Phase diagram of the Bose-Hubbard model on complex networks

Halu, Arda; Ferretti, Luca; Vezzani, A.; Bianconi, Ginestra

doi:10.1209/0295-5075/99/18001

Cited by 31 publications

(37 citation statements)

References 54 publications

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“…In fact it was shown that the classical Ising model [18][19][20], the percolation transition [21] the epidemic spreading [22] change significantly when the second moment k(k − 1) of the degree distribution diverges. Recently it is becoming clear that these effects of the topology apply also to quantum critical phenomena [23] and are found in the mean-field solution of both the Random Transverse Ising Model [15] and the Bose-Hubbard model [24]. An open problem, that we approach in this paper is to what extent mean-field results are indicative of the behavior of the critical phenomena in quenched networks.…”

Section: Introductionmentioning

confidence: 89%

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks

Bianconi¹

2012

J. Stat. Mech.

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A road map to understand the relation between the onset of the superconducting state with the particular optimum heterogeneity in granular superconductors is to study a Random Tranverse Ising Model on complex networks with a scale-free degree distribution regularized by and exponential cutoff p(k) ∝ k −γ exp [−k/ξ]. In this paper we characterize in detail the phase diagram of this model and its critical indices both on annealed and quenched networks. To uncover the phase diagram of the model we use the tools of heterogeneous mean-field calculations for the annealed networks and the most advanced techniques of quantum cavity methods for the quenched networks. The phase diagram of the dynamical process depends on the temperature T , the coupling constant J and on the value of the branching ratiowhere k is the degree of the nodes in the network. For fixed value of the coupling the critical temperature increases linearly withwhich diverges with the increasing cutoff value ξ for value of the γ exponent γ ≤ 3. This result suggests that the fractal disorder of the superconducting material can be responsible for an enhancement of the superconducting critical temperature. At low temperature and low couplings T ≪ 1 and J ≪ 1, instead, we observe a different behavior for annealed and quenched networks. In the annealed networks there is no phase transition at zero temperature while on quenched network we observe a Griffith phase dominated by extremely rare events and a phase transition at zero temperature. The Griffiths critical region, nevertheless, is decreasing in size with increasing value of the cutoff ξ of the degree distribution for values of the γ exponents γ ≤ 3. The interplay between disorder and superconductivity has attracted the interest of physics in the last decades. Disorder is expected to compete with superconductivity by enhancing the electrical resistivity of a system. In this situation by increasing the random disorder, the system undergoes a superconductor-insulator phase transition [1]. The theoretical explanation of the interplay between disorder and superconductivity is still a problem of intense debate for granular superconductors. results is guaranteed to be valid in quenched networks we have to approach the problem by the recently introduced quantum cavity method and by the mapping of this solution to the random polymer problem. We find that the phase diagram on annealed networks is significantly different from the phase diagram of the quenched networks as long as the second moment of the degree distribution remains finite. In particular we found a phase transition at zero temperature not predicted by the mean-field approach and a replica symmetry broken phase a low temperatures. Instead, as the second moment of the degree distribution diverges k(k − 1) → ∞ the mean-field predictions approach the quenched solution. Therefore we identify the second moment of the degree distribution as a key quantity characterizing the disorder of the topology of the network. As the second moment of the degr...

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Section: Introductionmentioning

confidence: 89%

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks

Bianconi¹

2012

J. Stat. Mech.

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“…Complex networks model systems as diverse as the brain and the internet; however, up till now they have been obtained in quantum systems by explicitly enforcing complex network structure in their quantum connections [2][3][4][5][6][7], e.g. entanglement percolation on a complex network [4].…”

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“…However, at the most basic level we can first ask, are quantum systems inherently complex? Must we impose complexity on quantum systems to obtain it [2][3][4][5][6][7], or is there a regime in which complexity naturally emerges, even in ground states of regular lattice models? In this Letter we show that emergent complexity can be well quantified in the simplest of 1D lattice quantum simulator models in terms of complexity measures around QCPs in direct analogy to similar measurements on the brain; moreover we establish a much-needed new set of tools for quantifying the complexity of far-from-equilibrium quantum dynamics.…”

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confidence: 99%

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

et al. 2017

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We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of EEG/fMRI measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z2, mean field superfluid/Mott insulator, and a BKT crossover.Classical statistical physics has developed a powerful set of tools for analyzing complex systems, chief among them complex networks, in which connectivity and topology predominate over other system features [1]. Complex networks model systems as diverse as the brain and the internet; however, up till now they have been obtained in quantum systems by explicitly enforcing complex network structure in their quantum connections [2][3][4][5][6][7], e.g. entanglement percolation on a complex network [4]. In contrast, complexity measures on the brain observe emergent complexity arising out of, e.g., a regular array of EEG electrodes placed on the scalp, via an adjacency matrix formed from the classical mutual information calculated between them [8]. We apply the quantum generalization of this measure, an adjacency matrix of the quantum mutual information calculated on quantum states [9], to well known quantum many-body models on regular 1D lattices, and uncover emergent quantum complexity which clearly identifies quantum critical points (QCPs) [10,11]. Quantum mutual information bounds two-point correlations from above [12], measurable in a precise and tunable fashion in e.g. atom interferometry in 1D Bose gases [13], among many other quantum simulator architectures. Using matrix product state (MPS) computational methods [14,15], we demonstrate rapid finite size-scaling for both transverse Ising and Bose-Hubbard models, including Z 2 , mean field, and BKT quantum phase transitions.As we move toward more and more complex quantum systems in materials design and quantum simulators, involving a hierarchy of scales, diverse interacting components, and a structured environment, we expect to observe long-lived dynamical features, fat-tailed distributions, and other key identifiers of complexity [16][17][18]. Such systems include quantum simulator technologies based on ultracold atoms and molecules [19], trapped ions [20], and Rydberg gases [21], as well as superconducting Josephson-junction based nanoelectromechanical systems in which different quantum subsystems form compound quantum machines with both electrical and mechanical components [22]. A key area in which we have taken a first step beyond phase diagrams and ground state properties is non-equilibrium quantum dynamics, where critical exponents and renormalization group theory are only weakly applicable at best, e.g. in the KibbleZurek mechanism,...

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“…Extensive research has been addressed to the characterization of quantum walks on complex networks (see Figure 4) [41,42,50] and to quantum transport [108]. Other critical phenomena studied on networks are the quantum Ising model, the Bose Hubbard model, Anderson localization, Bose-Einstein condensation between others [43][44][45][46][47][48][49]. It has been found that network structure strongly affect the phase diagram of quantum dynamical processes.…”

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confidence: 99%

Interdisciplinary and physics challenges of network theory

Bianconi¹

2015

EPL

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135

148

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-Network theory has unveiled the underlying structure of complex systems such as the Internet or the biological networks in the cell. It has identified universal properties of complex networks, and the interplay between their structure and dynamics. After almost twenty years of the field, new challenges lie ahead. These challenges concern the multilayer structure of most of the networks, the formulation of a network geometry and topology, and the development of a quantum theory of networks. Making progress on these aspects of network theory can open new venues to address interdisciplinary and physics challenges including progress on brain dynamics, new insights into quantum technologies, and quantum gravity.Introduction. -Network theory has emerged almost twenty years ago, as a new field for characterizing interacting complex systems, such as the Internet, the biological networks of the cell, and social networks. It is now time to reflect on the maturity of the field, indicating the main results obtained so far and the big challenges that lie ahead.Initially, the physics perspective, in particular the statistical mechanics approach, has dominated the field of Network Theory [1][2][3][4][5][6][7]. This point of view has played a central role to characterize the universal properties of the structure of complex networks. It has been found that despite the diversity of complex networks, ranging from the Internet to the protein interaction networks in the cell, most networks obey universal properties: they are small-world [9], they are scale-free [8], and they have a non trivial community structure [7]. Moreover, over the years, special attention has been addressed to the interplay between network structure and dynamics [10,11]. In fact phase diagrams of critical phenomena and dynamical processes change drastically when the dynamics is defined on complex networks. Complex networks are responsible for significant changes in the critical behaviour of percolation, Ising model, random walks, epidemic spreading, synchronization, and controllability of networks [10][11][12][13].The need to characterize complex systems, to extract relevant information from them, and to understand how dynamical processes are affected by network structure, has never been more severe than in the XXI century when we are witnessing a Big Data explosion in social sciences, information and communication technologies and in biology.

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Phase diagram of the Bose-Hubbard model on complex networks

Cited by 31 publications

References 54 publications

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

Interdisciplinary and physics challenges of network theory

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Phase diagram of the Bose-Hubbard model on complex networks

Cited by 31 publications

References 54 publications

Enhancement ofTcin the superconductor–insulator phase transition on scale-free networks

Enhancement ofTcin the superconductor–insulator phase transition on scale-free networks

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

Interdisciplinary and physics challenges of network theory

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Product

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About

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks

Enhancement ofT_cin the superconductor–insulator phase transition on scale-free networks