Dynamic Kibble-Zurek scaling framework for open dissipative many-body systems crossing quantum transitions

We study the quantum dynamics of many-body systems, in the presence of dissipation due to the interaction with the environment, under Kibble-Zurek (KZ) protocols in which one Hamiltonian parameter is slowly, and linearly in time, driven across the critical value of a zero-temperature quantum transition. In particular we address whether, and under which conditions, open quantum systems can develop a universal dynamic scaling regime similar to that emerging in closed systems. We focus on a class of dissipative mechanisms the dynamics of which can be reliably described through a Lindblad master equation governing the time evolution of the system's density matrix. We argue that a dynamic scaling limit exists even in the presence of dissipation, the main features of which are controlled by the universality class of the quantum transition. This requires a particular tuning of the dissipative interactions, the decay rate $u$ of which should scale as $u\sim t_s^{-\kappa }$ with increasing the time scale $t_s$ of the KZ protocol, where the exponent $\kappa =z/(y_{\mu }+z)$ depends on the dynamic exponent $z$ and the renormalization-group dimension $y_{\mu }$ of the driving Hamiltonian parameter. Our dynamic scaling arguments are supported by numerical results for KZ protocols applied to a one-dimensional fermionic wire undergoing a quantum transition in the same universality class of the quantum Ising chain, in the presence of dissipative mechanisms which include local pumping, decay, and dephasing.

Figure 1

Rescaled correlation $\lambda P(x,t)$, fixing $x/\lambda =1$, for the unitary dynamics of the Kitaev quantum wire in the thermodynamic limit, as a function of the scaling time variable $\tau$. Here we fix the scaling variable associated with the initial time, ${\tau }_i=-10$. Lines with different styles are for various values of the length scale $\lambda$, from 4 to 16, as indicated in the legend; these correspond to increasing the time scales as $t_s={\lambda }^2$. The upper right inset shows a magnification of the data for $4.4<\tau <5.6$, while the lower left inset displays rescaled correlations as a function of $1/\lambda$, for fixed $\tau =4.8$ (arrow in the upper inset), supporting an $O\left({\lambda }^{-1}\right)$ approach to the asymptotic KZ scaling limit. Analogous results are obtained for other values of the scaling variables $x/\lambda$ and ${\tau }_i$, and for the correlations $C(x,t)$ and $G(x,t)$.

Figure 2

Rescaled correlations $\lambda P(x,t)$ (a), $\lambda C(x,t)$ (b), and ${\lambda }^{2}G(x,t)$ (c), at fixed $x/\lambda =1$ (results for other values of $x/\lambda$ show analogous behaviors), for the unitary dynamics of the Kitaev quantum wire in the thermodynamic limit, as a function of the scaling variable $\tau$. Different line styles stand for various values of the length scale $\lambda$, from 4 to 16, analogously to Fig. 1 (see legend). Data belonging to one of the two color sets correspond to a given initial Hamiltonian parameter ${\overline{\mu }}_i<0$, which is kept fixed and equal to either ${\overline{\mu }}_i=-0.1$ (black circles) or ${\overline{\mu }}_i=-0.5$ (red squares). The insets in the three panels display rescaled correlations as a function of $1/\lambda$, for both cases of ${\overline{\mu }}_i$ presented in the main frames, at the $\tau$ value indicated by the blue arrow.

Figure 3

Rescaled correlations $\lambda P(x,t)$ [(${\mathrm{a}}_1$), (${\mathrm{b}}_1$), (${\mathrm{c}}_1$)] and $\lambda C(x,t)$ [(${\mathrm{a}}_2$), (${\mathrm{b}}_2$), (${\mathrm{c}}_2$)], at fixed rescaled distance $x/\lambda =1$ and initial rescaled time ${\tau }_i=-10$, for the dissipative Kitaev quantum wire in the thermodynamic limit, as a function of the scaling variable $\tau$. Analogous scaling behaviors are observed for other values of $x/\lambda$ and ${\tau }_i$, with different asymptotic scaling functions of $\tau$. Each panel in one of the three columns refers to a specific type of dissipation mechanism [see Eq. (31)]: decay [(${\mathrm{a}}_1$), (${\mathrm{a}}_2$), (${\mathrm{a}}_3$)], pumping [(${\mathrm{b}}_1$), (${\mathrm{b}}_2$), (${\mathrm{b}}_3$)], and dephasing [(${\mathrm{c}}_1$), (${\mathrm{c}}_2$), (${\mathrm{c}}_3$)]. The color code stands for three rescaled dissipative couplings: $\gamma =0.1$ (black), $\gamma =1$ (red), and $\gamma =10$ (green). Different line styles are for various values of $\lambda$, from 8 to 16 (see legend). (${\mathrm{a}}_3$), (${\mathrm{b}}_3$), (${\mathrm{c}}_3$) Rescaled correlations as a function of $1/\lambda$, up to $\lambda ={10}^2$, for $\tau =6$ (arrows in the panels above) and a given value of $\gamma$ for each panel (see figure).

Figure 4

Same kind of analysis as in Fig. 3, but fixing the initial Hamiltonian parameter ${\overline{\mu }}_i$, rather than the initial time ${\tau }_i$. We again show results for $x/\lambda =1$. In all the panels we kept the rescaled dissipation strength fixed and equal to $\gamma =0.5$. The color code refers to ${\overline{\mu }}_i=-0.1$ (black) and to ${\overline{\mu }}_i=-0.5$ (red), while different line styles stand for various values of the length scale $\lambda$, from 10 to 40. [(${\mathrm{a}}_3$), (${\mathrm{b}}_3$), (${\mathrm{c}}_3$)] Rescaled correlations $\lambda P(x,t)$ (filled symbols) and $\lambda C(x,t)$ (empty symbols) as a function of $1/\lambda$, for ${\overline{\mu }}_i=-0.1$ (circles) or ${\overline{\mu }}_i=-0.5$ (squares), at fixed $\tau =4$ (arrows in the above panels). Panels in the three columns refer to incoherent decay [(${\mathrm{a}}_1$), (${\mathrm{a}}_2$), (${\mathrm{a}}_3$)], pumping [(${\mathrm{b}}_1$), (${\mathrm{b}}_2$), (${\mathrm{b}}_3$)], and dephasing [(${\mathrm{c}}_1$), (${\mathrm{c}}_2$), (${\mathrm{c}}_3$)].

Figure 5

Same analysis as in Fig. 4, but for the rescaled correlation ${\lambda }^{2}G(x,t)$. We have computed it the presence of either incoherent decay or pumping, since with dephasing we were only able to compute two-point observables (while density-density correlations are a four-point observable) [8].

Figure 6

Rescaled correlation $\lambda P(x,t)$, fixing $x/\lambda =1$ and ${\overline{\mu }}_i=-0.5$, for the dissipative Kitaev quantum wire in the thermodynamic limit, as a function of the scaling time variable $\tau$. The dissipation has been chosen in the form of incoherent particle losses, with a coupling $u$ entering the Lindblad master equation (4) that is kept fixed and equal to $u=0.2$. Different lines stand for various values of the length scale $\lambda$, from 20 to 100 (see legend). Note that the various curves start from different initial rescaled times ${\tau }_i=\lambda {\overline{\mu }}_i$.

Figure 7

Rescaled correlations $LP(x,t)$ (a), $LC(x,t)$ (b), and $L^{2}G(x,t)$ (c), fixing $x/L=1/4$ (results for other values of $x/L$ show analogous behaviors), for the dissipative Kitaev quantum wire with a finite length $L$, as a function of the scaling variable $\tau$. The color code corresponds to several values of the inverse KZ speed $\upsilon$, while different line styles stand for various system sizes $L$ (see legends). Here we fix the scaling variables associated to the initial time (${\tau }_i=-10$) and to the dissipation (${\gamma }_L=1$), which has been chosen in the form of incoherent particle losses. The inset in panel (c) displays the rescaled correlation $L^{2}G(x,t)$ as a function of $1/L$ (data up to $L=512$), for fixed $\upsilon =0.1$ and $\tau =-0.5$ (arrow in the main panel).

Figure 8

Same as in Fig. 7, but for the correlation $C(x,t)$, for dissipation given by decay (a), pumping (b), or dephasing (c). The color code corresponds to several values of ${\gamma }_L$, while different line styles stand for various values of $L$ (see legends). Black curves are for ${\gamma }_L=0$ [in panels (b) and (c) we replot the same curve corresponding to $L=256$, for reference]. Here we fix ${\tau }_i=-10$ and $\upsilon =1$.

Figure 9

The rescaled correlation $LP(x,t)$ with $x/L=1/4$ as a function of $\tau$ (results for other values of $x/L$ show analogous behaviors), fixing the initial Hamiltonian parameter ${\overline{\mu }}_i=-0.1$ [(${\mathrm{a}}_1$), (${\mathrm{a}}_2$), black curves], $-0.2$ [(${\mathrm{b}}_1$), (${\mathrm{b}}_2$), red curves], and $-0.5$ [(${\mathrm{c}}_1$), (${\mathrm{c}}_2$), blue curves]. Different line styles stand for various $L$ (see legend). (${\mathrm{a}}_2$), (${\mathrm{b}}_2$), (${\mathrm{c}}_2$) Magnifications of panels (${\mathrm{a}}_1$), (${\mathrm{b}}_1$), and (${\mathrm{c}}_1$), for $6\le \tau \le 9$. Here we fix $\upsilon =0.1$, and ${\gamma }_L=1$ with dissipation given by incoherent decay. Note that the three continuous curves, corresponding to the largest size $L=256$ and different ${\overline{\mu }}_i$, are plotted in each of the six panels and cannot be distinguished on the scale of the plots reported.