Consider the simple case of two quantum harmonic oscillators that are coupled together. Solving the Schrodinger equation for this system requires us to shift to a basis where the oscillators are uncoupled, i.e., the Hamiltonian is diagonal. In this basis, if one of the uncoupled oscillators has a vanishing frequency, we identify it to be a zero-mode, or quite simply, a free particle. The presence of a zero-mode brings forth so many interesting features to the system, that are typically less explored:
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Entropy divergence : Entanglement entropy is the standard measure of quantum entanglement in bipartite systems. As it turns out, the presence of a zero mode in a wide variety of quantum systems including coupled oscillators, hydrogen atom (which is basically an entangled electron-proton system), lattice-regularized field theories (network of entangled oscillators), etc., all lead to a divergence in the entanglement entropy. The cause of divergence can be attributed to the fact that zero modes are non-normalizable, due to which there is an infinite and continuous spectrum of possible energy values that the particle can have. Physically, entropy measures ignorance to the information that helps describe the system, and since there are infinite possible values for the energy of a zero mode, it makes sense for the entropy to diverge.
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UV-IR duality : The entanglement entropy of a field enclosed in a sub-region, unlike black-hole entropy, is divergent. While this divergence is usually attributed to the ultraviolet (high energy) limit, we have shown that there is a more general criterion for entropy divergence --- the generation of zero-modes. This is made possible through an inherent scaling symmetry of the entanglement entropy that seemingly connects the UV and the IR (infrared). Such a scaling symmetry has been shown to exist in simple quantum systems such as the coupled oscillators all the way up to quantum fields in asymptotically flat and non-flat space-times.
- Quantum criticality : Zero modes are also associated with quantum criticality, as shown by the closing of the gap in the lower levels of the entanglement spectrum for a network of coupled oscillators. The degeneracy in the entanglement spectrum can be another explanation for why the entanglement entropy diverges.
