Entanglement and level crossings in frustrated ferromagnetic rings

We study the entanglement content of a class of mesoscopic tunable magnetic systems. The systems are closed finite spin-1/2 chains with ferromagnetic nearest-neighbor interactions frustrated by antiferromagnetic next-nearest-neighbor interactions. The finite chains display a series of level crossings reflecting the incommensurate physics of the corresponding infinite-size chain. We present dramatic entanglement signatures characterizing these unusual level crossings. We focus on multispin and global measures of entanglement rather than only one-spin or two-spin entanglements. We compare and contrast the information obtained from these measures to that obtained from traditional condensed-matter quantities such as correlation functions.

Figure 1

(Color online) [(a)–(c)] Various partitions of closed chain for calculating bipartite entanglement entropy. Spins belonging to partition $A$ $\left(B\right)$ are represented by open (full) circles. (d) Geometry for exploiting the sublattice entanglement $S_{\text{s}\text{.l}\text{.}}$, e.g., in an experimental realization. In this setup $S_{\text{s}\text{.l}\text{.}}$ is simply the entanglement between inner and outer rings. Dashed and full lines represent $J_1$ and $J_2$ interactions, respectively.

Figure 2

(Color online) Correlation functions and related quantities for $N=16$. (a) Correlation function at $\overline{\beta }=2$. (b) The structure factor, $S\left(q\right)$, at $\overline{\beta }=2$. (c) Position of the maximum of $S\left(q\right)$, as a function of $\overline{\beta }$. (d) Concurrences between two spins at distance $r=2,3,8$. (e) Entropy of entanglement between partition consisting of sites $\left(i,i+r\right)$ and the remaining $N-2$ sites. Shown also for $r=2$, 3, 8. In (c)–(e), nonlinear horizontal scale highlights the region near $\overline{\beta }=4$.

Figure 3

(Color online) 16-spin and 20-spin rings. Top: half-block and sublattice entanglements, $S_{1/2}$ (upper dashed), $S_{\text{s}\text{.l}\text{.}}$ (solid), $S_{2/4}$ (lower dashed, magnified), against $\overline{\beta }$ in the singlet region. For the $N=20$ curves, smaller values of $\overline{\beta }$ are omitted to focus on the level crossings. Middle and bottom: derivatives of $S_{1/2}\left(\overline{\beta }\right)$ and $S_{\text{s}\text{.l}\text{.}}\left(\overline{\beta }\right)$, respectively.

Figure 4

(Color online) Fidelity $F_{ϵ}\left(\overline{\beta }\right)$ with $ϵ={10}^{-3}$, and its derivative. Since $F$ is generally very close to unity, it is more convenient to plot $1-F$ on a logarithmic scale. Note that the discontinuity at the leftmost level crossing is pronounced in the fidelity but almost invisible in its derivative.

Figure 5

(Color online) Left panel shows sublattice entanglement entropy for $N=12$. Center (rightmost) panel shows evolution of $S_{\text{s}\text{.l}\text{.}}$ after a quench, without (with) dissipation. Time is expressed in units ${\left|J_1\right|}^{-1}$. Initial values are ${\overline{\beta }}_i=0.5$ (solid), ${\overline{\beta }}_i=3.5$ (dotted), and ${\overline{\beta }}_i=3.85$ (dashed). Final value is ${\overline{\beta }}_f=2.0$ in each case. Damping constant in rightmost panel is shown for $\gamma =0.5$ for all three ${\overline{\beta }}_i$. For ${\overline{\beta }}_i=0.5$, the thick solid line has smaller damping, $\gamma =0.1$. In the presence of dissipation, $S_{\text{s}\text{.l}\text{.}}$ relaxes to the ground-state value when the initial state has same lattice momentum $\left(k=\pi \right)$ but not when the initial state has different lattice momentum $\left(k=0\right)$.