Quantum critical behaviors and decoherence of weakly coupled quantum Ising models within an isolated global system

We discuss the quantum dynamics of an isolated composite system consisting of weakly interacting many-body subsystems. We focus on one of the subsystems, $\mathcal{S}$, and study the dependence of its quantum correlations and decoherence rate on the state of the weakly-coupled complementary part $\mathcal{E}$, which represents the environment. As a theoretical laboratory, we consider a composite system made of two stacked quantum Ising chains, locally and homogeneously weakly coupled. One of the chains is identified with the subsystem $\mathcal{S}$ under scrutiny, and the other one with the environment $\mathcal{E}$. We investigate the behavior of $\mathcal{S}$ at equilibrium, when the global system is in its ground state, and under out-of-equilibrium conditions, when the global system evolves unitarily after a quench of the coupling between $\mathcal{S}$ and $\mathcal{E}$. When $\mathcal{S}$ develops quantum critical correlations in the weak-coupling regime, the associated scaling behavior crucially depends on the quantum state of $\mathcal{E}$, whether it is characterized by short-range correlations (analogous to those characterizing disordered phases in closed systems), algebraically decaying correlations (typical of critical systems), or long-range correlations (typical of magnetized ordered phases). In particular, different scaling behaviors, depending on the state of $\mathcal{E}$, are observed for the decoherence of the subsystem $\mathcal{S}$, as demonstrated by the different power-law divergences of the decoherence susceptibility that quantifies the sensitivity of the coherence to the interaction with $\mathcal{E}$.

Figure 1

Sketch of a system made of two stacked Ising chains, weakly coupled by local and homogenous interactions controlled by a parameter $\kappa$. One of the chains represents the subsystem $\mathcal{S}$, while the other one plays the role of environment $\mathcal{E}$.

Figure 2

Scaling behavior of the decoherence factor $Q$ of the chain $\mathcal{S}$ in the critical regime ($g=1$), when it is coupled with a disordered environment $\mathcal{E}$ ($g_e=2$). The data are plotted versus $K_D=\kappa L^{1/2}$. PBC are used. The results are consistent with the scaling relation (20). The inset shows that the scaling corrections at fixed $K_D=1$ decay as $L^{-1}$ (the straight line is only meant to guide the eye), consistently with the RG arguments reported in the text.

Figure 3

Scaling plot of the ratio $R_{\kappa }$, defined in Eq. (22), versus $K_D=\kappa L^{1/2}$, for $g=1$ and $g_e=2$. Data appear to converge toward a single asymptotic curve with increasing $L$, supporting the scaling behavior reported in Eq. (20). In the inset we show data for $K_D=1$: scaling corrections apparently behave as $L^{-3/4}$ (the straight line is only meant to guide the eye), as predicted by RG arguments; see text.

Figure 4

The $g\text{–}\kappa$ phase diagram when the environment is disordered in the absence of coupling, i.e., for $g_e>g_{\mathcal{I}}=1$. We report the estimates of the critical points for $g_e=4$ and some values of $\kappa$. In the inset we report analogous data for $g_e=2$. The uncertainty on the estimates is smaller than, or at the most of the order of, the size of the symbols. The critical lines are obtained by fitting the data for the smallest values of $\kappa$ (those along the full lines) to $g_c\left(\kappa \right)=1+b{\kappa }^2$, obtaining $b\approx 0.14$ for $g_e=4$, and $b\approx 1.29$ gor $g_e=2$. These results fully support the RG prediction, cf. Eq. (23). The coefficient $b$ rapidly increases when $g_e$ approaches the critical value $g_{\mathcal{I}}=1$, consistently with the asymptotic behavior $b\left(g_e\right)\sim {(g_e-1)}^{-5/2}$; see Eq. (42).

Figure 5

Scaling plot of $\chi /L^{1-\eta }$, using the 2D Ising exponent $\eta =1/4$, versus $R_{\xi }$. For each $L$, $\kappa$, and $g_e$, data are obtained by varying $g$ around the critical point $g_c\left(\kappa \right)$. They show the scaling behavior reported in Eq. (15).

Figure 6

The decoherence function $Q$ of the subsystem $\mathcal{S}$ for $g=g_e=1$ (correspondingly, $r=r_e=0$). The data show an excellent FSS when plotted versus $K=\kappa L^{7/4}$, confirming the RG analysis. The inset shows the data at fixed $K=1$: size corrections apparently decay as $L^{-2}$, as predicted by the RG arguments (the line is only drawn to guide the eye).

Figure 7

Scaling behavior of the ratio $R_{\kappa }$, defined in Eq. (22), for $g=g_e=1$ (corresponding to $r=r_e=0$). The data show an asymptotic FSS when plotted versus $K=\kappa L^{7/4}$, confirming the RG analysis. The inset shows the data at fixed $K=1.5$: size corrections decay as $L^{-3/4}$, as expected from the RG arguments (the line is only drawn to guide the eye).

Figure 8

Plot of the ratio $R_{\mathrm{\Delta }}$ defined in Eq. (46) for $\kappa =0.2$. Top: results for $s=1$; bottom: results for $s=0$. Results for $s=1$ show a clear crossing point for $g\approx 1.31$, indicating the presence of a transition. For $s=0$ the ratio $R_{\mathrm{\Delta }}$ is always smaller than 1 and does not show any crossing, indicating that no transition occurs.

Figure 9

Some estimates of the critical points $g_c\left(\kappa \right)$ for $s\equiv r_e/r=2$ and $s=1$ (inset). The behavior for small $\kappa$ is in agreement with the scaling prediction, Eq. (40). The full lines are obtained by fitting the data of $g_c\left(\kappa \right)$ for small values of $\kappa$ ($\kappa \lesssim 0.03$) to $g_c\left(\kappa \right)=1+w{\kappa }^{\varepsilon }$ with $\varepsilon =4/7$. We obtain $w\approx 0.53$ for $s=2$ and $w\approx 0.76$ for $s=1$. Note that $w$ decreases with increasing $s$, as expected on the basis of the asymptotic behavior (43).

Figure 10

Decoherence function $Q$ versus $K_O=\kappa L^{y_h}$, for $g=g_{\mathcal{I}}=1$ and $g_e=0.5$ (ordered environment). In the inset, we show data for $K_O=0.6$: size corrections decay as $L^{-2}$ in agreement with RG arguments (the line is drawn to guide the eye).

Figure 11

Scaling plot of the ratio $R_{\kappa }$, defined in Eq. (22), versus $K_O=\kappa L^{y_h}$, for $g=g_{\mathcal{I}}=1$ and $g_e=0.5$ (ordered environment). In the inset, we show data at fixed $K_O=0.8$: size corrections decay as $L^{-3/4}$ in agreement with RG arguments (the line is drawn to guide the eye).

Figure 12

Plot of the ratio $E_{\mathrm{ex}}/{\mathrm{\Delta }}_{\mathcal{I}}\left(L\right)$—the energy exchange is defined in Eq. (58)—and of the decoherence factor $Q$, defined in Eq. (59), versus $\mathrm{\Theta }$. ${\mathrm{\Delta }}_{\mathcal{I}}\left(L\right)\sim L^{-z}$ is the gap of the Ising chain (2) at $g=g_{\mathcal{I}}$. For each $L$, we fix $g$, $g_e$, and $\kappa$, so that the equilibrium scaling variables defined in Eq. (31) take the values $W=1$, $W_e=0.5$, and $K=10$ (therefore, as $L$ increases, $g,g_e\to 1$ and $\kappa \to 0$). The collapse of the results for different sizes on a single curve supports the dynamic FSS relations Eqs. (62) and (64).