Person:
Tempesta, Piergiulio

First Name

Piergiulio

Last Name

Tempesta

URI

https://hdl.handle.net/20.500.14352/74979

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Físicas

Department

Física Teórica

Area

Matemática Aplicada

Identifiers

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Now showing 1 - 10 of 16

Group entropies: from phase space geometry to entropy functionals via Group Theory
(Entropy, 2018) Jensen, Henrik Jeldtoft; Tempesta, Piergiulio
The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.
Critical behavior of su(1/1) supersymmetric spin chains with long-range interactions
(Physical review E, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a general class of su(1 / 1) supersymmetric spin chains with long-range interactions which includes as particular cases the su (1 / 1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1 / 1 ) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low- energy excitations and the low-temperature behavior of the free energy, hich coincides with that of a (1 + 1)-dimensional conformal field theory (CFT) with central charge c = 1 when the chemical potential lies in the critical interval (0, ε (π)), ε (p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1 + 1) –dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c = 1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1 / 1) elliptic chain.
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
(Journal of statistical mechanics : theory and experiment, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m − k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).
A duality principle for the multi-block entanglement entropy of free fermion systems
(Scientific reports, 2017) Carrasco, J. A.; Finkel Morgenstern, Federico; González López, Artemio; Tempesta, Piergiulio
The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set A of sites of the subsystem considered and the set K of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets A and K to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets A and K consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and r-partite information generalizing the known ones for the single block case.
Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions
(Physical review E, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Singularity confinement for matrix discrete Painleve equations
(Nonlinearity, 2014) Cassatella-Contra, Giovanni A; Mañas Baena, Manuel; Tempesta, Piergiulio
We study the analytic properties of a matrix discrete system introduced by Cassatella and Manas (2012 Stud. Appl. Math. 128 252-74). The singularity confinement for this system is shown to hold generically, i.e. in the whole space of parameters except possibly for algebraic subvarieties. This paves the way to a generalization of Painleve analysis to discrete matrix models.
Formal groups and Z-entropies
(Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, 2016) Tempesta, Piergiulio
We shall prove that the celebrated Renyi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Renyi. A rucial aspect is that every Z-entropy is composable (Tempesta 2016 Ann. Phys. 365, 180-197. (doi: 10.1016/j.aop.2015.08.013)). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non- trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.
A new entropy based on a group-theoretical structure
(Annals of physics, 2016) Curado, Evaldo M. F.; Tempesta, Piergiulio; Tsallis, Constantino
A multi-parametric version of the nonadditive entropy S_q is introduced. This new entropic form, denoted by S_a,b,r, possesses many interesting statistical properties, and it reduces to the entropy S_q for b=0, a=r:=1−q (hence Boltzmann–Gibbs entropy S_BG for b=0, a=r→0). The construction of the entropy S_a,b,r is based on a general group-theoretical approach recently proposed by one of us, Tempesta (2016). Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of S_a,b,r with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy S_a,b,r can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles NN of the system, or even stabilizes, by increasing NN, to a limiting value. This paves the way to the use of this entropy in contexts where the size of the phase space does not increase as fast as the number of its constituting particles (or subsystems) increases.
The Lazard formal group, universal congruences and special values of zeta functions
(Transactions of the american mathematical society, 2015) Tempesta, Piergiulio
A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group [31]-[33]. Their role in the theory of L–genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann–Hurwitz–type zeta functions.
Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems.
(Symmetry integrability and geometry: methods and applications (SIGMA), 2016) Tondo, Giorgio; Tempesta, Piergiulio
In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized Stackel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As an application, we recover the Haantjes manifolds for the rational Calogero model with three particles and for the Benenti systems.