just getting into this one
http://arxiv.org/PS_cache/cond-mat/pdf/0703/0703037v1.pdf
However, this is not the whole story. We must also take into account that there may
be a nearest neighbor valence bond connecting regions A and B. Whether or not this
is present depends both on the parity of r and on whether the soliton is in region A or
B. If r is even, then this valence bond is present when the soliton is in A. Conversely,
if r is odd, then it is present when the soliton is in region B. When this valence bond
is present, it contributes an additional ln 2 to S. The probability of it being present is
p for r even and 1 − p for r odd. Adding this extra term we obtain:
S(r) = −p ln p − (1 − p) ln(1 − p) + [1/2 + (−1)r(p − 1/2)] ln 2. (3.5)
As remarked earlier, when J′K = 0 and R is odd, the impurity site is unentangled
with the rest of the chain which contains only nearest neighbor valence bonds, between
Quantum Impurity Entanglement 16
sites 2i and 2i+1. The only source of entanglement between A and B is a valence bond
from site r and r + 1 when r is even. Thus
S = (1/2)[1 + (−1)r] ln 2.
For most eigenfunctions we can neglect the perturbation due to this
coupling. The only important exception to this statement are the low energy states, with
wave-lengths of order the system size and energies of O(v/r). We can expect these to be
strongly perturbed by the coupling to subsystem B and consequently expect S(T) to be
strongly modified from the thermal entropy, Sth(T) at low temperatures. However, when
T ≫ v/r, we expect these states to make a negligible contribution to S(T) since the
density of states is much larger at higher energies. Therefore we expect S(T) to approach
Sth(T) when T ≫ v/r. This argument suggests that this should occur regardless of R.
We also expect this correspondance to hold regardless of boundary conditions. The
thermal entropy for region A becomes independent of whether it is calculated with pbc
or obc when T ≫ v/r. Similarly the entanglement entropy becomes independent of
whether region A is part of a system, A + B which obeys pbc or obc at T ≫ v/r.
In order to provide numerical evidence that S(T) indeed does approach Sth, the
thermal entropy, we have performed exact calculations for both quantities on an XX
spin chain of length R = 100. The finite T entanglement entropy is calculated for a
subsystem, A, of size r within a total system A+B of size R obeying periodic boundary
conditions. We have also calculated the thermal entropy, Sth(T), for a system of size
r also with periodic boundary conditions. We then compare S(T)/r to Sth(T)/r. Our
results are shown in Fig. E1 where both entropies are plotted per unit length of the
subsystem. Four different sub-system sizes of r = 2, 20, 50, 98 are considered and in all
cases do we observe excellent agreement with the thermal entropy at sufficiently high
temperature. In Fig. E2 is shown the finite temperature entanglement entropy for a
R = 100 site XX spin chain for two different subsystem sizes of r = 2, 98. At T = 0 the
two entanglement entropies are identical and the difference between the two is shown
as the solid line, quickly approaching 0 as T → 0.
love to see progress......
these are all o7 publications
enjoy
