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arXiv:1101.3654 · Physical Review A

Quantum phase transitions in fully connected spin models: An entanglement perspective

Michele Filippone, Sébastien Dusuel, Julien Vidal

Published Feb 25, 2011• 52 citations
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Abstract

We consider a set of fully connected spin models that display first- or second-order transitions and for which we compute the ground-state entanglement in the thermodynamical limit. We analyze several entanglement measures (concurrence, R\'enyi entropy, and negativity) and show that, in general, discontinuous transitions lead to a jump of these quantities at the transition point. Interestingly, we also find examples where this is not the case.

DOI
10.1103/physreva.83.022327

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We consider a set of fully connected spin models that display first- or second-order transitions and for which we compute the ground-state entanglement in the thermodynamical limit. We analyze several entanglement measures (concurrence, R'enyi entropy, and negativity) and show that, in general, discontinuous transitions lead to a jump of these quantities at the transition point. Interestingly, we also find examples where this is not the case.
We show that, for an exactly solvable quantum spin model, a discontinuity in the first derivative of the ground-state concurrence appears in the absence of a quantum phase transition. It is opposed to the popular belief that the nonanalyticity property of ground-state concurrence can be used to determine quantum phase transitions. We further point out that the analyticity property of the ground-state concurrence in general can be more intricate than that of the ground-state energy. Thus there is no one-to-one correspondence between quantum phase transitions and the nonanalyticity property of the concurrence. Moreover, we show that the von Neumann entropy, as another measure of entanglement, cannot reveal quantum phase transitions in the present model. Therefore, in order to link with quantum phase transitions, some other measures of entanglement are needed.
First order quantum phase transitions (1QPTs) are signalled, in the thermodynamic limit, by discontinuous changes in the ground state properties. These discontinuities affect expectation values of observables, including spatial correlations. When a 1QPT is crossed in the vicinity of a second order one, due to the correlation length divergence of the latter, the corresponding ground state is modified and it becomes increasingly difficult to determine the order of the transition when the size of the system is finite. Here we show that, in such situations, it is possible to apply finite size scaling (FSS) to entanglement measures, as it has recently been done for the order parameters and the energy gap, in order to recover the correct thermodynamic limit (Campostrini et al 2014 Phys. Rev. Lett. 113 070402). Such a FSS can unambiguously discriminate between first and second order phase transitions in the vicinity of multicritical points even when the singularities displayed by entanglement measures lead to controversial results.
We introduce a measure called reduced entropy of sublattice to quantify entanglement in spin, electron, and boson systems. By analyzing this quantity, we reveal an intriguing connection between quantum entanglement and quantum phase transitions in various strongly correlated systems: the local extremes of reduced entropy and its first derivative as functions of the coupling constant coincide, respectively, with the first and second order transition points. Exact numerical studies merely for small lattices reproduce several well-known results, demonstrating that our scenario is quite promising for exploring quantum phase transitions.
The entanglement entropy S\mathcal{S}S is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension S\mathcal{S}S generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the QQQ-state quantum Potts chain for Q>4Q>4Q>4 and calculate S\mathcal{S}S across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point S\mathcal{S}S shows a jump, which is expected to vanish for $Q\ensuremath{\rightarrow}{4}^{+}$. This jump is calculated in leading order as $\mathrm{\ensuremath{\Delta}}\mathcal{S}=lnQ[1\ensuremath{-}4/Q\ensuremath{-}2/(QlnQ)+O(1/{Q}^{2})]$.