Wigner Crystallization of Electrons in a One-Dimensional Lattice: A Condensation in the Space of States.

We study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021).JPAMB51751-811310.1088/1751-8121/aba144]. The transition takes place at a critical value r_{s}_{c} of the usual dimensionless parameter r_{s} (radius of the volume available to each electron divided by effective Bohr radius) for which we are able to provide rigorous lower and upper bounds. For large screening length these bounds can be expressed in a closed analytical form. Demanding Monte Carlo simulations allow to estimate r_{s}_{c}≃2.3±0.2 at lattice filling 3/10 and screening length 10 lattice constants. This value is well within the rigorous bounds 0.7≤r_{s}_{c}≤4.3. Finally, we show that if screening is removed after the thermodynamic limit has been taken, r_{s}_{c} tends to zero. In contrast, in a bare unscreened Coulomb potential, Wigner crystallization always takes place as a smooth crossover, not as a quantum phase transition.

Absence of small-world effects at the quantum level and stability of the quantum critical point.

The small-world effect is a universal feature used to explain many different phenomena like percolation, diffusion, and consensus. Starting from any regular lattice of N sites, the small-world effect can be attained by rewiring randomly an O(N) number of links or by superimposing an equivalent number of new links onto the system. In a classical system this procedure is known to change radically its critical point and behavior, the new system being always effectively mean-field. Here, we prove that at the quantum level the above scenario does not apply: when an O(N) number of new couplings are randomly superimposed onto a quantum Ising chain, its quantum critical point and behavior both remain unchanged. In other words, at zero temperature quantum fluctuations destroy any small-world effect. This exact result sheds new light on the significance of the quantum critical point as a thermodynamically stable feature of nature that has no analogy at the classical level and essentially prevents a naive application of network theory to quantum systems. The derivation is obtained by combining the quantum-classical mapping with a simple topological argument.

Statistical mechanics of random geometric graphs: Geometry-induced first-order phase transition.

Random geometric graphs (RGGs) can be formalized as hidden-variables models where the hidden variables are the coordinates of the nodes. Here we develop a general approach to extract the typical configurations of a generic hidden-variables model and apply the resulting equations to RGGs. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a nontrivial satisfaction problem: Given N nodes, a domain D, and a desired average connectivity 〈k〉, find, if any, the distribution of nodes having support in D and average connectivity 〈k〉. We find out that, in the thermodynamic limit, nodes are either uniformly distributed or highly condensed in a small region, the two regimes being separated by a first-order phase transition characterized by a O(N) jump of 〈k〉. Other intermediate values of 〈k〉 correspond to very rare graph realizations. The phase transition is observed as a function of a parameter a∈[0,1] that tunes the underlying geometry. In particular, a=1 indicates a rigid geometry where only close nodes are connected, while a=0 indicates a rigid antigeometry where only distant nodes are connected. Consistently, when a=1/2 there is no geometry and no phase transition. After discussing the numerical analysis, we provide a combinatorial argument to fully explain the mechanism inducing this phase transition and recognize it as an easy-hard-easy transition. Our result shows that, in general, ad hoc optimized networks can hardly be designed, unless to rely to specific heterogeneous constructions, not necessarily scale free.

Fluctuation analysis in complex networks modeled by hidden-variable models: necessity of a large cutoff in hidden-variable models.

It is becoming more and more clear that complex networks present remarkable large fluctuations. These fluctuations may manifest differently according to the given model. In this paper we reconsider hidden-variable models which turn out to be more analytically treatable and for which we have recently shown clear evidence of non-self-averaging, the density of a motif being subject to possible uncontrollable fluctuations in the infinite-size limit. Here we provide full detailed calculations and we show that large fluctuations are only due to the node-hidden variables variability while, in ensembles where these are frozen, fluctuations are negligible in the thermodynamic limit and equal the fluctuations of classical random graphs. A special attention is paid to the choice of the cutoff: We show that in hidden-variable models, only a cutoff growing as N(λ) with λ ≥ 1 can reproduce the scaling of a power-law degree distribution. In turn, it is this large cutoff that generates non-self-averaging.

Duality between equilibrium and growing networks.

In statistical physics any given system can be either at an equilibrium or away from it. Networks are not an exception. Most network models can be classified as either equilibrium or growing. Here we show that under certain conditions there exists an equilibrium formulation for any growing network model, and vice versa. The equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically, but even for any finite system size. The required conditions are satisfied in random geometric graphs in general and causal sets in particular, and to a large extent in some real networks.