Quasiperiodic quantum heat engines with a mobility edge

Steady-state thermoelectric machines convert heat into work by driving a thermally generated charge current against a voltage gradient. In this work, we propose a new class of steady-state heat engines operating in the quantum regime, where a quasiperiodic tight-binding model that features a mobility edge forms the working medium. In particular, we focus on a generalization of the paradigmatic Aubry-André-Harper (AAH) model, known to display a single-particle mobility edge that separates the energy spectrum into regions of completely delocalized and localized eigenstates. Remarkably, these two regions can be exploited in the context of steady-state heat engines as they correspond to ballistic and insulating transport regimes. This model also presents the advantage that the position of the mobility edge can be controlled via a single parameter in the Hamiltonian. We exploit this highly tunable energy filter, along with the peculiar spectral structure of quasiperiodic systems, to demonstrate large thermoelectric effects, exceeding existing predictions by several orders of magnitude. This opens the route to a new class of highly efficient and versatile quasiperiodic steady-state heat engines, with a possible implementation using ultracold neutral atoms in bichromatic optical lattices.

Figure 1

Schematic of the thermoelectric heat engine. We engineer the thermodynamic properties of the reservoirs in such a way to use the thermal current to drive electrons against a chemical potential bias.

Figure 2

An efficient thermoelectric device can be obtained through the use of an energy filter in the central system, blocking the transport at certain energies. The temperature bias drives particle (hole) transport above (below) the chemical potential, leading to zero net electric current in the presence of particle-hole symmetry. Finite electric current and, consequently, output power are instead obtained by differentiating the dynamics of the particles at energies above and below the chemical potential.

Figure 3

Eigenenergy spectra of GAAH systems with $b=(\sqrt{5}+1)/2, \lambda =-0.8$, and $\phi =0$ as a function of $\alpha$, for a chain of $N=987$ sites. The IPR of the corresponding eigenstate is shown by a color map, with green for extended and blue for completely localized states. The red line represents the mobility edge $E_c$ given by Eq. (22), which separates localized from delocalized states. The dashed black line indicates the value of $\alpha$ chosen in Sec. 4.

Figure 4

Spectra for single GAAH wires of length $N=987$, generated with (a) $\lambda =-0.8t,\alpha =0.792,\phi =0$ (dashed vertical line in Fig. 3) and (c) $\lambda =-1.4t,\alpha =0.330,\phi =0$. The mobility edge is shown by the red line. [(b)–(d)] The transmission functions associated respectively to the first and second configurations, averaged at every energy over 40 values of the phase $\phi$, as described in the text. Conduction is clearly possible only at energies that support extended eigenstates.

Figure 5

(a) Electric conductance and (b) thermal conductance as a function of chemical potential at fixed temperature $T=0.1(t/k_B)$.

Figure 6

(a) Seebeck factor and (b) thermoelectric figure of merit as a function of chemical potential at fixed temperature $T=0.1(t/k_B)$.

Figure 7

(a) Maximum power and (b) efficiency at maximum power as a function of chemical potential at fixed temperature $T=0.1$ and bias $\mathrm{\Delta }T=0.01(t/k_B)$. Blue circles mark the points of absolute maximum power.

Figure 8

(a) Absolute maximum power and (b) corresponding efficiency at maximum power as a function of temperature with $\mathrm{\Delta }T/T=0.1$. The solid black lines show results obtained by optimizing the power only over values of $\mu$ that give rise to a particle current flowing from the hot to the cold bath, $J_N>0$. The dashed blue lines are instead obtained by restricting the maximization to $J_N<0$. The chemical potential yielding this maximum power is shown in the inset.

Figure 9

(a) Absolute maximum power and (b) efficiency at maximum power, as in Fig. 8 but using a clean (i.e., nondisordered) wire as a working medium. Identical values for positive and negative current are obtained at symmetric chemical potentials relative to the center of the conducting region (inset).

Figure 10

(a) The transmission function for the setup in the same configuration as in the main text computed with NEGF (green lines), overlapped by a series of boxcar approximations (black lines) following its profile. Comparison of (b) the electric conductance, (c) the Seebeck coefficient, and (d) efficiency at maximum power obtained from the calculated transmission (solid green line) and from the boxcar approximation (dashed black lines).

Figure 11

Dependence of the transport properties on the system-bath coupling strength $\gamma$. (a) Electrical conductance as a function of the distance between the chemical potential and the mobility edge, obtained for multiple choices of $\gamma$. All curves have the same form and differ only in their magnitude. (b) Figure of merit around the mobility edge for different $\gamma$. Since the three coefficients show the same behavior, their combination is $\gamma$ independent and the different curves completely overlap. We adopt the same quasiperiodic chain used in the main text, fixing the temperature at $T=0.1t/k_B$.

Figure 12

(a) Efficiency at maximum power as a function of the system-bath coupling, for a fixed chemical potential, $\mu -E_c=1.57t$. Since the Onsager coefficients are modified by the same prefactor when $\gamma$ is changed, the efficiency remains constant. (b) Maximum power transferred by the machine at the same values of $\gamma$. It is evident that it is possible to tune the system-bath coupling in such a way to optimize power without changing efficiency.

Figure 13

Comparison between the transport coefficients computed through the Landauer-Büttiker integrals of Eq. (12) (solid black line), and the approximated forms in Eqs. (B7, B8, B9) (dashed blue line). (a) Electrical conductance. (b) Seebeck coefficient. The parameters of the system are the same as in the main text, but with a weak coupling of $\gamma =0.01t$. The proportionality factor of 0.06 in panel (a) is a free parameter, which encapsulates the microscopic details of the eigenfunctions that are neglected in the approximations (B7, B8, B9).

Figure 14

(a) Electrical conductance and (b) Seebeck coefficient at low temperature $T=0.1t/k_B$ and $\gamma =0.01t$. The quantities are computed through the exact Landauer-Büttiker integrals (solid black line), and the analytical formulas in Eqs. (B10) and (B11) (dashed blue line). The proportionality factor of 6.0 in panel (a) is a free parameter, which reflects the fractal structure of the transmission function that is neglected in the boxcar approximation.