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.
Jiaju Zhang, Pasquale Calabrese, Marcello Dalmonte, M. A. Rajabpour
SciPost Phys. Core 2, 007 (2020) ·
published 7 May 2020
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· pdf
We carry out a comprehensive comparison between the exact modular Hamiltonian
and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional
critical quantum spin chains. As a warm-up, we first illustrate how the trace
distance provides a more informative mean of comparison between reduced density
matrices when compared to any other Schatten $n$-distance, normalized or not.
In particular, as noticed in earlier works, it provides a way to bound other
correlation functions in a precise manner, i.e., providing both lower and upper
bounds. Additionally, we show that two close reduced density matrices, i.e.
with zero trace distance for large sizes, can have very different modular
Hamiltonians. This means that, in terms of describing how two states are close
to each other, it is more informative to compare their reduced density matrices
rather than the corresponding modular Hamiltonians. After setting this
framework, we consider the ground states for infinite and periodic XX spin
chain and critical Ising chain. We provide robust numerical evidence that the
trace distance between the lattice BW reduced density matrix and the exact one
goes to zero as $\ell^{-2}$ for large length of the interval $\ell$. This
provides strong constraints on the difference between the corresponding
entanglement entropies and correlation functions. Our results indicate that
discretized BW reduced density matrices reproduce exact entanglement entropies
and correlation functions of local operators in the limit of large subsystem
sizes. Finally, we show that the BW reduced density matrices fall short of
reproducing the exact behavior of the logarithmic emptiness formation
probability in the ground state of the XX spin chain.
