Topological and dynamical features of periodically driven spin ladders

Studies of periodically driven one-dimensional many-body systems have advanced our understanding of complex systems and stimulated promising developments in quantum simulation. It is hence of interest to go one step further, by investigating the topological and dynamical aspects of periodically driven spin ladders as clean quasi-one-dimensional systems with spin-spin interaction in the rung direction. Specifically, we find that such systems display subharmonic magnetization dynamics reminiscent to that of discrete time crystals (DTCs) at finite system sizes. Through the use of generalized Jordan-Wigner transformation, this feature can be attributed to the presence of corner Majorana $\pi$ modes (MPMs), which are of topological origin, in the systems' equivalent Majorana lattice. Special emphasis is placed on how the coupling in the rung direction of the ladder prevents degeneracy from occurring between states differing by a single spin excitation, thus preserving the MPM-induced $\pi /T$ quasienergy spacing of the Floquet eigenstates in the presence of parameter imperfection. This feature, which is absent in their strict one-dimensional counterparts, may yield fascinating consequences in future studies of higher dimensional Floquet many-body systems.

Figure 1

Schematic of various terms in Eqs. (1) and (3) when written in terms of Majorana operators. There, each term is a product of all Majorana operators contained in its respective dotted line.

Figure 2

(a) Phase diagram of the periodically driven Ising spin chain Eq. (1) in the 1D limit, i.e., $N_x=1$. (b) Stroboscopic magnetization dynamics at $N_x=1, J_y=0.6\pi /T$, and $h=0.8\pi /T$. (c) The subharmonic response of the system's average magnetization at $\mathrm{\Omega }=\pi /T, J_x=0.05\pi /T, J_y=0.6\pi /T$, and various lattice configurations. There, the system is initialized in a product state $|\uparrow \cdots \uparrow \rangle$ and the average magnetization is evolved up to $2000T$. (d) Perturbation-induced degeneracy splitting may break the $\pi /T$ quasienergy spacing structure.

Figure 3

The four corner spectral functions (with $\chi =16$ and $\epsilon =0.01$) plotted as a function of (we set $T=2$) [(a) and (c)] $h$ and [(b) and (d)] $J\equiv J_y$. The system size is taken as $4\times 2$ and the other systems parameters are $J_x=0.05\pi /2$, [(a) and (c)] $J_y=0.3\pi$ and [(b) and (d)] $h=0.4\pi$.

Figure 4

Quasienergy levels (and their $\pi /T$-shifted values) associated with system configurations [(a) and (b)] $N_x,N_y=1,4$ and [(c) and (d)] $N_x,N_y=2,2$. We take [(a) and (c)] a solvable parameter value of $h=\pi /T$ associated with perfect $2T$ magnetization dynamics and [(b) and (d)] a perturbed value of $h=0.85\pi /T$.

Figure 5

The stroboscopic magnetization dynamics [(a) and (c)] and its associated power spectrum (b, d) under the initial state of $|\uparrow \downarrow \uparrow \cdots \uparrow \rangle$. (a), (b) and (c), (d) correspond to the configurations $N_x,N_y=4\times 2$ and $1\times 8$, respectively. All parameter values are taken as $J_x=0.05\pi /T, h=0.85\pi /T$, and $J_y=2/T$.

Figure 6

[(a) and (b)] The subharmonic response of the system's average magnetization at $\mathrm{\Omega }=\pi /T$ and various lattice configurations under two different choices of initial states (a) $|\uparrow \downarrow \uparrow \cdots \uparrow \rangle$ and (b) $|\nearrow \cdots \nearrow \rangle$. [(c) and (d)] The stroboscopic evolution of the average magnetization in the ${\widehat{n}}_{\pi /8}$ direction and its associated power spectrum for the initial state $|\nearrow \cdots \nearrow \rangle$. There, green/black, magenta/cyan, and red/blue respectively correspond to system sizes $1\times 16, 4\times 4$, and $8\times 2$. In all panels, the other systems are chosen as $J_x=0.05\pi /T, J_y=0.6\pi /T$, and [(c,d)] $h=1.225\pi /T$.

Figure 7

[(a)–(c)] The stroboscopic evolution of the system's average magnetization in the ${\widehat{n}}_{\pi /8}$ direction for a spin ladder of size (a) $8\times 2$, (b) $4\times 4$, and (c) $1\times 16$. Blue (red) points are evaluated at odd (even) integer multiples of $T$, and the system is initially prepared in the $|\nearrow \cdots \nearrow \rangle$. [(d)–(f)] The associated power spectrum. System parameters are chosen as $J_x=0.05\pi /T, J_y=0.6\pi /T$, and $h=\pi /T$.

Figure 8

The stroboscopic evolution of the system's average magnetization in the ${\widehat{n}}_{\pi /8}$ direction for spin ladders of size [(a) and (d)] $8\times 2$, [(b) and (e)] $4\times 4$, and [(c) and (f)] $1\times 16$ at [(a)–(c)] $h=0.9\pi /T$ and [(d)–(f)] $1.1\pi /T$. Blue (red) points are evaluated at odd (even) integer multiples of $T$, and the system is initially prepared in the $|\nearrow \cdots \nearrow \rangle$. Other system parameters are the same as those in Fig. 7.