Finite Tensor Research Articles

Nonlinear elastic effects in phase field crystal and amplitude equations: Comparison toab initiosimulations of bcc metals and graphene

We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic one- and two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative $K'=4$ for bcc. A two-dimensional generalization suggests $K'_{2D}=5$. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical non-linearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain.

On Unimodular Finite Tensor Categories

Let C be a finite tensor category with simple unit object, let Z(C) denote its monoidal center, and let L and R be a left adjoint and a right adjoint of the forgetful functor U:Z(C)→C⁠. We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is self-dual, where 1∈C is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

Stress tensor for a scalar field in a spatially varying background potential: Divergences, “renormalization”, anomalies, and Casimir forces

Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporal-spatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth- and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for $z<0$ and rises like $z^\alpha$, $\alpha>0$, for $z>0$. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, $\alpha=1$ and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

Interphase Models for Nanoparticle-Polymer Composites

AbstractIn this work, the interphase effects of nanoparticle enhanced polymer composite materials are investigated. Three interphase models based on finite size representative volume element (RVE) are proposed to estimate the effective material properties of nanoparticle-polymer composites, including” (1) The finite Eshelby tensor based shell model, (2) An improved double inclusion model, and (3) A modified Hashin-Shtrikman model. These micromechanics models are employed to analyze the interphase phase effects on the effective material properties of nanoparticle-polymer composites. One crucial factor that influences the strength of nanoparticle enhanced polymer composites is the material properties of the interphase between nanoparticles and the corresponding polymer matrix. These three improved micromechanics models are designed to incorporate interphase effects. Using the interphase models, the authors estimate the effective Young’s and shear moduli for a set of four material systems of silica nanoparti...

Eilenberg–Watts Theorem for 2-categories and quasi-monoidal structures for module categories over bialgebroid categories

We prove Eilenberg–Watts Theorem for 2-categories of the representation categories C-Mod of finite tensor categories C. For a consequence we obtain that any autoequivalence of C-Mod is given by tensoring with a representative of some class in the Brauer–Picard group BrPic(C). We introduce bialgebroid categories over C and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor–Zelinsky sequence exists in this setting. In this context, for a symmetric bialgebroid category A, we interpret the middle cohomology group appearing in the third level of the latter sequence. We obtain a group of quasi-monoidal structures on the representation category A-Mod.

The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on [formula omitted]-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress and the displacement. The discrete spaces for the normal stress τii, the shear stress τij and the displacement ui are span{1,xi}, span{1,xi,xj} and span{1}, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, the element is nothing else but the 1D Raviart–Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element are 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. These elements are the simplest element for any space dimension.The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved. Numerical tests in 2D and 3D are presented to show a superiority of the new elements over others, as a superconvergence is exhibited and proved.

Design of thin simplified cloak with finite and small dynamic range constitutive tensors

A thin simplified cloak is proposed with constitutive tensors $$\epsilon $$ (or μ) which are spatially variant along the radial and angular directions, and invariant along the axial direction. The proposed simplified material properties $$\epsilon $$ (or μ) are free of infinity value at inner boundary and also, the cloak is perfectly impedance matched with the free space at the outer boundary. The practical realization of proposed simplified media is possible with two-dimensional metamaterials. The dynamic range of each of the spatially variant simplified constitutive tensor is reduced in contrast to the earlier reported cloak. Numerical results also confirm that the cloaking performance remains good for proposed thin cloak even with small dynamic range of material properties.

Enhanced Casimir effect for doped graphene

We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression for the finite temperature polarization tensor with a chemical potential. It is found that doping leads to a strong enhancement of the Casimir force reaching almost $60\%$ in quite realistic situations. This result should be important for planning and interpreting the Casimir measurements, especially taking into account that the Casimir interaction of undoped graphene is rather weak.

A Trace for Bimodule Categories

We study a 2-functor that assigns to a bimodule category over a finite \(\Bbbk \)-linear tensor category a \(\Bbbk \)-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories.

Hilbert Basis Theorem and Finite Generation of Invariants in Symmetric Tensor Categories in Positive Characteristic

In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in the case of semisimple categories and more generally in the case of categories with fiber functors to the characteristic $p > 0$ Verlinde category of $SL_{2}$. We also construct a symmetric finite tensor category $\sVec_{2}$ over fields of characteristic $2$ and show that it is a candidate for the category of supervector spaces in this characteristic. We further show that $\sVec_{2}$ does not fiber over the characteristic $2$ Verlinde category of $SL_{2}$ and then prove the conjecture for any category fibered over $\sVec_{2}$.

C*-Non-Linear Second Quantization

We construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22(3):1550001, 2015). We prove that the inductive limit C*-algebra is factorizable and has a natural localization given by a family of C*-sub-algebras each of which is localized on a bounded Borel subset of $${\mathbb{R}}$$ . Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C*-algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras.

Crossed extensions of the corepresentation category of finite supergroup algebras

We present explicit examples of finite tensor categories that are C2-graded extensions of the corepresentation category of certain finite-dimensional non-semisimple Hopf algebras.

Pivotal Tricategories and a Categorification of Inner-Product Modules

This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider (bi)module categories over pivotal tensor categories with additional structure on the inner homs. These inner-product module categories are related to Frobenius algebras and lead to the notion of *-Morita equivalence for pivotal tensor categories. In particular they allow to transport the pivotal structure to the categories of endofunctors. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.

On the Brauer–Picard group of a finite symmetric tensor category

Let Cn denote the representation category of finite supergroup ∧kn⋊Z/2Z. We compute the Brauer–Picard group BrPic(Cn) of Cn. This is done by identifying BrPic(Cn) with the group of braided tensor autoequivalences of the Drinfeld center of Cn and studying the action of the latter group on the categorical Lagrangian Grassmannian of Cn. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.

Counterterms for static Lovelock solutions

In this paper, we introduce the counterterms that remove the non-logarithmic divergences of the action in third order Lovelock gravity for static spacetimes. We do this by defining the cosmological constant in such a way that the asymptotic form of the metric have the same form in Lovelock and Einstein gravities. Thus, we employ the counterterms of Einstein gravity and show that the power law divergences of the action of Lovelock gravity for static spacetimes can be removed by suitable choice of coefficients. We find that the dependence of these coefficients on the dimension in Lovelock gravity is the same as in Einstein gravity. We also introduce the finite energy-momentum tensor and employ these counterterms to calculate the finite action and mass of static black hole solutions of third order Lovelock gravity. Next, we calculate the thermodynamic quantities and show that the entropy calculated through the use of Gibbs-Duhem relation is consistent with the obtained entropy by Wald’s formula. Furthermore, we find that in contrast to Einstein gravity in which there exists no uncharged extreme black hole, third order Lovelock gravity can have these kind of black holes. Finally, we investigate the stability of static charged black holes of Lovelock gravity in canonical ensemble and find that small black holes show a phase transition between very small and small black holes, while the large ones are stable.

Module categories over affine group schemes

Let k be an algebraically closed field of characteristic p\ge 0 . Let G be an affine group scheme over k . We classify the indecomposable exact module categories over the rigid tensor category \mathrm {Coh}_f(G) of coherent sheaves of finite dimensional k -vector spaces on G , in terms of (H,\psi) - equivariant coherent sheaves on G . We deduce from it the classification of indecomposable geometrical module categories over Rep (G) . When G is finite, this yields the classification of all indecomposable exact module categories over the finite tensor category Rep (G) . In particular, we obtain a classification of twists for the group algebra k[G] of a finite group scheme G . Applying this to u(\mathfrak {g}) , where \mathfrak {g} is a finite dimensional p -Lie algebra over k with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.

The pivotal cover and Frobenius–Schur indicators

In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius–Schur (FS) indicators in “non-pivotal” settings. For an object V∈Cpiv, the (n,r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the “adjoint object” in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Frechet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γ_c(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C^∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗A, where A is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)_0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.

Constructing generalized Cesàro formulas for finite plane strains

The problem of finding the displacement vector from a system of nonlinear differential equations which includes displacement gradient components is studied. Expressions on the right side of this system for certain parameter values have the kinematic sense of Lagrange and Euler finite strain tensors. The task is to construct generalized Cesaro formulas for finite strains. The construction of the solution consists of two stages (algebraic and differential), and the second is performed for space whose dimension is greater than or equal to two. An algorithm for the inversion of the original system is proposed, and analytical constructions for the case of two-dimensional space are performed. The problem is solved at the first (algebraic) stage, i.e., an exact analytical expression for the displacement vector components is derived through the known finite strain tensor and an unknown scalar function having the kinematic sense of rotation. Necessary conditions for the existence of this relationship are formulated.