Emergence of integer quantum Hall effect from chaos

Abstract

We present an analytic microscopic theory showing that in a large class of spin-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 (he) 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 he, 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 he, i.e., it jumps by unity whenever he decreases, passing through each critical value. This intriguing phenomenon is not triggered by the like 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, for the first time, 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)].

Controlling symmetry and localization with an artificial gauge field in a disordered quantum system

“…Engineering spin-orbit-coupled dynamical Floquet systems (e.g., using internal-state-dependent optical potentials) would allow, for example, to study the symplectic symmetry class, where the Anderson metal-insulator transition is expected to occur in dimensions as low as two. This also opens an avenue for the study of fascinating phenomena, like quantum Hall effect 39 , Floquet topological insulators and artificial magnetism.…”

Self-duality triggered dynamical transition

“…which characterizes how the wavepacket spreads in the course of time. Like QKR systems [1,2,11,12], this probe provides an important link between SM models and condensed matter systems. Notably, when E(ψ ψ ψ t ) remains bounded as t → ∞, the wavepacket does not spread at long time and remains localized in a finite region in momentum space; no quantum transport arises and this simulates a quantum insulator in condensed matter physics.…”

“…As shown below, thanks to this difference the SM model exhibits a much more intriguing phenomenology than the ordinary Maryland model. On the other hand, the SM model crucially differs from the spinful QKR [11,12] in that its free Hamiltonian between kicks is linear in n 1 , while for the spinful QKR the free Hamiltonian is quadratic in n 1 . It is well known that kicked systems with a quadratic free Hamiltonian may exhibit a chaotic behavior (in some sense), which cannot be observed with a linear one.…”

“…Such variants did not command major changes of the general picture of dynamical localization. A surprising result has been instead observed with a recent 2D variant endowed with spin 1/2 [11,12]. By tuning the Planck constant e , a sequence of dynamical transitions arise.…”

Quantum simulation of disordered systems with cold atoms

“…While this is conceivable [67] it can be very difficult in practice. However, if this could be done, it will open a wealth of new possibilities, as recent theoretical suggestions including complex spin-orbit-coupled Hamiltonians allow the realization of puzzling systems displaying momentum-space topological insulator properties and opening ways to the realization of a Quantum Hall physics quantum simulator [68,69,49,50].…”