Symmetry-protected many-body Aharonov-Bohm effect

TL;DR

This work demonstrates a genuine many-body Aharonov-Bohm effect on the edge of two-dimensional symmetry-protected topological states with global Z_N symmetry. It develops both a low-energy field theory using a non-chiral Luttinger liquid and a fully regularized lattice edge model with non-onsite Z_N symmetry, showing how gauge flux twists shift the edge spectrum in a manner determined by the Z_N class. The twisted-edge spectra reproduce conformal-field-theory predictions and encode SPT invariants related to group cohomology 3-cocycles, while the twisted boundary construction exposes edge anomalies for nontrivial SPTs. This dual framework advances gauging non-onsite symmetries in bosonic systems and suggests extensions to broader symmetry classes and fermionic settings.

Abstract

It is known as a purely quantum effect that a magnetic flux affects the real physics of a particle, such as the energy spectrum, even if the flux does not interfere with the particle's path - the Aharonov-Bohm effect. Here we examine an Aharonov-Bohm effect on a many-body wavefunction. Specifically, we study this many-body effect on the gapless edge states of a bulk gapped phase protected by a global symmetry (such as $\mathbb{Z}_{N}$) - the symmetry-protected topological (SPT) states. The many-body analogue of spectral shifts, the twisted wavefunction and the twisted boundary realization are identified in this SPT state. An explicit lattice construction of SPT edge states is derived, and a challenge of gauging its non-onsite symmetry is overcome. Agreement is found in the twisted spectrum between a numerical lattice calculation and a conformal field theory prediction.

Figures (3)

• Figure 1: (a) and (c): Single- and many-body wavefunctions upon flux insertion, respectively. (b) and (d): Flux effect captured by twisted boundary conditions showing the associated branch cut.
• Figure 2: Spectrum of the SPT Hamiltonian Eq. (\ref{['eq:Hamiltonian lattice']}) with respect to the lowest energy $E^{(p)}_{N,0}$, on a ring as a function of the lattice momentum $k \in \mathbb{Z}$. First few primary states labeled by $(n,m)$. (a) Spectrum of $H^{(p=1)}_{2}$ with $\lambda^{(p=1)}_{2} = 0.82$ and $M = 20$ sites. (b) Spectrum of $H^{(p=1,2)}_{3}$ with $\lambda^{(p=1,2)}_{3} =0.26$ and $M = 12$ sites. The values of $\lambda^{(p)}_{N}$ above guarantee a proper normalization so that states in the same conformal tower separated by $\delta k = \pm 1$ are integer spaced (up to finite size effects) [see Ref. Henkel].
• Figure 3: Spectrum of the twisted SPT Hamiltonian with respect to the lowest energy $E^{(p)}_{N,0}$ on a ring as a function of the lattice momentum $\tilde{k}$, with the same values of $\lambda^{(p)}_{N}$ as in Fig. (\ref{['fig:spectrum untwisted SPT']}). First few primary states labeled by $(n,m)$. (a) Spectrum of $\tilde{H}^{(1)}_{2}$ with $M=20$ sites. (b) Spectrum of $\tilde{H}^{(1)}_{3}$ (+) and $\tilde{H}^{(2)}_{3}$ ($\times$) with $M=12$ sites. (c) Comparison between $\tilde{\Delta}^{(1)}_{2}$ (circles) and numerical results (+) plotted as a function of the momentum $\tilde{\mathcal{P}}^{(1)}_{2}$. All points are two-fold degenerate. Red circles represent primary states, while the remaining points account for descendant states in the CFT spectrum. (d) Comparison between $\tilde{\Delta}^{(1)}_{3}$ (circles) and data points (+) plotted in terms of the momentum $\tilde{\mathcal{P}}^{(1)}_{3}$. Same for $\tilde{\Delta}^{(2)}_{3}$ (squares) and data points ($\times$) plotted in terms of the momentum $\tilde{\mathcal{P}}^{(2)}_{3}$.