Rank Symmetric Tensor Research Articles

Hall-like behaviour of higher rank Chern-Simons theory of fractons

Fracton phases of matter constitute an interesting point of contact between condensed matter and high-energy physics. The limited mobility property of fracton quasi-particles finds applications in many different contexts, including quantum information, spin liquids, elasticity, hydrodynamics, gravity and holography. In this paper we adopt a field theoretical approach to investigate the three dimensional action of a rank-2 symmetric tensor field invariant under the covariant fracton symmetry. The theory appears as a non-topological higher rank generalization of the ordinary Chern-Simons model, depending only on the traceless part of the tensor gauge field. After defining a field strength, a rank-2 traceless “electric” field and a “magnetic” vector field are identified, in analogy with the standard Chern-Simons ones. Once matter is introduced, a Hall-like behaviour with fractonic features emerges. In particular, our model shows a Hall-like dipole current, together with a vectorial “flux-attachment” relation for dipoles. This gives a possible starting point for a fracton-vortex duality. A gauge-fixing term is then introduced, from which propagators are computed and the counting of the degrees of freedom is performed. Finally, the energy-momentum tensor is shown to be conserved and the integrated energy density is proved to be zero, which reminds the topological nature of the standard Chern-Simons model.

Maxwell theory of fractons

We show that the main properties of the fracton quasiparticles can be derived from a generalized covariant Maxwell-like action. Starting from a rank-2 symmetric tensor field ${A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, we build a partially symmetric rank-3 tensor field strength ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$ which obeys a kind of Bianchi identity. The most general action invariant under the covariant ``fracton'' transformation ${\ensuremath{\delta}}_{\text{fract}}{A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)={\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Lambda}}(x)$ consists of two independent terms: one describing linearized gravity (LG) and the other referable to fractons. The whole action can be written in terms of ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, and the fracton part of the invariant Lagrangian writes as ${F}^{2}(x)$, in analogy with Maxwell theory. The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton literature, and the field equations of motion, which have the same form as the covariant Maxwell equations (${\ensuremath{\partial}}^{\ensuremath{\mu}}{F}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\mu}}(x)=0$), can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Amp\`ere laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the ``Bianchi identity'' for the tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, as in Maxwell theory. In the covariant generalization of the fracton theory, the equations describing the fracton limited mobility, i.e., the charge and dipole conservation, are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, we increase the known analogies between LG and fracton theory by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.

The Thermal Conductivity Tensor of β‐Ga2O3 from 300 to 1275 K

AbstractLaser‐flash measurements of the thermal conductivity of Czochralski‐grown β‐Ga2O3 single crystals are reported perpendicular to four independent crystallographic lattice planes , (010), (001), and (100) between room temperature and 1200 K. λ is represented by a symmetric tensor of rank 2 and has four independent components. Tensor components W m−1 K−1 and W m−1 K−1 (values for 300 K) can be directly measured. W m−1 K−1 and W m−1 K−1, instead, are obtained by combining W m−1 K and W m−1 K. The crystallographic ‐axis is the direction with the highest thermal conductivity, λ22. The thermal conductivity assumes its lowest value (11.73 W m−1 K−1) in a direction close to the crystallographic ‐axis. However, this direction does not coincide with , and changes with temperature.

A note on higher rank descriptions of massless and massive spin-1 particles

The Maxwell theory can be written as a first order model with the help of a two-form auxiliary field, such master action allows the proof of duality between 1-form and D−3 forms. Here we show that the replacement of the two-form auxiliary field by an arbitrary (non symmetric) rank-2 tensor leads to a new massless spin-1 dual theory in terms of a partially antisymmetric rank-3 tensor. In the massive spin-1 case we have a non symmetric generalization of the massive two-form theory (Kalb–Ramond). The coupling of the massive non symmetric spin-1 model to matter fields is investigated via master actions.We also show that massive models with severe discontinuity in their massless limit can also be obtained from Kaluza–Klein dimensional reduction of massless higher rank tensors which become Stueckelberg fields after the reduction.

An Anisotropic Continuum Damage Mechanics Model for the Simulation of Yield Surface Evolution and Ratcheting

A new anisotropic continuum damage model was presented on the basis of irreversible thermodynamics. Second rank symmetric damage tensor, kinematic hardening tensor, and damage potential surface were introduced. The yield surface in this study has been modeled to distort in the stress space by the evolution of the damage tensor. The new model was applied in yield surface characterization and led to a well description of the subsequent yield surfaces. This model is then applied to predict experimental results of the cyclic nonproportional loading paths obtained by Kang et al. [2004], Hassan et al. [2002] and Taleb and Cailletaud [2011]. It is demonstrated that the new model can predict the yield surface distortion and the ratcheting strains of multiaxial nonproportional cyclic loading with acceptable accuracy and by using less material constants with respect to the previous models.

More exact results on chiral gauge theories: The case of the symmetric tensor

We study dynamics of $SU(N\ensuremath{-}4)$ gauge theories with fermions in rank-2 symmetric tensor and $N$ antifundamental representations, by perturbing supersymmetric theories with anomaly-mediated supersymmetry breaking. We find the $SU(N)\ifmmode\times\else\texttimes\fi{}U(1)$ global symmetry is dynamically broken to $SO(N)$ for $N\ensuremath{\ge}17$, a different result from conjectures in the literature. For $N<17$, the theory initially flows to a superconformal fixed point, but is diverted by the soft masses, which act as a relevant perturbation.

Hamiltonian analysis and positivity of a new massive spin-2 model

Recently a new model has been proposed to describe free massive spin-2 particles in D dimensions in terms of a non symmetric rank-2 tensor e μν and a mixed symmetry tensor B μ[αβ]. The model is invariant under linearized diffeomorphisms without Stueckelberg fields. It resembles a spin-2 version of the topologically massive spin-1 BF model (Cremmer–Scherk model). Here we apply the Dirac–Bergmann procedure in order to identify all Hamiltonian constraints and perform a complete counting of degrees of freedom. In D = 3 + 1 we find 5 degrees of freedom corresponding to helicities ±2, ±1, 0 as expected. The positivity of the reduced Hamiltonian is proved by using spin projection operators. We have also proposed a parent action that establishes the duality between the Fierz–Pauli and the new model. The equivalence between gauge invariant correlation functions of both theories is demonstrated.

Bootstrapping Heisenberg magnets and their cubic instability

We study the critical $O(3)$ model using the numerical conformal bootstrap. In particular, we use a recently developed cutting-surface algorithm to efficiently map out the allowed space of CFT data from correlators involving the leading $O(3)$ singlet $s$, vector $\phi$, and rank-2 symmetric tensor $t$. We determine their scaling dimensions to be $(\Delta_{s}, \Delta_{\phi}, \Delta_{t}) = (0.518942(51), 1.59489(59), 1.20954(23))$, and also bound various OPE coefficients. We additionally introduce a new "tip-finding" algorithm to compute an upper bound on the leading rank-4 symmetric tensor $t_4$, which we find to be relevant with $\Delta_{t_4} < 2.99056$. The conformal bootstrap thus provides a numerical proof that systems described by the critical $O(3)$ model, such as classical Heisenberg ferromagnets at the Curie transition, are unstable to cubic anisotropy.

Counting Tensor Rank Decompositions

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

On Lorentz-invariant bispin-2 theories

We investigate a Lorentz invariant action which is quadratic in two rank-2 symmetric tensor fields in Minkowski spacetime. We apply a scalar-vector-tensor decomposition to two tensor fields by virtue of 3-dimensional rotation-invariance of Minkowski spacetime and classify theories with seven degrees of freedom based on the Hamiltonian analysis. We find two new theories, which cannot be mapped from the linearized Hassan-Rosen bigravity. In these theories, the new mass interactions can be allowed thanks to the transverse diffeomorphism invariance of action.

The dimensional reduction of linearized spin-2 theories invariant under transverse diffeomorphisms

Here we perform the Kaluza–Klein dimensional reduction from D+1 to D dimensions of massless Lagrangians described by a symmetric rank-2 tensor and invariant under transverse differmorphisms (TDiff). They include the linearized Einstein–Hilbert theory, linearized unimodular gravity and scalar tensor models. We obtain simple expressions in terms of gauge invariant field combinations and show that unitarity is preserved in all cases. After fixing a gauge, the reduced model becomes a massive scalar tensor theory. We show that the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of the massless sector of consistent massive scalar tensor models. We discuss some subtleties when eliminating Stückelberg fields directly at action level as gauge conditions. A non local connection between the massless sector of the scalar tensor theory and the pure tensor TDiff model leads to a parametrization of the non conserved source which naturally separates spin-0 and spin-2 contributions in the pure tensor theory. The case of curved backgrounds is also investigated. If we truncate the non minimal couplings to linear terms in the curvature, vector and scalar constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff) cases. We prove that our linearized massive scalar tensor models admit those curved background extensions.

Fierz\u2013Pauli theory reloaded: from a theory of a symmetric tensor field to linearized massive gravity

Modifying gravity at large distances by means of a massive graviton may explain the observed acceleration of the Universe without Dark Energy. The standard paradigm for Massive Gravity is the Fierz–Pauli theory, which, nonetheless, displays well known flaws in its massless limit. The most serious one is represented by the vDVZ discontinuity, which consists in a disagreement between the massless limit of the Fierz–Pauli theory and General Relativity. Our approach is based on a field-theoretical treatment of Massive Gravity: General Relativity, in the weak field approximation, is treated as a gauge theory of a symmetric rank-2 tensor field. This leads us to propose an alternative theory of linearized Massive Gravity, describing five degrees of freedom of the graviton, with a good massless limit, without vDVZ discontinuity, and depending on one mass parameter only, in agreement with the Fierz–Pauli theory.

Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

We show that if a homogeneous polynomial f in n variables has Waring rank $$n+1$$ , then the corresponding projective hypersurface $$f=0$$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of f. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.

Emergent Elasticity in Amorphous Solids.

The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of elasticity from local mechanical constraints offers a new paradigm for systems with no broken symmetry, analogous to emergent gauge theories of quantum spin liquids. Specifically, our U(1) rank-2 symmetric tensor gauge theory of elasticity translates to the electromagnetism of fractonic phases of matter with the stress mapped to electric displacement and forces to vector charges. We corroborate our theoretical results with numerical simulations of soft frictionless disks in both two and three dimensions, and experiments on frictional disks in two dimensions. We also present experimental evidence indicating that force chains in granular media are subdimensional excitations of amorphous elasticity similar to fractons.

More on dual actions for massive spin-2 particles

Here we start from a dual version of Vasiliev’s first order action for massless spin-2 particles (linearized first order Einstein–Hilbert) and derive, via Kaluza–Klein dimensional reduction from D + 1 to D dimensions, a set of dual massive spin-2 models. This set includes the massive ‘BR’ model, a spin-2 analogue of the spin-1 Cremmer–Scherk model. In our approach the linearized Riemann curvature emerges from a solution of a functional constraint. In D = 2 + 1 the BR model can be written as a linearized version of a new bimetric model for massive gravitons. We also have a new massive spin-2 model, in arbitrary dimensions, invariant under linearized diffeomorphisms. It is given in terms of a non symmetric rank-2 tensor and a mixed symmetry tensor.

Massless Spin-2 Rank-3 Tensor Field in de Sitter Universe

From the conformal transformation, a massless spin-2 field must be presented by a rank-3 mixed symmetric tensor field. A massless spin-2 rank-3 mixed symmetric tensor field in de Sitter ambient space formalism is studied in this paper, which may be regarded as the linearized gravity or its dual. Recalling the field equation, the possible physical solutions of the field equation are calculated. The solutions can be written in terms of a rank-2 symmetric tensor field and a scalar field. We discuss that under what conditions this tensor field can be associated with the de Sitter and conformal groups unitary representations. Finally, for the physical solution, the quantum field operator and the two-point function are presented.

Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics

In this work, based on Eringen’s theory of nonlocal anisotropic elasticity, the three-dimensional nonlocal anisotropic elasticity of generalized Helmholtz type is developed. The derivation of a new three-dimensional nonlocal anisotropic kernel, which is the Green function of the three-dimensional anisotropic Helmholtz equation, enables to capture anisotropic length scale effects by means of a length scale tensor, which is a symmetric tensor of rank two. The derived nonlocal kernel function possesses up to six internal characteristic lengths on the Angstrom-scale. The presented theory of nonlocal elasticity possesses the appropriate property to be a generalized continuum theory of Angstrom-mechanics, since the range of its validity and applicability is up to the Angstrom-scale. The connection between the theory of nonlocal anisotropic elasticity and lattice theory is established. The tensor function of nonlocal elastic moduli as well as the nonlocal kernel function is given in terms of the Hessian matrix in the lattice approach. In the framework of the considered theory, the modeling of dislocations in anisotropic materials taking into consideration anisotropic dislocation core effects is presented. Important dislocation key formulas, namely the anisotropic Peach–Koehler stress formula, the Peach–Koehler force and the anisotropic Blin’s formula, are derived. A major tool used in deriving the expression of anisotropic Blin’s formula is Kirchner’s so-called $${\varvec{F}}$$-tensor, which is here generalized toward nonlocal anisotropic elasticity. The main feature and advantage of the derived fields, compared with the corresponding ones in classical anisotropic elasticity, is that they are free of singularities. Numerical applications to straight dislocations in bcc Fe are given, revealing the ability and advantage of the considered theory to describe adequately nonsingular anisotropic stress and self-energy fields capturing the effects of anisotropy on the Angstrom-scale.

Non-proportional stress and stress-strain controlled paths cyclic loading modeling by using anisotropic continuum damage model

In this study, the effect of developed anisotropic damage on the ratcheting is studied. Damage occurs during extra loading in an anisotropic manner. A modified anisotropic continuum damage model of Murakami is developed in this paper and combined with nonlinear kinematic hardening model of Chaboche. The constitutive relations for anisotropic damage of elastoplastic materials are developed based on irreversible thermodynamics. 2nd rank symmetric damage tensor, the kinematic hardening tensor and tensor of damage potential surface movement are specified according to the internal state manifold which is consistent to the thermodynamic state of solids. The symmetrization of effective stress tensor is carried out by the proposed method of Cordebois and Sidoroff (1982). The proposed yield surface distortion model consists of equivalent effective stress and it distorts in the stress space by evolution of damage parameter. Thus, the induced anisotropy is due to plastic deformation and damage evolution. Combination of the nonlinear kinematic hardening model of Chaboche and the model of anisotropic continuum damage is applied to cyclic loading modeling and the ratcheting strains are predicted under various loading paths.

Erratum to: Initial conditions and degrees of freedom of non-local gravity

We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are characterized by, respectively, two and four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.

On Lorentz-invariant spin-2 theories

We construct Lorentz-invariant massless/massive spin-2 theories in flat spacetime. Starting from the most generic action of a rank-2 symmetric tensor field whose Lagrangian contains up to quadratic in first derivatives of a field, we investigate the possibility of new theories by using the Hamiltonian analysis. By imposing the degeneracy of the kinetic matrix and the existence of subsequent constraints, we classify theories based on the number of degrees of freedom and constraint structures and obtain a wider class of Fierz-Pauli theory as well as massless and partially massless theories, whose scalar and/or vector degrees of freedom are absent. We also discuss the relation between our theories and known massless and massive spin-2 theories.