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I have two questions:

  1. Let us imagine that we have an integer quantum Hall system with electric Hall conductance as $\sigma_\text{H}$ on a two-dimensional (spatial) torus with size $L_1\times L_2$. If we uniformly insert a unit $2\pi$-flux into the torus (which is the minimal flux on a torus allowed by Dirac's quantization), namely the flux density being $2\pi/(L_1L_2)$, the total additional charge, due to the quantum Hall effect, will be $\sigma_\text{H}/(e^2/\hbar)\in\mathbb{Z}$. My first question is whether this system with a unit flux inserted still has a unique gapped ground state with a finite gap in the thermodynamic limit?
  2. If we consider a fractional quantum Hall system with $\sigma_\text{H}$ on the torus. After insertion of the unit flux, the additional charge $\sigma_\text{H}/(e^2/\hbar)\notin\mathbb{Z}$ will be fractional. However, we know that the total charge number should be an integer since the elementary particle here is electrons. Therefore, there must be some other excitations to cancel the fractional part of this fractional charge. The only possibility I can think of is the quasihole or quasiparticle in the fractional quantum Hall system. Since these excitations are local, my second question is how these local excitations do the job if we have translational symmetry, or whether my understanding is actually wrong?
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