Topological exchange statistics in one dimension

The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this approach gives only trivial or near-trivial exchange statistics in one dimension because two-body coincidences are excluded from configuration space. Instead, we include these path-ambiguous singular points and consider configuration space as an orbifold. This orbifold topological approach allows unified analysis of exchange statistics in any dimension and predicts possibilities for anyons in one-dimensional systems, including non-Abelian anyons obeying alternate strand groups. These results clarify the nontopological origin of fractional statistics in one-dimensional anyon models.

Figure 1

Depiction of the three types of generator relations (9) for ${\pi }_1^{*}\left(\mathcal{Q}\right)$ when $N=4$ as strand diagrams, read from the bottom: (a) the self-inverse relation ${\sigma }_1^2=1$ that is broken for the braid group generators; (b) the Yang-Baxter relation ${\sigma }_1{\sigma }_2{\sigma }_1={\sigma }_2{\sigma }_1{\sigma }_2$ that is broken for the traid group generators; (c) the locality relation ${\sigma }_1{\sigma }_3={\sigma }_3{\sigma }_1$ that is broken for the fraid group generators.

Figure 2

Graphical comparison of the actions of $O\left(S_N\right)\sim S_N$ and ${\pi }_1^{*}\left(\mathcal{Q}\right)\sim S_N$ on ${\mathcal{X}}_2$ for $N=3$. (a) The relative configuration space for three distinguishable particles on a line with coordinates $(x_1,x_2,x_3)$. The horizontal coordinate is $z_1=(x_1-x_2)/\sqrt{2}$ and the vertical coordinate is $z_2=(x_1+x_2-x_3)/\sqrt{6}$. The black lines are the two-body coincidence locus ${\mathrm{\Delta }}_2$ and they section $\mathcal{X}$ into six alcoves ${\mathcal{Y}}_{\omega }\in {\pi }_0\left({\mathcal{X}}_2\right)$ where the particles have different position orders $x_{{\omega }_1}<x_{{\omega }_2}<x_{{\omega }_3}$. (b) Reflections across each of the three colored lines are orthogonal transformations $O_s$ of $\mathcal{X}$ that represent the passive permutation of particle labels. For example, $O_{\left(12\right)}$ is a reflection across the vertical line (red) that permutes the labels of particle 1 and particle 2 and $O_{\left(23\right)}$ is a reflection across the diagonal line with positive slope (green) that permutes the labels of particles 2 and 3. The double-sided arrows (red) connect points $x_{\omega }\in p^{-1}\left(q\right)$ and indicate how the ${\mathcal{Y}}_{\omega }\in {\pi }_0\left({\mathcal{X}}_2\right)$ are permuted by $O_{\left(12\right)}$. (c) The relative components of the orbifold $\mathcal{Q}$ and a based loop ${\sigma }_1={\pi }_1^{*}\left(\mathcal{Q}\right)$ starting and ending at manifold point $q$. This path realizes an exchange ${\sigma }_1$ of the first two particles. (d) The based loop ${\sigma }_1$ in $\mathcal{Q}$ is lifted to six paths in ${\mathcal{X}}_2$ that start at $x_{\omega }$ and end at $x_{{\omega }^{\prime }}$. For visual clarity, half of the path lifts are represented as dashed lines. These six lifts of element ${\sigma }_1={\pi }_1^{*}\left(\mathcal{Q}\right)$ define a map ${\tilde{\sigma }}_1$ on ${\mathcal{X}}_2$ that permutes the ${\mathcal{Y}}_{\omega }\in {\pi }_0\left({\mathcal{X}}_2\right)$.

Figure 3

Depictions of six different elements of $S_4\left(S^1\right)=\mathbb{Z}\wr S_4={\mathbb{Z}}^4⋊S_4$ mentioned in the text. In these strand diagrams, the ring has been cut flattened into a line and the gray dashed lines on either side are the cut. The elements depicted include: ${\sigma }_4$, the pairwise exchange of the first and fourth particle (relative to the cut) constructed from $S_N$ generators as in first line of (13a); $t_1$, one of the four generators of ${\mathbb{Z}}^4$ subgroup of $S_4\left(S^1\right)$; its inverse $t_1^{-1}$; the around-the-back operator ${\sigma }_0$ defined in (13b) and simplified using ${\sigma }_i^2=1$; the $\zeta$ operators defined in (13c) corresponding to a shift in the center-of-mass by $\pi /2$ around the ring; and ${\zeta }^{-1}$ its inverse.

Figure 4

Depiction of the universal cover $\tilde{\mathcal{Q}}=\tilde{\mathcal{X}}$ and several useful domains for $N=2$ and $\mathcal{M}=S^1$. The black dots are all lifts of the same point in $\mathcal{Q}$ and $\mathcal{X}$, the black lines are the lifts of the coincidence locus ${\mathrm{\Delta }}_2$ to $\tilde{\mathcal{Q}}$. The red square (a) is one choice for the fundamental domain isomorphic to the torus $\mathcal{X}=T^2$. Each congruent square would then be labeled by a pair of integers in ${\pi }_1\left(\mathcal{X}\right)=Z^2$ describing the path to fundamental domain. The green triangle (b) and purple square (c) are equivalent alternate choices for the fundamental domain isomorphic to the Möbius band $\mathcal{Q}$ or ${\mathcal{Q}}_2$. The blue rectangle (d) is the double cover of $\mathcal{Q}$ or ${\mathcal{Q}}_2$. It is the smallest torus ${\overline{T}}^2=S_{\mathrm{rel}}^1\times S_{\mathrm{com}}^1$ for which there exists separable and single-valued center-of-mass and relative coordinates for indistinguishable particles. Although it has the same area in configuration space, it is not equivalent to the torus $\mathcal{X}=T^2$. The yellow square (e) is the double cover of $\mathcal{X}$ that is the smallest torus for which exist single-valued center-of-mass and relative coordinates for distinguishable particles.

Figure 5

Two views of a depiction of the universal cover $\tilde{\mathcal{Q}}=\tilde{\mathcal{X}}$ and several useful domains for $N=3$ and $\mathcal{M}=S^1$. The black dots are all lifts of the same point in $\mathcal{Q}$ and $\mathcal{X}$, the black planes in the lower figure are the lifts of the coincidence locus ${\mathrm{\Delta }}_2$ to $\tilde{\mathcal{Q}}$. The five highlighted domains are equivalent to the five domains depicted in Fig. 4: The red cube is a fundamental domain congruent to $\mathcal{X}=T^3$, the green 3-orthoscheme tetrahedron and purple triangular prism are equivalent alternate choices for the fundamental domain isomorphic to the Penrose triangle $\mathcal{Q}$ or ${\mathcal{Q}}_2$. The blue rhombic prism is the minimal separable torus for indistinguishable particles ${\overline{T}}^3=T_{\mathrm{rel}}^2\times S_{\mathrm{com}}^1$ and the yellow rhombic prism is the minimal separable torus for distinguishable particles.