Parafermionic Algebra Research Articles

A duality between vertex superalgebras [formula omitted] and [formula omitted] and generalizations to logarithmic vertex algebras

We introduce a certain subalgebra F‾ of the Clifford vertex superalgebra (bc system), which is completely reducible (as a LVir(−2,0)–module) and C2–cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF(1)). We show that there is an interesting duality between SF(1) and F‾, given by the fact that F‾ can be equipped with the structure of a SF(1)–module and vice versa.Using the decomposition of F‾ and a free-field realization from [3], we decompose Lk(osp(1|2)) at the critical level k=−3/2 as a module for Lk(sl(2)). The decomposition of Lk(osp(1|2)) is exactly the same as that of the N=4 superconformal vertex algebra with central charge c=−9, denoted by V(2). Using the duality between F‾ and SF(1), we prove that Lk(osp(1|2)) and V(2) are in a duality of the same type. As an application, we construct and classify all irreducible Lk(osp(1|2))–modules in the category O and the category R, which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra N−3/2(osp(1|2)) as a N−3/2(sl(2))–module.Extending this example, we introduce for each p≥2 a non-conformal vertex algebra Anew(p) and show that Anew(p) is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra Vnew(p), which is isomorphic to the logarithmic vertex algebra V(p) as a module for slˆ(2).We also conjecture the existence of the conformal vertex algebra V(sp(2n)) which is in a duality with the affine vertex algebra L−n−1/2(osp(1|2n)).

Generalized parafermions of orthogonal type

There is an embedding of affine vertex algebras Vk(gln)↪Vk(sln+1), and the coset Ck(n)=Com(Vk(gln),Vk(sln+1)) is a natural generalization of the parafermion algebra of sl2. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter W∞-algebra of type W(2,3,…). In this paper, we consider an analogous structure of orthogonal type, namely Dk(n)=Com(Vk(so2n),Vk(so2n+1))Z2. We realize this algebra as a one-parameter quotient of the two-parameter even spin W∞-algebra of type W(2,4,…), and we classify all coincidences between its simple quotient Dk(n) and the algebras Wℓ(so2m+1) and Wℓ(so2m)Z2. As a corollary, we show that for the admissible levels k=−(2n−2)+12(2n+2m−1) for soˆ2n the simple affine algebra Lk(so2n) embeds in Lk(so2n+1), and the coset is strongly rational. As a consequence, the category of ordinary modules of Lk(so2n+1) at such a level is a braided fusion category.

Majorana zero modes in a quantum wire platform without Rashba spin-orbit coupling

We propose a platform for engineering helical fermions in a hybridized double-quantum-wire setup. When our setup is proximity coupled to an $s$-wave superconductor it can become a class $D$ topological superconductor exhibiting Majorana zero modes. The goal of this proposal is to expand the group of available Hamiltonians to those without strong Rashba spin-orbit interactions which are essential to many other approaches. Furthermore, we show that there exist electron-electron interactions that stabilize fractional excitations which obey $\mathbb{Z}^{\,}_{3}$ parafermionic algebra.

An exact symmetry in \u03bb-deformed CFTs

We consider λ-deformed current algebra CFTs at level k, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large k expansion, β-function. We find that this is covariant under a remarkable exact symmetry involving the coupling λ, the level k and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov C -function at two-loops in the large k expansion, as exact func- tions of the deformation parameter. Finally, we extend the above results to λ-deformed parafermionic algebra coset CFTs which interpolate between exact coset CFTs in the UV and a symmetric coset space in the IR.

De Finetti Theorems for Braided Parafermions

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

Parafermion algebra with bilinear relation

In this paper, we find the parafermion algebra of order [Formula: see text] expressed in terms of bilinear relation (commutator form). We find the correct Hamiltonian of parafermion of order [Formula: see text] by using the parasupersymmetry. Finally, we discuss the comparison of specific heats for several parafermions of order [Formula: see text].

Z4 parafermions in one-dimensional fermionic lattices

Parafermions are emergent excitations which generalize Majorana fermions and are potentially relevant to topological quantum computation. Using the concept of Fock parafermions, we present a mapping between lattice $\mathbb{Z}_4$ parafermions and lattice spin-$1/2$ fermions which preserves the locality of operators with $\mathbb{Z}_4$ symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide range of parameters, and discuss its signatures in the fermionic spectral function.

Measurement and control of a Coulomb-blockaded parafermion box

Parafermionic zero modes are fractional topologically protected quasiparticles expected to arise in various platforms. We show that Coulomb charging effects define a parafermion box with unique access options via fractional edge states and/or quantum antidots. Basic protocols for the detection, manipulation, and control of parafermionic quantum states are formulated. With those tools, one may directly observe the dimension of the zero-mode Hilbert space, prove the degeneracy of this space, and perform on-demand digital operations satisfying a parafermionic algebra.

Generalized parafermions and nonlocal Josephson effect in multilayer systems

We theoretically investigate the effects of backscattering and superconducting proximity terms between the edges of two multi-layer fractional quantum Hall (FQH) systems. While the different layers are strongly interacting, we assume that tunneling between them is absent. Studying the boundaries between regions gapped by the two mechanisms in an $N$-layer system, we find $N$ localized zero-mode operators realizing a generalized parafermionic algebra. We further propose an experiment capable of probing imprints of the generalized parafermionic bound states. This is done by coupling different superconducting contacts to different layers, and examining the periodicity of the Josephson effect as a function of the various relative superconducting phases. Remarkably, even if we apply a phase difference between the superconductors in one layer, we induce a Josephson current at the other layers due to inter-layer interactions. Furthermore, while the Josephson effect is commonly used to probe only charged degrees of freedom, the possibility of independently controlling the superconducting phase differences between the layers allows us to find imprints of the neutral modes of the underlying multi-layer system. In particular, we propose two configurations, one of which is capable of isolating the signal associated with the charge modes, while the other probes the neutral modes.

Planar Para Algebras, Reflection Positivity

We define a planar para algebra, which arises naturally from combining planar algebras with the idea of $\mathbb{Z}_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under para isotopy. For each $\mathbb{Z}_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), that we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity, by relating the two reflections through the string Fourier transform.

Modified Pauli matrix and spin algebra from the Z2-decomposition of the parafermionic Fock space

In this paper the Z2-decomposition of the bosonic Fock space and the projection operators for each subspace are considered. Using these, we obtain two kinds of step operators obeying trilinear relation which resembles the parafermion algebra. The Z2-decomposition of the parafermionic Fock space is also investigated and a new kind of finite-dimensional algebra obeying the trilinear relation which resembles the parafermion algebra of order p. Finally, the modified Pauli matrices and the modified spin algebra are introduced from the Z2-decomposition of the parafermionic Fock space.

Two more solutions for the parafermionic chiral algebra [formula omitted] with the dimension of the principal parafermionic fields [formula omitted], [formula omitted], [formula omitted

In this paper, which is the second one in a series of two papers, we shall present two more solutions, non-minimal ones, for the Z3 parafermionic chiral algebra with Δψ=Δψ+=8/3, ψ(z), ψ+(z) being the principal parafermionic fields.

Parafermionic chiral algebra [formula omitted] with the dimension of the principal parafermion fields [formula omitted], [formula omitted], [formula omitted

We analyze, and prove, the associativity of the new Z3 parafermionic chiral algebra which has been announced some time ago, with principal parafermionic fields having the conformal dimension Δψ=8/3. In doing so we have developed a new method for analyzing the associativity of a given chiral algebra of parafermionic type, the method which might be of a more general significance than a particular conformal field theory studied in detail in this paper. Still, even in the context of our particular chiral algebra, of Z3 parafermions with Δψ=8/3, the new method allowed us to give a proof of associativity which we consider to be complete.

Parafermionic Liouville field theory and instantons on ALE spaces

In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on $\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.

$\Bbb {Z}_2$ x $\Bbb {Z}_2$ graded superconformal algebra of parafermionic type

We present a new conformal algebra. It is $\Bbb {Z}_2$ x $\Bbb {Z}_2$ graded and generated by three N = 1 superconformal algebras coupled to each other by nontrivial relations of parafermionic type. The representation theory and unitary models of the algebra are briefly discussed. We also conjecture the existence of infinite series of parafermionic algebras containing many N = 1 or N = 2 superconformal subalgebras.

Deformed parafermionic algebra from single-band tight-binding dynamics

The quantum dynamics of a driven single-band tight-binding model with different boundary conditions is considered. The relation between the Hamiltonian describing the single-band tight-binding dynamics and the Hamiltonian of a discrete-charge mesoscopic quantum circuit is elucidated. It is shown that the former Hamiltonian, with Dirichlet boundary conditions, can be considered as a realization of the deformed parafermionic polynomial algebras.

Algebraic approach to parafermionic conformal field theories

Parafermionic conformal field theories are considered on a purely algebraic basis. The generalized Jacobi type identity is presented. Systems of free fermions coupled to each other by nontrivial parafermionic type relations are studied in detail. A new parafermionic conformal algebra is introduced, it describes the sl(2|1)2/u(1)2 coset system.

Parasupersymmetry and [formula omitted]-fold supersymmetry in quantum many-body systems. II: Third order

Based on the general formalism of parafermionic algebra and parasupersymmetry proposed previously by us, we explicitly construct third-order parafermionic algebra and multiplication law, and then realize third-order parasupersymmetric quantum systems. We find some novel features in the third-order, namely, the emergence of a fermionic degree of freedom and of a generalized parastatistics. We show that for one-body cases the generalized Rubakov–Spiridonov model can be constructed also in our framework and find that it admits a generalized 3-fold superalgebra. We also find that a three-body system can have third-order parasupersymmetry where three independent supersymmetries are folded. In both cases, we also investigate the new concept of quasi-parasupersymmetry introduced by us and find that those of order (3, 3) are indeed realized under less restrictive conditions than (ordinary) parasupersymmetric cases.

Parasupersymmetry and [formula omitted]-fold supersymmetry in quantum many-body systems. I: General formalism and second order

We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed algebra. Within our formulation, we show generically that every parasupersymmetric quantum system of order p consists of N -fold supersymmetric pairs with N ⩽ p and thus has weak quasi-solvability and isospectral property. We also propose a new type of non-linear supersymmetries, called quasi-parasupersymmetry, which is less restrictive than parasupersymmetry and is different from N -fold supersymmetry even in one-body systems though the conserved charges are represented by higher-order linear differential operators. To illustrate how our formulation works, we construct second-order parafermionic algebra and three simple examples of parasupersymmetric quantum systems of order 2, one is essentially equivalent to the one-body Rubakov–Spiridonov type and the others are two-body systems in which two supersymmetries are folded. In particular, we show that the first model admits a generalized 2-fold superalgebra.

On scaling fields in Z N Ising models

We study the space of scaling fields in the $Z_N$ symmetric models with the factorized scattering and propose simplest algebraic relations between form factors induced by the action of deformed parafermionic currents. The construction gives a new free field representation for form factors of perturbed Virasoro algebra primary fields, which are parafermionic algebra descendants. We find exact vacuum expectation values of physically important fields and study correlation functions of order and disorder fields in the form factor and CFT perturbation approaches.