In 2013, Atala and co-workers directly measured the Zak phase — the Berry phase accumulated by a Bloch eigenstate when its quasimomentum traverses the one-dimensional Brillouin zone — of topological bands in a dimerized optical lattice, using a Ramsey-type interferometric protocol with ultracold atoms, and they outlined the next steps of this program: realizing domain walls in the dimerized lattice to observe topologically protected edge states and the fractional charge bound to them, extending the interferometry to two-dimensional bands (where the Zak-phase variation gives the band's topological density and Chern numbers), to degenerate multiband manifolds (non-Abelian Berry phases), and to periodically driven (Floquet) bands. Subsequent experiments realized the two-dimensional interferometric extension, multiband tomography, and Floquet-topology measurements. This problem records the surviving open core of the source program with a single common objective — completing the direct interferometric characterization of topological band structure in optical lattices in the two settings the method has not reached: (i) direct observation of the fractionalized particle number bound to a domain wall between the two topologically distinct dimerizations in a filled band, and (ii) measurement of the non-Abelian (Wilczek-Zee) Berry phase of a genuinely degenerate Bloch-band manifold.

Background

A particle in a periodic potential has energy eigenstates organized into Bloch bands, labeled by a quasimomentum kk in the Brillouin zone (the quasimomentum space of the periodic system). When the quasimomentum of a Bloch eigenstate is transported around a closed loop, the eigenstate acquires a geometric Berry phase in addition to any dynamical phase. For the loop that traverses the full one-dimensional Brillouin zone, this Berry phase is called the Zak phase (J. Zak, 1989); it is defined modulo 2π2\pi and, for a band with inversion or chiral symmetry, takes the quantized values 00 or π\pi. The canonical band structure in which the Zak phase is topologically meaningful is the Su-Schrieffer-Heeger (SSH) model: a one-dimensional chain with alternating strong and weak bonds (two possible dimerizations), originally introduced for the polymer polyacetylene. The two dimerizations are topologically distinct — the Zak phase of the filled lower band is 00 in one and π\pi in the other — and the Zak phase of each band equals π\pi times the winding number of the band's eigenvector around the origin as kk crosses the zone.

A key physical consequence, going back to Jackiw and Rebbi (1976), is that a defect interpolating between two topologically distinct regimes binds a mid-gap (zero-mode) state, and when the band is filled the defect carries a fractional particle number: in the ideal continuum limit, one half fermion per spin species sits on the soliton. In the dimerized chain, a domain wall between the two dimerizations therefore binds a topologically protected edge state and, for a filled band, a fractionalized charge — the one-dimensional avatar of polyacetylene solitons.

Atala et al. (2013) realized this physics with ultracold atoms: a bosonic optical superlattice (two optical lattices of commensurate periods, giving an alternating strong/weak-bond chain whose relative phase, the lattice-phase parameter ϕ\phi, is tunable) implements the dimerized band structure. They measured the Zak phase directly by Ramsey interferometry: a superposition of two momentum states, coupled by lattice modulation so that the atomic quasimomentum moves around a closed loop in the Brillouin zone, acquires a phase shift between the two interferometer arms that equals the Zak phase and is read out from the final state populations. They observed the predicted quantized jump between 00 and π\pi across the dimerization transition and extracted the winding number, establishing direct, state-resolved measurement of band topology without relying on edge-state or transport proxies. In their conclusions they identified the open directions of the program: using single-site optical control to build domain walls and directly study edge states and fractional charges at the interface of the two topologically distinct phases; extending the technique to two-dimensional systems, where the change of the Zak phase across the Brillouin zone gives the topological density of the band and hence the Chern number (the integer invariant counting the total Berry flux through the zone) and the π\pi Berry flux of Dirac points; measuring non-Abelian Berry phases in degenerate bands, as occur in quantum-spin-Hall systems (two-dimensional topological-insulator bands whose counter-propagating edge states are protected by time-reversal symmetry); and applying the method to Floquet (periodically driven) systems, where band topology is organized by quasi-energy and stroboscopic micromotion. The proposal of Abanin, Kitagawa, Bloch and Demler (2013) had already developed the two-dimensional interferometric extension in theory.

Two technical notions are needed for the remaining open part. The Wilczek-Zee holonomy is the matrix-valued generalization of the Berry phase acting on a set of degenerate states transported around a loop: each path-ordered integral of the non-Abelian Berry connection is a unitary matrix mixing the degenerate subspace, and its eigenvalues are gauge-invariant observables. A 'filled band' means every quasimomentum state of the band is occupied by one particle per spin mode, which in optical lattices requires either fermionic atoms or hard-core bosons (bosons whose on-site repulsion is so strong that two of them never share a lattice site) prepared well below the band gap.

Problem Statement

Complete the two unrealized components of the interferometric Zak-phase program of Atala et al. (2013) in optical lattices, whose common objective is the direct measurement of topological band-structure properties that the original one-dimensional, non-degenerate, undriven protocol did not reach:

(1) Domain-wall fractionalization in a filled band. Engineer a sharp domain wall between the two topologically distinct dimerizations of a one-dimensional optical superlattice, load the lower band with fermions (or hard-core bosons) to near complete filling, and directly measure the topologically protected mid-gap edge state and the fractionalized particle number (fractional quantum number) bound to the domain wall — the half-integer charge per spin species in the ideal limit, with the deviation quantified by the band parameters and the wall profile.

(2) Non-Abelian Berry phase in degenerate Bloch bands. Extend the interferometric measurement so that it acts within a genuinely degenerate manifold of Bloch states (for example a Kramers-degenerate doublet, a pair of Bloch states forced to be degenerate by time-reversal symmetry for half-integer spin, in a quantum-spin-Hall-type band) and measure the non-Abelian (Wilczek-Zee) Berry phase — the matrix-valued holonomy acquired by the degenerate multiplet around a loop in the Brillouin zone, characterized by its gauge-invariant eigenvalues — rather than the scalar Zak phase.

Both measurements are to be performed at a level of control where the systematic uncertainties identified in the source experiment (lattice-phase control of the superlattice and residual magnetic-field fluctuations) are reduced below the scale of the effect being measured, and in an actual lattice realization of the two settings, with the measured observables quantitatively compared to the band theory of the realized lattice.

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