Molecular Control of Floquet Topological Phase in Non-adiabatic Thouless Pumping

Abstract

Non-adiabatic Thouless pumping of electrons is studied in the framework of topological Floquet engineering, particularly with a focus on how atomistic changes to chemical moieties control the emergence of the Floquet topological phase. We employ real-time time-dependent density functional theory to investigate the extent to which the topological invariant, the winding number, is impacted by molecular-level changes to trans-polyacetylene. In particular, several substitutions to trans-polyacetylene are studied to examine different effects on the electronic structure, including the mesomeric effect, inductive effect, and electron conjugation effect. Maximally localized Wannier functions are employed to relate the winding number to the valence bond description by expressing the topological pumping as the transport dynamics of the localized Wannier functions. By further exploiting the gauge invariance of the quantum dynamics in terms of the minimal particle–hole excitations, the topological pumping of electrons can be also represented as a cyclic transition among the bonding and antibonding orbitals. Having connected the topological invariant to the chemical concepts, we demonstrate molecular-level control of the emergence of the Floquet topological phase, presenting an exciting opportunity for the intuitive engineering of molecular systems with such an exotic topological phase.

Real-Time Time-Dependent Density Functional Theory for Simulating Nonequilibrium Electron Dynamics

“…The winding number, being equal to the integrated particle current over the periodic time T , can be given in terms of the energy spectrum of this effective Floquet Hamiltonian, ε i (quasienergy), or equivalently in terms of nonadiabatic Aharonov-Anandan geometric phase of the Floquet states. Conveniently for electronic structure theory, the winding number can be expressed in terms of time-dependent Bloch states, u n ( k , t ), as , W = 1 2 π ∑ n normalO normalc normalc . [ ∫ B Z d k ⟨ u n false( k , t = T false) | i ∂ k | u n false( k , t = T false) ⟩ − ∫ B Z d k ⟨ u n false( k , t = 0 false) | i ∂ k | u n false( k , t = 0 false) ⟩ ] One may note that the winding number can be expressed analogously to the static Chern insulator, as W = C ≡ 1 2 π ∫ 0 T normald t …”

“…While the double and single bond Wannier centers move in opposite directions, having more electrons in the double bonds than in the single bonds results in a net unidirectional transport overall. This interpretation of the topological property in terms of the valence bond model can be used to predict the emergence of the Floquet topological phase with respect to molecular changes to trans-polyacetylene as discussed in detail in ref .…”

“…Topological matter has become a very active topic of research in the past decade, and RT-TDDFT simulation can contribute greatly to its development, especially for connecting various model Hamiltonian descriptions to real chemical systems. There are growing efforts to develop a molecular-level understanding of topological materials from the perspective of chemistry, , and the gauge transformation was utilized here to rationalize how the Floquet topological phase emerges in the framework of well-recognized chemistry concepts . Although RT-TDDFT simulation provides fundamental molecular-level understanding, quantitative prediction of the specific conditions, such as the electric field amplitudes/frequencies, remains challenging.…”

“…The second example here illustrates how RT-TDDFT can advance our fundamental understanding, in this case, of how certain physical properties are determined by the topological character of the time-dependent Hamiltonian. We focus on a particular work that connects the mathematical formulation of nonadiabatic Thouless pumping to a chemically intuitive description on the molecular level. , One of the earliest realizations that topological characteristics of the Hamiltonian govern certain dynamical properties came from Thouless in 1983 when the quantized pumping phenomenon was predicted . This seminal work on quantized particle transport in a one-dimensional system showed that the quantization derives from the topology of the underlying quantum-mechanical Hamiltonian.…”

“…The winding number, being equal to the integrated particle current over the periodic time T , can be given in terms of the energy spectrum of this effective Floquet Hamiltonian, ε i (quasienergy), or equivalently in terms of nonadiabatic Aharonov-Anandan geometric phase of the Floquet states. Conveniently for electronic structure theory, the winding number can be expressed in terms of time-dependent Bloch states, u n ( k , t ), as , One may note that the winding number can be expressed analogously to the static Chern insulator, as where C would be the first Chern number of the Floquet states, and the generalized Berry curvature is given by F n ( k , t ) = i [⟨∂ t u n ( k , t )|∂ k u n ( k , t )⟩ – ⟨∂ k u n ( k , t )|∂ t u n ( k , t )⟩] . Using the Blount identity , the winding number can be expressed in terms of the time-dependent MLWFs, w n ( r , t ), as where the position operator here is defined according to the formula given by Resta for extended periodic systems and L is the lattice length of the unit cell.…”