Dynamical theory for one-dimensional fermions with strong two-body losses: Universal non-Hermitian Zeno physics and spin-charge separation

We study an interacting one-dimensional gas of spin-1/2 fermions with two-body losses. The dynamical phase diagram that characterizes the approach to the stationary state displays a wide quantum Zeno region, identified by a peculiar behavior of the lowest eigenvalues of the associated non-Hermitian Hamiltonian. We characterize the universal dynamics of this Zeno regime using an approximation scheme based on an effective decoupling of charge and spin degrees of freedom, where the latter effectively evolve according to a non-Hermitian Heisenberg Hamiltonian. We present detailed results for the time evolution from initial states with one particle per site with either incoherent or antiferromagnetic spin order, showing how peculiar charge properties witnessed by the momentum distribution function build up in time.

Figure 1

Imaginary part (with opposite sign) of the eigenvalues of the non-Hermitian Hamiltonian $\tilde{H}$ in the sector with $N=L=8$ and $S_z=0$ for $U=0$ (left) and $\hslash \gamma =0.1$ (right). Green dashed vertical lines mark the boundaries of the MF region, while red dot-dashed lines mark the QZ one.

Figure 2

Left: The parameter space and the QZ region marked by $\left|\xi \right|/t>8.0$; the markers indicate the points of the phase diagram for which the Lindblad dynamics in Eq. (1) has been numerically studied. Right: Universal QZ dynamics of the density of the gas for an initial Néel state according to the Lindblad equation (1): Different markers correspond to different points in the QZ region of the parameter space. The collapse is obtained by rescaling time with $\gamma$. Simulations are performed for $L=6$.

Figure 3

Left: The parameter space and the QZ region marked by $\left|\xi \right|/t>8.0$; the markers indicate the points of the phase diagram for which the Lindblad dynamics in Eq. (1) has been numerically studied. Right: Universal QZ dynamics of the density of the gas for an initial Néel state according to the Lindblad equation (1): Different markers correspond to different points in the QZ region of the parameter space. The collapse is obtained by rescaling time with $\mathrm{\Gamma }$ in Eq. (5). Simulations are performed for $L=6$.

Figure 4

Sketch of a loss process taking place on $|{\mathrm{\Psi }}_{\mathrm{N}}\rangle$ at site $j=3$. After the action of the jump operator $L_3^{\prime }$, the state of the system is proportional to that written at the bottom. In this state, space correlations are created only for the fermions with spin down (highlighted by red), implying the value ${\delta }_{{\mathrm{\Psi }}_0}=1/2$ in (12) for the antiferromagnetic state $|{\mathrm{\Psi }}_{\mathrm{N}}\rangle$. Right: Dynamics of the number of particles $n\left(\tau \right)$ for an initial $|{\mathrm{\Psi }}_{\mathrm{N}}\rangle$ state. The dashed curves for $L=10$ and 12 are full simulations of the effective master equation for the hard-core states (6). The RE-DSC produces a very good description of the process, and the ${\tau }^{-1/3}$ is a fitting function.

Figure 5

Dynamics of the number of particles $n\left(\tau \right)$ (left) and of the spin correlation function $g^{\left(2\right)}\left(\tau \right)$ (right) for an initial ordered Néel state. Dashed lines are obtained with numerical simulations of the master equation for $L=10$ and 12. Solid green lines indicate the theoretical predictions obtained with our RE-DSC theory.

Figure 6

Momentum distribution function $n_k\left(\tau \right)$ computed from the full numerical solution of the Lindblad master equation for $L=10$ (squares) and 12 (triangles) at different times and the predictions of (RE-DSC) Eq. (12) (dashed lines). Left: Time evolution from the infinite-temperature spin state. Right: Time evolution from the Néel spin state.

Figure 7

Dynamics of the number of particles $n\left(\tau \right)$ (left) and of the spin correlation function $g^{\left(2\right)}\left(\tau \right)$ (right) for an initial maximally mixed spin state. Dashed lines are obtained with numerical simulations of the master equation for $L=10$ and 12. Solid green lines indicate the theoretical predictions obtained with our MF-DSC theory.

Figure 8

Non-Hermitian evolution for the spin-singlet projection operator for (left) a maximally mixed spin state and (right) an ordered Néel state. Dashed lines: MPS data of the non-Hermitian Heisenberg evolution. Dotted lines: Long-time behavior.

Figure 9

Time evolution of the effective spin chemical potential ${\beta }_s\left(\tau \right)$ (orange-dashed lines) compared with the time integral of the squared density (blue solid lines) for a maximally mixed spin state (left) and an ordered Néel state (right). Dotted lines represent the theoretical trends.

Figure 10

Time evolution of the parameter ${\delta }_{{\mathrm{\Psi }}_0}\left(\tau \right)$ for the maximally mixed state (blue) and the Néel state (red).