Update page 'Partial Overlaps of Many Body Wavefunctions'
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# Partial Overlaps of Many-Body Wavefunctions
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## Description
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Eigenstates are mutually orthogonal when integrated over the whole space.
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One can however ask "how" orthogonal states are, by looking at the overlap
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computed over a fraction of space (for example, particles confined to one
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side of the system).
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Expectation: all pairs of states are mutually orthogonal, but some are more
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orthogonal than others.
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Formally, one can start from
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\[PO_{ab} (s) = \int_0^{L-s} dx \psi^\dagger_a \psi_b\]
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with $s$ between $0$ and $L$.
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For which systems can this partial overlap be computed exactly?
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For integrable systems?
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Is this a smarter way to distinguish localized systems from delocalized ones?
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Is this a way to define a physically relevant "distance" between wavefunctions?
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(for example, $dPO/ds$ around $s = 0$, or any other measure really).
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This is complementary to entanglement entropy, but in a sense much more physical.
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## Applications
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- relaxation dynamics in locally quenched problems
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- better understanding of physical operator matrix elements
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