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Massimo Ostilli

Publications and source records attributed to Massimo Ostilli.

At least 19 recordsLinked to original sources
arXiv2026

Rigorous existence and location of quantum phase transitions in lattice Hamiltonian systems

We extend the analysis of the class of quantum phase transitions (QPTs) that can be interpreted as condensations in state space, first introduced in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021)], by generalizing the arguments of [M. Ostilli and C. Presilla, Phys. Rev. Lett. 127, 040601 (2021)] to prove the existence and determine the location (via simple bounds) of QPTs in general one-parameter lattice Hamiltonians. Unlike our original formulation, this extension also encompasses second-order QPTs, for which we provide the explicit example of the transverse-field Ising model. Our analysis suggests that, under conditions typically satisfied in physical contexts, any QPT taking place in lattice systems can be interpreted as a condensation in state space.

quant-ph↗
arXiv2026

Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach

Partitioning large networks into stable clusters of synchronized nodes is a challenging task. Recent approaches based on spectral analysis can provide exact results on specific dynamics but remain unfeasible for very large networks. Moreover, within a stochastic framework, it is unclear which dynamics should be chosen to study synchronization. Here we propose an unbiased and scalable method based on the message-passing algorithm. By exploiting the collective behavior emerging across critical points of an effective Ising-like model, we identify dynamically coherent clusters of synchronized nodes and illustrate the approach on some large real-world networks. We find that, unlike continuous-time dynamics, abrupt desyncrhronization occurs even in simple graphs, without the need to invoke higher order interactions. However, when noise is included, the transition to synchronization becomes smoother and proceeds through the formation of plateaus, albeit at the cost of requiring larger coupling strengths.

physics.soc-ph↗
arXiv2025

Topologically protected synchronization in networks

In a graph, we say that two nodes are topologically equivalent if their sets of first neighbors, excluding the two nodes, coincide. We prove that nonlinearly coupled heterogeneous oscillators located on a group of topologically equivalent nodes can get easily synchronized when the group forms a fully connected subgraph (or combinations thereof), regardless of the status of all the other oscillators. More generally, any change occurring in the remainder of the graph will not alter the synchronization status of the group. Typically, the group can synchronize when $k^{(\mathrm{OUT})}\leq k^{(\mathrm{IN})}$, $k^{(\mathrm{IN})}$ and $k^{(\mathrm{OUT})}$ being the common internal and outgoing degree of each node in the group, respectively. Simulations confirm our rigorous analysis and suggest that groups of topologically equivalent nodes act as independent pacemakers.

cond-mat.dis-nn↗
arXiv2023

Finite temperature quantum condensations in the space of states: a new perspective for quantum annealing

In nature, everything occurs at finite temperature and quantum phase transitions (QPTs) cannot be an exception. Nevertheless, they are still mainly discussed and formulated at zero temperature. We show that the condensation QPTs recently introduced at zero temperature can naturally be extended to finite temperature just by replacing ground state energies with corresponding free energies. We illustrate this criterion in the paradigmatic Grover model and in a system of free fermions in a one-dimensional inhomogeneous lattice. In agreement with expected universal features, the two systems show structurally similar phase diagrams. Last, we explain how finite temperature condensation QPTs can be used to construct quantum annealers having, at finite temperature, output-probability exponentially close to 1 in the system size. As examples we consider again the Grover model and the fermionic system, the latter being well within the reach of present heterostructure technology.

quant-ph↗
arXiv2023

Finite temperature quantum condensations in the space of states: General Proof

We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently introduced and discussed at zero temperature~(Ostilli and Presilla 2021 \textit{J. Phys. A: Math. Theor.} \textbf{54} 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space $\mathcal{H}$ into two subspaces, $\mathcal{H}_\mathrm{cond}$ and $\mathcal{H}_\mathrm{norm}$, such that, in the thermodynamic limit, $\dim \mathcal{H}_\mathrm{cond}/ \dim \mathcal{H} \to 0$ and the free energies of the system restricted to these subspaces cross each other for some value of the Hamiltonian parameters, then, the system undergoes a first-order quantum phase transition driven by those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temperature of the system. We also show that the critical surface has universal features at high and low temperatures.

quant-ph↗
arXiv2021

Ground-state-energy universality of noninteracting fermionic systems

When noninteracting fermions are confined in a $D$-dimensional region of volume $\mathrm{O}(L^D)$ and subjected to a continuous (or piecewise continuous) potential $V$ which decays sufficiently fast with distance, in the thermodynamic limit, the ground state energy of the system does not depend on $V$. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in $D$ dimensions. We find that this universality property holds under a quite mild condition on $V$, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.

cond-mat.stat-mech↗
arXiv2021

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 J. Phys.~A \textbf{54}, 055005 (2021). 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 ${r_s}_{c}\simeq 2.3 \pm 0.2$ at lattice filling $3/10$ and screening length 10 lattice constants. This value is well within the rigorous bounds $0.7\leq {r_s}_{c}\leq 4.3$. 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.

cond-mat.stat-mech↗
arXiv2021

First-order quantum phase transitions as condensations in the space of states

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian $H=K+gV$, where $K$ and $V$ are two non commuting operators acting on the space of states $\mathbb{F}$, we may always write $\mathbb{F}=\mathbb{F}_\mathrm{cond} \oplus \mathbb{F}_\mathrm{norm}$ where $\mathbb{F}_\mathrm{cond}$ is the subspace spanned by the eigenstates of $V$ with minimal eigenvalue and $\mathbb{F}_\mathrm{norm}=\mathbb{F}_\mathrm{cond}^\perp$. If, in the thermodynamic limit, $M_\mathrm{cond}/M \to 0$, where $M$ and $M_\mathrm{cond}$ are, respectively, the dimensions of $\mathbb{F}$ and $\mathbb{F}_\mathrm{cond}$, the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, $ε$, coincides with the smaller between $ε_\mathrm{cond}$ and $ε_\mathrm{norm}$, the ground state energies per particle of the system restricted to the subspaces $\mathbb{F}_\mathrm{cond}$ and $\mathbb{F}_\mathrm{norm}$, respectively: $ε=\min\{ε_\mathrm{cond},ε_\mathrm{norm}\}$. It may then happen that, as a function of the parameter $g$, the energies $ε_\mathrm{cond}$ and $ε_\mathrm{norm}$ cross at $g=g_\mathrm{c}$. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace $\mathbb{F}_\mathrm{cond}$) and a normal phase (system spread over the large subspace $\mathbb{F}_\mathrm{norm}$)....

quant-ph↗
arXiv2020

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 $\mathcal{O}(N)$ 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 $\mathcal{O}(N)$ 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 analogous 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.

cond-mat.stat-mech↗
arXiv2020

1D Three-state mean-field Potts model with first- and second-order phase transitions

We analyze a three-state Potts model built over a lattice ring, with coupling $J_0$, and the fully connected graph, with coupling $J$. This model is effectively mean-field and can be exactly solved by using transfer-matrix method and Cardano formula. When $J$ and $J_0$ are both ferromagnetic, the model has a first-order phase transition which turns out to be a smooth modification of the known phase transition of the traditional mean-field Potts model ($J_0=0$), despite, as we prove, the connected correlation functions are now non zero, even in the paramagnetic phase. Furthermore, besides the first-order transition, there exists also a hidden continuous transition at a temperature below which the symmetric metastable state ceases to exist. When $J$ is ferromagnetic and $J_0$ antiferromagnetic, a similar antiferromagnetic counterpart phase transition scenario applies. Quite interestingly, differently from the Ising-like two-state case, for large values of the antiferromagnetic coupling $J_0$, the critical temperature of the system tends to a finite value.

cond-mat.stat-mech↗
arXiv2019

Critical states in Political Trends. How much reliable is a poll on Twitter? A study by means of the Potts Model

In recent years, Twitter data related to political trends have tentatively been used to make predictions (poll) about several electoral events. Given $q$ candidates for an election and a time-series of Twitts (short messages), one can extract the $q$ mean trends and the $q(q+1)/2$ Twitt-to-Twitt correlations, and look for the statistical models that reproduce these data. On the base of several electoral events and assuming a stationary regime, we find out the following: i) the maximization of the entropy singles out a microscopic model (single-Twitt-level) that coincides with a $q$-state Potts model having suitable couplings and external fields to be determined via an inverse problem from the two sets of data; ii) correlations decay as $1/N_{eff}$, where $N_{eff}$ is a small fraction of the mean number of Twitts; iii) the simplest statistical models that reproduce these correlations are the multinomial distribution (MD), characterized by $q$ external fields, and the mean-field Potts model (MFP), characterized by one coupling; iv) remarkably, this coupling turns out to be always close to its critical value. This results in a MD or MFP model scenario that discriminates between cases in which polls are reliable and not reliable, respectively. More precisely, predictions based on polls should be avoided whenever the data maps to a MFP because anomalous large fluctuations (if $q=2$) or sudden jumps (if $q\geq 3$) in the trends might take place as a result of a second-order or a first-order phase transition of the MFP, respectively.

cond-mat.stat-mech↗
arXiv2018

Phase transitions and gaps in quantum random energy models

By using a previously established exact characterization of the ground state of random potential systems in the thermodynamic limit, we determine the ground and first excited energy levels of quantum random energy models, discrete and continuous. We rigorously establish the existence of a universal first order quantum phase transition, obeyed by both the ground and the first excited states. The presence of an exponentially vanishing minimal gap at the transition is general but, quite interestingly, the gap averaged over the realizations of the random potential is finite. This fact leaves still open the chance for some effective quantum annealing algorithm, not necessarily based on a quantum adiabatic scheme.

cond-mat.stat-mech↗
arXiv2017

Thermalization of noninteracting quantum systems coupled to blackbody radiation: A Lindblad-based analysis

We study the thermalization of an ensemble of $N$ elementary, arbitrarily-complex, quantum systems, mutually noninteracting but coupled as electric or magnetic dipoles to a blackbody radiation. The elementary systems can be all the same or belong to different species, distinguishable or indistinguishable, located at fixed positions or having translational degrees of freedom. Even if the energy spectra of the constituent systems are nondegenerate, as we suppose, the ensemble unavoidably presents degeneracies of the energy levels and/or of the energy gaps. We show that, due to these degeneracies, a thermalization analysis performed by the popular quantum optical master equation reveals a number of serious pathologies, possibly including a lack of ergodicity. On the other hand, a consistent thermalization scenario is obtained by introducing a Lindblad-based approach, in which the Lindblad operators, instead of being derived from a microscopic calculation, are established as the elements of an operatorial basis with squared amplitudes fixed by imposing a detailed balance condition and requiring their correspondence with the dipole transition rates evaluated under the first-order perturbation theory. Due to the above-mentioned degeneracies, this procedure suffers a basis arbitrariness which, however, can be removed by exploiting the fact that the thermalization of an ensemble of noninteracting systems cannot depend on the ensemble size. As a result, we provide a clear-cut partitioning of the thermalization time into dissipation and decoherence times, for which we derive formulas giving the dependence on the energy levels of the elementary systems, the size $N$ of the ensemble, and the temperature of the blackbody radiation.

quant-ph↗
arXiv2017

Thermalization of the Lipkin-Meshkov-Glick model in blackbody radiation

In a recent work, we have derived simple Lindblad-based equations for the thermalization of systems in contact with a thermal reservoir. Here, we apply these equations to the Lipkin-Meshkov-Glick model (LMG) in contact with a blackbody radiation and analyze the dipole matrix elements involved in the thermalization process. We find that the thermalization can be complete only if the density is sufficiently high, while, in the limit of low density, the system thermalizes partially, namely, within the Hilbert subspaces where the total spin has a fixed value. In this regime, and in the isotropic case, we evaluate the characteristic thermalization time analytically, and show that it diverges with the system size in correspondence of the critical points and inside the ferromagnetic region. Quite interestingly, at zero temperature the thermalization time diverges only quadratically with the system size, whereas quantum adiabatic algorithms, aimed at finding the ground state of same system, imply a cubic divergence of the required adiabatic time.

quant-ph↗
arXiv2016

Fermi's golden rule for $N$-body systems in a black-body radiation

We review the calculation of Fermi's golden rule for a system of $N$-body dipoles, magnetic or electric, weakly interacting with a blackbody radiation. By using the magnetic or electric field-field correlation function evaluated in the 1960s for the black body radiation, we deduce a general formula for the transition rates and study its limiting, fully coherent or fully incoherent, regimes.

quant-ph↗
arXiv2015

Criticality and Chaos in Systems of Communities

We consider a simple model of communities interacting via bilinear terms. After analyzing the thermal equilibrium case, which can be described by an Hamiltonian, we introduce the dynamics that, for Ising-like variables, reduces to a Glauber-like dynamics. We analyze and compare four different versions of the dynamics: flow (differential equations), map (discrete-time dynamics), local-time update flow, and local-time update map. The presence of only bilinear interactions prevent the flow cases to develop any dynamical instability, the system converging always to the thermal equilibrium. The situation is different for the map when unfriendly couplings are involved, where period-two oscillations arise. In the case of the map with local-time updates, oscillations of any period and chaos can arise as a consequence of the reciprocal "tension" accumulated among the communities during their sleeping time interval. The resulting chaos can be of two kinds: true chaos characterized by positive Lyapunov exponent and bifurcation cascades, or marginal chaos characterized by zero Lyapunov exponent and critical continuous regions.

cond-mat.dis-nn↗
arXiv2015

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

Random geometric graphs (RGG) 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 RGG. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a non trivial satisfaction problem: Given $N$ nodes, a domain $\mathcal{D}$, and a desired average connectivity $\langle k\rangle$, find - if any - the distribution of nodes having support in $\mathcal{D}$ and average connectivity $\langle k\rangle$. 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 $\mathop{O}(N)$ jump of $\langle k\rangle$. Other intermediate values of $\langle k\rangle$ correspond to very rare graph realizations. The phase transition is observed as a function of a parameter $a\in[0,1]$ 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 anti-geometry 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.

cond-mat.dis-nn↗
arXiv2014

Fluctuations analysis in complex networks modeled by hidden variable models. Necessity of a large cut-off 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 re-consider 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 cut-off: we show that in hidden-variable models, only a cut-off growing as $N^λ$ with $λ\geq 1$ can reproduce the scaling of a power-law degree distribution. In turn, it is this large cut-off that generates non-self-averaging.

cond-mat.dis-nn↗