Corner contribution to percolation cluster numbers in three dimensions

Abstract

In three-dimensional critical percolation we study numerically the number of clusters, $N_{\Gamma}$, which intersect a given subset of bonds, $\Gamma$. If $\Gamma$ represents the interface between a subsystem and the environment, then $N_{\Gamma}$ is related to the entanglement entropy of the critical diluted quantum Ising model. Due to corners in $\Gamma$ there are singular corrections to $N_{\Gamma}$, which scale as $b_{\Gamma} \ln L_{\Gamma}$, $L_{\Gamma}$ being the linear size of $\Gamma$ and the prefactor, $b_{\Gamma}$, is found to be universal. This result indicates that logarithmic finite-size corrections exist in the free-energy of three-dimensional critical systems.Comment: 6 pages, 7 figures. arXiv admin note: text overlap with arXiv:1210.467

Entanglement of skeletal regions

“…The (disorder-averaged) EE of certain skeletal regions for the bond-diluted quantum Ising model displays universal logarithmic corrections [15,16]. These can be obtained applying Cardy-Peschel formula, thus sharing similar structures with the EE in z = 2 Lifshitz theory.…”

“…For one-dimensional (1d) systems, single-site entanglement has been shown to provide a diagnostic of quantum phase transitions [9][10][11], and has found successful use in the context of quantum impurity models [12,13]. In higher dimensions, although studies have been performed on certain specific models/subregions [14][15][16][17][18], much less is known about entanglement measures defined for skeletal regions.…”

Quantum multicritical point in the two- and three-dimensional random transverse-field Ising model

“…Finally, we note that our SDRG investigations can be extended in several directions. Here, we mention the characterization of the entanglement entropy 24,25,[37][38][39][40] , transverse correlations 51 , boundary critical exponents 52 , the impact of long-range interactions 53,54 , as well as the dynamical singularities in the disordered and ordered Griffiths phases 26,27,42 .…”

“…formally corresponding to a diverging z dynamical exponent, characteristic of ultraslow dynamics, a hallmark of IDFPs. Both the percolation and generic QCPs have been studied either analytically or numerically to high precision in 2 and 3D, including the quantum entanglement properties 24,25,[37][38][39][40] . The last missing piece from a complete understanding of the RTIM phase diagram in Fig.…”

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

“…In the following section we show, that in the random cluster representation S Γ is simply given by the mean number of clusters in the optimal sets which are crossed by Γ. This type of problem has already been considered by two of us in the case of the non-random Potts model both for Q = 1, representing percolation 28,29 and for general values of Q ≤ 4 30 . Repeating the reasoning applied in these papers we show that the dominant term of S Γ represents the area law to which there are logarithmic corrections at the critical point due to corners and these are calculated by conformal techniques.…”