We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets $A$ and $\overline{A}$ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on $A$ and $\overline{A}$. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy $S_A$ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where $S_A\left(L\right)\propto L^{\beta }$, with $\beta$ being intermediate between $\beta =0$ (area law) and $\beta =1$ (BP state inducing volume law). For this model, the dispersion relation has a “zigzag” structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\beta$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability $C/r^{d+\sigma }$, where $r$ is the distance length between distinct sites and $d=1$. We introduce and test an order-$N$ Monte Carlo algorithm and we determine as a function of $\sigma$ the critical value $C_c$ at which percolation occurs. The critical exponents in the range $0<\sigma <1$ are reported. Our analysis is in agreement, up to a numerical precision $\approx {10}^{-3}$, with the mean-field result for the anomalous dimension $\eta =2-\sigma$, showing that there is no correction to $\eta$ due to correlation effects. The obtained values for $C_c$ are compared with a known exact bound, while the critical exponent $\nu$ is compared with results from mean-field theory, from an expansion around the point $\sigma =1$ and from the $\varepsilon$-expansion used with the introduction of a suitably defined effective dimension $d_{\mathrm{eff}}$ relating the long-range model with a short-range one in dimension $d_{\mathrm{eff}}$. We finally present a formulation of our algorithm for bond percolation on general graphs, with order $N$ efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension $d$ with $\sigma >0$.

We study renormalization group (RG) fixed points of scalar field theoin order to get access to the corresponding RG beta functions, we derive general multicomponent beta functionals β_V and β_Z in the aforementioned upper critical dimensions, most of which are novel. The field theories we analyze have N = 2 (polygons), N = 3 (Platonic solids) and N =4 (hyper-Platonic solids) field components. The main results of this analysis include a new candidate universality class in three physical dimensions based on the symmetry group \( {\mathbb{D}}_5 \) of the Pentagon. Moreover we find new Icosahedron fixed points in d < 3, the fixed points of the 24-Cell, multi-critical O(N) and ϕⁿ-Cubic universality classes.

Universality is one of the key concepts in understanding critical phenomena. However, for interacting inhomogeneous systems described by complex networks, a clear understanding of the relevant parameters for universality is still missing. Here we discuss the role of a fundamental network parameter for universality, the spectral dimension. For this purpose, we construct a complex network model where the probability of a bond between two nodes is proportional to a power law of the nodes' distances. By explicit computation we prove that the spectral dimension for this model can be tuned continuously from 1 to infinity, and we discuss related network connectivity measures. We propose our model as a tool to probe universal behavior on inhomogeneous structures and comment on the possibility that the universal behavior of correlated models on such networks mimics the one of continuous field theories in fractional Euclidean dimensions.

Based on conformal symmetry we propose an exact formula for the four-point connectivities of Fortuin-Kasteleyn clusters in the critical Ising model when the four points are anchored to the boundary. The explicit solution we found displays logarithmic singularities. We check our prediction using Monte Carlo simulations on a triangular lattice, showing excellent agreement. Our findings could shed further light on the formidable task of the characterization of logarithmic conformal field theories and on their relevance in physics.