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Modern Physics Letters A

FRACTIONAL SPIN ON THE SPHERE AND THE BREAKING OF ROTATIONAL INVARIANCE

Giuseppe Bimonte, Giovanni Sparano

Published Feb 28, 1991• 0 citations
Publisher

Abstract

Topological considerations, in apparent disagreement with operatorial ones, seem to suggest that only integer and half-integer spins are possible on a sphere. In this letter we show that fractional spin is indeed possible on a sphere, but only by breaking the rotational invariance. As we will show in a subsequent paper, this is an example of the fact that the quantizations individuated by the covering space are those that preserve the continuous classical symmetries.

DOI
10.1142/s0217732391000531

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Topological considerations, in apparent disagreement with operatorial ones, seem to suggest that only integer and half-integer spins are possible on a sphere. In this letter we show that fractional spin is indeed possible on a sphere, but only by breaking the rotational invariance. As we will show in a subsequent paper, this is an example of the fact that the quantizations individuated by the covering space are those that preserve the continuous classical symmetries.
This chapter gives a brief introduction to rotations in two-dimensional space and finds that rotations in two-dimensional space have some special properties. In particular, spin and statistics have peculiarities not possible in three- or higher-dimensional space—spin is not quantized in integer or half-integer units of h and hence the particles are not necessarily bosons or fermions but may obey any statistics and were thus called anyons by Wilczek. In fact, the concept of fractional statistics had been already put forward by Leinaas and Myrheim in the late 1970s. They found that the origin of the fractional statistics lies in the peculiar topological properties of the configuration space of many identical particles. This space is doubly connected in three or more dimensions but is multiply connected in two dimensions. Later Goldin, Menikoff and Sharp reached a similar conclusion using a completely different method based on the rigorous study of the unitary representations of current algebra and diffeomorphism groups.
The spin quantum number of particles with fractional spin is neither an integer nor a half-integer. Quasiparticles knownas anyons are the most familiar examples of such particles. However, despite the considerable interest in anyons dueto their potential applications in quantum computation, a systematic analysis within the framework of quantummechanics addressing the fundamental question of whether elementary particles with fractional spin can exist is stilllacking. Two major obstacles are the requirement that spin operators satisfy an su(2) algebra and the spin-statisticstheorem. Focusing on spin-1/4 particles, we show that the first obstacle can be overcome by assuming that their spinoperators obey a deformed su(2) algebra with unequal structure constants. This symmetry breaking leads to a uniquerotational symmetry axis and imposes that the two particles forming a bound state have aligned symmetry axes andidentical spin projections. The second obstacle can be overcome by assuming that the binding interaction acts betweenparticles of different species, aligning their spins and confining their relative motion to the plane perpendicular to thesymmetry axis. Under these conditions, the braid operators describing the interchange of two spin-1/4 particles can berepresented in terms of the spin-component operator associated with the symmetry axis, closely paralleling theconventional spin-statistics connection. These results establish a framework in which spin-1/4 particles can beconsistently incorporated into quantum mechanics. Finally, we discuss the potential relevance of the proposedframework to theories beyond the Standard Model in which quarks are regarded as composite particles.
We introduce a solvable spin-rotational and time-reversal invariant spin-1 model in two dimensions. Depending on parameters, the ground state is an equal-weight superposition of all valence loops called "resonating valence loop" (RVL) or an equal-weight superposition of all valence bonds known as "resonating valence bond" (RVB). In RVL, ends of open loops are deconfined spinons of spin-1/2 that cannot be obtained by simple combinations of spin-1 -- a phenomenon of fractionalization; while in RVB, all quasiparticles carry an integer spin, hence no fractionalization. RVL and RVB are spin liquids with identical topological order but different spin-rotational and time-reversal symmetry quantum numbers of quasiparticles. We propose that quantized symmetry quantum number gives a systematic way to (partially) classify phases with identical topological order in dimensions greater than one.
In this chapter we consider the unit sphere S2\mathbb S^2S2 of the Euclidean space R3\mathbb R^3R3 with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on S2\mathbb S^2S2 and allow us to build a quantization of the two-sphere S2\mathbb S^2S2. They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of S2\mathbb S^2S2 and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.
In this work we study the crystal symmetry fractionalization in chiral spin liquids with the chiral-semion topological order. We show that if such a chiral spin liquid is realized in a two-dimensional lattice model with odd number of spin-12\frac1221​ per unit cell and the state preserves spin rotation symmetry and translation symmetries, the semion excitation must carry both half-integer spin and fractional crystal quantum numbers. As a result, only a unique symmetry enriched topological phase can be realized in chiral spin liquids in a spin-12\frac1221​ kagome lattice model. These fractional symmetry quantum numbers are confirmed numerically using ground state wave functions obtained with the density matrix renormalization group method.