Symmetry-protected many-body Aharonov-Bohm effect

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 wave function. 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 analog of spectral shifts, the twisted wave function, 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.

Figure 1

(a) and (c) Single- and many-body wave functions 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. (15) with respect to the lowest energy $E_{N,0}^{\left(p\right)}$, on a ring as a function of the lattice momentum $k\in \mathbb{Z}$. The first few primary states are labeled by $(n,m)$. (a) Spectrum of $H_2^{(p=1)}$ with ${\lambda }_2^{(p=1)}=0.82$ and $M=20$ sites. (b) Spectrum of $H_3^{(p=1,2)}$ with ${\lambda }_3^{(p=1,2)}=0.26$ and $M=12$ sites. The values of ${\lambda }_N^{\left(p\right)}$ 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. [17]).

Figure 3

Spectrum of the twisted SPT Hamiltonian with respect to the lowest energy $E_{N,0}^{\left(p\right)}$ on a ring as a function of the lattice momentum $\tilde{k}$, with the same values of ${\lambda }_N^{\left(p\right)}$ as in Fig. 2. The first few primary states are labeled by $(n,m)$. (a) Spectrum of ${\tilde{H}}_2^{\left(1\right)}$ with $M=20$ sites. (b) Spectrum of ${\tilde{H}}_3^{\left(1\right)}$ (+) and ${\tilde{H}}_3^{\left(2\right)}$ ($\times$) with $M=12$ sites. (c) Comparison between ${\tilde{\Delta }}_2^{\left(1\right)}$ (circles) and numerical results (+) plotted as a function of the momentum ${\tilde{\mathcal{P}}}_2^{\left(1\right)}$. All points are twofold-degenerate. Red circles represent primary states, while the remaining points account for descendant states in the CFT spectrum. (d) Comparison between ${\tilde{\Delta }}_3^{\left(1\right)}$ (circles) and data points (+) plotted in terms of the momentum ${\tilde{\mathcal{P}}}_3^{\left(1\right)}$. Same for ${\tilde{\Delta }}_3^{\left(2\right)}$ (squares) and data points ($\times$) plotted in terms of the momentum ${\tilde{\mathcal{P}}}_3^{\left(2\right)}$.