  Numerical studies of the reduced density matrix of a gapped spin-1/2
Heisenberg antiferromagnet on a two-leg ladder find that it has the same form
as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic
chain at a finite temperature determined by the spin gap of the ladder. We
investigate this interesting result by considering a model of free fermions on
a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin
systems on a ladder with a gapped ground state using exact solutions and
several controlled approximations. We calculate the reduced density matrix and
the entanglement entropy for a leg of the ladder (i.e. cut made between the
chains). In the fermionic system we find the exact form of the reduced density
matrix for one of the chains and determine the entanglement spectrum
explicitly. Here we find that in the weak tunneling limit of the ladder the
entanglement entropy of one chain of the gapped ladder has a simple and
universal form dictated by conformal invariance. In the case of the spin
system, we consider the strong coupling limit by using perturbation theory and
get the reduced density matrix by the Schmidt decomposition. The entanglement
entropies of a general gapped system of two coupled conformal field theories
(in 1+1 dimensions) is discussed using the replica trick and scaling arguments.
We show that 1) for a system with a bulk gap the reduced density matrix has the
form of a thermal density matrix, 2) the long-wavelength modes of one subsystem
(a chain) of a gapped coupled system are always thermal, 3) the von Neumann
entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap
plays the role of effective temperature.
