SciPost Phys. 12, 195 (2022) ·
published 14 June 2022
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· pdf
Consider a generic quantum spin chain that can be mapped to free quadratic
fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary
boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian
after JW transformation. Using ancillary sites and enlarging the Hamiltonian we
first introduce a bigger quadratic Hamiltonian. Then we diagonalize this
enlarged Hamiltonian in its most generic form and show that all the states are
degenerate because of the presence of a zero mode. The eigenstates of the
original spin chain with boundary magnetic fields can be derived after
appropriate projection. We study in-depth the properties of the eigenstates of
the enlarged Hamiltonian. In particular, we find: 1) the eigenstates in
configuration bases, 2) calculate all the correlation functions, 3) find the
reduced density matrices, 4) calculate the entanglement entropy. We show that
the generic eigenstate of the enlarged Hamiltonian (including the eigenstates
of the original spin chain) breaks the parity number symmetry and consequently
one needs to take care of some technicalities regarding the calculation of the
reduced density matrix and entanglement entropy. Interestingly we show that the
entanglement structure of these eigenstates is quite universal and independent
of the Hamiltonian. We support our results by applying them to a couple of
examples.
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