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arXiv:quant-ph/0506209 · Physical Review A

Entangling power of permutation-invariant quantum states

Vladislav Popkov, Mario Salerno, Gunter Markus Schutz

Published Sep 22, 2005• 38 citations
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Abstract

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 $n$ sites in a system of length $L$ 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 $C$ is a function depending only on magnetization.

DOI
10.1103/physreva.72.032327

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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 \ensuremathσlog2[2\ensuremathπen(L\ensuremath−n)∕L]+C\ensuremath{\sigma}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}[2\ensuremath{\pi}en(L\ensuremath{-}n)∕L]+C\ensuremathσlog2​[2\ensuremathπen(L\ensuremath−n)∕L]+C, where \ensuremathσ\ensuremath{\sigma}\ensuremathσ is the on-site spin and CCC is a function depending only on magnetization.
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 $\frac{1}{2}{\mathrm{log}}_{2}[n(L\ensuremath{-}n)∕L]C(y)$, where $C(y)=2\mathrm{\ensuremath{\pi}}e(\frac{1}{4}\ensuremath{-}{y}^{2})$.
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Bipartite entanglement in the ground state of a chain of NNN quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and noncritical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem SA{S}_{A}SA​ consists of a comb of LLL spins, spaced ppp sites apart. Our results are thus not restricted to a simple area law, but contain nonlocal information, parametrized by the spacing ppp. For the XXXXXX model we calculate the von Neumann entropy analytically when $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and investigate its dependence on LLL and ppp. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.
We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.
In this paper we develop a new approach to the investigation of the bi-partite entanglement entropy in integrable quantum spin chains. Our method employs the well-known replica trick, thus taking a replica version of the spin chain model as starting point. At each site i of this new model we construct an operator T-i which acts as a cyclic permutation among the n replicas of the model. Infinite products of T-i give rise to local operators, precursors of branch-point twist fields of quantum field theory. The entanglement entropy is then expressed in terms of correlation functions of such operators. Employing this approach we investigate the von Neumann and Renyi entropies of a particularly interesting quantum state occurring as a limit (in a compact convergence topology) of the antiferromagnetic XXZ quantum spin chain. We find that, for large sizes, the entropy scales logarithmically, but not conformally.
Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. In this limit, the entropy of the block approaches a constant. The limiting entropy is a function of the anisotropy and of the magnetic field. The entropy reaches minima at product states and increases boundlessly at phase transitions.