Fine structure of heating in a quasiperiodically driven critical quantum system

We study the heating dynamics of a generic one-dimensional critical system when driven quasiperiodically. Specifically, we consider a Fibonacci drive sequence comprising the Hamiltonian of uniform conformal field theory (CFT) describing such critical systems and its sine-square deformed counterpart. The asymptotic dynamics is dictated by the Lyapunov exponent which has a fractal structure embedding Cantor lines where the exponent is exactly zero. Away from these Cantor lines, the system typically heats up fast to infinite energy in a nonergodic manner where the quasiparticle excitations congregate at a small number of select spatial locations resulting in a buildup of energy at these points. Periodic dynamics with no heating for physically relevant timescales is seen in the high-frequency regime. As we traverse the fractal region and approach the Cantor lines, the heating slows enormously and the quasiparticles completely delocalize at stroboscopic times. Our setup allows us to tune between fast and ultraslow heating regimes in integrable systems.

Figure 1

Top: Uniform Hamiltonian $\mathcal{H}$ and the SSD Hamiltonian $\tilde{\mathcal{H}}$. Bottom: Fibonacci quasiperiodic drive. The time evolution follows the relation (2). At every Fibonacci step $n$, the time evolution involves $F_{n+1}$ unitary operators, comprising $F_{n-1}$ times $U_0$ and $F_n$ times $U_1$.

Figure 2

Phase diagram obtained from the Lyapunov exponent [Eq. (10)] for the Fibonacci drive. The high-frequency regime around point $\left(a\right)$ corresponds to non-heating phases whereas the bright regions, for instance around point $\left(c\right)$, correspond to heating phases. For clarity, only two Cantor lines where the Lyapunov exponent is strictly zero are shown in this phase diagram. To elucidate the fractal structure of the phase diagram, we zoom successively across a representative horizontal cut indicated by the red dashed lines. As we approach the Cantor points embedded in this cut, the system starts to manifest slow heating concomitant with an increasing stroboscopic delocalization of the quasiparticle excitations. The three points marked $\left(a\right)$, $\left(b\right)$, and $\left(c\right)$ represent these three regimes and will be extensively discussed in the rest of the paper.

Figure 3

Top: Flow of the conformal mapping ${\tilde{z}}_N$ when driven by the Fibonacci sequence with $T=\tilde{T}$ for up to $N=1000$ and $L=100$. With reference to the phase diagram: (a1) $T=1$, in the high-frequency regime the system escapes heating and such a flow is dense on the unit circle. (b1) $T=55.101$, in the slow heating regime, the excitation alternate between a large number of recurring regions. (c1) $T=78$, as the system strongly heats up, the excitations localize in the system and ${\tilde{z}}_N$ only has a few recurring points. Bottom: Stroboscopic time evolution of the logarithm of the total energy $E(t=NT)$, as a function of the number of iterations $N$. (a2) In the non-heating high-frequency regime the energy only oscillates, (b2) in the fractal regime the energy fluctuates and increases very slowly, and (c2) in the heating regime the energy grows exponentially fast with minimal fluctuations. The red dashed lines correspond to the heating rates of the exponentially growing energy at long time, given by the Lyapunov exponent $\left(10\right)$.

Figure 4

Flow of the Fibonnacci trace map in the high-frequency phase, for $L=100$ and $T=\tilde{T}=1$, for 500 iterations of the trace map. In this region, the trace map (17) stays bounded within the central part of the manifold (19) determined by the invariant (18) for a very large number of iterations. In the other regimes, this map will typically be unbounded, apart from a set of parameters of measure 0.

Figure 5

Top: Stroboscopic quasiparticle evolution as a function of both space and time predicted by the CFT for a variety of initial conditions. (a1) High-frequency phase, $T/L=0.01$, where the quasiparticle excitations evolve ballistically in an effective curved space-time. (b1) Fractal region, $T/L=0.55101$. Beyond a transient time of approximately 400 stroboscopic steps, the quasiparticle trajectories collapse onto a a unique trajectory independent of the initial conditions. The quasiparticles stroboscopically explore more and more regions of space as one approaches the Cantor line. (c1) High Lyapunov region, $T/L=0.78$. Here, for all initial conditions and after very few time steps the time evolution collapses onto a single trajectory alternating between a few fixed points. Bottom: A comparison between CFT prediction for the Loschmidt echo [Eq. (9)] for the quasiperiodically driven CFT and the analogous quantity for free fermions hopping on a lattice, given by Eq. (27), for the three different regimes. In the high-frequency regime (a2) both completely agree. In the strongly heating (c2) regime they agree for a large number of stroboscopic steps. In the slow-heating regime (b2), because of the fractal structure of the phase diagram, the numerical results strongly depend on the number of lattice sites.

Figure 6

Lyapunov exponent as a function of the spectrum $E$ and the parameter $\lambda$, using the correspondence (21). This figure exactly reproduces the figure for the spectrum $E\left(\lambda \right)$ of the Fibonacci quasicrystal, which can be found in Ref. [56]. Here, regions of nonzero Lyapunov exponent are dense for $\lambda >0$ and correspond to gaps in the spectrum. The spectrum itself consists in regions of 0 Lyapunov exponent, of measure zero.

Figure 7

Effective deformation $f_N\left(x\right)$ given by (C4), for $L=100$, $\frac{T}{L}=0.01$, $N\in \{1,20\}$. The orange curve corresponds to the effective deformation of the periodic drive, $f\left(x\right)=1-\frac{1}{2}cos\left(\frac{2\pi x}{L}\right)$.

Figure 8

Möbius phase diagrams obtained from the Lyapunov exponent given by Eq. (10) for the Fibonacci quasiperiodic drive between $\mathcal{H}$ and ${\mathcal{H}}_{\text{Möb}}\left(\theta \right)$ defined by Eq. (D1) for (a) $\theta =0.3$, (b) $\theta =0.55$, (c) $\theta =0.7$.