Quantum engine based on many-body localization

Many-body-localized (MBL) systems do not thermalize under their intrinsic dynamics. The athermality of MBL, we propose, can be harnessed for thermodynamic tasks. We illustrate this ability by formulating an Otto engine cycle for a quantum many-body system. The system is ramped between a strongly localized MBL regime and a thermal (or weakly localized) regime. The difference between the energy-level correlations of MBL systems and of thermal systems enables mesoscale engines to run in parallel in the thermodynamic limit, enhances the engine's reliability, and suppresses worst-case trials. We estimate analytically and calculate numerically the engine's efficiency and per-cycle power. The efficiency mirrors the efficiency of the conventional thermodynamic Otto engine. The per-cycle power scales linearly with the system size and inverse-exponentially with a localization length. This work introduces a thermodynamic lens onto MBL, which, having been studied much recently, can now be considered for use in thermodynamic tasks.

Figure 1

Schematic of MBL engine. We formulate an Otto engine cycle for a many-body quantum system that exhibits an MBL phase. We illustrate the engine with a spin chain (green dots and black arrows). A random disorder potential (jagged red line) localizes the particles. Particles interact and hop between sites (horizontal red arrows). Consider strengthening the interactions and the hopping frequency. The system transitions from strong localization to a thermal phase or to weak localization. The engine thermalizes with a hot bath (flames) and with a cold bath (ice cube). The cold bath has a small bandwidth $W_{\mathrm{b}}$, to take advantage of small energy gaps' greater prevalence in the highly localized regime.

Figure 2

Qubit toy model for the MBL Otto cycle. A qubit models two “working levels” in the MBL Otto engine's many-body spectrum. The energy eigenstates $|E_t^{\left(1\right)}\rangle$ and $|E_t^{\left(2\right)}\rangle$ span the “working subspace.” The gap $E_t^{\left(2\right)}-E_t^{\left(1\right)}$ begins at size ${\delta }_{\mathrm{GOE}}$ during a successful trial. The gap shrinks to ${\delta }_{\mathrm{MBL}}$, then returns to ${\delta }_{\mathrm{GOE}}$. In addition to changing the gap, each Hamiltonian tuning changes the eigenstates' functional forms. The displacement ${\delta }_{\mathrm{displ}}$ is included for generality. The blue text marks the times $t=0,\tau ,...,{\tau }^{\prime \prime \prime }$ at which the strokes begin and end during a work-outputting trial. The spectator level $|E_t^{\left(3\right)}\rangle$ fails to impact the engine's efficiency. The cold bath has too narrow a bandwidth $W_{\mathrm{b}}$ to couple $|E_t^{\left(3\right)}\rangle$ to any other level. If the engine begins any trial on the top green line, the engine remains on that line throughout the trial. Zero net work is outputted.

Figure 3

Otto engine cycle for a mesoscale MBL system. Two energies in the many-body spectrum capture the cycle's basic physics. The engine can be regarded as beginning each trial in an energy eigenstate drawn from a Gibbs distribution. The red dot represents the engine's starting state in some trial of interest. During stroke 1, $H_{\mathrm{meso}}\left(t\right)$ is tuned from “thermal” to MBL. During stroke 2, the engine thermalizes with a cold bath. $H_{\mathrm{meso}}\left(t\right)$ returns from MBL to thermal during stroke 3. Stroke 4 resets the engine, which thermalizes with a hot bath. The tunings (strokes 1 and 3) map onto the thermodynamic Otto cycle's isentropes. The thermalizations (strokes 2 and 4) map onto isochores. The engine outputs work $W_1$ and $W_3$ during the tunings and absorbs heat $Q_2$ and $Q_4$ during thermalizations. MBL gap statistics' lack of level repulsion enhances the cycle; the engine “slides down” the lines that represent tunings, losing energy outputted as work.

Figure 4

Three classes of diabatic transitions. Hops to arbitrary energy levels, modeled with general adiabatic perturbation theory (APT), plague the ETH regime. Landau-Zener transitions and fractional-Landau-Zener transitions plague the many-body-localized regime.

Figure 5

Average per-cycle power $〈W_{\mathrm{tot}}〉$ (top) and efficiency ${\eta }_{\mathrm{MBL}}$ (bottom) as functions of the cold-bath bandwidth $W_{\mathrm{b}}$. Each red dot represents an average over 1000 disorder realizations of the random-field Heisenberg Hamiltonian (32). The slanted blue lines represent the analytical predictions (13) and (15) of Sec. 3c. When $W_{\mathrm{b}}\ll 〈\delta 〉$ (in the gray shaded region), the engine operates in the regime of interest. Here, $〈W_{\mathrm{tot}}〉$ and ${\eta }_{\mathrm{MBL}}$ vary linearly with $W_{\mathrm{b}}$, as predicted. The error bars are smaller than the numerical-data points.

Figure 6

Average per-cycle work as a function of tuning speed. We numerically simulated 995 disorder realizations of the random-field Heisenberg Hamiltonian (32) for a system of $N=8$ sites (red dots). The results are compared to the analytical estimate (13) for the adiabatic work output (blue line) and an empirical straight-line fit $W_{\mathrm{tot}}=W_0-W_1{\left(v{\delta }_{-}\right)}^{1/3}/W_{\mathrm{b}}$ (black line). Errors in the estimate of the mean, computed as $\left(\text{sample}\text{standard}\text{deviation}\right)/\sqrt{(}\text{\#}\text{realizations})$, lead to error bars smaller than the numerical-data points.

Figure 7

Magnitude $|〈Q_2〉|$ of the average heat absorbed during cold thermalization (stroke 2) as a function of the cold-bath bandwidth $W_{\mathrm{b}}$ (a), the cold-bath temperature $T_{\mathrm{C}}$ (b), and the hot-bath temperature $T_{\mathrm{H}}=1/{\beta }_{\mathrm{H}}$ (c). The blue lines represent the magnitude of the analytical prediction (A10). See Sec. 6 for other parameters and definitions. The analytics match the numerics' shapes, and the agreement is fairly close, in the appropriate limits (where $\frac{W_{\mathrm{b}}}{〈\delta 〉}\ll 1$ and $T_{\mathrm{C}}/〈\delta 〉\ll 1$, in the gray shaded regions). The analytics systematically underestimate $|〈Q_2〉|$ at fixed $W_{\mathrm{b}}$, due to the small level repulsion at finite $N$. The analytical prediction (A10) substantially underestimates $|〈Q_2〉|$ when the cold-bath bandwidth is large, $W_{\mathrm{b}}\gtrsim 〈\delta 〉$. Such disagreement is expected: the analytics rely on $\frac{W_{\mathrm{b}}}{〈\delta 〉}\ll 1$, neglecting chains of small gaps: ${\delta }_j^{\prime },{\delta }_{j+1}^{\prime },\cdots <W_{\mathrm{b}}$. Such chains proliferate as $W_{\mathrm{b}}$ grows. A similar reason accounts for the curve's crossing the origin in (b): we analytically compute $〈Q_2〉$ only to second order in $T_{\mathrm{C}}/〈\delta 〉$.

Figure 8

Average heat $〈Q_4〉$ absorbed during hot thermalization (stroke 4) as a function of the cold-bath bandwidth $W_{\mathrm{b}}$ (a), the cold-bath temperature $T_{\mathrm{C}}$ (b), and the hot-bath temperature $T_{\mathrm{H}}=1/{\beta }_{\mathrm{H}}$ (c). The blue lines represent the analytical prediction (A13), to lowest order in $T_{\mathrm{C}}$, with the ${\beta }_{\mathrm{H}}$ dependence of $〈Q_4〉$, too small a correction to include in Eq. (A13): $〈Q_4〉\approx W_{\mathrm{b}}-\frac{2ln2}{{\beta }_{\mathrm{C}}}+\frac{{\left(W_{\mathrm{b}}\right)}^2}{2〈\delta 〉}{e}^{-N{\left({\beta }_{\mathrm{H}}\mathcal{E}\right)}^2/4}$. See Sec. 6 for other parameters and definitions. The analytics' shapes agree with the numerics, and the fit is fairly close, in the appropriate limits (where $e^{-{\beta }_{\mathrm{C}}{W}_{\mathrm{b}}}\ll 1, \frac{1}{{\beta }_{\mathrm{C}}〈\delta 〉}\ll 1$, and $\frac{W_{\mathrm{b}}}{〈\delta 〉}\ll 1$, in the gray shaded regions). The predictions underestimate $〈Q_4〉$; see the Fig. 7 caption. (c) suggests that the numerics deviate significantly from the analytics: the numerics appear to depend on ${\beta }_{\mathrm{H}}$ via a linear term absent from the $〈Q_4〉$ prediction. This seeming mismatch appears symptomatic of finite sample and system sizes.

Figure 9

Per-cycle power $〈W_{\mathrm{tot}}〉$ as a function of the cold-bath bandwidth $W_{\mathrm{b}}$ (a), the cold-bath temperature $T_{\mathrm{C}}$ (b), and the hot-bath temperature $T_{\mathrm{H}}=1/{\beta }_{\mathrm{H}}$ (c). The blue lines represent the analytical prediction $〈W_{\mathrm{tot}}〉\approx W_{\mathrm{b}}-\frac{2ln2}{{\beta }_{\mathrm{C}}}$: Eq. (A14), to first order in $\frac{W_{\mathrm{b}}}{〈\delta 〉}$ and in $\frac{1}{{\beta }_{\mathrm{C}}〈\delta 〉}$. The analytics largely agree with the numerics in the appropriate regime: $\frac{W_{\mathrm{b}}}{〈\delta 〉}\ll 1$ and $\frac{T_{\mathrm{C}}}{〈\delta 〉}\ll 1$ (in the gray shaded region). Outside that regime, the analytics underestimate $〈W_{\mathrm{tot}}〉$; see Fig. 7 for an analysis. (c) suggests that the numerics depend on ${\beta }_{\mathrm{H}}$ via a linear term absent from the analytical prediction; see the caption of Fig. 8.

Figure 10

Efficiency ${\eta }_{\mathrm{MBL}}$ as a function of the cold-bath bandwidth $W_{\mathrm{b}}$ (a), the cold-bath temperature $T_{\mathrm{C}}$ (b), and the hot-bath temperature $T_{\mathrm{H}}=1/{\beta }_{\mathrm{H}}$ (c). The blue lines represent the analytical predictions (A18) and (A19). (c) shows the leading-order ${\beta }_{\mathrm{H}}$ dependence of ${\eta }_{\mathrm{MBL}}$, a correction too small to include in Eq. (A19): $1-\frac{W_{\mathrm{b}}}{2〈\delta 〉}{e}^{-N{\left({\beta }_{\mathrm{H}}\mathcal{E}\right)}^2/4}$. See Sec. 6 for other parameters and definitions. The analytics agree with the numerics fairly well in the appropriate regime ($\frac{W_{\mathrm{b}}}{〈\delta 〉}\ll 1, \frac{T_{\mathrm{C}}}{〈\delta 〉}\ll 1$, and $\sqrt{N}{T}_{\mathrm{H}}\mathcal{E}\ll 1$). The analytics underestimate ${\eta }_{\mathrm{MBL}}$; see the Fig. 7 caption.

Figure 11

Level-spacing distribution for ${10}^6$ disorder realizations of the random-field Heisenberg model at disorder width $h=20$ and system size $N=8$ (blue line). The vertical black line shows the estimate of the level-repulsion parameter ${\delta }_{-}$.

Figure 12

Energies of a cold-thermalized many-body-localized system. We illustrate our implementation of cold thermalization with this example chain of six energies. The cold bath has a bandwidth of size $W_{\mathrm{b}}$, depicted in green.