Dynamics of heat and mass transport in a quantum insulator

The real-time evolution of two pieces of quantum insulators, initially at different temperatures, is studied when they are glued together. Specifically, each subsystem is taken as a Bose-Hubbard model in a Mott insulator state. The process of temperature equilibration via heat transfer is simulated in real time using the minimally entangled typical thermal states algorithm. The analytic theory based on quasiparticle transport is also given.

Figure 1

Dynamical evolution of the variance of the particle number on each site. The initial state is chosen with an inhomogeneous temperature profile in order to induce a heat current. The black curves are obtained for a tensor product initial state, where the left and right parts of the system are both at thermal equilibrium but at different temperatures, ${\beta }_{L}U=6,{\beta }_{R}U=8.$ The red curves correspond to ”smoothly” glued initial conditions, as described in the main text. For $tU=2,$ the cusp in the black curve is a remnant of boundary effects occurring when tunneling between the two subsystems is abruptly turned on at $t=0.$ The short-range fluctuations on the curves are of statistical origin, an unavoidable effect of the METTS approach. They vanish when the number of METTS states tends to infinity.

Figure 2

Local thermometer based on the particle number variance. (a) Local variance on the various sites for two smoothly glued Mott insulators, with inverse temperatures ${\beta }_{L}U=8$ (left part) and ${\beta }_{R}U=12$ (right part). The temperature profile is given by Eq. (2), with a (smooth) interface size $s=3$ (black solid line), $s=7$ (red, dashed line), and $s=10$ (blue, dot-dashed line). (b) Local variance vs the (inverse) local temperature, Eq. (2). Except for small differences, the three curves collapse, proving that the local variance—with a good approximation—depends only on the local temperature, regardless of the sharpness of the temperature profile. This proves that the local variance can be used as a reliable thermometer in an inhomogeneous system.

Figure 3

Dynamical evolution of the average particle number $〈n_i〉$ and average squared particle number $\langle n_i^2\rangle$ after two Mott insulators at different temperatures ${\beta }_{L}U=6,{\beta }_{R}U=8$ are sharply connected at time $t=0.$ Each subsystem has initially 35 particles on 35 sites. Note the spreading of the central perturbation observed both on the average density $〈n_i〉$ (lower, black curves in all the panels) and on the average squared density $\langle n_i^2\rangle$ (higher, red lines in all the panels).

Figure 4

Energy currents associated with the data of Fig. 3. The current initially starts flowing at the connecting point, $i=0$, between the two samples at different temperatures. It then propagates on both sides, reaching sites $i=\pm 5$ around $tU=25,$ and site $i=10$ later around $tU=50.$

Figure 5

Spatiotemporal development of the energy current for the data represented in Fig. 3. Observe the ballistic spreading of the perturbation, initially born at the connecting point, $i=0$, between the two Mott insulators at different temperatures. See text for further discussion.

Figure 6

Comparison of quasiexact numerical METTS simulations (left column) with the Bogoliubov theory (right). From top to bottom: mean site occupation $\langle n_i\rangle$, variance $\langle n_i^2\rangle -{\langle n_i\rangle }^2$, quasiparticle density $n_i^{\left(\mathrm{qp}\right)}$, and quasihole density $n_i^{\left(\mathrm{qh}\right)}$. The dashed white (black) straight lines correspond to ballistic propagation of quasiparticles (quasiholes) from the connection point between the two subsamples at the maximal allowed velocity, i.e., $x^{\left(\mathrm{qp}\right)}\left(t\right)=\pm 4Jt [x^{\left(\mathrm{qh}\right)}\left(t\right)=\pm 2Jt]$. See text for a detailed discussion.

Figure 7

Comparison of the currents computed from the numerical METTS simulations (left column) with the Bogoliubov theory (right). The currents are obtained by numerical calculations for the same parameter values as in Fig. 6. From top to bottom: mass current, energy current, heat (variance) current as defined in Eq. (57), and quasihole current as defined in Eqs. (51) (the quasiparticle current coincides with the energy current). See text for discussion.

Figure 8

Comparison of the average particle density $\langle n_i\rangle$ (lower curves) and the average squared density $\langle n_i^2\rangle$ (upper curves) at time $tU=60,$ for the quasiexact METTS simulations (solid black lines) and the Bogoliubov theory. The raw Bogoliubov predictions are the dotted green curves; rescaled results, taking into account finite-size effects, are shown as dashed red lines and agree very well with the METTS simulations. See text for discussion of the various features.

Figure 9

Comparison of the mass and energy currents for the same data as in Fig. 8. The energy current is predicted by the Bogoliubov theory to depend only on the quasiparticle current, see Eq. (51). It displays the characteristic “semicircle” shape predicted by our Bogoliubov theory, Eq. (55). In contrast, the mass current is the difference between the quasiparticle and quasihole “semicircles,” see Eq. (51), and it presents two maxima propagating at the maximum velocity of the quasiholes $\pm 2J.$

Figure 10

Particle number variance at initial time $t=0$ and at $tU=60,$ for two smoothly glued Mott insulators with (inverse) temperatures ${\beta }_{L}U=6, {\beta }_{R}U=8,$ and $J/U=0.05.$ The interface size, Eq. (2), is $s=15$. Solid lines correspond to quasiexact numerical results using the METTS method, while dashed curves are theoretical estimates. A single parameter—required to take into account finite-size effects—has been used to adjust the numerical and theoretical curves. The sharpest (black) step corresponds to the initial time $t=0.$ The smoother (red) curve corresponds to $tU=60.$ Note that the real-time evolution strongly smooths the initial site-to-site fluctuations. At $tU=60,$ the theoretical prediction is almost indistinguishable from the quasiexact numerical simulation.

Figure 11

Mass (upper panel) and energy (lower panel) currents at $tU=60$ for the same data as in Fig. 10. Numerical results (solid black curves) are compared with the predictions of the Bogoliubov theory (dashed red curves). While the magnitude of the currents is not predicted properly, the overall shape is faithfully predicted by the theory.

Figure 12

Particle number variance at initial time $t=0, tU=10$, and at $tU=60$ for two smoothly glued Mott insulators with (inverse) temperatures ${\beta }_{L}U=8, {\beta }_{R}U=10$, and $J/U=0.05.$ The interface size, Eq. (2), is $s=15$. Solid lines correspond to quasiexact numerical results using the METTS method, while dashed curves are theoretical estimates. A single parameter—required to take into account finite-size effects—has been used to adjust the numerical and theoretical curves. The sharpest (black) step corresponds to the initial time $t=0.$ The smoother (red and green) curves corresponds to $tU=10$ and 60. The real-time evolution strongly smooths the initial site-to-site fluctuations. At $tU=60,$ the theoretical prediction is almost indistinguishable from the quasiexact numerical simulation.

Figure 13

Mass (upper panel) and energy (lower panel) currents at $tU=60$ for the same data as in Fig. 12. Numerical results (solid black curves) are compared with the predictions of the Bogoliubov theory (dashed red curves). The agreement is very satisfactory.