Strong-disorder renormalization group for periodically driven systems

Quenched randomness can lead to robust nonequilibrium phases of matter in periodically driven (Floquet) systems. Analyzing transitions between such dynamical phases requires a method capable of treating the twin complexities of disorder and discrete time-translation symmetry. We introduce a real-space renormalization group approach, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.

Figure 1

Top: A generic configuration of the disordered periodically driving Ising chain in the Majorana fermion representation. Circles represent Majorana fermion operators, single lines represent bilinear couplings ${\epsilon }_i$ with ${\epsilon }_i\in [-\pi /2,\pi /2]$, and double lines represent bilinear couplings $\pi +{\epsilon }_j$ with ${\epsilon }_j$ in the same range. The chain has domains with couplings near 0 (0 domains) and with couplings near $\pi$ ($\pi$ domains). Bottom: We can factor out the exact $\pi$ pulses to form one long-ranged exact $\pi$ pulse spanning the $\pi$ domain, with all couplings now in the range $[-\pi /2,\pi /2]$.

Figure 2

Decimation inside a 0 domain or a $\pi$ domain follows the same rule. (a) Within a 0-domain, a strong bond $\mathrm{\Omega }$ is renormalized via the rule $\tilde{J}=J_L{J}_R/tan\mathrm{\Omega }$. (b) Within a $\pi$ domain, the long-ranged $\pi$ pulse does not affect the unitary rotation, giving the same virtual coupling. We can view decimation within a $\pi$ domain as decimation toward a $\pi$ bond by replacing the long-ranged $\pi$ pulse with a product of short-ranged ones.

Figure 3

(a) An exact transformation of each domain wall reveals that we can view the $\pi$ and 0 puddles as being connected by second-order terms (see below for details). We will never decimate these because $J_R\mathrm{\Omega }<J_R$ and $J_L\mathrm{\Omega }<J_L$, so at every step $J_L$ or $J_R$ dominate. (b) The chain after the exact transformations of factoring out $\pi$-pulses and rotating the Majoranas at every domain wall. Approximating $cos\mathrm{\Omega }\approx 1,sin\mathrm{\Omega }\approx \mathrm{\Omega }$ shows that the puddles are decoupled at leading order.

Figure 4

(a) An even-length domain, after it has been decimated down to only two bonds, can be rotated into this two-chain form without decimating. (b) When decimating the central bond in the setup above, every three-step pathway through the decimated bond gets renormalized according to $M_L{M}_R/tan\mathrm{\Omega }\approx M_L{M}_R/\mathrm{\Omega }$, where $M_{L,R}$ are the bonds to the left and right, respectively. This type of decimation occurs with odd-length domains.

Figure 5

After the 0 and $\pi$ domains have been decimated, we are left with a configuration of similar form to the chain in step one. As we decimate the central couplings of each puddle, we renormalize to the chain in step two, and so on. This leads to two linked chains.

Figure 6

Cartoon of the four possible phases and their transitions as deduced from the strong-disorder RG flow to two decoupled Majorana chains near 0 and $\pi$ quasienergy. The phases are controlled by the dimerization of these chains ${\delta }_{0,\pi }$. The PM has both chains trivially dimerized; the SG has the 0 chain topologically dimerized and the $\pi$ chain trivially dimerized; the $\pi \mathrm{SG}$ has the 0 chain trivially dimerized and the $\pi$ chain topologically dimerized; and the $\pi \mathrm{SG}$ has both chains topologically dimerized (insets). Both chains can be independently tuned to Ising-class criticality (yellow and red lines), resulting in a twice-Ising multicritical point (orange star).