Torsional Monopoles and Torqued Geometries in Gravity and Condensed Matter

Abstract

Torsional degrees of freedom play an important role in modern gravity theories as well as in condensed matter systems where they can be modeled by defects in solids. Here we isolate a class of torsion models that support torsion configurations with a localized, conserved charge that adopts integer values. The charge is topological in nature and the torsional configurations can be thought of as torsional 'monopole' solutions. We explore some of the properties of these configurations in gravity models with non-vanishing curvature, and discuss the possible existence of such monopoles in condensed matter systems. To conclude, we show how the monopoles can be thought of as a natural generalization of the Cartan spiral staircase.The analogy between geometry and defects in gravity theories and in theories of elasticity in solids is an old and well developed field of study [1][2][3][4][5]. Disclinations and dislocations in crystals are defects in the ordered lattice which carry finite curvature and torsion respectively. Transporting a particle around a disclination (dislocation) produces a non-zero rotation (translation) by the end of the cycle. Dislocations are particularly interesting because, while sources of curvature are ubiquitous in the natural universe, the effects of torsion in a gravitational context are thus far negligible experimentally [6]. In solids, however, dislocations affect many important material properties and are present even in the cleanest materials. Thus condensed matter systems can provide useful laboratories for the study of torsion.As we will show, the defects that we will describe cannot be described by the classical geometric theory of elasticity. Instead, these defects may occur in materials described by micropolar elasticity theory [7]. Micropolar elasticity theory (or Cosserat elasticity) is a simple extension of classical elasticity to include local orientational degrees of freedom of the constituent particles/molecules of the elastic medium. The defect we investigate, which we dub a torsional 'monopole' (TM), is a defect that does not require a lattice deformation, but a deformation texture in the local rotational degrees of freedom. Such defects could exist in biological or granular systems (two common systems described by micropolar elasticity) and may affect solids with a strong coupling between orbital electronic motion and local spin or orbital degrees of freedom. Although TMs are not very complicated objects, as evidenced by the structure shown in Fig. 1, there are some subtle issues that require a careful treatment. We present a general treatment of these defects in a gravitational context in flat and curved space and then give an explicit construction of a TM and its relation to defects in solids and the "Cartan spiral staircase." We begin with a simple formulation of our construction in flat space. To isolate the purely torsional degrees of freedom in the absence of other geometric constructs, we will begin with three highly constraining FIG. 1. A cross-section through the o...

Lattice field theory with torsion

“…This means that the torsion can stand in for the axial chemical potential. We can also study many other torsioninduced and dislocation-induced phenomena found in earlier works [2][3][4][5][6][7][19][20][21][22][23].…”

A gauge-theoretical approach to elasticity with microrotations

“…One can approach this along the lines proposed in Randono & Hughes (2011). As a first step, we can replace the ansatz (3.1) by taking instead of (2.23) the orthogonal matrix that is a product of N factors The dot denotes the usual matrix product.…”

“…The geometric constraint (1.1) of the vanishing curvature rules out the disclinations. It was shown in Randono & Hughes (2011) that dislocations are also not relevant to the point-like soliton solutions. The solitons of this type are mentioned by Unzicker (2002), who calls them a 'Shankar monopole' (Shankar 1977) that describes a point-like defect in a A-phase superfluid Helium-3, (Mermin 1979;Vollhardt & Wölfle 1990;Volovik 2003).…”

A-geometrical approach to topological insulators with edge dislocations

“…Real materials are imperfect and contain topological defects such as dislocations [15,18,35], disclinations [19,20] and gauge fields induced by strain [21,22]; therefore, a natural question is to formulate the physics of topological insulators in the presence of such defects [8]. These topological defects can be analyzed using the coordinate transformation method given in [26] which modifies the Hamiltonian for a topological insulator with a defect by the metric tensor and the spin connection [30][31][32][33][34].…”

“…Therefore, we expect that the chiral excitations [29] will be sensitive to such defects. 3 In this paper, we will introduce the tangent space approach used in differential geometry [24,33,34] to study the propagation of electrons for a space-dependent coordinate [26]. We find that the continuum representation of the edge dislocation [26] generates a spin connection [30][31][32] which is controlled by the Burger vector.…”