Emergence of integer quantum Hall effect from chaos

We present an analytic microscopic theory showing that in a large class of spin-$\frac{1}{2}$ quasiperiodic quantum kicked rotors, a dynamical analog of the integer quantum Hall effect (IQHE) emerges from an intrinsic chaotic structure. Specifically, the inverse of the Planck's quantum ($h_e$) and the rotor's energy growth rate mimic the “filling fraction” and the “longitudinal conductivity” in conventional IQHE, respectively, and a hidden quantum number is found to mimic the “quantized Hall conductivity.” We show that for an infinite discrete set of critical values of $h_e$, the long-time energy growth rate is universal and of order of unity (“metallic” phase), but otherwise vanishes (“insulating” phase). Moreover, the rotor insulating phases are topological, each of which is characterized by a hidden quantum number. This number exhibits universal behavior for small $h_e$, i.e., it jumps by unity whenever $h_e$ decreases, passing through each critical value. This intriguing phenomenon is not triggered by the likes of Landau band filling, well known to be the mechanism for conventional IQHE, and far beyond the canonical Thouless-Kohmoto-Nightingale-Nijs paradigm for quantum Hall transitions. Instead, this dynamical phenomenon is of strong chaos origin; it does not occur when the dynamics is (partially) regular. More precisely, we find that a topological object, similar to the topological theta angle in quantum chromodynamics, emerges from strongly chaotic motion at microscopic scales, and its renormalization gives the hidden quantum number. Our analytic results are confirmed by numerical simulations. Our findings indicate that rich topological quantum phenomena can emerge from chaos and might point to a new direction of study in the interdisciplinary area straddling chaotic dynamics and condensed matter physics. This work is a substantial extension of a short paper published earlier by two of us [Y. Chen and C. Tian, Phys. Rev. Lett. 113, 216802 (2014)].

Figure 1

Schematic representation of Planck-IQHE for small $h_e$ in spin-$\frac{1}{2}$ quasiperiodic QKR. Red line: for an infinite discrete set of critical values of $h_e$ the rotor's energy increases linearly at long times, i.e., ${lim}_{t\to \infty }\frac{E\left(t\right)}{t}={\sigma }^{*}=\mathcal{O}\left(1\right)$, which is a characteristic of metals; for other values the rotor's energy saturates at long times, i.e., ${lim}_{t\to \infty }\frac{E\left(t\right)}{t}=0$, which is a characteristic of insulators. Blue line: the insulator is characterized by a hidden quantum number ${\sigma }_{\mathrm{H}}^{*}$; this number jumps by unity whenever $h_e^{-1}$ increases passing through a critical value. (The sharp transitions are smeared at finite times.)

Figure 2

A spin-$\frac{1}{2}$ particle moving on a ring (yellow dashed lines) subjected to a train of pulsed external potential (kicking) switched on at integer times $t=s=0,1,2,\cdots$ (red wavy lines), with the time rescaled by the kicking period. The potential depends on both the angular position and the spin of the particle, and its profile is modulated in time with a single modulation frequency $\tilde{\omega }$. Between two successive potential pulses, i.e., $s^{+}\le t\le {(s+1)}^{-}$, both the angular momentum and the spin polarization of the particle are conserved (left column, with the arrow of the dashed line representing the direction of rotation); upon kicking, i.e., at time $t={(s+1)}^{+}$, the particle undergoes an abrupt change in the angular momentum and a flip of spin simultaneously (right column). The superscripts “$\pm$” stand for the time right before (after) kicking.

Figure 3

(a) A quasiperiodic QKR with one modulation frequency is equivalent to a $2\mathrm{D}$ strictly periodic QKR. The latter corresponds to a quantum motion in the $N\equiv (n_1,n_2)$ space under the influence of a periodic driving force. $n_1$ is the genuine angular momentum and $n_2$ virtual, canonically conjugate to the modulation parameter. The properties of localization in the $N$ space are governed by interference between the advanced (solid line) and the retarded (dashed line) quantum amplitudes. (b) When the coherent propagation of the advanced and the retarded quantum amplitudes is chaotic in both $n_1$ and $n_2$ directions, mappings from the $N$ space (with its boundary identified as the same point) to the supersymmetry $\sigma$-model space of unitary symmetry $\simeq H^2\times S^2$ arise. These mappings are classified by the homotopy group $\mathbb{Z}$.

Figure 4

The quantum stochastic web in $N$ space as a possible physical picture for the critical metallic phase. The link supports ballistic motion while the node is the origin of quantum stochasticity. Inset: this web is topologically equivalent to a graph made up of links and nodes.

Figure 5

(a) The RG flow line structure of two-parameter RG equations. The fixed points (solid circles) at ${\tilde{\sigma }}_{\mathrm{H}}=n$ ($n\in \mathbb{Z}$) correspond to insulating phases and at ${\tilde{\sigma }}_{\mathrm{H}}=n+\frac{1}{2}$ to metal-insulator transitions. (b) The analytic prediction for the critical $h_e$ values is made by letting the unrenormalized transport parameter ${\sigma }_{\mathrm{H}}$ (solid line) be half-integers. Namely, the solid and dashed lines intersect at the critical $h_e$ values. (c) Long-time simulations confirm the transitions with the critical values in excellent agreement with analytic predictions, and the critical points are evenly spaced along the $h_e^{-1}$ axis. They also confirm that the critical metallic phase has an energy growth rate order of unity. The simulation results of $E\left(t\right)/t$ are for $t=1.2\times {10}^6$ and for the specific forms of $H_0={(-i{h}_e{\partial }_{{\theta }_1})}^{\alpha }$, with the exponent $\alpha =2$ (black solid line) and $\alpha =4$ (red dashed line), respectively, and $V$ given by Eqs. (172), (173), and (174). Simulations confirm that the critical value is robust against changing $\alpha$.

Figure 6

(a) Simulations of the quantum evolution (5) show that at short times the rotor's energy grows linearly irrespective of the value of $h_e$. (b) The numerical (solid circles) and analytical (solid line) results for the rate of this short-time energy growth are in excellent agreement.

Figure 7

Further numerical evidences of strong chaoticity. (a) Simulations of the equivalent $2\mathrm{D}$ evolution show that the system exhibits short-time diffusion in both $n_1$ and $n_2$ directions, i.e., both $E_1\left(t\right)$ (solid lines) and $E_2\left(t\right)$ (dashed lines) grow linearly at short times. (b) The Floquet operator $\widehat{U}$ governing the $2\mathrm{D}$ evolution has a chaotic quasienergy spectrum when the angular momentum lattice is small. The level spacing distribution (histograms) obeying the Wigner surmise of unitary type (dashed line). These results are irrespective of the value of $h_e$.

Figure 8

Analytic prediction for the $\mu -h_e^{-1}$ phase diagram (top panel). Simulation results of the energy growth rate at $t={10}^5$ for $H_0={(-i{h}_e{\partial }_{{\theta }_1})}^{\alpha }$, with $\alpha =2$ (middle panel) and 4 (bottom panel). The dashed lines are the phase boundaries predicted analytically.

Figure 9

(a) Simulations of $2\mathrm{D}$ dynamics show that the long-time diffusion rates of rotor metal in $n_1$ (blue solid line) and $n_2$ directions (red dashed line) are different, indicating the anisotropicity of the $2\mathrm{D}$ model. For this model $H_0={(-i{h}_e{\partial }_{{\theta }_1})}^2$ and $V\left(\mathrm{\Theta }\right)$ is given by Eqs. (172, 173, 174). (b) The geometric mean of the two diffusion rates. The results are for $t={10}^4$.

Figure 10

The same as Fig. 9 but for $H_0={(-i{h}_e{\partial }_{{\theta }_1})}^4$.

Figure 11

(a) For $\tilde{\omega }=2\pi$ the rotor's energy saturates following a transient growth, irrespective of the value of $h_e$. The representative simulation results are shown by the solid lines with $h_e^{-1}=0.77$ (blue), $2.13$ (green), and $3.42$ (red), respectively. These energy profiles are in sharp contrast to those represented by the dashed lines for which the values of $h_e$ are the same but $\tilde{\omega }=\frac{2\pi }{\sqrt{5}}$. The latter exhibits linear growth in the entire course of time since the system is at the critical metallic phase. (b) The motion in the $n_2$ direction is ballistic for $\tilde{\omega }=2\pi$ (solid) while diffusive for $\tilde{\omega }=\frac{2\pi }{\sqrt{5}}$ (dashed). Two lines with the same color correspond to the same value of $h_e$.

Figure 12

(a) The metallic growth at the topological transition for $\frac{\tilde{\omega }}{2\pi }=\frac{1}{\sqrt{5}}$ (black) is replaced by a crossover from a linear energy growth to saturation, when $\frac{1}{\sqrt{5}}$ is approximated by a rational number $p/q$. This number is set to $1/2$ (green), $4/9$ (light blue), $17/38$ (dark blue), $72/161$ (red), and $305/682$ (pink) from bottom to top. Here, $h_e^{-1}=2.13$. (b) Simulation results confirm the scaling (196) for the saturation value of $E\left(t\right)$. For each value of $h_e^{-1}$ we choose several different values of $p/q$. From left to right, the four clusters of data points are for $p/q=4/9,17/38,72/161$, and $305/682$, respectively. (c) Simulations of the equivalent $2\mathrm{D}$ systems show that the saturation of $E_1\left(t\right)$ at large times (red solid) is associated with the restoration of regular dynamics in the $n_2$ direction manifesting in an asymptotic ballistic growth of $E_2\left(t\right)$ (blue dashed). Here, $h_e^{-1}=2.13$ and $p/q=4/9$.

Figure 13

Numerical simulations show that when $d_3=0.8(1-cos{\theta }_1-cos{\theta }_2)$ [cf. Eq. (173)] is replaced by $d_3=0.8$ and other conditions of the model used for obtaining Fig. 5 are not changed, the system is always insulting and no transition occurs, in agreement with analytic predictions.

Figure 14

Simulation results of $E\left(t\right)/t$ at different times. The center of these profiles is located at $h_e^{-1}=2.19$, corresponding to the central peak in Fig. 5.