Disordered ultracold atomic gases in optical lattices: A case study of Fermi-Bose mixtures

We present a review of properties of ultracold atomic Fermi-Bose mixtures in inhomogeneous and random optical lattices. In the strong interacting limit and at very low temperatures, fermions form, together with bosons or bosonic holes, composite fermions. Composite fermions behave as a spinless interacting Fermi gas, and in the presence of local disorder they interact via random couplings and feel effective random local potential. This opens a wide variety of possibilities of realizing various kinds of ultracold quantum disordered systems. In this paper we review these possibilities, discuss the accessible quantum disordered phases, and methods for their detection. The discussed quantum phases include Fermi glasses, quantum spin glasses, “dirty” superfluids, disordered metallic phases, and phases involving quantum percolation.

Figure 1

(Color online) Quantum phase diagrams of Fermi-Bose mixtures in a homogeneous optical lattice as functions of ${\tilde{\mu }}^{\mathrm{b}}$ and $\alpha =U∕V$, for ${\rho }_{\mathrm{f}}=0.4$ and $J∕V=0.02$. Roman numbers denote the total number of particles that form the composite and a bar means that the composite is formed by bosonic holes rather than bosons. (a) Diagram of composites where the filled small (blue) dots symbolize fermions, large (red) dots symbolize bosons, and empty (red) dots, bosonic holes. The subindex A (R) indicates attractive (repulsive) composites interactions. (b) Detailed quantum phase diagram of fermionic composites. The subindices denote here different phases: DW (density wave), FL (fermi liquid), SF (superfluid), and FD (fermionic domains). The strongly correlated phases for small but finite $J$ are surrounded by characteristic lobs [53], beyond which bosons become superfluid. Therefore there are thin regions of bosonic superfluid between the various composite phases [112]. The general explanation of the figure appears in Sec. 3A while the numerical calculations of some of the possible phases are reported, for the case of sites B, in Sec. 4B. A detailed explanation of the phases exhibited by composites $\overline{\mathit{II}}$ (B sites), composites $\mathit{I}$ (A sites), and composites $\mathit{II}$ (C sites) in the presence of disorder can be found in Secs. 3C, 3D, 3E, respectively.

Figure 2

(Color online) Tunneling, $d_{ij}$, and nearest-neighbor couplings, $K_{ij}$, of type $\overline{\mathit{II}}$ composites as a function of disorder ${\Delta }_{ij}^{\mathrm{b}}$, and for different boson-fermion interactions, $\alpha$, in the case ${\mu }_i^{\mathrm{f}}=0$.

Figure 3

(Color online) Schematic phase diagram of type $\overline{\mathit{II}}$ fermionic composites for low disorder $({\Delta }_{ij}^{\mathrm{b}}⪡1,\alpha )$ and vanishing fermionic on-site energy $({\mu }_i^{\mathrm{f}}=0)$ as a function of the ratio between NN interactions and tunneling for the composites.

Figure 4

(Color online) Tunneling, $d_{ij}$, and nearest-neighbor couplings, $K_{ij}$, of type $\overline{\mathit{II}}$ composites as a function of disorder ${\Delta }_{ij}^{\mathrm{b}}$, and for different boson-fermion interactions, $\alpha$, in the case ${\mu }_i^{\mathrm{f}}={\mu }_i^{\mathrm{b}}={\mu }_{i}V$.

Figure 5

(Color online) Ratio of nearest-neighbor coupling and tunneling $K_{ij}∕d_{ij}$ as a function of disorder ${\Delta }_{ij}^{\mathrm{b}}$ for $\alpha =0.7$ and $J∕V=0.01$ in the case of type $\mathit{I}$ composite fermions.

Figure 6

(Color online) Schematic representation of connecting bonds in type $\mathit{I}$ composite systems. The bonds are separated in two types: (i) the weak bonds in which two composites do not interact and (ii) the strong bonds where only one composite can stay. The short lines represent the bonds and the crossing points of the bonds are the lattice sites.

Figure 7

(Color online) Tunneling, $d_{ij}$, and nearest-neighbor couplings, $K_{ij}$, of type $\mathit{II}$ composites as a function of disorder ${\Delta }_{ij}^{\mathrm{b}}$ and for different boson-fermion interactions, $\alpha$, in the case ${\mu }_i^{\mathrm{f}}=0$.

Figure 8

Probability of having one fermion and zero boson at each lattice site for the $N=100$ sites in a Fermi-Bose mixture with $N_{\mathrm{b}}=60$, $N_{\mathrm{f}}=40$, $J_{\mathrm{b}}∕V=J_{\mathrm{f}}∕V=0.02$ and in the presence of harmonic traps for bosons and fermions characterized by ${\omega }_{\mathrm{b}}={10}^{-7}$ and ${\omega }_{\mathrm{f}}=5\times {10}^{-7}$, respectively. The interaction between fermions and bosons is (a) $\alpha =0$ (independent bosonic MI and Fermi gas), (b) $\alpha =0.5$ (Fermi liquid), (c) $\alpha =1$ (ideal Fermi gas), and (d) $\alpha =10$ (fermionic insulator domain).

Figure 9

Dynamical crossover from the Fermi gas to the Fermi glass phases. The parameters are the same as in Fig. 8c. (a) Variance of the number of fermions per lattice site as a function of the amplitude of the disorder $\Delta$. (b) Probability of having one composite (one fermion and zero boson) at each lattice site for the $N$ sites in the absence of disorder and (c) after ramping up adiabatically diagonal disorder with amplitude $\Delta =5\times {10}^{-4}$.

Figure 10

Dynamical crossover from the fermionic domain insulator to a disordered insulating phase. The parameters correspond to Fig. 8d. (a) Variance of the number of fermions per lattice site as a function of the amplitude of the disorder. (b) Probability of having one composite (one fermion and zero boson) in each lattice site in the absence of disorder and (c) after ramping up adiabatically the disorder with amplitude $\Delta =3\times {10}^{-4}$.

Figure 11

Almeida-Thouless plane in the reduced variables $kT∕K$, $h∕K$, and $H∕K$. The constraint on the magnetization (41) has been effectively neglected for this, as it is a parameter in the experiment as well.