Recent studies have shown that logarithmic divergence of entanglement entropy as a function of the size of a subsystem is a signature of criticality in quantum models. We demonstrate that the ground-state entanglement entropy of
n sites for the ferromagnetic Heisenberg spin-
21 chain of the length
L in a sector with fixed magnetization
y per site grows as $\frac{1}{2}{\mathrm{log}}_{2}[n(L\ensuremath{-}n)∕L]C(y)$, where $C(y)=2\mathrm{\ensuremath{\pi}}e(\frac{1}{4}\ensuremath{-}{y}^{2})$.
Block entanglement entropy in the ground state of a quantum spin chain is investigated. The spins have Kitaev-type nearest-neighbor interaction, of strength
Jx or
Jy, through either
x or
y components of the spins on alternating bonds, along with a transverse magnetic field
h. An exact solution is obtained through Jordan-Wigner fermionization, and it exhibits a macroscopically degenerate ground state for
h=0 and a nondegenerate ground state for $h\ensuremath{\ne}0$ for all interaction strengths. For a chain of
N spins, we study the block entropy of a partition of
L contiguous spins. The block entanglement entropy needs the eigenvalues of the
2L-dimensional reduced density matrix. We employ an efficient method that reduces this problem to evaluating eigenvalues of an
L-dimensional matrix, which enables us to calculate easily the block entanglement for large-
N chains numerically. The entanglement entropy grows as
logL at the degeneracy point
h=0 and only for
Jx=Jy. For $h\ensuremath{\ne}0$, the entropy becomes independent of the size, thus obeying the area law. For ${J}_{x}\ensuremath{\ne}{J}_{y}$, the block entropy shows a nonmonotonic behavior for
L<N/2.
We present a perturbative method to compute the ground state entanglement entropy for interacting systems. We apply it to a collective model of mutually interacting spins in a magnetic field. At the quantum critical point, the entanglement entropy scales logarithmically with the subsystem size, the system size, and the anisotropy parameter. We determine the corresponding scaling prefactors and evaluate the leading finite-size correction to the entropy. Our analytical predictions are in perfect agreement with numerical results.
We compute the entropy of entanglement of two blocks of
L spins at a distance
d in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size
L=1 and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both
L and
d. This formula is in excellent agreement with numerical results.
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c eff , defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.
We study the entanglement entropy of a block of contiguous spins in excited states of spin chains. We consider the XY model in a transverse field and the XXZ Heisenberg spin-chain. For the latter, we developed a numerical application of algebraic Bethe Ansatz. We find two main classes of states with logarithmic and extensive behavior in the dimension of the block, characterized by the properties of excitations of the state. This behavior can be related to the locality properties of the Hamiltonian having a given state as ground state. We also provide several details of the finite size scaling. Entanglement entropy of excited states 2 1.