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arXiv:quant-ph/0404026 · Physical Review A

Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model

Vladislav Popkov, Mario Salerno

Published Jan 3, 2005• 91 citations
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Abstract

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-$\frac{1}{2}$ 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 $C(y)=2\mathrm{\ensuremath{\pi}}e(\frac{1}{4}\ensuremath{-}{y}^{2})$.

DOI
10.1103/physreva.71.012301

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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 nnn sites for the ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chain of the length LLL in a sector with fixed magnetization yyy per site grows as 12log2[n(L\ensuremath−n)∕L]C(y)\frac{1}{2}{\mathrm{log}}_{2}[n(L\ensuremath{-}n)∕L]C(y)21​log2​[n(L\ensuremath−n)∕L]C(y), where C(y)=2\ensuremathπe(14\ensuremath−y2)C(y)=2\mathrm{\ensuremath{\pi}}e(\frac{1}{4}\ensuremath{-}{y}^{2})C(y)=2\ensuremathπe(41​\ensuremath−y2).
We investigate the von Neumann entanglement entropy as function of the size of a subsystem for permutation invariant ground states in models with finite number of states per site, e.g., in quantum spin models. We demonstrate that the entanglement entropy of nnn sites in a system of length LLL generically grows as $\ensuremath{\sigma}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}[2\ensuremath{\pi}en(L\ensuremath{-}n)∕L]+C$, where $\ensuremath{\sigma}$ is the on-site spin and CCC is a function depending only on magnetization.
It is known that the entropy of a block of spins of size L embedded in an infinite pure critical spin chain diverges as the logarithm of L with a prefactor fixed by the central charge of the correspondent conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an ‘effective’ central charge. By computing the entanglement entropy for a family of models which includes the N -states random Potts chain and the Z N clock model, we give some definitive answer to recent conjectures about the behaviour of this effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.
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.
By Lanczos exact diagonalization and the infinite time-evolving block decimation (iTEBD) technique, the two-site entanglement as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization in the antiferromagnetic Heisenberg (AFH) model under an external field are investigated. With increasing external field, the small size system shows some distinct upward magnetization stairsteps, accompanied synchronously with some downward two-site entanglement stairsteps. In the thermodynamic limit, the two-site entanglement, as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization are calculated, and the critical magnetic field h(c) = 2.0 is determined exactly. Our numerical results show that the quantum entanglement is sensitive to the subtle changing of the ground state, and can be used to describe the magnetization and quantum phase transition. Based on the discontinuous behavior of the first-order derivative of the entanglement entropy and fidelity per site, we think that the quantum phase transition in this model should belong to the second-order category. Furthermore, in the magnon existence region (h < 2.0), a logarithmically divergent behavior of block entanglement which can be described by a free bosonic field theory is observed, and the central charge c is determined to be 1.
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.