Noise-driven dynamics and phase transitions in fermionic systems

We study abrupt changes in the dynamics and/or steady state of fermionic noise-driven systems produced by small changes in the system parameters. Specifically, we consider fermionic systems whose dynamics is described by master equations that are quadratic (and, under certain conditions, quartic) in creation and annihilation operators. We analyze phase transitions in the steady state as well as “dynamical transitions.” The latter are characterized by abrupt changes in the rate at which the system asymptotically approaches the steady state. We illustrate our general findings with relevant examples of fermionic (and, equivalently, spin) systems and show that they can be realized in ion chains.

Figure 1

The poles $z^0,z^{\pm }$ [see Eq. (83)] are plotted for $J=1,\gamma =\pm 0.1$. As $B$ is changed from 0 to 20, the poles $z^{+}\left(z^{-}\right)$ for positive anisotropy $\gamma =+0.1$ move along the blue (red) solid curves, and $z^{+}$ crosses the contour at the critical value $B=2J$. For negative $\gamma =-0.1,z^{-}$ crosses at $B=2J$. At the crossing, the integral Eq. (82) changes nonanalytically.

Figure 2

Asymptotic decay rate $\Delta$ [see Eq. (78)] of the $XY$ chain (12) for different anisotropy parameters $\gamma$ as a function of the magnetic field in the limits $N\to \infty$ and $g\to 0$. A phase transition in $\Delta$ is visible at $B=2J$ for $\gamma \ne 0$.

Figure 3

Asymptotic decay rate $\Delta$ [see Eq. (78)] of the $XY$ chain (12) for different coupling strengths $g,\gamma =1$, and $N=100$ as a function of the magnetic field $B$. For $g\le 0.1{\left(J/\hslash \right)}^{0.5}$, the results agree with the limit of weak coupling $g\to 0$ [see Eq. (89)].

Figure 4

Asymptotic decay rate $\Delta$ [see Eq. (89)] of the $XY$ chain (12) for different system sizes $N$ and $\gamma =1,g=0.01{\left(J/\hslash \right)}^{0.5}$ as a function of the magnetic field $B$. For $N\ge 50$, the thermodynamic limit is reached, except for small variations at the phase transition $B=2J$.

Figure 5

Evolution of the magnetization $\langle {\sigma }_z^j\rangle$ in time starting from the system ground state of the $XY$ chain (12) for different magnetic fields $B,g=0.01{\left(J/\hslash \right)}^{0.5}$, and $\gamma =1$. The magnetization decreases exponentially in time.

Figure 6

The ADRs $\Delta$ [see Eq. (89)] of the $XY$ chain (12) for $\gamma =1$, for $g=0.01{\left(J/\hslash \right)}^{0.5}$, and for $g\to 0$ (result of perturbation theory) as a function of the magnetic field $B$ are compared with the late-time decoherence rates extracted from Fig. 5. The agreement between the ADR and the late-time decoherence rate shows the validity of our calculations for finite times.