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We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength JI) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength JII). In the absence of the internetwork coupling (J = 0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (JI≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as JI is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of JI. The nonmonotonic change of the critical value of JII is also compared with the result from the numerical RG calculation.

In a number of classical statistical-physical models, there exists acharacteristic dimensionality called the upper critical dimension abovewhich one observes the mean-field critical behavior. Instead of constructinghigh-dimensional lattices, however, one can also considerinfinite-dimensional structures, and the question is whether this mean-fieldcharacter extends to quantum-mechanical cases as well. We thereforeinvestigate the transverse-field quantum Ising model on the globally couplednetwork and on the Watts-Strogatz small-world network by means of quantum MonteCarlo simulations and the finite-size scaling analysis. We confirm that bothof the structures exhibit critical behavior consistent with the mean-fielddescription. In particular, we show that the existing cumulant methodhas difficulty in estimating the correct dynamic critical exponent andsuggest that an order parameter based on the quantum-mechanicalexpectation value can be a practically useful numerical observable todetermine critical behavior when there is no well-defined dimensionality.

In a number of classical statistical-physical models, there exists acharacteristic dimensionality called the upper critical dimension abovewhich one observes the mean-field critical behavior. Instead of constructinghigh-dimensional lattices, however, one can also considerinfinite-dimensional structures, and the question is whether this mean-fieldcharacter extends to quantum-mechanical cases as well. We thereforeinvestigate the transverse-field quantum Ising model on the globally couplednetwork and on the Watts-Strogatz small-world network by means of quantum MonteCarlo simulations and the finite-size scaling analysis. We confirm that bothof the structures exhibit critical behavior consistent with the mean-fielddescription. In particular, we show that the existing cumulant methodhas difficulty in estimating the correct dynamic critical exponent andsuggest that an order parameter based on the quantum-mechanicalexpectation value can be a practically useful numerical observable todetermine critical behavior when there is no well-defined dimensionality.

In a number of classical statistical-physical models, there exists acharacteristic dimensionality called the upper critical dimension abovewhich one observes the mean-field critical behavior. Instead of constructinghigh-dimensional lattices, however, one can also considerinfinite-dimensional structures, and the question is whether this mean-fieldcharacter extends to quantum-mechanical cases as well. We thereforeinvestigate the transverse-field quantum Ising model on the globally couplednetwork and on the Watts-Strogatz small-world network by means of quantum MonteCarlo simulations and the finite-size scaling analysis. We confirm that bothof the structures exhibit critical behavior consistent with the mean-fielddescription. In particular, we show that the existing cumulant methodhas difficulty in estimating the correct dynamic critical exponent andsuggest that an order parameter based on the quantum-mechanicalexpectation value can be a practically useful numerical observable todetermine critical behavior when there is no well-defined dimensionality.

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength JI) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength JII). In the absence of the internetwork coupling (J = 0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (JI≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as JI is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of JI. The nonmonotonic change of the critical value of JII is also compared with the result from the numerical RG calculation.

In a number of classical statistical-physical models, there exists acharacteristic dimensionality called the upper critical dimension abovewhich one observes the mean-field critical behavior. Instead of constructinghigh-dimensional lattices, however, one can also considerinfinite-dimensional structures, and the question is whether this mean-fieldcharacter extends to quantum-mechanical cases as well. We thereforeinvestigate the transverse-field quantum Ising model on the globally couplednetwork and on the Watts-Strogatz small-world network by means of quantum MonteCarlo simulations and the finite-size scaling analysis. We confirm that bothof the structures exhibit critical behavior consistent with the mean-fielddescription. In particular, we show that the existing cumulant methodhas difficulty in estimating the correct dynamic critical exponent andsuggest that an order parameter based on the quantum-mechanicalexpectation value can be a practically useful numerical observable todetermine critical behavior when there is no well-defined dimensionality.

In a number of classical statistical-physical models, there exists acharacteristic dimensionality called the upper critical dimension abovewhich one observes the mean-field critical behavior. Instead of constructinghigh-dimensional lattices, however, one can also considerinfinite-dimensional structures, and the question is whether this mean-fieldcharacter extends to quantum-mechanical cases as well. We thereforeinvestigate the transverse-field quantum Ising model on the globally couplednetwork and on the Watts-Strogatz small-world network by means of quantum MonteCarlo simulations and the finite-size scaling analysis. We confirm that bothof the structures exhibit critical behavior consistent with the mean-fielddescription. In particular, we show that the existing cumulant methodhas difficulty in estimating the correct dynamic critical exponent andsuggest that an order parameter based on the quantum-mechanicalexpectation value can be a practically useful numerical observable todetermine critical behavior when there is no well-defined dimensionality.