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arXiv:1612.04789

Entanglement of heterogeneous free fermion chains

Yuchi He

Published Dec 14, 2016• 0 citations
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Abstract

We calculate the ground state entanglement entropy between two heterogeneous parts of a free fermion chain. The two parts could be XX chains with different parameters or an XX half chain connected with a quantum Ising half chain. It is shown that logarithmic behavior holds if the two parts are conformally critical. In other cases, area law holds with abundant convergence behaviors. In particular, when XX chain at Lifshitz point is connected with a conformally or Lifshitz critical part, entanglement entropy converges algebraically with a fractional subleading index.

DOI
10.48550/arxiv.1612.04789

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We calculate the ground state entanglement entropy between two heterogeneous parts of a free fermion chain. The two parts could be XX chains with different parameters or an XX half chain connected with a quantum Ising half chain. It is shown that logarithmic behavior holds if the two parts are conformally critical. In other cases, area law holds with abundant convergence behaviors. In particular, when XX chain at Lifshitz point is connected with a conformally or Lifshitz critical part, entanglement entropy converges algebraically with a fractional subleading index.
The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area law is proven to hold-or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.
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The half-chain entanglement entropy (EE) of the one-dimensional XXXXXX (isotropic XYXYXY) 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 XXXXXX chain and QICs: (1) the EE of the XXXXXX chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the XXXXXX chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the XXXXXX 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 XXXXXX 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 XXXXXX 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 LLL is the length of the spin chain. We find that the fitted parameters for the leading term lnLlnLlnL 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.
For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.