arXiv subjects

Shun-Jen Cheng

Publications and source records attributed to Shun-Jen Cheng.

At least 19 recordsLinked to original sources

arXiv2026

Categorical Equivalences of Finite W-Superalgebras and Clifford Twists

Associated with an even nilpotent element $e$ in a basic classical Lie superalgebra $\mathfrak{g}$, we study, in full generality, two constructions of finite $W$-superalgebras, defined via Whittaker models and isotropic subspaces, respectively. We prove that both formulations are independent of the various choices made in their constructions, thereby yielding, for a fixed good grading for $e$, at most two isomorphism classes of $W$-superalgebras. In the case when there are two non-isomorphic versions, we establish that they differ precisely by a Clifford extension. Consequently, when the two $W$-superalgebras are non-isomorphic, their module categories are equivalent up to a Clifford twist. Building on this equivalence and utilizing the Skryabin equivalence, we classify their irreducible representations in terms of generalized Whittaker modules over $\mathfrak{g}$

math.RT↗

arXiv2026

Simple character formulas for finite $W$-superalgebras of type $A$

We establish a canonical basis character formula for the irreducible modules in arbitrary parabolic BGG-type categories, including the category of finite-dimensional modules, for finite $W$-superalgebras of type $A$. These categories categorify the tensor product modules of irreducible polynomial representations and their duals over a quantum group of type $A$. Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig's dual canonical basis in the tensor product modules. Our formula provides a uniform generalization of several character formulas in BGG categories for Lie superalgebras and for $W$-algebras of type $A$.

math.RT↗

arXiv2025

Whittaker modules and representations of finite $W$-algebras of queer Lie superalgebras

We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$. We also obtain an analogue of Losev-Shu-Xiao decomposition for the finite $W$-superalgebras $U(\mathfrak q(n), E)$ of $\mathfrak q(n)$ associated to an odd nilpotent element $E\in \mathfrak q(n)_{\bar{1}}$. As an application, we establish several equivalences of categories of Whittaker $\mathfrak q(n)$-modules and analogues of BGG category of $U(\mathfrak q(n), E)$-modules. In particular, we reduce the multiplicity problem of Verma modules over $U(\mathfrak q(n), E)$ to that of the Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$.

math.RT↗

arXiv2025

Machine-learning-enabled methodology for the ab-initio simulations of sub-$\mu$m-wide nanoribbons

Simulation of mesoscopic nanostructures is a central challenge in condensed matter physics and device applications. First-principles methods provide accurate electronic structures but are computationally prohibitive for large systems, while empirical band theories are efficient yet limited by parameter fitting that neglects wavefunction information and often yields non-transferable parameters. We propose a methodology that bridges these approaches, achieving first-principles-level reliability with computational efficiency through a machine-learning-enabled tight-binding framework. Our approach starts with Wannier tight-binding (WTB) parameters from small nanostructures, which serve as training data for machine learning (ML). To remove the gauge freedom of Wannier functions that obscures size- and geometry-dependent parameter trends, we construct gauge-independent (GI) bases and transform the WTB model into a gauge-independent WTB (GI-WTB) model. This enables robust parameter fitting and ML prediction of parameter variations, yielding the machine-learning GI-WTB (ML-GI-WTB) model. Applied to MoS2 armchair-edge nanoribbons, the ML-GI-WTB model shows excellent agreement with first-principles results and enables reliable simulations of sub-$\mu$m-wide nanoribbons. This framework provides a scalable tool for predicting electronic properties of realistic nanostructures beyond the reach of conventional first-principles methods.

cond-mat.mtrl-sci↗

arXiv2024

Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimes\Lambda(\theta)$, where $\theta$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $\chi$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $\chi$. In the case when $\chi$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.

math.RT↗

arXiv2024

Super duality for Whittaker modules and finite $W$-algebras

We establish a super duality as an equivalence between Whittaker module categories over a pair of classical Lie algebra and Lie superalgebra in the infinite-rank limit. Building on this result and utilizing the Losev-Shu-Xiao decomposition, we obtain a super duality which is an equivalence between module categories over a pair of finite $W$-algebras and $W$-superalgebras at the infinite-rank limit.

math.RT↗

arXiv2024

Signatures of valley drift in the diversified band dispersions of bright, gray, and dark excitons in MoS2 monolayers under uni-axial strains

We present a comprehensive theoretical investigation of the strain-modulated excitonic properties of uni-axially strained transition-metal dichalcogenide monolayers (TMD-MLs) by solving the Bethe-Salpeter equation (BSE) established on the basis of first principles. We show that imposing an uni-axial strain onto a MoS_$2$ monolayers leads to the diversified band dispersions of the bright exciton (BX), gray exciton (GX), and dark exciton (DX) states, as a consequence of the competitive interplay between strain-induced valley drift (VD) and momentum-dependent electron-hole exchange interaction (EHEI). While the band dispersions of BX doublet in the light-accessible small reciprocal area remain almost unchanged against strain, the band dispersion of DX is reshaped by an increasing uni-axial strain from a parabola to a Mexican-hat-like profile, featured with unusual sign-reversal of the heavy effective mass and strain-activated brightness. In contrast, the effective mass of GX is drastically lightened by uni-axial strain and remains always positive. We show that the strain-diversified exciton band dispersions leads to the distinct exciton diffusivities and angle-resolved optical patterns of BX, GX, and DX in a strained TMD-ML, suggesting the feasibility of {\it spatially} resolving spinallowed and -forbidden excitons in exciton transport experiments and angle-resolved optical spectroscopies.

physics.app-ph↗

arXiv2024

Efficient light upconversion via resonant exciton-exciton annihilation of dark excitons in few-layer transition metal dichalcogenides

In this work, we report a pronounced light upconversion in few-layer transition metal dichalcogenides. Our joint theory-experiment study attributes the upconversion photoluminescence to a resonant exciton-exciton annihilation involving a pair of dark excitons with opposite momenta, followed by the spontaneous emission of upconverted bright excitons, which can have a high upconversion efficiency. Additionally, the upconversion photoluminescence is generic in MoS2, MoSe2, WS2, and WSe2, showing a high tuneability from green to ultraviolet light.

cond-mat.mtrl-sci↗

arXiv2023

Enhanced photo-excitation and angular-momentum imprint of gray excitons in WSe$_{2}$ monolayers by spin-orbit-coupled vector vortex beams

A light beam can be spatially structured in the complex amplitude to possess orbital angular momentum (OAM), which introduces a new degree of freedom alongside the intrinsic spin angular momentum (SAM) associated with circular polarization. Moreover, super-imposing two twisted lights with distinct SAM and OAM produces a vector vortex beam (VVB) in non-separable states where not only complex amplitude but also polarization are spatially structured and entangled with each other. In addition to the non-separability, the SAM and OAM in a VVB are intrinsically coupled by the optical spin-orbit interaction and constitute the profound spin-orbit physics in photonics. In this work, we present a comprehensive theoretical investigation, implemented on the first-principles base, of the intriguing light-matter interaction between VVBs and WSe$_{2}$ monolayers (WSe$_{2}$-MLs), one of the best-known and promising two-dimensional (2D) materials in optoelectronics dictated by excitons, encompassing bright exciton (BX) as well as various dark excitons (DXs). One of the key findings of our study is the substantial enhancement of the photo-excitation of gray excitons (GXs), a type of spin-forbidden dark exciton, in a WSe$_2$-ML through the utilization of a twisted light that possesses a longitudinal field associated with the optical spin-orbit interaction. Our research demonstrates that a spin-orbit-coupled VVB surprisingly allows for the imprinting of the carried optical information onto gray excitons in 2D materials, which is robust against the decoherence mechanisms in materials. This observation suggests a promising method for deciphering the transferred angular momentum from structured lights to excitons.

cond-mat.mes-hall↗

arXiv2023

Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras

We study the Whittaker category $\mathcal N(\zeta)$ of the Lie superalgebra $\mathfrak g$ for an arbitrary character $\zeta$ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of $\mathfrak g$. We prove that the Backelin functor from either the integral subcategory or any strongly typical block of the BGG category to the Whittaker category sends irreducible modules to irreducible modules or zero. The category $\mathcal N(\zeta)$ provides a suitable framework for studying finite $W$-superalgebras associated with an even principal nilpotent element. For the periplectic Lie superalgebras $\mathfrak{p}(n)$, we formulate the principal finite $W$-superalgebras $W_\zeta$ and establish a Skryabin-type equivalence. For a basic classical and a strange Lie superalgebras, we prove that the category of finite-dimensional modules over a given principal finite $W$-superalgebra $W_\zeta$ is equivalent to $\mathcal N(\zeta)$ under the Skryabin equivalence, for a non-singular character $\zeta$. As a consequence, we give a super analogue of Soergel's Struktursatz for a certain Whittaker functor from the integral BGG category $\mathcal O$ to the category of finite-dimensional modules over $W_\zeta$.

math.RT↗

arXiv2023

The key role of non-local screening in the environment-insensitive exciton fine structures of transition-metal dichalcogenide monolayers

In this work, we present a comprehensive theoretical and computational investigation of exciton fine structures of WSe$_2$-monolayers, one of the best known two-dimensional (2D) transition-metal dichalcogenides (TMD's), in various dielectric-layer environments by solving the first-principles-based Bethe-Salpeter equation. While the physical and electronic properties of atomically thin nano-materials are normally sensitive to the variation of surrounding environment, our studies reveal that the influence of dielectric environment on the exciton fine structures of TMD-ML's is surprisingly limited. We point out that the non-locality of Coulomb screening plays a key role to suppress the factor of dielectric environment and drastically shrink the fine structure splittings between bright exciton (BX) states and various dark exciton (DX) states of TMD-ML's. The intriguing non-locality of screening in 2D materials can be manifested by the measurable {\it non-linear} correlation between the BX-DX splittings and exciton binding energies with varying the surrounding dielectric environments. The revealed environment-insensitive exciton fine structures of TMD-ML's suggest the robustness of prospective dark-exciton-based opto-electronics against the inevitable variation of inhomogeneous dielectric environment.

cond-mat.mes-hall↗

arXiv2022

Whittaker categories of quasi-reductive Lie superalgebras and quantum symmetric pairs

We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character $ζ$ of the nilpotent radical, the associated Backelin functor $Γ_ζ$ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category $\mathcal O$. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $Γ_ζ$ and its target category, respectively, categorify a $q$-symmetrizing map and the corresponding $q$-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$.

math.RT↗

arXiv2022

Twisted-light-induced exciton wave packets in transition-metal dichalcogenide monolayers

We present a comprehensive theoretical investigation of the photo-generated excitons in transition-metal dichalcogenide monolayers (TMD-ML's) by Laguerre-Gaussian beams, a celebrated kind of twisted lights (TL's) carrying quantized orbital angular momenta (OAM). We show that the photo-excitation of TL incident to a TMD-ML leads to the formation of spatially localized exciton wave packets, constituted by the superposition of finite-momentum exciton states determined by the intriguing interplay between the multiple degrees of freedom of the optical and excitonic subsystems. Consequently, the TL-induced exciton wave packets yield profound directional photo-luminescences whose polar-angle-dependences are encoded by the transferred optical OAM and azimuthal angle part, despite OAM-irrelevant, optically resolves the exchange-split longitudinal and transverse exciton bands. Interestingly, the application of linearly polarized TL onto a valley-excitonic system mimics an exciton multiplexer allowing for selectively detecting the individual valley-mixed exciton bands, which are normally hardly measured spectrally.

cond-mat.mes-hall↗

arXiv2022

Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras

We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $\Gamma_\zeta$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $\Gamma_\zeta$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $\Gamma_\zeta$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.

math.RT↗

arXiv2021

Inherently high valley polarizations of momentum-forbidden dark excitons in transition-metal dichalcogenide monolayers

High degree of valley polarization of optically active excitons in transition-metal dichalcogenide monolayers (TMD-MLs) is vital in valley-based photonic applications but known to be likely spoiled by the intrinsic electron-hole exchange interactions. In this study, we present a theoretical investigation of the valley and optical properties of finite-momentum dark excitons in WSe$_2$-MLs by solving the density-functional-theory(DFT)-based Bethe-Salpeter equation (BSE) under the guidance of symmetry analysis. %We reveal that, in general, finite-momentum excitons are actually well immune from the exchange-induced valley depolarization, except for those with specific exciton momenta coincident with the $3σ_v$ and $3C_2'$ symmetries in the $D_{3h}$ point group of TMD-MLs. We reveal that, unlike the bright exciton inevitably subjected to electron-hole exchange interaction, inter-valley finite-momentum dark excitons in WSe$_2$-MLs are well immune from the exchange-induced valley depolarization and inherently highly valley-polarized under the enforcement of the crystal symmetries. More importantly, the superior valley polarizations of the inter-valley dark excitons in WSe$_2$-MLs are shown almost fully transferable to the optical polarization in the phonon-assisted photo-luminescences because of the native suppression of exchange-induced depolarization in the second-order optical processes. The analysis of phonon-assisted photo-luminescences accounts for the recently observed brightness, high degree of optical polarization and long lifetime of the inter-valley dark exciton states in tungsten-based TMD-MLs.

cond-mat.mes-hall↗

arXiv2020

Blocks and characters of $G(3)$-modules of non-integral weights

We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the general linear Lie superalgebras $\mathfrak{gl}(1|1)$, $\mathfrak{gl}(2|1)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(3|2)$.

math.RT↗

arXiv2020

Symmetry governed valley-pseudospin textures of the full-zone excitonic bands of transition-metal dichalcogenide monolayers

Preserving a high degree of valley polarization of excitons in photo-excited transition-metal dichalcogenide monolayers (TMD-MLs) is desirable for the valley-based photonic applications, but widely recognized as a hard task hindered by the intrinsic electron-hole exchange interaction. In this study, we present a comprehensive investigation of valley-polarized finite-momentum excitons in WSe$_2$-MLs over the entire Brillouin zone by solving the density-functional-theory(DFT)-based Bethe-Salpeter equation (BSE) under the guidance of symmetry analysis. We reveal that finite-momentum excitons are actually in general well immune from the exchange-induced valley depolarization, except for those with specific exciton momenta directionally coincide with the axes associated with the $3σ_v$ and $3C_2'$ symmetries in TMD-MLs. Governed by the symmetries, the valley pseudo-spin texture of the full-zone exciton band in the momentum space is locally featured by individual skyrmion-like structures where highly valley-polarized finite-momentum exciton states are centred. Remarkably, we show that the high degrees of valley polarizations of the finite-momentum exciton states are excellently well transferable to the optical polarizations in the resulting phonon-assisted photo-luminescences, suggesting the prospective usefulness of those inter-valley excitons in valley-based photonics.

cond-mat.mes-hall↗

arXiv2020

Representation theory of a semisimple extension of the Takiff superalgebra

We study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel subalgebras. We further compute all extension groups between two finite-dimensional simple objects and prove that all non-principal blocks in the finite-dimensional module category are Koszul.

math.RT↗