Pulse and quench induced dynamical phase transition in a chiral multiferroic spin chain

Quantum dynamics of magnetic order in a chiral multiferroic chain is studied. We consider two different scenarios: ultrashort terahertz excitations or a sudden electric field quench. Performing analytical and numerical exact diagonalization calculations, we trace the pulse induced spin dynamics and extract quantities that are relevant to quantum information processing. In particular, we analyze the dynamics of the system chirality, the von Neumann entropy, and the pairwise and many-body entanglement. If the characteristic frequencies of the generated states are noncommensurate, then a partial loss of pair concurrence occurs. Increasing the system size, this effect becomes even more pronounced. Many-particle entanglement and chirality are robust and persist in the incommensurate phase. To analyze the dynamical quantum transitions for the quenched and pulsed dynamics we combined the Weierstrass factorization technique for entire functions and the Lanczos exact diagonalization method. For a small system we obtained analytical results including the rate function of the Loschmidt echo. Exact numerical calculations for a system up to 40 spins confirm phase transition. Quench-induced dynamical transitions have been extensively studied recently. Here we show that related dynamical transitions can be achieved and controlled by appropriate electric field pulses.

Figure 1

Chirality ${\kappa }_i$ as a function of the time for two-, three-, four-, and five-excitation initial states for an $L=30$ spin chain with periodic boundary conditions. The duration of the applied pulse is $\varepsilon \approx 0.3$, and its strength is $d_1=0.5$. The spin-exchange couplings and the initial electric-field strength are $J_1=-J_2=-1.0$ and $d_0=0.05$, respectively. Note that $J_2$ sets the energy units.

Figure 2

Time dependence of the averaged two-tangle (top) and the von Neumann entropy bottom) for $L=18,20,22,30,40$ (right column) and $L=4$ (left column) chain sizes. In all cases the system is in a two-excitation ground state. The duration of the applied pulse is $\varepsilon \approx 0.3$, and the strength $d_1=0.5$. The spin-exchange couplings and initial electric-field strength are $J_1=-J_2=-1.0$ and $d_0=0.05$, respectively.

Figure 3

The time dependence of the von Neumann entropy for (a) $L=4$ and (b) $L=20$ for different pulse strengths $d_1$. In all cases the initial state is a state with $L/2$ excitations (i.e., $S^z=0$ sector). The duration of the applied pulse is $\varepsilon \approx 0.3$, and the strength $d_1=0.5$. The spin-exchange couplings and initial electric-field strength are $J_1=-J_2=-1.0$ and $d_0=0.05$, respectively.

Figure 4

Quench protocol applied to a four-spin system in a two-excitation ground state. (a) Zeros ${\theta }_k=i{t}_k$ for different $k$ and $J_1=-J_2=-1.0, B=0.25, d_0=0.05$. Green squares, blue circles, black diamonds, and red points correspond to $d_1=2.5, d_1=3.5, d_1=5.49$, and $d_1=100.0$, respectively. (b) Rate function and Schmidt gap for the parameters corresponding to black diamonds in (a). The Schmidt gap acquires the maximum at times $t$ when the rate function becomes zero. (c) Rate function and Schmidt gap for the parameters corresponding to red points in (a).

Figure 5

Pulse applied to the four-spin system in a two-excitation ground state. (a) Zeros ${\theta }_k=i{t}_k$ for different $k$ and $J_1=-J_2=-1.0, B=0.25, d_0=0.001$. Green squares, red points, and black diamonds correspond to $d_1=0.3, d_1=0.5554$, and $d_1=0.8$ respectively. In the case of $d_1=0.5554$ (red points) $\text{Re}\left({\theta }_k\right)\approx 0$. (b) Rate function and Schmidt gap for $d_1=0.5554$.

Figure 6

Time evolution of the rate function $l\left(t\right)$ and Schmidt gap $\mathrm{\Delta }\left(t\right)$ for a periodic chain of size $L=22$. The peaks of the rate function at time $t^{*}\approx 9.41,28.36,47.20,\cdots$ correspond to the dynamical phase transitions. The inset shows a zoom into the cusp region at one of the nonanalytic points. The parameters are $J_1=-J_2=-1.0$. The electric fields $d_1$ and $d_0$ here are $d_1=0.44,d_0=0.057$.

Figure 7

The rate function and the Schmidt gap for a periodic chain with size $L=18$ and $J_1=-J_2=-1.0$. The pattern of the Schmidt gap and the rate function are complementary to each other. For $L=18$ the electric fields $d_1$ and $d_0$ are chosen as $d_1=2.34,d_0=0.098$, respectively.

Figure 8

The diagram of the quench-parameter pairs $(d_0+d_1)\to d_0$ for the $L=22$ site system in the two-excitation sector. The parameter region (yellow area), for which the Schmidt gap acquires zeros in time, defines pairs of quench parameters for which the dynamical phase transition occurs. We also show the quench-parameter pairs for which the rate function $l\left(t\right)$ has singularities (red asterisks and circles). The asterisks correspond to the quench between the different phases, while the circles represent quench within the same phase. Further system parameters are $J_1=-J_2=-1.0$.

Figure 9

The time evolution of the equal-time spin-spin ${\langle S_i^z{S}_{i+j}^z\rangle }_t$ (top) and the chirality ${\langle {\kappa }_i{\kappa }_{i+j}\rangle }_t={\langle {({\vec{S}}_i\times {\vec{S}}_{i+1})}^z{({\vec{S}}_{i+j}\times {\vec{S}}_{i+j+1})}^z\rangle }_t$ (bottom) correlation functions for the case shown in Fig. 6. Insets show the correlation functions vs distance $j$ (profiles of the cuts) at the singularities $t^{*}$ (indicated by dashed lines) and at local minima (indicated by dash-dotted lines) of the rate function $l\left(t\right)$.

Figure 10

The spin-spin $\langle S_i^z{S}_{i+j}^z\rangle$ (top) and the chirality $\langle {\kappa }_i{\kappa }_{i+j}\rangle =\langle {({\vec{S}}_i\times {\vec{S}}_{i+1})}^z{({\vec{S}}_{i+j}\times {\vec{S}}_{i+j+1})}^z\rangle$ (bottom) correlation functions vs $d_0$ for the $L=22$ spin system in a two-excitation ground state, $J_1=-1.0$. Insets show the correlation functions vs distance $j$ (profiles of the cuts) at $d_0=0.0$, 0.076, 0.5, 1.0 indicated by dashed lines in the main plot. One can also identify the level-crossing transition exhibited in the finite jump in the correlation functions at the transition point $d_0\approx 0.076$.