Open Heisenberg chain under boundary fields: A magnonic logic gate

We study the spin transport in the quantum Heisenberg spin chain subject to boundary magnetic fields and driven out of equilibrium by Lindblad dissipators. An exact solution is given in terms of matrix product states, which allows us to calculate exactly the spin current for any chain size. It is found that the system undergoes a discontinuous spin-valve-like quantum phase transition from ballistic to subdiffusive spin current, depending on the value of the boundary fields. Thus, the chain behaves as an extremely sensitive magnonic logic gate operating with the boundary fields as the base element.

Figure 1

Schematic drawing of the dissipators $D_{L,R}\left(\rho \right)$ (black solid arrows) and the boundary fields $h$ (red dashed arrows) acting on the first and last spins of the chain. In (a) the fields are in the same direction as the dissipators $\left(h>0\right)$, and in (b) the fields act in directions opposite to the dissipators $\left(h<0\right)$.

Figure 2

Spin current $J$ for $f=1,\theta =0$, and $h=0$. (a) $J$ vs $\gamma$ for different sizes $N$. (b) $J$ vs $N$ for different values of $\gamma$. The dotted black line has a slope of $-2$.

Figure 3

Spin current $J$ as a function of the boundary fields $h$ with $f=1$ and $\theta =0$. (a) $N^{2}J$ vs $h$ for $\gamma =1$ and different values of $N$. (b) $J/\gamma$ vs $h$ for $N=100$ and different values of $\gamma$ around ${\gamma }^{*}=1/N=0.01$. (c) and (d) $J/\gamma$ vs $h$ for $\gamma ={10}^{-5}$ and different values of $N$. The dashed lines in (d) correspond to Eq. (24).

Figure 4

Spin current $J$ vs $h$ when $f<1$, computed using the exact diagonalization of Eq. (2) for $N=6$. (a) $\gamma ={10}^{-5}$. (b) $\gamma =1$. The dotted black lines correspond to the MPS solution when $f=1$.

Figure 5

Small size effects in the spin current. (a) $J/\gamma$ vs $Nh$ for $\gamma ={10}^{-5}$ and different values of $N$. (b) $J/\gamma$ vs $Nh$ for $N=15$ and different values of $\gamma$.

Figure 6

$J$ vs $h$ for $N=500$ and different values of $\theta$ (as defined in Fig. 1). (a) $\gamma ={10}^{-4}$ and (b) $\gamma =1$.

Figure 7

Magnon density profile $\langle n_i\rangle =(1+\langle {\sigma }_i^z\rangle )/2$ for different values of $h$, with $N=500,\gamma ={10}^{-5}$, and $\theta =0$. (a) $h<0$ near the plateau transition [see Fig. 3]. (b) $h>0$.