arXiv subjects

Francesco Toppan

Publications and source records attributed to Francesco Toppan.

At least 19 recordsLinked to original sources

arXiv2026

Signature of paraparticles: a minimal Gedankenexperiment

Paraparticles beyond bosons and fermions can be exchanged via either the braid group (anyons, existing up to $D=2$ space dimensions) or the permutation group; in the latter case the space dimensions are not limited. Besides being predicted, anyons have been experimentally detected. The situation differs for paraparticles exchanged via the permutation group ("permutation-group parastatistics").The first test to detect their theoretical signature was published in 2021 (for $Z_2\times Z_2$-graded parafermions; it was soon followed by a second paper proving the detectability of $Z_2\times Z_2$-graded parabosons). Later on, two further papers proved theoretical signatures of permutation-group parastatistics. These works demonstrate that, in certain situations, a long-held belief on the "conventionality of parastatistics" argument can be evaded: some measurements of permutation-group paraparticles cannot be recovered from ordinary bosons/fermions. The main question now is how to experimentally detect or engineer in the laboratory such paraparticles. For this aim a minimal setup for the theoretical test is here provided: a Gedankenexperiment (a simplified version of the two tests published in 2021) which, essentially, is a flow chart of logical operations. The key point is to present, to experimentalists, the necessary steps to be simulated/realized in the laboratory (possibly, by manipulating qudits). In this minimal setup, the detection/engineering of paraparticles is mapped into a chirality test. The mathematical setting is based on $Z_2\times Z_2$-graded color Lie (super)algebras and derived mathematical structures.

quant-ph↗

arXiv2025

Integrable $\mathbb{Z}_2^2$-graded super-Liouville Equation and Induced $\mathbb{Z}_2^2$-graded super-Virasoro Algebra

We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied to derive a $\mathbb{Z}_2^2$-graded extension of the super-Liouville equation based on a $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2).$ The mathematical tools employed in this work are a $\mathbb{Z}_2^2$-graded version of the zero-curvature formalism and of the Polyakov's soldering procedure. It is demonstrated that both methods yield the same $\mathbb{Z}_2^2$-graded super-Liouville equation. An algebraic construction of solutions to the resulting equations is also presented, together with their B\"acklund transformations. Furthermore, three distinct new $\mathbb{Z}_2^2$-graded extensions of the super-Virasoro algebra are obtained via Hamiltonian reduction of the WZNW currents defined for $\mathbb{Z}_2^2$-$\mathfrak{osp}(1|2).$

math-ph↗

arXiv2025

Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework

We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups $Z_3^2$, $Z_2^p\times Z_3^2$ for $p=1,2,3$, and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg-Wyler's color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.

math-ph↗

arXiv2025

Braided Majorana qubits as a minimal setting for Topological Quantum Computation?

I point out that a possible minimal setting to realize Kitaev's proposal of a Topological Quantum Computation which offers topological protection from decoherence could in principle be realized by braided Majorana qubits. Majorana qubits and their braiding were introduced in Nucl. Phys. B 980, 115834 (2022) and further analyzed in J. Phys. A: Math. Theor. 57, 435203 (2024). Braided Majorana qubits implement a Gentile-type parastatistics with at most $s-1$ excited states accommodated in a multiparticle sector (the integer $s=2,3,4,\ldots$ labels quantum group reps at roots of unity). It is argued that braided Majorana qubits could play, for topological quantum computers, the same role as standard bits for ordinary computers and as qubits for "ordinary" quantum computers.

quant-ph↗

arXiv2024

On the detectability of paraparticles beyond bosons and fermions

In this paper I present the state of the art concerning the theoretical detectability (and the open challenges for the experimental detectability) of a special class of paraparticles beyond bosons and fermions. The particles under considerations, obeying a parastatistics, are mutually exchanged via the permutation group and can exist in any space dimension (the anyons, which transform under the more general braid group, cannot exist in more than two space dimensions).

math-ph↗

arXiv2024

On braid statistics versus parastatistics

I report the recent advances in applying (graded) Hopf algebras with braided tensor product in two scenarios: i) paraparticles beyond bosons and fermions living in any space dimensions and transforming under the permutation group; ii) physical models of anyons living in two space-dimensions and transforming under the braid group. In the first scenario simple toy models based on the so-called $2$-bit parastatistics show that, in the multiparticle sector, certain observables can discriminate paraparticles from ordinary bosons/fermions (thus, providing a counterexample to the widespread belief of the "conventionality of parastatistics" argument). In the second scenario the notion of (braided) Majorana qubit is introduced as the simplest building block to implement the Kitaev's proposal of a topological quantum computer which protects from decoherence.

hep-th↗

arXiv2024

On the Classification of the L\'evy-Leblond Spinors

The first-order L\'evy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the "square roots" of the Schr\"odinger equation in ($1+d$) dimensions and admit spinor solutions. In this paper we show how to extend to the L\'evy-Leblond spinors the real/complex/quaternionic classification of the relativistic spinors (which leads to the notions of Dirac, Weyl, Majorana, Majorana-Weyl, Quaternionic spinors). Besides the free equations, we also consider the presence of potential terms. Applied to a conformal potential, the simplest $(1+1)$-dimensional LLE induces a new differential realization of the $osp(1|2)$ superalgebra in terms of differential operators depending on the time and space coordinates.

math-ph↗

arXiv2024

Integrable $\mathbb{Z}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories

In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$-graded extensions of classical, two-dimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and Sinh-Gordon models; they are based on $\mathbb{Z}_2^2$-graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$-graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$-graded Sinh-Gordon model is derived from an affine $\mathbb{Z}_2^2$-graded color Lie algebra, mimicking a procedure originally introduced by Babelon-Bonora to derive the ordinary Sinh-Gordon model. The color Lie algebras under considerations are: the $6$-generator $\mathbb{Z}_2^2$-graded $sl_2$, the $\mathbb{Z}_2^2$-graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$-graded Virasoro algebra obtained from a Hamiltonian reduction.

math-ph↗

arXiv2024

Volichenko-type metasymmetry of braided Majorana qubits

This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, mixed-bracket Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the Volichenko algebras defined in 1990 by Leites-Serganova as metasymmetries which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators. In a previous paper braided $Z_2$-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at roots of unity and realizes, for a given integer $s=2,3,4,\ldots$, an interpolation between ordinary Majorana fermions (recovered at $s=2$) and bosons (recovered in the $s\rightarrow \infty$ limit); it implements a parastatistics where at most $s-1$ indistinguishable particles are accommodated in a multi-particle sector. The structures discussed in this work are: - the quantum group interpretation of the roots of unity truncations recovered from reps of the quantum superalgebra ${\cal U}_q({{osp}}(1|2))$; - the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products; - the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras; - the $s\rightarrow \infty$ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators; - (meta)symmetries of ordinary differential equations given by matrix Schr\"{o}dinger equations in $0+1$ dimension induced by the braided creation/annihilation operators; - in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system as a ternary algebra.

math-ph↗

arXiv2023

Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics

In a recent paper (Balbino-de Freitas-Rana-FT, arXiv:2309.00965) we proved that the supercharges of the supersymmetric quantum mechanics can be statistically transmuted and accommodated into a $Z_2^n$-graded parastatistics. In this talk I derive the $6=1+2+3$ transmuted spectrum-generating algebras (whose respective $Z_2^n$ gradings are $n=0,1,2$) of the ${\cal N}=2$ Superconformal Quantum Mechanics. These spectrum-generating algebras allow to compute, in the corresponding multiparticle sectors of the de Alfaro-Fubini-Furlan deformed oscillator, the degeneracies of each energy level. The levels induced by the $Z_2\times Z_2$-graded paraparticles cannot be reproduced by the ordinary bosons/fermions statistics. This implies the theoretical detectability of the $Z_2\times Z_2$-graded parastatistics.

hep-th↗

arXiv2023

The parastatistics of braided Majorana fermions

This paper presents the parastatistics of braided Majorana fermions obtained in the framework of a graded Hopf algebra endowed with a braided tensor product. The braiding property is encoded in a $t$-dependent $4\times 4$ braiding matrix $B_t$ related to the Alexander-Conway polynomial. The nonvanishing complex parameter t defines the braided parastatistics. At $t = 1$ ordinary fermions are recovered. The values of $t$ at roots of unity are organized into levels which specify the maximal number of braided Majorana fermions in a multiparticle sector. Generic values of $t$ and the $t =-1$ root of unity mimick the behaviour of ordinary bosons.

hep-th↗

arXiv2022

First quantization of braided Majorana fermions

A ${\mathbb Z}_2$-graded qubit represents an even (bosonic) "vacuum state" and an odd, excited, Majorana fermion state. The multiparticle sectors of $N$, braided, indistinguishable Majorana fermions are constructed via first quantization. The framework is that of a graded Hopf algebra endowed with a braided tensor product. The Hopf algebra is ${U}({\mathfrak {gl}}(1|1))$, the Universal Enveloping Algebra of the ${\mathfrak{gl}}(1|1)$ superalgebra. A $4\times 4$ braiding matrix $B_t$ defines the braided tensor product. $B_t$, which is related to the $R$-matrix of the Alexander-Conway polynomial, depends on the braiding parameter $t$ belonging to the punctured plane ($t\in {\mathbb C}^\ast$); the ordinary antisymmetry property of fermions is recovered for $t=1$. For each $N$, the graded dimension $m|n$ of the graded multiparticle Hilbert space is computed. Besides the generic case, truncations occur when $t$ coincides with certain roots of unity which appear as solutions of an ordered set of polynomial equations. The roots of unity are organized into levels which specify the maximal number of allowed braided Majorana fermions in a multiparticle sector. By taking into account that the even/odd sectors in a ${\mathbb Z}_2$-graded Hilbert space are superselected, a nontrivial braiding with $t\neq 1$ is essential to produce a nontrivial Hilbert space described by qubits, qutrits, etc., since at $t=1$ the $N$-particle vacuum and the antisymmetrized excited state encode the same information carried by a classical $1$-bit.

hep-th↗

arXiv2021

Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

The "$10$-fold way" refers to the combined classification of the $3$ associative division algebras (of real, complex and quaternionic numbers) and of the $7$, ${\mathbb Z}_2$-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the $10$-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in ${\mathbb Z}_2\times{\mathbb Z}_2$-graded physics (classical and quantum invariant models, parastatistics) we classify the associative ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebras and show that $13$ inequivalent cases have to be added to the $10$-fold way. Our scheme is based on the "alphabetic presentation of Clifford algebras", here extended to graded superdivision algebras. The generators are expressed as equal-length words in a $4$-letter alphabet (the letters encode a basis of invertible $2\times 2$ real matrices and in each word the symbol of tensor product is skipped). The $13$ inequivalent ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebras are split into real series ($4$ subcases with $4$ generators each), complex series ($5$ subcases with $8$ generators) and quaternionic series ($4$ subcases with $16$ generators).

math-ph↗

arXiv2021

Inequivalent quantizations from gradings and ${\mathbb Z}_2\times {\mathbb Z}_2$ parabosons

This paper introduces the parastatistics induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded algebras. It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type (so far, in the literature, only parastatistics induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superalgebras and producing parafermions have been considered). It is shown how to detect ${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosons in the multi-particle sector of a quantum model. The difference with respect to a system composed by ordinary bosons is spotted by measuring some selected observables on certain given eigenstates. The construction of the multi-particle states is made through the appropriate braided tensor product. The application of ${\mathbb Z}_2$- and ${\mathbb Z}_2\times {\mathbb Z}_2$- gradings produces $9$ inequivalent multi-particle Hilbert spaces of a $4\times 4$ matrix oscillator. The ${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosonic Hilbert space is one of them.

hep-th↗

arXiv2021

Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

This paper presents the classification, over the fields of real and complex numbers, of the minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras and Lie superalgebras spanned by $4$ generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Poincar\'e superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed ${\mathbb Z}_2\times{\mathbb Z}_2$-graded (super)algebras. We mention, in particular, the notion of ${\mathbb Z_2}\times{\mathbb Z}_2$-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further byproduct we point out that, contrary to ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superalgebras, a theory invariant under a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

math-ph↗

arXiv2020

${\mathbb Z}_2\times {\mathbb Z}_2$-graded parastatistics in multiparticle quantum Hamiltonians

The recent surge of interest in ${\mathbb Z}_2\times {\mathbb Z}_2$-graded invariant mechanics poses the challenge of understanding the physical consequences of a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded symmetry. In this paper it is shown that non-trivial physics can be detected in the multiparticle sector of a theory, being induced by the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded parastatistics obeyed by the particles. The toy model of the ${\cal N}=4$ supersymmetric/ ${\mathbb Z}_2\times {\mathbb Z}_2$-graded oscillator is used. In this set-up the one-particle energy levels and their degenerations are the same for both supersymmetric and ${\mathbb Z}_2\times{\mathbb Z}_2$-graded versions. Nevertheless, in the multiparticle sector, a measurement of an observable operator on suitable states can discriminate whether the system under consideration is composed by ordinary bosons/fermions or by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded particles. Therefore, ${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics has experimentally testable consequences. Furthermore, the ${\mathbb Z}_2\times {\mathbb Z}_2$-grading constrains the observables to obey a superselection rule. As a technical tool, the multiparticle sector is encoded in the coproduct of a Hopf algebra defined on a Universal Enveloping Algebra of a graded Lie superalgebra with a braided tensor product.

hep-th↗

arXiv2019

The octonionically-induced ${\cal N}=7$ exceptional $G(3)$ superconformal quantum mechanics

Both the ${\cal N}=7$ superconformal quantum mechanics possessing the exceptional $G(3)$ Lie superalgebra as dynamical symmetry and its associated deformed oscillator with $G(3)$ as spectrum-generating superalgebra are presented. This superconformal quantum mechanics, uniquely defined up to similarity transformations, is obtained via the octonionically-induced "quasi-nonassociative" method employed to derive the exceptional ${\cal N}=8$ $F(4)$ model. To construct the $G(3)$ theories, the covariant embedding of the $7$-dimensional representation of the Lie algebra $g_2$ within the $8\times 8$ matrices spanning the $Cl(0,7)$ Clifford algebra is derived. The Hilbert space of the $G(3)$ deformed oscillator is given by a $16$-ple of square-integrable functions of a real space coordinate. The spectrum of the theory is computed.

hep-th↗

arXiv2019

A three-dimensional superconformal quantum mechanics with $sl(2|1)$ dynamical symmetry

We construct a three-dimensional superconformal quantum mechanics (and its associated de Alfaro-Fubini-Furlan deformed oscillator) possessing an $sl(2|1)$ dynamical symmetry. At a coupling parameter $β\neq 0$ the Hamiltonian contains a $\frac{1}{r^2}$ potential and a spin-orbit (hence, a first-order differential operator) interacting term. At $β=0$ four copies of undeformed three-dimensional oscillators are recovered. The Hamiltonian gets diagonalized in each sector of total $j$ and orbital $l$ angular momentum (the spin of the system is $\frac{1}{2}$). The Hilbert space of the deformed oscillator is given by a direct sum of $sl(2|1)$ lowest weight representations. The selection of the admissible Hilbert spaces at given values of the coupling constant $β$ is discussed. The spectrum of the model is computed. The vacuum energy (as a function of $β$) consists of a recursive zigzag pattern. The degeneracy of the energy eigenvalues grows linearly up to $E\sim β$ (in proper units) and quadratically for $E>β$. The orthonormal energy eigenstates are expressed in terms of the associated Laguerre polynomials and the spin spherical harmonics. The dimensional reduction of the model to $d=2$ produces two copies (for $β$ and $-β$, respectively) of the two-dimensional $sl(2|1)$ deformed oscillator. The dimensional reduction to $d=1$ produces the one-dimensional $D(2,1;α)$ deformed oscillator, with $α$ determined by $β$.

hep-th↗