Melting of the critical behavior of a Tomonaga-Luttinger liquid under dephasing

Strongly correlated quantum systems often display universal behavior as, in certain regimes, their properties are found to be independent of the microscopic details of the underlying system. An example of such a situation is the Tomonaga-Luttinger liquid description of one-dimensional strongly correlated bosonic or fermionic systems. Here, we investigate how such a quantum liquid responds under dissipative dephasing dynamics and, in particular, we identify how the universal Tomonaga-Luttinger liquid properties melt away. Our study, based on adiabatic elimination, shows that dephasing first translates into the damping of the oscillations present in the density-density correlations, a behavior accompanied by a change in the Tomonaga-Luttinger liquid exponent. This first regime is followed by a second one characterized by the diffusive propagation of featureless correlations as expected for an infinite temperature state. We support these analytical predictions by numerically exact simulations carried out using a number-conserving implementation of the matrix product states algorithm adapted to open systems.

Figure 1

Spin correlations along the $z$ direction ${\mathcal{C}}_d\left(t\right)$ for a system of length $L=2000$, filling $\overline{n}=1/8$, initial interaction strength $V=J$, and dissipative strength $\hslash \gamma =5J$ corresponding to ${\hslash }^{2}D=0.2J^2$. The curves represent (i) $Dt=0$, (ii) $Dt=0.34$, (iii) $Dt=0.95$, (iv) $Dt=2.44$, (v) $Dt=11.24$, (vi) $Dt=62.98$, and (vii) $Dt=995.54$. The inset: the same labeling as in the main figure, curves (i)–(vi) are ordered from top to bottom, curve (vii) is not shown. The spatial oscillations of the Tomonaga-Luttinger correlations are damped under the effect of dephasing. The correlations are obtained by numerically solving the set of differential equations, Eq. (5).

Figure 2

Spatial scaling of the spin correlations along the $z$ direction ${\mathcal{C}}_d\left(t\right)$ for short to intermediate times. The same system parameters as in Fig. 1 are used. In (a), the numerical solution (solid line) is compared to the analytical solution (circle markers) Eq. (6). In (b) and (c), the numerical solution (solid line) is compared to the simplified expression Eq. (8) (diamond markers).

Figure 3

Appearance, at longer times, of a second regime characterized by spatially uniform correlations. The same system parameters as in Fig. 1 are used. The curves are labeled in order of appearance at $d=1$ from top to bottom. The value of the correlation plateau scales as $t^{-1/2}$, and, beyond the propagation front, the correlations scale as $d^{-2}$. For $Dt=11.24$, one sees that, in this second regime, the correlations obtained from Eq. (6) (circle markers) also agree perfectly with the numerical solution (solid line). In addition, for $Dt=995.54$, the value on the correlation plateau is shown to be in very good agreement with the approximated expression found in Eq. (9) (horizontal dashed line).

Figure 4

Evolution of the spin correlations along the $z$ direction ${\mathcal{C}}_d\left(t\right)$ as a function of time for $3\le d\le 50$ (from the right, top to bottom). The same system parameters as in Fig. 1 are used. For each distance, a circle marks the time $t_{max}$ at which the correlations are maximum. This maximum corresponds to the passage of the propagation front separating the regions displaying uniform and algebraically decaying correlations. The inset: the front propagates diffusively as $d\sim \sqrt{D{t}_{max}}$.

Figure 5

Local density $\langle {\widehat{n}}_l\rangle$ versus site number $l$ for different times: $tJ=0$ (blue dashed line), $tJ=3\hslash$ (orange dashed-dotted line), and $tJ=5\hslash$ (violet solid line). Panels (a) and (b) are for filling $\overline{n}=1/8(L=88,N=11),V=0$, and $V=J$; (c) and (d) are for filling $\overline{n}=1/4(L=84,N=21),V=0$, and $V=J$. For all panels, $\hslash \gamma =0.05J$ corresponding to ${\hslash }^{2}D=20J^2$. The amplitude of the density oscillations are damped due to dephasing. Results obtained via MPS simulations with $\chi =8000,dtJ=0.01\hslash$, and cutoff $\varepsilon ={10}^{-7}$.

Figure 6

Rescaled local density ${\langle {\widehat{n}}_l\rangle }_{\mathrm{rscd}}$ versus site number $l$, for different times: $tJ=0$ (blue dashed line), $tJ=3\hslash$ (orange dashed-dotted line), and $tJ=5\hslash$ (violet solid line). Panels (a) and (b) are for filling $\overline{n}=1/8(L=88,N=11),V=0$, and $V=J$; (c) and (d) are for filling $\overline{n}=1/4(L=84,N=21),V=0$, and $V=J$. For all panels $\hslash \gamma =0.05J$. For all system parameters and times considered, the bulk of the evolved profiles can be collapsed into the corresponding initial density profile. Results obtained via MPS simulations using the same parameters as in Fig. 5.

Figure 7

Density-density correlation $\langle {\widehat{n}}_l{\widehat{n}}_{l+d}\rangle -\langle {\widehat{n}}_l\rangle \langle {\widehat{n}}_{l+d}\rangle$ as a function of distance $d(l=37)$ for $L=88,N=11,V=J$, and $\hslash \gamma =0.05J$. Times considered: $tJ=0$ (blue dashed line), $tJ=3\hslash$ ($Dt=60$, orange dashed-dotted line), $tJ=4\hslash$ ($Dt=80$, yellow dotted line), and $tJ=5\hslash$ ($Dt=100$, violet solid line). The inset: the spatial oscillations of the Tomonaga-Luttinger correlations are damped under the effect of dephasing. Results obtained via MPS simulations using the same parameters as in Fig. 5.

Figure 8

Absolute value of the initial density-density correlations $|\langle {\widehat{n}}_l{\widehat{n}}_{l+d}\rangle -\langle {\widehat{n}}_l\rangle \langle {\widehat{n}}_{l+d}\rangle |$ as a function of distance $d$ for three different locations. Panels (a) and (b) are for filling $\overline{n}=1/8(L=88,N=11),V=0$, and $V=J$; (c) and (d) are for filling $\overline{n}=1/4(L=84,N=21),V=0$, and $V=J$. As well known from Tomonaga-Luttinger theory, the exponent $\eta$ in $d^{-1/\eta }$ is a function of the filling and interaction strength. Results obtained from MPS simulations using the same parameters as in Fig. 5.

Figure 9

Absolute value of the density-density correlations $|\langle {\widehat{n}}_l{\widehat{n}}_{l+d}\rangle -\langle {\widehat{n}}_l\rangle \langle {\widehat{n}}_{l+d}\rangle |$ as a function of distance $d$ for $\overline{n}=1/8(L=88,N=11)$ at three times: $tJ=0$ (blue dashed line), $tJ=3\hslash$ (orange dashed-dotted line), and $tJ=5\hslash$ (violet solid line). Panel (a): $V=0$, panel (b): $V=J$. For all panels, $\hslash \gamma =0.05J$. Results obtained via MPS simulations using the same parameters as in Fig. 5.

Figure 10

Absolute value of the density-density correlations $|\langle {\widehat{n}}_l{\widehat{n}}_{l+d}\rangle -\langle {\widehat{n}}_l\rangle \langle {\widehat{n}}_{l+d}\rangle |$ as a function of distance $d$ for $\overline{n}=1/4(L=84,N=21)$ at three times: $tJ=0$ (blue dashed line), $tJ=3\hslash$ (orange dashed-dotted line), and $tJ=5\hslash$ (violet solid line). Panel (a): $V=0$, panel (b): $V=J$. For all panels, $\hslash \gamma =0.05J$. Results obtained via MPS simulations using the same parameters as in Fig. 5.