Fermionic Integral Research Articles

Generalization to d-dimensions of a fermionic path integral for exact enumeration of polygons on hypercubic lattices

The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree 2(d-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2(d-1)$$\\end{document}, so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible.

Anomalous U(1) extension of the Standard Model

We present a set of example models in which the Standard Model (SM) symmetry group is extended by a new abelian symmetry. This additional symmetry appears anomalous in the effective low-energy theory; however, the anomalies cancel out when massive chiral fermions not present in the effective low-energy theory are taken into account. These chiral fermions under the new abelian gauge group, are chosen to be vector-like under the SM symmetries, and reside in the same representations as quarks and leptons. This allows us to quantitatively determine the magnitude of tree-level interactions between three vector bosons induced in low-energy effective field theory by the integration of chiral heavy fermions. We also examine the perturbativity constraints of the theory and the ultraviolet cut-off. We conclude by highlighting possible extensions of our work.

Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-Volkenborn integral including the Volkenborn integral and p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some various identities, formulas and relations related to several combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. By using generating functions, we give relations between the Lah numbers, the Bernoulli numbers, the Euler numbers and the Laguerre polynomials. We also give further comments and remarks on these functions, numbers and integral formulas related to q-type operators potentially used to solve problems in the areas such as physics, quantum mechanics, quantum systems and the others. In addition, we provide some tables containing some of the p-adic integral formulas obtained in this paper.

Some identities involving Bernoulli, Euler and degenerate Bernoulli numbers and their applications

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable.

Conformal bridge transformation, mathcal{PT} - and supersymmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcal{PT} -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcal{PT} -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

A note on p-adic Raabe formulas

The Hurwitz-type Euler (or the alternating Hurwitz) zeta function is defined by the following series for Re(s) > 0 and x ∈ ℂ\\(-∞, 0]. It can be analytically continued to the entire s-plane without any pole. In this note, we give a new proof of an analogue of Raabe's 1843 formula for p-adic Hurwitz-type Euler zeta functions by modifying Cohen and Friedman's method. We also use this method to construct the Raabe formulas for the nth derivative of p-adic Diamond-Euler Log Gamma functions. In addition, we obtain some identities for the fermionic p-adic integration, from which, some classical formulas for Euler polynomials have been deduced.

Some Explicit Expressions for Twisted Catalan-Daehee Numbers

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by using p-adic integrals or generating functions.

Robust quantum transport at particle-hole symmetry

We study quantum transport in disordered systems with particle-hole symmetric Hamiltonians. The particle-hole symmetry is spontaneously broken after averaging with respect to disorder, and the resulting massless mode is treated in a random-phase representation of the invariant measure of the symmetry group. We compute the resulting fermionic functional integral of the average two-particle Green's function in a perturbation theory around the diffusive limit. The results up to two-loop order show that the corrections vanish, indicating that the diffusive quantum transport is robust. On the other hand, the diffusion coefficient depends strongly on the particle-hole symmetric Hamiltonian we choose to study. This reveals a connection between the underlying microscopic theory and the classical long-scale metallic behaviour of these systems.

New construction of type 2 degenerate central Fubini polynomials with their certain properties

Kim et al. (Proc. Jangjeon Math. Soc. 21(4):589–598, 2018) have studied the central Fubini polynomials associated with central factorial numbers of the second kind. Motivated by their work, we introduce degenerate version of the central Fubini polynomials. We show that these polynomials can be represented by the fermionic p-adic integral on mathbb{Z}_{p}. From the fermionic p-adic integral equations, we derive some new properties related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind.

Fermionic path integrals and correlation dynamics in a 1D XY system

We derive time dependent correlation functions in an one dimensional XY spin model with the use of generating functionals, the latter being defined as path integrals over fermionic coherent states. We focus on the proper construction of the aforementioned integrals in order to avoid the inconsistencies usually encountered in the literature. The static limit of our results successfully reproduces the known ones, confirming the validity of our construction and allowing for further investigation of the dynamics. In the same context, we examine the case of a general driven transverse magnetic field, for which case we derive formulae for the equal-time correlation functions and confirm the consistency of our results with the Kibble–Zurek mechanism.

Phase diagram and dynamics of the SU(N) symmetric Kondo lattice model

In heavy-fermion systems, the competition between the local Kondo physics and intersite magnetic fluctuations results in unconventional quantum critical phenomena which are frequently addressed within the Kondo lattice model (KLM). Here we study this interplay in the SU($N$) symmetric generalization of the two-dimensional half-filled KLM by quantum Monte Carlo simulations with $N$ up to 8. While the long-range antiferromagnetic (AF) order in SU($N$) quantum spin systems typically gives way to spin-singlet ground states with spontaneously broken lattice symmetry, we find that the SU($N$) KLM is unique in that for each finite $N$ its ground-state phase diagram hosts only two phases -- AF order and the Kondo-screened phase. The absence of any intermediate phase between the $N=2$ and large-$N$ cases establishes adiabatic correspondence between both limits and confirms that the large-$N$ theory is a correct saddle point of the KLM fermionic path integral and a good starting point to include quantum fluctuations. In addition, we determine the evolution of the single-particle gap, quasiparticle residue of the doped hole at momentum $(\pi,\pi)$, and spin gap across the magnetic order-disorder transition. Our results indicate that increasing $N$ modifies the behavior of the coherence temperature: while it evolves smoothly across the magnetic transition at $N=2$ it develops an abrupt jump -- of up to an order of magnitude -- at larger but finite $N$. We discuss the magnetic order-disorder transition from a quantum-field-theoretic perspective and comment on implications of our findings for the interpretation of experiments on quantum critical heavy-fermion compounds.

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

Ordinary and degenerate Euler numbers and polynomials

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Disordered fermions, extra dimensions, and a solvable Yang-Mills theory

Generalizing disorder couplings of the SYK model by means of SU(N) matrices we formulate a lattice model of fermions in d+1 dimensions. Integration of fermions yields an effective theory of Yang-Mills fields in d dimensions, the latter approaching the standard Yang-Mills theory in the case of heavy fermions and the classical limit of vanishing coupling constant of the theory. Quantum mechanically, the theory is solved using large N approximation of the dual effective theory of Hermitian matrices in d dimensions. The theory is asymptotically free and confines the color. In case of massless fermions the emerging theory is an asymptotic safe QCD theory. We discuss also the relationship of this theory to the SYK model.

Fermionic Schwinger model for spinning Feshbach–Villars particle in magnetic field

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using fermionic coherent state path integral. The fermionization of the charge-spin symmetry is presented following the Schwinger recipe. The explicit calculations are done in the free case and magnetic interaction using the Foldy–Wouthuysen tranformation. The spectrum and wave functions are deduced.

Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials

In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on mathbb{Z}_{p} and generating functions. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

Some identities of special numbers and polynomials arising from p-adic integrals on mathbb{Z}_{p}

In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic p-adic integrals on mathbb{Z}_{p}.

Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

The Boole polynomials associated with the p-adic gamma function

We set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on Zp. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.