Semiclassical theory of strong localization for quantum thermalization

We introduce a semiclassical theory for strong localization that may arise in the context of many-body thermalization. As a minimal model for thermalization we consider a few-site Bose-Hubbard model consisting of two weakly interacting subsystems that can exchange particles. The occupation of a subsystem ($x$) satisfies in the classical treatment a Fokker-Planck equation with a diffusion coefficient $D\left(x\right)$. We demonstrate that it is possible to deduce from the classical description a quantum breaktime $t^{*}$ and, hence, the manifestations of a strong localization effect. For this purpose it is essential to take the geometry of the energy shell into account and to make a distinction between different notions of phase-space exploration.

Figure 1

Illustration of the system and its phase space. (a) We consider $N=60$ bosons in a four-site system that is formed by weakly coupling a trimer and a monomer subsystems. The trimer consists of three strongly coupled sites. The system is described by the Hamiltonian of Eq. (2). (b) The phase space of the system is divided into cells that are labeled by $r=(x,\varepsilon )$. A given cell $r_0$ (circled and colored in red) overlaps with energy surfaces $E_1<E<E_2$, forming a region that we call its “energy shell” (strictly speaking, we display the projection of a high-dimensional energy shell onto the two-dimensional plane). Each surface $E$ overlaps with many cells, as shown schematically for $E_1$ (the white spaces between the cells are for visual purposes only). Note that the width in $\varepsilon$ of the energy surface shrinks to zero for $x\to N$, where the trimer-monomer coupling term Eq. (4) vanishes. A single classical trajectory explores a zero-thickness energy surface $E$, and can visit at most ${\mathrm{\Omega }}_E$ cells. To be precise, only a fraction ${\mathcal{F}}^{\text{cl}}$ is explored, because ${\mathrm{\Omega }}_E$ counts not only cells that belong to the chaotic sea, but also cells that reside in quasi-integrable regions. A semiclassical cloud that starts at $r_0$ explores a larger volume that includes all the accessible cells within the finite-thickness energy shell. (c) An abstract illustration of the high-dimensional energy surfaces. Each surface is associated with a quantum eigenstate $E_{\alpha }$. The number of surfaces ${\mathcal{N}}_E$ that participate in the dynamics (overlapping with the red cell) might be much smaller than the number of cells ${\mathrm{\Omega }}_E$ that intersect a given energy surface.

Figure 2

The quantum localization effect. Panels (a) and (b) display the two-dimensional saturation profile $P_{\infty }\left(r\right|r_0)$ for the semiclassical and the quantum simulations, respectively. The initial condition $r_0$ corresponds to having all particles in the trimer ($x_0=60$) with ${\varepsilon }_0=1.181$. The nonlinear color scale encodes the probability from low (blue) to high (red). The semiclassical simulation in (a) provides the determination of the dynamically accessible volume of the energy shell. The quantum simulation exhibits strong localization. In panel (c) the distribution is projected onto $x$ space. The quantum (red) simulation features a peak at the initial $x_0$, unlike the semiclassical (blue) simulation that reflects phase-space ergodicity, as implied by the agreement with the (black) normalized density of states. Panel (d) displays the spreading length $L_{\infty }$ for different values of $x_0$, at the same ${\varepsilon }_0$ as in panels (a–c). Note that for small $x_0$ the energy shell gets wider, and hence $L_{\infty }$ becomes higher. The abnormally low values of the semiclassical spreading (blue symbols) for $x_0\le 24$ indicate a lack of classical ergodicity, and hence are of no interest for us. Our objective is to provide a theory for the quantum spreading (red symbols), where strong localization shows up in the range $25\le x_0\le 29$ and for $x_0\ge 56$. Details of the simulations can be found in Appendix pp2.

Figure 3

Signatures of localization in the LDOS. Panels (a–c) compare the quantum LDOS (gray) with its semiclassical counterpart (black). The calculation is done for the same preparation as in Fig. 2, that has the energy ${\varepsilon }_0=1.181$, and for two other preparations with the same energy but with different initial occupations ($x_0=26,44,60$). The resolution has been improved by integrating the density $\rho \left(E\right)$ over an energy range that corresponds to 50 level spacings. Quantum ergodicity is reflected in panel (b), where the LDOS matches well the semiclassical envelope. Quantum localization is reflected in panels (a) and (c). The vertical axis has been zoomed in (a, c) and hence the peaks are chopped. Note also the reduced range of the horizontal axis in panel (c). Panel (d) provides a sharper view of panel (c). It displays the bare probabilities $p_{\alpha }$ instead of the smoothed density $\rho \left(E\right)$, and uses a log scale for the vertical axis. The quantum symbols are color-coded according to the value of ${\langle x\rangle }_{\alpha }$. The semiclassical LDOS is the black line. We observe that localization is present both in $E$ and in $x$. The localization in $x$ is reflected as sparsity: there are few low-lying blue points that correspond to small ${\langle x\rangle }_{\alpha }$ values, and many high-lying red points that correspond to large values.

Figure 4

The quantum spectrum. (a) The unperturbed states $\left|r\right.〉=\left|x,\varepsilon \right.〉$ are color-coded according to ${\mathcal{F}}^{\text{qm}}$. Red color (high ${\mathcal{F}}^{\text{qm}}$) implies quantum ergodicity, as in the LDOS of Fig. 3. By contrast, blue color (low ${\mathcal{F}}^{\text{qm}}$) indicates strong quantum localization, as in Fig. 3. (b) The perturbed states $\left|E_{\alpha }\right.〉$ are color-coded according to $\text{var}{\left(x\right)}_{\alpha }$, and positioned according to $({〈x〉}_{\alpha },{〈\varepsilon 〉}_{\alpha })$. The low-variance states (blue) have significant overlaps only with unperturbed states for which $x\approx {\langle x\rangle }_{\alpha }$, while the large-variance states (red) correspond to microcanonical states within the chaotic sea.

Figure 5

Breaktime determination. The functions ${\mathrm{\Omega }}_t^{\text{cl}}$, ${\mathrm{\Omega }}_t^{\text{sc}}$, ${\mathrm{\Omega }}_t^{\text{qm}}$, and ${\mathcal{N}}_t^{\text{qm}}$ are plotted versus time (see legend) for a preparation that has an initial occupation $x_0=55$ with energy ${\varepsilon }_0=1.181$. For further numerical details see Appendix pp3. The saturation values are indicated by dotted horizontal lines. The semiclassical estimate for the breaktime, based on Eq. (1), is determined by the intersection of the dashed line with ${\mathrm{\Omega }}_t^{\text{cl}}/{\mathrm{\Omega }}_E$.

Figure 6

Semiclassical prediction of strong localization. The data points in the present figure are based on simulations of the type presented in Fig. 5, with the same ${\varepsilon }_0$, but for different values of $x_0$ [same simulations as in Fig. 2]. (a) The classical ergodicity measure is calculated for each $x_0$. Our interest is focused in the range of $x_0$, where ${\mathcal{F}}^{\text{cl}}$ indicates a nearly ergodic classical motion. Note that 100% ergodicity cannot be reached because each energy surface contains inaccessible quasiregular regions. A secondary test for ergodicity is the agreement between the exploration-spreading ratio (squares) and it ergodic value (line) which is implied by Eq. (17). (b) The scaled breaktime $t^{*}/t_H$ is deduced from Eq. (1) via the procedure that has been illustrated in Fig. 5. (c) The quantum ergodization measure ${\mathcal{F}}^{\text{qm}}$ and the dynamical localization measure ${\mathcal{F}}^{\text{s}}$ for different initial conditions. The horizontal red and blue lines mark the ergodic values ${\mathcal{F}}_{\text{erg}}^{\text{qm}}=1/3$ and ${\mathcal{F}}_{\text{erg}}^{\text{s}}=2/3$, respectively, that are attained for simulations with $30\le x_0\le 55$. The prediction for ${\mathcal{F}}^{\text{s}}$ is based on the semiclassical breaktime estimate Eq. (19). The deviation of actual ${\mathcal{F}}^{\text{s}}$ from the predicted value at small $x_0$ is apparently related to remnants of quasi-integrability.

Figure 7

Various volumes that appear in the semiclassical analysis: the long-time volume explored by a single ergodic trajectory (black circles); the saturation spreading-volume of a semiclassical cloud (red squares); and the total volume of the energy shell (blue diamonds). We show the $x_0$ dependence of these volumes and use the $x_0$-independent volume of the energy surface as a reference. The calculations are done for the same parameters that were specified in Fig. 6, where the derived classical ergodicity measures are displayed.

Figure 8

Histograms of the quantum LDOS ergodicity measure $\mathcal{F}={\mathcal{N}}_{\infty }/{\mathcal{N}}_E$ (solid red) and of the inverse-LDOS ergodicity measure $\mathcal{F}={\mathrm{\Omega }}_{\infty }/{\mathrm{\Omega }}_E$ (dashed blue), which is defined by reversing the roles of $\mathcal{H}$ and ${\mathcal{H}}_0$. The latter characterizes the ergodicity of the $E$ eigenstates in $r$ space. The value $\mathcal{F}=1/3$ is typical for the quantum-ergodic eigenstates. The distributions refer to states within the energy window that is used in all our simulations.