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 nn-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}ℓ−2
for large length of the interval \ellℓ.
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.
Gaussian, exponential, and power-law decay of time-dependent correlation functions in quantum spin chains
