{'fraction_non_alphanumeric': 0.03447732997481108, 'fraction_numerical': 0.006769521410579345, 'mean_word_length': 4.481018119068162, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantum quenches and thermalization on scale-free graphs

We show that after a quantum quench of the parameter controlling the number of particles in a Fermi-Hubbard model on scale free graphs, the distribution of energy modes follows a power law dependent on the quenched parameter and the connectivity of the graph. This paper contributes to the literature of quantum quenches on lattices, in which, for many integrable lattice models the distribution of modes after a quench thermalizes to a Generalized Gibbs Ensemble; this paper provides another example of distribution which can arise after relaxation. We argue that the main role is played by the symmetry of the underlying lattice which, in the case we study, is scale free, and to the distortion in the density of modes.


Introduction
There has been recent interest in the effective thermal dynamics following a quantum quench in spin chains [1]. The dynamics out of equilibrium of quantum systems [3] has received a great amount of attention [4,5,6,7]. It became clear that after a quantum quench, many observables at equilibrium after a quantum quench are distributed according to a Generalized Gibbs Ensemble (GGE) [2,1,7,8]. In a previous work, we studied quantum quenches in a Fermi-Hubbard model which does not conserve the number of particles [9]. We studied the energy of the excitations, which are invariant under time evolution after the quench, and found that these are distributed according to a GGE. There, the temperature is associated with the gap in the spectrum, which is due to the coupling of non-conserving number of particles term. A similar phenomenon happens in quantum liquids [10]. In this paper we explore a similar approach on a different type of underlying interaction network. Several classical statistical models have been studied on complex networks [11]. Complex networks have become an area of tremendous recent interest since the discoveries of the small world and scale free properties in many realistic networks. A small world network is characterized by short network distance a high clustering coefficient. Several reviews of the subjects are now available [12,13,14]. Important applications of these techniques are spreading of diseases [15] and syncronization [16] on complex networks. Watts and Strogatz demonstrated that the two small-world characteristics can be obtained from a regular network by rewiring or adding a few long-range links shortcuts, which connect otherwise distant nodes [17]. A regular network intrinsically already has a high clustering coefficient, but has large network distance. Thus, few shortcuts can reduce the distance exponentially unaffecting the clustering coefficient. The scale-free property is characterized by an algebraic degree distribution: where the degree variable k measures the number of links of node in the network, P (d 1) ≈ d γ , and where γ is the algebraic scaling exponent. Barabási and Albert discovered the scale-free property and also proposed growth and preferential attachment as the two basic mechanisms responsible for the scale-free property. Here, growth requires that the numbers of nodes and links increase with time and preferential attachment means that when a new node is added to the network, the probability that it connects to an existing node is proportional to the number of links that this node has already had. In this paper, we consider the properties of quantum quenches on Barabási-Albert type of graphs. It is important to understand what kind of distributions can arise in a quantum quench. Although per se the study of quantum phenomena on complex networks might not be physically relevant, it is important from the theoretical point of view. While GGE is a quite common example of distribution arising, here we show that another type of distribution arise.
2 Set up of the model

Scale-free graphs
In this section we recall the growth algorithm used in the preferential attachment model introduced by Barabási and Albert [20]. The growth algorithm (preferential attachment) is parametrized by a single parameter, M . The starting graph is a single node, with no edges. Then, at each step, a new node is added, with M edges. The edges are attached at random to the previous existing nodes, with a probability proportional to the degree of the node. If d i is the degree of the vertex i, at each edge is attached to a node i with probability p i = di i di . As it is well known, these graphs are scale-free, i.e. for N 1, the degree distribution is a power-law, P (d 1) ≈ d α , with α exponent of the power law. Another property, here important to mention, is that scale-free graphs are ultra-small: the average distance between two nodes goes as ∼ log(log(N )), where N is the number of nodes in the network. Notably, Bose-Einstein condensation appear in growing networks if preferential attachment growth is generalized with fitness [18].

Hamiltonian and Quantum Quench
Here we want to recollect the formalism introduced in [9]. The Hamiltonian we will consider in the present paper is the following Fermi-Hubbard model: where a i (a † i ) is the annihilation (creation) fermionic operator that annihilate (create) a particle in the vertex i of the background graph Γ M . The matrices A are, respectively, the adjacency matrix of Γ M and its antisymmetrized form. In the present paper, the adjacency matrix will be the one of a scale-free graph built using the Barabási-Albert growth algorithm. The sum runs over all the N nodes of the graph Γ M , where M is the connectivity parameter introduced previously. The coupling J is the tunneling of the particles between two connected sites and λ controls the strength of the Hamiltonian terms that do not conserve the number of particles. The physical properties are independent from time-scaling if we perform a sudden quantum quench, thus we can measure excitations in units of J [1] . In particular, we introduced a notion of particle (with an associated discrete labeling k, (k)) given in terms of ladder operators η k . Once the notion of particle that the detector measures is established, we can determine the energy distribution (number of particles with momentum k) of the ground state of the system For graphs with discrete translational invariance, this is associated with the Fourier transform vectors e ikx , but for graphs without particular symmetry this identification is lost. The notion of particle η k , together with its dispersion relation (k) will be defined in terms of a test Hamiltonian The momentum distribution (2) that the detector measures is given by the overlap between the ground state of the system and the eigenstates states of the test Hamiltonian H test . The quantum quench we are going to perform is given by from the ground state of H(Γ M , λ), |GS ,. After the quench. The system H(Γ M , 0) will see the ground state |GS as an excited state, and thus it makes sense to calculate the spectrum-density n( (k)). The hopping Hamiltonian can be written as The eigenmodes of H test , labeled by an integer k, and with energy (k), define our notion of particle. These are created and annihilated by the operators η † k and η k , and are the excitations that the detector measures, and that we will calculate. Therefore, we need to compute and calculate the distribution. As we will see, n(k) ≈ (k) γ , where γ is the exponent we will study. The two Hamiltonians have the same number of nodes, thus their Hilbert states overlap (coincident). [2] If the system were at equilibrium with an external bath at temperature T , we would have In the case of an n-dimensional tori, we found in [9] for the same model we study here, that: where λ/2 plays the role of the temperature. The spectrum of the adjacency matrix of a scale-free network was studied in [21]. It is known that eigenvalues are distributed according to a power law, while for the eigenvectors an analytical form is still lacking. However, the components of the eigenvectors are strongly localized at the hubs. As a matter of fact, in our numerical calculation we will neglect the role of the shape of the Fermi surface, approximating it with a sphere. This implies that a small error is made on the temperature, that can become more and more relevant for λ ≈ 1. Thus, our result is valid in the limit |λ| 1.

Results
Here we show that the distribution of the modes in the ground state follows a power law. Since analytical techniques are lacking, we evaluated eqn. (5) numerically. all the numerical results are obtained with a number of nodes N = 700. We can tackle such a big quantum system, due to the Bogoliubov transformation which diagonalizes this model; the quantum dynamics of this system is in fact restricted to a Hilbert space of size 2N , instead of the 2 N of an ordinary spin system. [2] The Hamiltonian (1) is a quadratic model, hence, it can be diagonalized as by means of a Bogoliubov transformation of the fundamental particle operators, a † i , a j . In turn, these are related by another Bogoliubov transformations to the operators η, η † . Then, the operators η, η † will be connected to the ψ, ψ † by the Bogoliubov transformation that is the composition of the Bogoliubov transformations that relate ψ, ψ † to a, a † and a, a † to η, η † . It can be written formally as where α kq and β kq are the Bogoliubov coefficents. In Fig. 2 we plot log(n(k)) against log( (k)). The functional dependence appears, numerically, to be of the form: where γ is weakly dependent on λ, in the range λ ∈ [0, 0.2] and takes value γ ∈ [−0.85, −0.95]. Tables 1 and 2 show the values obtained numerically by fixing M and λ and fitting the power law, together with the error, of the parameters f and γ. In Fig. 4 and Fig. 5 we plot the functional dependence of the constant in front of the Zipf's law, by fixing the values of M and λ. These appear to be both convex functions of the parameters. The power law is rather stable over several order of magnitudes, log( (k)) ∈ [−2, 2], although the density of points is higher in log( (k)) 0. For λ ∈ [0.2, 1], the distribution changes shape in log( (k)) < 0). Thus, restoring the units, the power law for density of excitations is valid only in the limit λ J 1. It is clear from this analysis that the power law exponent, in its domain of validity, is independent from the connectivity parameter M .

Conclusions
In this paper we discussed quantum quenches of a Fermi-Hubbard model on scale-free graphs. A previous analysis of this quench protocol was performed in [9], and solved analytically for the case of n-dimensional torii, showing that the Generalized Gibbs Ensemble emerges (GGE). The quenched parameter, λ, controls the conservation of the particle number. For non-integrable systems, it is a known fact that the expectation value of several observables after a quantum quench, are similar to those calculated on a thermal state, i.e. a GGE. The observables we considered were the density of eigenmodes calculated over the ground state of the unquenched Hamiltonian. In this work, we have shown that the spectrum of the excitations for scale free graphs are distributed according to a power law. We analyzed the functional dependence of the two parameters of the power law on the topological properties of the scale-free graph, the connectivity, and the quenched parameter. We found that while small values of λ this distribution is a power law, for higher values of the quenched parameter (λ > 0.2), the distribution changes shape and the power law behavior is lost.
The outcome of our analysis shows that the underlying symmetry of the graph does indeed contribute the to the shape of the distribution of modes. While GGE naturally emerges in the case of translational invariant lattices, we have given evidences for a counter example, in which a power law distribution for the modes spectral density arises. Due to the lack of analytical understanding of the eigenmodes for scale free graphs, we have reached the conclusion using a numerical approach. We have focused on a Fermi-Hubbard model for a quite simple, technical reason. The scale free properties of graphs generated using a preferential attachment become evident for large graphs, e.g. for the case N → ∞. We thus had to focus on a model which can be diagonalized numerically for large graphs. For the Fermi-Hubbard model considered in the present paper, we could take advantage of a Bogoliubov transformation, which allows to tackle the diagonalization process on a matrix of size proportional to the number of nodes, and not exponentially increasing, as standard in quantum mechanics.  Table 1 Table of γ and f as function of M , for fixed λ = 0.05 and N = 700.