#aubryandreharpermodel

Aubry Andre Harper Model

A recent study shown that quantised currents can continue in disordered materials when the protective energy gaps that were thought to be necessary are erased, challenging topological protection in quantum systems. A driven Aubry-André-Harper (AAH) chain, a basic model for quasiperiodic systems, demonstrates this paradoxical persistence, which could revolutionise quantum state preparation and allow new investigations of odd quantum phenomena.

The durability of topological properties like precise, quantised currents has been attributed to symmetry-forbidden transitions or energy gaps between occupied and empty quantum states. Thouless pumps, a key example of topological phases where slow time-periodic modulation causes quantised charge transport, were thought to break down in periodic systems if disorder decreased energy gaps. Research by Emmanuel Gottlob, Ulrich Schneider, Dan S. Borgnia, and Robert-Jan Slager shows that quasiperiodic systems don't always do this.

Knowing the Aubry-André-Harper (AAH) Model

The work focusses on the almost Mathieu operator, the one-dimensional Aubry-André-Harper (AAH) lattice. It is a basic model of quasiperiodic systems since its structure is orderly but not recurring in space. The main components of the AAH Hamiltonian are an onsite cosine potential ($\hat{V}(\varphi)$) and a nearest-neighbor hopping term ($\hat{J}$) that allows particles to traverse across neighbouring sites.

A crucial feature of the Aubry Andre Harper Model is the irrational parameter $\beta$. The hopping term's translation invariance is compromised by this irrationality, giving it peculiar quantum properties. For $|V/J| > 1$, the system's eigenfunctions demonstrate Anderson localisation, which means particles are limited to certain regions based on the ratio of potential strength $V$ to hopping strength $J$, even without extra disorder.

The researchers assess the AAH model's spectral properties using a “configuration-space” approach. Each lattice site $i$ is assigned a unit circle angle by the formula $\theta_i = 2\pi\beta i + \varphi \mod 2\pi$ in this abstract space Since $\beta$ is irrational and each site has a unique $\theta$ value, this mapping reliably populates the unit circle. This method is crucial because it avoids the lack of a Brillouin zone in quasiperiodic lattices, which is used to calculate topological invariants in periodic.

When there is no hopping ($J/V = 0$), the Aubry Andre Harper Model is dense on a specific energy interval. Similar onsite energies in surrounding sites provide resonances with a small hopping amplitude ($J/V \ll 1$). These resonances divide the continuous spectrum into discrete energy bands by creating pairs of spectral gaps. The AAH spectrum is a “nowhere-dense Cantor set,” defined by an unending, fractal hierarchy of these gaps, not continuous. The gap-labelling theorem reliably quantifies the IDoS inside these bands, where each gap has an integer Chern number.

Thouless Pumping and Unprecedented Robustness

Thouless pumping modulates a Hamiltonian slowly, continuously, and regularly. The AAH model achieves this by linearly raising phase $\varphi$ while keeping a constant pumping rate. This produces a quantised charge transfer proportional to the band's Chern number. In contrast to periodic systems, which only quantise charge per pump cycle, the quasiperiodic AAH model quantises time-independent current.

This study's discovery that quantised currents remain while chaos closes energy gaps is groundbreaking. Because local disorder convolves with the clean spectrum, unperturbed spectral gaps smaller than the disorder strength collapse in the AAH chain. Contrary to widespread assumption, quantised currents only break down at much higher critical disorder intensities and survive long after this gap closure. A band with a Chern number of +4 remained quantised even when the disorder strength was more than an order of magnitude bigger than the clean gaps.

A local picture based on Landau-Zener (avoided-crossing) transitions in the configuration space explains this remarkable robustness. When phase $\varphi$ changes, all localised states in the Aubry Andre Harper Model resonate with another state, possibly far distant in real space. If the pumping rate is slow enough (adiabatic), a Landau-Zener transition causes quantised transport.

The AAH spectrum's fractal nature (Cantor set) and Anderson localisation are crucial. If these resonances are ordered, the quantised current is unaffected by the phase winding varying between subsequent Landau-Zener processes in disorder. Quantisation breaks down only when the disorder disrupts this order and particles are back-reflected. To protect quantised currents from long-range tunnelling, a high pumping rate is necessary. This suggests that local resonance stability, not clean energy gap magnitude, limits robustness.

Prepare for New Quantum Technologies

The experimental implications of this discovery are significant. The researchers present a dependable method for preparing quantum computing with chosen Chern numbers that can be immediately applied to cold atom and photonic research. Such configurations can achieve the Aubry Andre Harper Model by superimposing optical lattices. As phase is pumped, atomic clouds can be spatially separated into Chern number-corresponding regions. This method is reliable even with small energy gaps around target bands.

By applying this protocol to a 2D lattice of 1D AAH chains, states with a well-defined “real” 2D Chern number can be created. The integer and fractional quantum Hall effects may open up new experimental avenues for studying quantum Hall physics in conventional condensed matter systems with flat bands, high Chern numbers, and non-local interactions.

In summary