Topological Defect

Topological defects or topological solitons are irregularities or disruptions that occur within continuous fields or ordered states of matter.

# Overview

Mathematically, topological defects occur when a solution of a system of partial differential equations or of a quantum field theory is homotopically distinct from the vacuum solution, that is the boundary conditions entail the existence of homotopically distinct solutions.

When solutions to the differential equations are topologically distinct, they can be classified by their homotopy class. Topological defects are not only stable against small perturbations, but cannot decay or be undone or be de-entangled, precisely because there is no continuous transformation that will map them (homotopically) to a nullhomotopic (uniform, or trivial) solution.

Topological defects are interesting since they reflect internal symmetries of conformal field theories. Consider a field theory described by a continuous symmetry group $G$ which is spontaneously broken to a subgroup $H \leq G$. Whether the theory admits topological defects depends on the homotopy of the space of cosets of $H$ in $G$ that is the order parameter of a medium, and is sometimes referred to as a vacuum manifold $\mathcal{M} = G / H$ (particularly in Quantum Field Theories).

 Classifying Topological Defects By Homotopy In Order Parameter Space Consider a closed loop around the defect. The order parameter field $\mathbf{u}$ changes as we move around the loop. The positions of the atoms around the loop with respect to their local "ideal" lattice drifts upwards continuously. This precisely corresponds to a loop around the order parameter space, corresponding to the hole in the torus. Moving the atoms slightly will deform the loop, but won't change the number of times the loop winds through or around the hole. Two loops which traverse the torus same number of times are equivalent. The equivalence classes are elements of the first homotopy group of the torus $\pi_1(\mathbf{T}) \cong \mathbb{Z} \times \mathbb{Z}$.

Whether the theory admits (invertible) topological defects depends on whether the vacuum manifold has nontrivial homotopy groups. If $G$ is a universal cover for $G/H$, it can be shown that,

$$\pi_n(G/H) = \pi_{n-1}(H)$$

Recall that a homotopy group $\pi_n$ of a topological space $X$ consists of equivalence classes of maps of spheres $S^n$ into $X$ (with a fixed basepoint $x_0 \in X$) where two maps are homotopy equivalent if they can be deformed into each other. Thus various types of defects in the medium can be classified by elements of homotopy groups of the order parameter space $X$ in different dimensions. Line defects correspond to elements of $\pi_1(X)$ as they can be enclosed by a loop, point defects correspond to elements of $\pi_2(X)$ as they can be englobed by a sphere, textures correspond to elements of $\pi_3(X)$ and so on. Defects that belong to the same equivalence class can be continuously deformed into each other.

 Multiplying two loops The product of two loops is given by starting from their intersection, traversing the first loop, and then traversing the second. The inverse of a loop is the same loop traveled backwards: compose the two and one can shrink them continuously back into nothing. The algebraic structure of homotopy groups of the order parameter space has a physical interpretation: if two defect lines coalesce, their combined homotopy class must be given by the loop enclosing both. This large loop can be deformed into two little loops, so the homotopy class of the coalesced line defect is the product of the homotopy classes of the individual defects.

The inverse of a loop $\gamma$ is just the loop which runs along the same path in the opposite direction. The identity element consists of the equivalence classes of loops which do not enclose a hole: they can all be contracted smoothly to a point (and thus to one another). Finally, multiplication law has direct physical implication: encircling two defect lines of strength $\mathbf{u}$ and $\mathbf{v}$ is completely equivalent to encircling one defect of strength $\mathbf{u} \times \mathbf{v}$.

 Entanglement Trail We can pass by if we leave a trail: is the connecting double line topologically trivial? Encircle the double line by a loop. The loop can be wiggled and twisted off the doble line, but it still encircles around the two legs of the defects $\alpha$, $\beta$.

 Defect Entanglement Can a defect line of class $\alpha$ pass by the defect line of class $\beta$, without getting entangled?

Furthermore, crossing defects with equivalence classes $\alpha$ and $\beta$ get entangled if and only if they are members of separate conjugacy classes of $\pi_1(X)$, that is if their equivalence classes do not commute: $\alpha\beta\alpha^{-1}\beta^{-1} \neq 1$. In physical sense, defects of commutative classes can pass through one another, while defects of non-commutative classes entangle. (Note that in real world systems commuting dislocations can fail to pass through one another for energetic reasons as well). Unfortunately, there are very few real world materials where defect entanglement matters: biaxial nematics and metallic glasses being the two examples. It was also applied in the style of the statistics of anyons in topological quantum computing, where, orbits of two anyons can braid around one another.