The half-chain entanglement entropy (EE) of the one-dimensional
XX (isotropic
XY) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed (
++) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the
XX chain and QICs: (1) the EE of the
XX chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the
XX chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the
XX chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the
XX chain with fixed-fixed (
++) BCs is the sum of the EE of a QIC with fixed-fixed (
++) BCs and that with free-free BCs; and (5) the EE of the
XX chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed (
++) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where
L is the length of the spin chain. We find that the fitted parameters for the leading term
lnL and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.
By means of free fermionic techniques we study the time evolution of the entanglement entropy
S(t) of a block of spins in the random transverse-field Ising chain after a sudden change of the parameters of the Hamiltonian. We consider global quenches, when the parameters are modified uniformly in space, as well as local quenches, when two disconnected blocks are suddenly joined together. For a noncritical final state, the dynamical entanglement entropy is found to approach a finite limiting value for both types of quenches. If the quench is performed to the critical state, the entropy grows for an infinite block as $\mathcal{S}(t)\ensuremath{\sim}\mathrm{ln}\mathrm{ln}t$. This type of ultraslow increase is explained through the strong disorder renormalization group method.
We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.
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.
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 the 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 the ground state. We also provide several details of the finite size scaling.