary mixtures in mixed dimensions

E. Malatsetxebarria, F. M. Marchetti, and M. A. Cazalilla

Phys. Rev. A 88, 033604 – Published 3 September 2013

Abstract

We study the stability of a Bose-Fermi system loaded into an array of coupled one-dimensional (1D) “tubes,” where bosons and fermions experience different dimensions: Bosons are heavy and strongly localized in the 1D tubes, whereas fermions are light and can hop between the tubes. Using the $^{174}$ Yb- $^{6}$ Li system as a reference, we obtain the equilibrium phase diagram. We find that, for both attractive and repulsive interspecies interaction, the exact treatment of 1D bosons via the Bethe ansatz implies that the transitions between pure fermion and any phase with a finite density of bosons can only be first order and never continuous, resulting in phase separation in density space. In contrast, the order of the transition between the pure boson and the mixed phase can either be second or first order depending on whether fermions are allowed to hop between the tubes or they also are strictly confined in 1D. We discuss the implications of our findings for current experiments on $^{174}$ Yb- $^{6}$ Li mixtures as well as Fermi-Fermi mixtures of light and heavy atoms in a mixed-dimensional optical-lattice system.

Received 26 April 2013

DOI:https://doi.org/10.1103/PhysRevA.88.033604

Authors & Affiliations

E. Malatsetxebarria¹, F. M. Marchetti², and M. A. Cazalilla^3,1

¹Centro de Física de Materiales CSIC-UPV/EHU and Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal, E-20018 San Sebastián, Spain
²Departamento de Física Teórica de la Materia Condensada & Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid 28049, Spain
³Physics Department, National Tsing Hua University, Hsinchu, Taiwan

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Vol. 88, Iss. 3 — September 2013

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Figure 1
Schematic representation of the system studied in this work, namely, a mixed-dimensionality Bose-Fermi system. The Bose-Fermi cloud is loaded in an anisotropic optical lattice that produces a two-dimensional $N_{y} \times N_{z}$ array of one-dimensional tubes of length $L$ . The fermions are light enough to hop between the tubes, whereas bosons, which are assumed to be much heavier, are confined in the 1D tubes. The bosonic and fermionic Wannier functions [ $φ_{R} (r_{⊥})$ and $w_{R} (r_{⊥})$ , respectively] are also displayed.Reuse & Permissions
Figure 2
Zero-temperature phase diagrams for a repulsive ( $a_{B F} > 0$ , top panel) and attractive ( $a_{B F} < 0$ , bottom panel) Bose-Fermi mixture with fixed dimensionless hopping amplitude $\tilde{t} = 37.8$ and fixed interaction parameter $ζ = 0.23$ (the parameters of this calculation correspond to a $^{174}$ Yb- $^{6}$ Li system, see main text). The diagrams are plotted vs the boson $μ_{B}^{1D} / E_{r}^{B}$ and fermion $μ_{F} / E_{r}^{B}$ chemical potentials, where $E_{r}^{B}$ is the boson recoil energy; note that $μ_{B}^{1D} / E_{r}^{B} ≪ 1$ , thus ensuring that the bosons remain confined to 1D throughout. The thin solid black lines correspond to second-order (i.e., continuous) phase transitions between either the vacuum and the pure boson and fermion phases or between the pure boson and the homogeneous mixed phase. The thick solid (red) line corresponds to a first-order transition between the pure fermion and the mixed phase. Second- and first-order lines meet at at the tricritical point at $μ_{F} = μ_{B} = 0$ (filled blue circle). The region with finite boson density is the light gray shaded region. In both cases, phase separation can only occur between mixed and pure fermion phases.Reuse & Permissions
Figure 3
Phase diagrams in the density plane $({\tilde{n}}_{B}, {\tilde{n}}_{F})$ , where the dimensionless units are introduced in Eqs. (24) and (25) and below. In order to obtain a more accurate estimate of the phase boundaries, we have used the parameters $\tilde{t} = 0.023$ and $ζ = 0.27$ . The corresponding phase diagrams in the chemical potential plane (not shown for brevity) display the same features and phase topology as the diagrams shown Fig. 2, which are computed for the experimentally relevant $^{174}$ Yb- $^{6}$ Li system, but for which the phase boundary in the density plane proved much harder to determine numerically. The top panel corresponds to a repulsive Bose-Fermi interactions, whereas the bottom panel corresponds to attractive interactions. In both cases, the system can either be in a uniform mixed phase (lightly gray shaded region) or, by undergoing a first-order transition, it can be in a phase-separated state (dark gray shaded region), where a pure fermion and mixed phase coexist. The dotted-dashed lines connect points on the first-order boundary with the same values of the chemical potentials.Reuse & Permissions
Figure 4
Zero-temperature phase diagrams for a repulsive ( $a_{B F} > 0$ , top panel) and attractive ( $a_{B F} < 0$ , bottom panel) Bose-Fermi $^{174}$ Yb- $^{6}$ Li mixture in 1D, i.e., for zero hopping amplitude $t = 0$ , and for $ζ = 0.23$ . Differently from the finite- $t$ case (cf. Fig. 2), the transition between the pure boson and mixed phase becomes first order (thick red solid line), meeting with the other two first-order transition lines (one between the pure fermion and mixed phase and the other between the pure fermion and the pure boson phases) at a triple point (filled violet square), where the three phases coexist. Bottom panel: For an attractive mixture there is neither a triple point nor a tricritical point, rather the first-order line crosses the second-order line at two critical end points (filled [red] diamond) delimiting the region of phase separation between the vacuum and a mixed phase.Reuse & Permissions
Figure 5
Dimensionless free-energy density $\tilde{f}$ plotted versus the dimensionless bosonic density ${\tilde{n}}_{B} = {\tilde{ρ}}_{B}^{0}$ for fixed dimensionless hopping strength $\tilde{t} = 0.023$ , fixed interaction parameter $ζ = 0.27$ , and a 1D repulsive Bose-Fermi system, corresponding to the density phase diagram shown in the top panel of Fig. 6. The values of the chemical potentials are fixed at the three different first-order transition lines near the triple point $(μ_{B}^{1D} / E_{r}^{B}, μ_{F} / E_{r}^{B}) ≃ (35.3, 11.2)$ , which describes the state where phase separation occurs between the three phases, i.e., pure boson, pure fermion, and mixed phase: $(1) ≃ (36.3, 11.4)$ describe phase separation between pure fermion and mixed phase; $(2) ≃ (33.9, 10.8)$ phase separation between pure fermion and pure boson; finally $(3) ≃ (38.1, 12.4)$ phase separation between mixed and pure boson.Reuse & Permissions
Figure 6
Phase diagrams in the density plane $({\tilde{n}}_{B}, {\tilde{n}}_{F})$ for $ζ = 0.27$ and zero hopping strength $t = 0$ (same remarks as for Fig. 3 apply). Top panel: For repulsive interactions in the 1D limit, phase separation occurs, not only between a pure fermion and a mixed phase, but also between a pure boson and pure fermion phase, as well as between a pure boson and a mixed phase. Bottom panel: In contrast, for the attractive case, there is no phase separation between pure phases, rather, it exists in a region of phase separation between the vacuum and a mixed phase.Reuse & Permissions

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