Relaxation times of dissipative many-body quantum systems

We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than $\sim 1/L$. In integrable systems with boundary dissipation one typically observes scaling of $\sim 1/L^3$, while in chaotic ones one can have faster relaxation with the gap scaling as $\sim 1/L$ and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.

Figure 1

The Liouvillian superoperator for the $\mathit{\text{XX}}$ chain with dephasing is equivalent to a non-Hermitian ladder Hamiltonian, with $\mathit{\text{XX}}$ interaction along upper and lower legs (thin lines) and an imaginary $z\text{−}z$ interaction due to dephasing along rungs (double lines).

Figure 2

The gap $g$ of the $\mathit{\text{XX}}$ model with dephasing $\gamma =0.5$. Shown is the exact gap (symbols), the full curve is the approximate formula (8), while the dotted curve shows the asymptotic expression $2{\pi }^2/\left(\gamma L^2\right)$. The approximate result (8) is accurate for $L$ larger than the transition point $L_{\mathrm{c}}:=\pi \sqrt{2}/\gamma$.

Figure 3

Phase diagram of the largest decay mode. (a) Dependence of gap $g/\gamma$ on dephasing and $L$. Full black line is the critical ${\gamma }_c$ (9) delimiting two phases. In (b) and (c) are shown expansion coefficients $|c_{k,j}|$ of the largest decay mode $x,x={\sum }_{k,j=1}^L{c}_{k,j}|k\rangle \langle j|$, showing an almost-diagonal dephasing-dominated phase with $g\sim 1/L^2$ in (b), and a delocalized phase with $g\sim 1/L^0$ in (c).

Figure 4

Complex eigenvalues of $\mathcal{L}$ for $\mathit{\text{XX}}$ model with bulk dephasing, $\gamma =1,L=50$, and one-particle sector. For $\gamma >{\gamma }_c$ there are of order $\sim L$ eigenvalues that are on the real axis and are separated from the bulk consisting of the remaining $\sim L^2$ complex eigenvalues. Real parts of the bulk eigenvalues are around $-4\gamma$ with the width being $\sim \gamma /L$.

Figure 5

The gap $g$ in the one-particle sector of the $\mathit{\text{XXZ}}$ chain with dephasing, $\gamma =1$. Symbols show results of numerical diagonalization for anisotropies $\mathrm{\Delta }=0.5$ and $2.0$, while the full line is $2{\pi }^2/L^2$, giving the asymptotic gap, which is equal to the one in the $\mathit{\text{XX}}$ chain with dephasing (8).

Figure 6

The largest nonzero eigenvalue $c_1$ of matrix $R$ (12) in the $\mathit{\text{XXZ}}$ chain with dephasing, determining the gap Eq. (13). Full line is $1/L^{0.8}$, suggesting the asymptotic dependence for $\mathrm{\Delta }>1$.

Figure 7

The largest nonzero eigenvalue in the even or odd sector for the $\mathit{\text{XXZ}}$ chain with dephasing, all for $L=8$. Insets show dependence of the critical ${\gamma }_c$ that determines validity of perturbation series. Left: Gapless regime of $\mathrm{\Delta }=0.5$ for which ${\gamma }_c$ depends on $L$ algebraically (dashed curve in the inset is $\sim 1/L^{1.2})$. Right: Gapped regime $\mathrm{\Delta }=2.0$ for which ${\gamma }_c\asymp exp(-kL)$ (inset).

Figure 8

Gap for the $\mathit{\text{XXZ}}$ chain with dephasing in the half-filling sector, $\gamma =1$. Full lines denote asymptotic $\sim 1/L^2$ scaling for all $\mathrm{\Delta }$.

Figure 9

Relative error of the gap calculated from perturbative $H_{\mathrm{eff}}$ (15), $\gamma =1$. For large $\mathrm{\Delta }$ the error does not go to zero, signifying that the second-order perturbation theory in large $\mathrm{\Delta }$ fails.

Figure 10

The gap $g$ in the one-particle sector of the $\mathit{\text{XX}}$ chain with an incoherent one-way hopping (16). The kink at $L=10$ is because the eigenvalue responsible for $g$ goes from being complex to real.

Figure 11

Left: Eigenvalues of $\mathcal{L}$ in the one-particle sector for the $\mathit{\text{XX}}$ chain with boundary dephasing of strength $\gamma =0.01$. Three crosses are eigenvalues that have the largest real part and give $g$ (18). Right: Numerical $g\left(\gamma \right)$ (symbols) for $L=40$ and prediction by Eq. (19) (full curve).

Figure 12

One-particle sector of the $\mathit{\text{XXZ}}$ chain with boundary dephasing. (a) Numerically calculated gap $g$ for small $\gamma =0.01$ and the theory [full curves, Eq. (21)]. (b) $g\left(\gamma \right)$ always scales as $\sim 1/L^3$; symbols are numerical values for two different sizes and $\mathrm{\Delta }$; full curves are Eq. (22).

Figure 13

Half-filling sector of the $\mathit{\text{XXZ}}$ chain with boundary dephasing. (a) The largest nonzero eigenvalue of $R$, Eq. (20), determining the gap for small $\gamma$ (13). Asymptotic behavior is denoted by dotted line (=139/$L^3)$ and full curve $\mathrm{[}\sim exp(-1.32L)]$. (b) The gap for nonsmall $\gamma =1$. Asymptotically, the scaling is the same as for small $\gamma$, namely the dotted line is $80/L^3$ $\left(\mathrm{\Delta }=0.5\right)$ and the full curve is $\sim exp(-1.32L)$ $\left(\mathrm{\Delta }=2.0\right)$.

Figure 14

The gap $g$ for the $\mathit{\text{XXZ}}$ model and boundary driving (23) with $\mu =0.1$ and $\mathrm{\Gamma }=1$. Straight dashed line suggests asymptotic scaling $g\propto 1/L^3$.

Figure 15

The $\mathit{\text{XXZ}}$ chain with magnetization driving and small dissipation $\mathrm{\Gamma }$. (a) The largest nonzero eigenvalue $c_1$ of $R$ [Eq. (24)] determining the gap for small $\mathrm{\Gamma }$. Full line suggests asymptotic scaling $\sim 1/L^3$. (b) Dependence of exact gap on $\mathrm{\Gamma }$ and perturbative $c_1$ from frame (a), all for $\mathrm{\Delta }=1.5$ (for $\mathrm{\Delta }=0.5$ behavior is similar). Deviations from $c_1$ begin at an $L$-independent $\mathrm{\Gamma }$.

Figure 16

The gap $g$ for the staggered $\mathit{\text{XXZ}}$ chain with boundary dephasing $\left(\gamma =1\right)$ in the one-particle sector. Dashed black lines are Eq. (22), having asymptotic decay $\sim 1/L^3$.

Figure 17

Staggered $\mathit{\text{XXZ}}$ chain with $\mathrm{\Delta }=0.5$ and weak boundary dephasing. (a) The largest nonzero eigenvalue of $R$, Eq. (20), determining the gap for small $\gamma$. Asymptotic behavior (full line) is $\approx 3.5/L$. (b) With increasing size the convergence radius ${\gamma }_c$ up to which perturbation theory holds decreases.

Figure 18

The gap $g$ for the staggered $\mathit{\text{XXZ}}$ chain with boundary dephasing $\left(\gamma =1\right)$ in the half-filled sector. The full black line in the main plot is $1/L$; in the inset it is $\propto exp(-1.5L)$. Full symbols are obtained by exact diagonalization, empty with tDMRG.

Figure 19

Staggered $\mathit{\text{XXZ}}$ chain with weak boundary magnetization driving. (a) The largest nonzero eigenvalue of $R$, Eq. (24), determining the gap for small $\mathrm{\Gamma }$, scales as $\sim 1/L$. (b) Agreement between perturbative $c_1$ from (a) with the exact gap $g\left(\mathrm{\Gamma }\right)$, all for $\mathrm{\Delta }=0.5$. Convergence radius ${\mathrm{\Gamma }}_c$ decreases with $L$.

Figure 20

The gap $g$ for the chaotic $\mathit{\text{XXZ}}$ model with staggered field (26) and boundary magnetization driving (23), $\mu =0.1,\mathrm{\Gamma }=1$. Full symbols are exact diagonalization while empty are obtained from tDMRG. Straight lines suggest $\propto 1/L$ and $\propto 1/L^2$ asymptotic behavior.

Figure 21

Gap $g$ for the tilted Ising model (27) with boundary dephasing of strength $\gamma =1$. Points are numerical calculations: full squares for exact diagonalization and empty squares for tDMRG.