Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity

Expectation values of twist fields and universal entanglement saturation of the free massive boson

The evaluation of vacuum expectation values (VEVs) in massive integrable quantum field theory (QFT) is a nontrivial renormalization-group "connection problem" -- relating large and short distance asymptotics -- and is in general unsolved. This is particularly relevant in the context of entanglement entropy, where VEVs of branch-point twist fields give universal saturation predictions. We propose a new method to compute VEVs of twist fields associated to continuous symmetries in QFT. The method is based on a differential equation in the continuous symmetry parameter, and gives VEVs as infinite form-factor series which truncate at two-particle level in free QFT. We verify the method by studying U(1) twist fields in free models, which are simply related to the branch-point twist fields. We provide the first exact formulae for the VEVs of such fields in the massive uncompactified free boson model, checking against an independent calculation based on angular quantization. We show that logarithmic terms, overlooked in the original work of Callan and Wilczek [Phys. Lett. B333 (1994)], appear both in the massless and in the massive situations. This implies that, in agreement with numerical form-factor observations by Bianchini and Castro-Alvaredo [Nucl. Phys. B913 (2016)], the standard power-law short-distance behavior is corrected by a logarithmic factor. We discuss how this gives universal formulae for the saturation of entanglement entropy of a single interval in near-critical harmonic chains, including log log corrections.Comment: V2: 37 pages, explications and references adde

The supersymmetric Ruijsenaars-Schneider model

An integrable supersymmetric generalization of the trigonometric Ruijsenaars-Schneider model is presented whose symmetry algebra includes the super Poincar\'e algebra. Moreover, its Hamiltonian is showed to be diagonalized by the recently introduced Macdonald superpolynomials. Somewhat surprisingly, the consistency of the scalar product forces the discreteness of the Hilbert space.Comment: v1: 11 pages, 1 figure. v2: new format, 5 pages, short section added at the end of the article addressing the problem of consistency of the scalar product (e.g., positivity of the weight functions and the normalization of the ground state wave function). To appear in Physical Review Letter

A quartet of fermionic expressions for M(k,2k±1)M(k,2k\pm1) Virasoro characters via half-lattice paths

We derive new fermionic expressions for the characters of the Virasoro minimal models $M(k,2k\pm1)$ by analysing the recently introduced half-lattice paths. These fermionic expressions display a quasiparticle formulation characteristic of the $\phi_{2,1}$ and $\phi_{1,5}$ integrable perturbations. We find that they arise by imposing a simple restriction on the RSOS quasiparticle states of the unitary models $M(p,p+1)$. In fact, four fermionic expressions are obtained for each generating function of half-lattice paths of finite length $L$, and these lead to four distinct expressions for most characters $\chi^{k,2k\pm1}_{r,s}$. These are direct analogues of Melzer's expressions for $M(p,p+1)$, and their proof entails revisiting, reworking and refining a proof of Melzer's identities which used combinatorial transforms on lattice paths. We also derive a bosonic version of the generating functions of length $L$ half-lattice paths, this expression being notable in that it involves $q$-trinomial coefficients. Taking the $L\to\infty$ limit shows that the generating functions for infinite length half-lattice paths are indeed the Virasoro characters $\chi^{k,2k\pm1}_{r,s}$.Comment: 29 pages. v2: minor improvements, references adde

Approche combinatoire des modèles minimaux en théorie des champs conformes : connexion avec les chemins sur réseau demi-entier

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2010-2011Une description des états Virasoro dans les modules irréductibles de plus haut poids des modèles minimaux M (p, p') en théorie des champs conformes est fournie par les chemins RSOS. Ces chemins sont issus des configurations des hauteurs sur une rangée du réseau des modèles statistiques exactement résolubles RSOS lors de l'évaluation de la probabilité locale d'une hauteur dans le régime III. Une seconde catégorie de chemins, définis sur un réseau demi-entier, a été proposée et élevée au rang de conjecture comme une description alternative des états Virasoro pour les modèles M (p, 2p + 1). L'avantage de cette seconde catégorie de chemins réside dans la formulation du poids qui ne dépend plus de la hauteur. L'analyse combinatoire de ces derniers est suffisamment simple pour permettre d'obtenir leurs fonctions génératrices qui sont, en vertu de la conjecture, équivalentes aux formules de caractères pour ces modèles minimaux. En nous intéressant davantage à ces chemins sur réseau demi-entier, nous avons découvert qu'il est également possible de les utiliser pour décrire les états Virasoro des modèles M (p + 1.2p+ 1). Nous présentons l'analyse combinatoire pour cette nouvelle classe ainsi que la dérivation des fonctions génératrices. Nous montrons également comment relier les chemins sur réseau demi-entier aux chemins RSOS au moyen d'une bijection qui préserve le poids. Par conséquent, nous validons la conjecture sur l'équivalence des chemins sur réseau demi-entier et des états Virasoro pour tous les modèles minimaux M(p + e, 2p-rl), e = 0,1

Universal scaling of the logarithmic negativity in massive quantum field theory

We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r. We show that the former saturates to a finite value, and that the latter tends to zero, as r -> ∞. We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large-r analysis of the negativity allows for the detection of bound states

Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.Comment: 18 page

Les polynômes de Macdonald dans le superespace et le modèle Ruijsenaars-Schneider supersymétrique

Tableau d'honneur de la Faculté des études supérieures et postdorales, 2014-2015La théorie des superpolynômes symétriques ([DLM03, DLM06]) est généralisée avec l’introduction d’une nouvelle base de superfonctions qui dépend de deux paramètres q et t. Cette nouvelle base, que l’on appelle les polynômes de Macdonald dans le superespace (ou simplement, les superpolynômes de Macdonald), généralise toutes les autres bases de superfonctions connues. Celles-ci sont retrouvées via différentes spécialisations (ou limites) de q et t. On démontre que les superpolynômes de Macdonald sont uniquement déterminés par les deux propriétés suivantes. Premièrement, ils se décomposent de façon triangulaire dans la base des superfonctions monomiales (par rapport à l’ordre de dominance entre les superpartitions). Deuxièmement, ils sont orthogonaux par rapport à un produit scalaire donné dans la base des superfonctions sommes de puissances et qui dépend de q, t. L’étape clef pour démontrer ce résultat est la connexion avec la théorie des polynômes non symétriques de Macdonald. En fait, il est montré que les superpolynômes de Macdonald sont également donnés par un processus de symétrisation particulier des polynômes non symétriques de Macdonald. Cette connexion peut être alors exploitée pour obtenir une famille d’opérateurs qui est diagonale dans la base des superpolynômes de Macdonald ainsi qu’une seconde relation d’orthogonalité donnée par l’évaluation d’un terme constant. Ces deux éléments, i.e. famille d’opérateurs et orthogonalité (analytique), permettent de relier les superpolynômes de Macdonald à un problème de mécanique quantique supersymétrique généralisant le modèle Ruijsenaars-Schneider (RS). L’hamiltonien de ce modèle est défini par l’anticommutateur d’une supercharge qui est le générateur de la transformation supersymétrique. La structure algébrique sous-jacente à ce modèle est l’algèbre de Poincaré supersymétrique (i.e. une algèbre de Lie graduée). Tous les états propres de l’hamiltonien sont donnés par le produit de la fonction d’onde de l’état du vide par les superpolynômes de Macdonald. L’intégrabilité du modèle est également démontrée.The theory of symmetric superpolynomials ([DLM03, DLM06]) is further extended with the introduction of a family of superpolynomials that depends upon two parameters, denoted by q and t. This new basis, that can be called Macdonald polynomials in superspace (or simply stated, Macdonald superpolynomials), generalizes all the previously discovered bases of superpolynomials. These are obtained by the evaluation (or by a limiting process) of the parameters q and t. It is proved that the Macdonald superpolynomials are uniquely defined by the two following properties. First, they decompose triangularly in the monomial basis (with respect to a certain ordering between superpartitions). Second, they are orthogonal with respect to a given scalar product evaluated in the power sum basis and which depends on q and t. The crucial step to prove this result is the connection between Macdonald superpolynomials and the theory of non-symmetric Macdonald polynomials. More precisely, it is showed that the Macdonald superpolynomials can be expressed by a certain symmetrizer acting on the non-symmetric analogue. Using this connection, a family of eigen-operators is obtained, which is diagonalized by the Macdonald superpolynomals basis. In addition, another orthogonality relation that involves a constant term evaluation (referred to as the analytic orthogonality) is obtained. These two elements, i.e. the eigen-operators and the orthogonality (analytic), link the Macdonald superpolynomials to a supersymmetric quantum mechanic model that generalizes the Ruijsenaars-Schneider (RS) model. The Hamiltonian of this model is naturally written as an anticommutator of a supercharge which is the generator of supersymmetric transformation. The underlying algebra of this model is the super Poincaré algebra (i.e. a graded Lie algebra). All the quantum states of the Hamiltonian are given as a product of the ground state function times Macdonald superpolynomials. Finally, the integrability of the supersymmetric RS model is demonstrated