Localization of interacting Fermi gases in quasiperiodic potentials

We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasiperiodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low-energy) Anderson localized and (high-energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasiperiodic potential. To discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point.

Figure 1

Logarithmic color-scale plot of the inverse participation ratio of the single-particle eigenstates $d_s/P$ as a function of the rescaled eigenstate index $j/M_s$ and of the quasidisorder intensity $V/E_{rs}$, i.e., the intensity of the two optical lattices. $M_s=L/d_s=610$ and $M_l=M_s/r=377$ are the number of periods of the short-period and of the long-period lattices, respectively; $E_{rs}$ is the recoil energy corresponding to the former. The ratio of the two optical-lattice periods is $d_l/d_s=r\cong 1.61803$, i.e., close to the golden ratio. The continuous horizontal (violet) segment indicates the index of the highest-occupied orbital for a density so that the short-period lattice is half filled, while the dashed (brown) horizontal segment the one so that the long-period lattice is fully filled. The vertical (red) bar with diagonal pattern indicates the Anderson localization transition of the states with index $j\simeq M_s/2$.

Figure 2

The main panel shows the participation ratio $P$ of the single-particle eigenstate labeled $j=M_s/2$ as a function of the quasidisorder strength $V/E_{rs}$, for different system sizes $L$. $d_s$ and $d_s$ are the period lengths of the short-period and of the long-period lattices, respectively. The vertical (red) bar with diagonal pattern indicates the location of the Anderson transition. The inset shows the scaling of the inverse participation ratio $d_s/P$ as a function of the inverse system size $d_s/L$, for two values of the quasidisorder strength: in one case $P$ saturates for large system sizes, whereas in the other case it diverges as $P\propto L$ (see continuous black line). These two values bracket the critical point, and they determine the width of the (red) bar in the main panel.

Figure 3

The lower panel shows the modulus of the expectation value of the many-body phase operator $\left|Z\right|$ as a function of the quasidisorder strength $V/E_{rs}$, for different system sizes $L$. Full and empty symbols correspond to QMC and numerical-integration (NI) data, respectively. The continuous curves are the empirical fitting functions (see text). The vertical (red) bar indicates the Anderson transition. The density $n=1/d_s$ is fixed so that the short-period lattice is half filled, the period-lengths ratio is $r\simeq 1.618$. The upper panel shows the empty symbols with connecting lines indicate the many-particle localization length $\lambda /d_s$ (left vertical axis), while the dashed curves indicate the (rescaled) derivative of $\left|Z\right|$ with respect to $V/E_{rs}$ (right vertical axis).

Figure 4

The main panel shows the modulus of the expectation value of the many-body phase operator $\left|Z\right|$ as a function of the quasidisorder strength $V/E_{rs}$, for the same system size $L=58d_s$ but different interaction strengths $\gamma$. The density $n=1/d_s$ is fixed so that the short-period lattice is half filled, the period-length ratio is $r\simeq 1.65$. The full line shows modified hyperbolic tangent (see text). The inset shows the finite-size scaling analysis of $\left|Z\right|$ for the noninteracting case $\gamma =0$ (full symbols) and for an interacting case with $\gamma =0.02$ (empty symbols). Continuous curves are cubic fitting functions shown to guide the eye.

Figure 5

Critical quasidisorder strength $V_c/E_{rs}$, which separates the metallic phase (yellow region) from the insulating phase (cyan region), as a function of the interaction strength $\gamma$. The density is $n=1/d_s$, the optical lattice period is $r\simeq 1.65$. The empty (blue) circles indicate the data obtained from the crossing of the $\left|Z\right|\left(V\right)$ curves for different system sizes; the full (red) squares are those obtained from the maximum of the derivative of these curves. The continuous (black) line $V_c/E_{rs}=1.03\left(3\right)\gamma +1.176\left(2\right)$ is a linear fit to the latter dataset.