From 9faec5bc0f004c37e121bf35b50631af0d4a37e6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jean-S=C3=A9bastien=20Caux?= Date: Wed, 8 May 2019 08:33:35 +0000 Subject: [PATCH] Update page 'Partial Overlaps of Many Body Wavefunctions' --- ...ial-Overlaps-of-Many-Body-Wavefunctions.md | 28 +++++++++++++++++++ 1 file changed, 28 insertions(+) create mode 100644 Partial-Overlaps-of-Many-Body-Wavefunctions.md diff --git a/Partial-Overlaps-of-Many-Body-Wavefunctions.md b/Partial-Overlaps-of-Many-Body-Wavefunctions.md new file mode 100644 index 0000000..ecbfdc8 --- /dev/null +++ b/Partial-Overlaps-of-Many-Body-Wavefunctions.md @@ -0,0 +1,28 @@ +# Partial Overlaps of Many-Body Wavefunctions + +## Description +Eigenstates are mutually orthogonal when integrated over the whole space. +One can however ask "how" orthogonal states are, by looking at the overlap +computed over a fraction of space (for example, particles confined to one +side of the system). + +Expectation: all pairs of states are mutually orthogonal, but some are more +orthogonal than others. + +Formally, one can start from +\[PO_{ab} (s) = \int_0^{L-s} dx \psi^\dagger_a \psi_b\] +with $s$ between $0$ and $L$. + +For which systems can this partial overlap be computed exactly? +For integrable systems? + +Is this a smarter way to distinguish localized systems from delocalized ones? + +Is this a way to define a physically relevant "distance" between wavefunctions? +(for example, $dPO/ds$ around $s = 0$, or any other measure really). + +This is complementary to entanglement entropy, but in a sense much more physical. + +## Applications +- relaxation dynamics in locally quenched problems +- better understanding of physical operator matrix elements \ No newline at end of file