Particle exchange statistics beyond fermions and bosons

TL;DR

The work extends quantum statistics beyond fermions and bosons by introducing paraparticles governed by an $R$-matrix that satisfies the Yang–Baxter equation. It develops a second-quantized formalism, derives generalized exclusion statistics, and presents exactly solvable 1D and 2D spin models where free paraparticles emerge as quasiparticles, with observable exchange statistics in 2D. The results show nontrivial exchange rotations in internal space, topological features in 2D, and a framework compatible with locality and Hermiticity, suggesting the possible existence of novel elementary paraparticles. This provides a new platform for exploring exotic quantum statistics and their potential realization in condensed matter and high-energy contexts.

Abstract

It is commonly believed that there are only two types of particle exchange statistics in quantum mechanics, fermions and bosons, with the exception of anyons in two dimension. In principle, a second exception known as parastatistics, which extends outside of two dimensions, has been considered but was believed to be physically equivalent to fermions and bosons. In this paper we show that nontrivial parastatistics inequivalent to either fermions or bosons can exist in physical systems. These new types of identical particles obey generalized exclusion principles, leading to exotic free-particle thermodynamics distinct from any system of free fermions and bosons. We formulate our theory by developing a second quantization of paraparticles, which naturally includes exactly solvable non-interacting theories, and incorporates physical constraints such as locality. We then construct a family of exactly solvable quantum spin models in one and two dimensions where free paraparticles emerge as quasiparticle excitations, and their exchange statistics can be physically observed and is notably distinct from fermions and bosons. This demonstrates the possibility of a new type of quasiparticle in condensed matter systems, and, more speculatively, the potential for previously unconsidered types of elementary particles.

Figures (10)

• Figure 1: The generalized exclusion statistics and free particle thermodynamics of paraparticles defined by the $R$-matrices in Ex. \ref{['ex:decoupled']}-\ref{['ex:1m1']} of Tab. \ref{['tab:Hilbert_series']}, and a comparison to ordinary fermions and bosons. (left) the level degeneracy $\{d_n\}_{n\geq 0}$; (right) thermal expectation value of the single-mode occupation number $\langle \hat{n}\rangle_\beta$.
• Figure 2: The 2D exactly solvable spin model on a $7\times 7$ lattice with open boundary conditions. Each black dot represents a 16 dimensional qudit on which the local operators $\hat{x}^\pm_{i,a},\hat{y}^\pm_{i,a}$ act, and each open circle represents a 64 dimensional auxiliary qudit on which the local operators $\hat{w}^\pm_{ab}$ act, for $w=u_L,u_R,v_L$, or $v_R$. Each colored triangle represents a 3-body interaction between qudits on its 3 vertices. In addition, we have 8-body interactions around every even plaquette (i.e. the white and gray plaquettes). Eq. \ref{['eq:JWT_string_2D']} gives example of a paraparticle operator $\hat{\psi}_{i,a}^\pm$ defined on the string $\Gamma$ (shown in purple), which is an MPO acting consecutively on all the purple dots.
• Figure 3: Illustration of paraparticle exchange in the 2D solvable spin model. The shaded square represents the 2D system with OBC as shown in Fig. \ref{['fig:mainlattice']}. $i$ and $j$ label the black site in the upper left and lower right corners of the 2D lattice, respectively, where paraparticles can be locally created and measured. The unitary exchange operator $\hat{E}_{ij}$ moves the paraparticles along the two colored paths, and the result of the exchange is given in Eq. \ref{['eq:Uij_action']}.
• Figure S1: Graphical representation of the CRs between the local spin operators $\{\hat{x}^\pm_{a},\hat{y}^\pm_{a}\}^m_{a=1}$ in Eq. \ref{['eq:XYQA']}. The matrix elements of each operator $\{\hat{x}^\pm_{a},\hat{y}^\pm_{a}\}^m_{a=1}$ is a tensor (represented by the triangles) with two quantum indices (e.g. the indices $q_1$ and $q_2$ shown in figure) and one auxiliary index (e.g. the index $a$), and the $R$-matrix (represented by a square) is a tensor with four auxiliary indices. Matrix multiplication goes from top to bottom in the quantum space and from left to right in the auxiliary space.
• Figure S2: The action of the operators $\hat{x}^\pm_a,\hat{y}^\pm_a$ and $\hat{n}$ on the basis states $|0\rangle$, $\{|1,b\rangle\}^m_{b=1}$, $|2\rangle$, for the spin model corresponding to the $R$-matrix in Ex. \ref{['ex:1m1']}.

Theorems & Definitions (18)

• Theorem 1
• Theorem S2.1
• Theorem S2.2
• Lemma S2.3
• proof
• Lemma S2.4
• proof
• Theorem S2.5
• Definition S5.1
• Remark S5.1