---
abstract: 'We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. The braid relations are used to calculate the quasiparticle’s spin in the fractional quantum Hall states on Riemann surfaces. The quasiparticle’s spin is found to be topological independent and satisfies physical restrictions.'
address: 'International School for Advanced Studies, SISSA, I-34014 Trieste, Italy '
author:
- Dingping Li
title: 'Intrinsic quasiparticle’s spin and fractional quantum Hall effect on Riemann surfaces'
---

The possibility of fractional statistics on two dimensional surfaces was discovered in Ref.[@lei]. When two fractional-statistics particles (anyons) are interchanged, the wave function changes by a phase $\exp(i\theta)$, where $\theta$ is neither $\theta =0$ (Bose statistics) nor $\theta =\pi$ (Fermion statistics). The quasiparticles in fractional quantum Hall systems (for a review on the fractional quantum Hall effect (FQHE), see Ref.[@girvin]) are anyons [@arovas; @halp] and this picture had been used to construct the hierarchical wave function in the FQHE [@halp]. Anyons may also find applications in some condensed matter systems [@sup]. On higher dimension spaces ($D>2$), only Fermion and Boson exist and the Fermion’s spin is an half-integer, the Boson’s spin is an integer. It will be very interesting to know what is the spin of anyons and the spin-statistics relation of anyons. The spin-statistics relation of anyons is generalized one if the spin $s$ equals to $s={\case\theta/2\pi}$. Anyons in various models, for example, in non-linear sigma models, Chern-Simons field theories and relativistic quantum field theories on $2D$ dimension spaces indeed have the generalized spin-statistics relation [@rdt; @fro]. Naturally, we will ask the question what is the spin-statistics relation of the quasiparticle in the FQHE. The recent discussion about the quasiparticle’s spin (QPS) can be found in Refs.[@lidp; @wen; @kiv]. Reference.[@lidp] calculated the QPS by analyzing the hierarchical wave function or by calculating the Berry phase of the quasiparticles moving in a closed path on the sphere. Reference.[@wen] obtained the QPS by analyzing the Ginzburg-Laudau-Chern-Simons (GLCS) theory of the FQHE on the sphere. On the other hand, Reference.[@kiv] calculated the QPS based on the GLCS theory on the disc geometry. However, the results in Refs.[@lidp; @wen; @kiv] are different from each other. The ambiguity of the QPS in the literature is due to lack of a good definition of the QPS.

In this paper, we will calculate the QPS by using braid relations of anyons on Riemann surfaces [@wu; @thou; @ein; @wend; @eina; @imbo; @bla]. We will show that the QPS calculated in the following is consistent with physical restrictions and is intrinsic in the sense of its topological independence. The results of the present paper is agreed with the results of Ref.[@lidp].

Let us consider $N$ spinning anyons on an oriented compact Riemann surface with $g$ handles[@eina; @imbo; @bla]. To define the anyon’s spin, we attach an oriented local frame to every particle. When a particle moves on a curved surface (the torus is a flat surface), the attached frame is parallel transported and a path-dependent frame-rotation is associated with the particle transport. Let us denote the clockwise $2\pi$ rotation of the frame attached to the particle by $R_{2\pi}$. The action of the operator $R_{2\pi}$ on the wave function will give a phase $\exp (i2\pi S)$. We define $S$ as the spin of the particle. $S$ is equal to $1/2$ for the electron by this definition. The braiding operators are $\sigma_i ,\, \rho_{n,i} ,\, \tau_{n,i}$, where $\sigma_i$ interchanges (clockwise) particle $i$ and particle $i+1$ and $\rho_{n,i} ,\, \tau_{n,i}$ take particle $i$ around noncontractable loops on the $m^{,}th$ handle. Here we use the same definition of operators $\rho_{n,i} ,\, \tau_{n,i}$ as Ref. [@eina]. The braid relations for spinning anyons on the Riemann surface with $g$ handles are

\[brarel\] $$\sigma_j\sigma_{j+1}\sigma_j
=\sigma_{j+1}\sigma_j\sigma_{j+1} ,
\label{brarel:1}$$ $$\tau_{m,j+1}= \sigma^{-1}_j \tau_{m,j}\sigma_j ,
\rho_{m,j+1}=\sigma^{-1}_j\rho_{m,j}\sigma_j ,
\label{brarel:2}$$ $$\rho^{-1}_{m,j}\tau_{m,j+1}\sigma^{-2}_j\rho_{m,j}
\sigma^2_j\tau^{-1}_{m,j+1}=\sigma^2_j ,
\label{brarel:3}$$

$$\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2
\sigma_1=R^{2(g-1)}_{2\pi}
\prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}.
\label{relation}$$

For the spinless anyons on the sphere, Equation (\[relation\]) becomes $\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2
\sigma_1=1 $, which had been derived in Ref. [@thou]. It expresses the fact that a closed (clockwise) loop of particle $1$ around all the other particles can be continuously shrunk to a point on the rear side of the sphere. Eq. (\[relation\]) is the generalization of the case of the spinless anyons on the sphere to the spinning anyons on general Riemann surfaces[@eina; @imbo]. When we deform the loop of the left side of Eq. (\[relation\]) to the loop of the right side of Eq. (\[relation\]), described by $\prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}$, the attached spin frame of is rotated ($4\pi (g-1)$ rotation), we obtain a phase $R^{2(g-1)}_{2\pi}$. We need to include this phase in the right side of the equation. However for the charged anyons in magnetic field, which is the case of quasiparticles in FQHE, the braid relation  (\[relation\]) should be changed. We also need to include the Aharonov-Bohm phase $\exp (2\pi iq\Phi )$ in the right of Eq. (\[relation\]) because the charged anyon interacts with the magnetic field, where $q$ is the anyon’s charge and $\Phi$ is the magnetic flux out of the surface. Thus in stead of Eq. (\[relation\]), for the charged anyons on the magnetic field, we have $$\begin{aligned}
\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2
\sigma_1 & = &
\exp (2\pi iq\Phi ) R^{2(g-1)}_{2\pi} \nonumber \\
& \times & \prod^g_n\rho^{-1}_{n,1}
\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1} .
\label{relation1}\end{aligned}$$ We will only consider the Abelian fractional statistics, which means that the representation of operator $\sigma_i$ is given by $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$, where ${\bf 1}_M$ is the $M \times M$ identity matrix. Inserting $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$ in Eq. (\[brarel:2\]) and Eq. (\[brarel:3\]), one obtains that $\tau_{m,j}=\tau_m , \, \rho_{m,j}=\rho_m ,\,
\tau_m \rho_m =\sigma^2\rho_m \tau_m $. These relations and Eq. (\[relation1\]) yield $$\begin{aligned}
\exp [2i(N-1)\theta] & = &
\exp [2\pi iq\Phi +4\pi(g-1)S]  \nonumber \\
& \times & \exp (-2ig\theta) .
\label{brarel3}\end{aligned}$$ If there are several kinds of anyons, we need to introduce mutual statistics[@mutual]. The mutual statistics $\theta_{i,j}$ means that when a particle of the $i^{,}{th}$ kind moves clockwise around a particle of the $j^{,}{th}$ kind, we get a phase $\exp (2\theta_{i,j})$. $\theta_{i,i}=\theta_i$ actually is the fractional statistics parameter of the particle of the $i^{,}{th}$ kind. The left hand of Eq. (\[brarel3\]) is a phase which is obtained by moving one particle (clockwise) around all other particles. If there exist other kinds of particles, instead of Eq. (\[brarel3\]), we have $$\begin{aligned}
\exp [2i(N_i-1)\theta_i
& + &
2i\sum_{j\not= i}^l N_j\theta_{i,j}]
=\exp (-2ig\theta_i) \nonumber \\
& \times &
\exp [2\pi iq_i\Phi +4\pi(g-1)S_i],
\label{mainre}\end{aligned}$$ where there are $l$ different kinds of particles and the spin of the particle of the $i^{,}{th}$ kind is $S_i$ and the charge is $q_i$. Eq. (\[mainre\]) gives a constraint on the parameters (numbers, statistics and spin ) of anyons that $$[((N_i-1+g)\theta_i
+  \sum_{j\not= i}^l N_j\theta_{i,j})
/ {\pi} ]
-q_i\Phi
 -  2(g-1)S_i
\label{const}$$ is an integer. Now we consider the FQHE on a surface with $g$ handles. We use the metric $ds^2=g_{z\bar z}dzd{\bar z}$ in complex coordinates. The volume form is $dv=[ig_{z\bar z} / 2]dz\wedge d{\bar z}
= g_{z\bar z}dx\wedge dy$. As in the case of the FQHE on the disc, sphere and torus, we apply a constant magnetic field on the surface. The natural generalization of the constant magnetic field to high genus Riemann surfaces[@bolte] is $F=Bdv=(\partial_zA_{\bar z}-
\partial_{\bar z}A_z)dz\wedge d{\bar z}$. Thus $ \partial_zA_{\bar z}-
\partial_{\bar z}A_z =ig_{z\bar z}B / 2$. The flux $\Phi$ is given by $2\pi \Phi = \int F =BV$, where $V$ is the area of the surface and we assume here $B>0$ ($\Phi >0$). The Hamiltonian of an electron on the surface under the magnetic field is given by the Laplace-Beltrami operator, $$\begin{aligned}
H & = & [1/ 2m \sqrt{g}]
(P_{\mu}-A_{\mu})g^{\mu \nu}\sqrt{g}(P_{\nu}-A_{\nu})
\nonumber \\
& = & [ g^{z\bar z} / m]
[(P_z-A_z)(P_{\bar z}-A_{\bar z}) \nonumber \\
& + &  (P_{\bar z}-A_{\bar z})(P_z-A_z)]  \\
\label{hamil}
& = & [2g^{z\bar z}/ m]
(P_z-A_z)(P_{\bar z}-A_{\bar z})+[B/ 2m] \nonumber\end{aligned}$$ where $g^{z\bar z}=[1 / g_{z\bar z}]$ and $P_{z}=-i\partial_z ,\, P_{\bar z}=-i\partial_{\bar z}$. The inner product of two wave functions is defined as $<\psi_1 | \psi_2 >=\int dv {\bar \psi_1 }\times \psi_2$. $H^{\prime}=[2g^{z\bar z} / m]
(P_z-A_z)(P_{\bar z}-A_{\bar z})$ is a positive definite hermitian operator Because $<\psi | H^{\prime} |\psi > \, \geq 0$ for any $\psi$. Thus if $H^{\prime} \psi =0$, $\psi$ satisfies $(P_{\bar z}-A_{\bar z})\psi =0$. The solutions of this equation are the ground states of the Hamiltonian $H$ or $H^{\prime}$, or the lowest Landau levels (LLL). The existence of the solutions of this equation is guaranteed by Riemann-Roch theorem[@griff]. The solutions belong to the holomorphic line bundle under the gauge field. Riemann-Roch theorem tells us that $h^0(L)-h^1(L)=deg(L)-g+1$, where $h^0(L)$ is the dimension of the holomorphic line bundle or the degeneracy of the ground states of the Hamiltonian $H$, $h^1(L)$ is the dimension of the holomorphic differential and $deg(L)$ is the degree of the line bundle which is equal to the first Chern number of the gauge field, or the magnetic flux out of the surface, $\Phi$. Because $deg(L) >2g-2$ (the magnetic field is very strong in the FQHE), $h^1(L)$ is equal to zero [@griff] and $h^0(L)=\Phi-g+1$. As a consistent check, $h^0(L)$ indeed gives the right degeneracy of the ground states in the case of a particle on the sphere and torus interacting with a magnetic-monopole field. In the case of high genus surfaces, Ref. [@avron] had discussed the degeneracy of the LLL for the leaky tori (see also Ref. [@asory] for the related discussions).

If the filling factor is $1 /  m$ in the state of the FQHE [@laughlin], we suppose to have relation $m(N+\Delta_g)=\Phi$. We assume that $\Delta_g$ is independent of $m$, and this assumption will be justified by the explicit constructions of some examples of Laughlin wave functions on high genus surfaces [@iengoli]. If $m=1$, we have an integer quantum Hall state and expect that the LLL are completely filled. In this case, $N=\Phi-\Delta_g$, and $N$ is also the degeneracy of the LLL, which should be equal to $\Phi-g+1$ (according to the discussions above). Thus $\Delta_g$ is equal to $g-1$. The relation between the electron numbers and the flux for the FQHE at the filling factor as $1 /  m$ is then $m(N+g-1)=\Phi$, which indeed gives correct results in the case of the Laughlin state on the sphere and torus [@haldane]. If $N_q$ quasiparticles are created, one has $m(N+\Delta_g)+N_q=\Phi$. The mutual statistics between the electron and quasiparticle is $2\theta_{mut}=2\pi$, the charge of the quasiparticle is $1 / m$ and the statistics parameters is $\pi / m$. We remark that, if $2\theta_{i,j}=2\pi, i\not= j$ in Eq. (\[const\]) or Eq. (\[mainre\]), we can simply omit the term $N_j\theta_{i,j}$. By applying Eq. (\[const\]) to those $N_q$ quasiparticles, one can show that $$[(N_q-1+g)/
m]+N- [\Phi / m] -2(g-1)S_q
\label{spin1}$$ is an integer. From Eq. (\[spin1\]) and the equation $m(N+\Delta_g)+N_q=\Phi$, the QPS turns out to be $S_q=[1 / 2m]+[n / 2(g-1)]$ ($n$ is an integer and $g \not= 1$ is assumed). Let us discuss how to fix $n$. We consider a cluster of particles which contains $n_i$ particles of the $i^{,}{th}$ kind with the mutual statistics $\theta_{i,j}$. By using the method developed in Ref. [@thou], we get the statistics, spin and charge of the cluster, $$S_c  = \sum_i [n_i(n_i-1)\theta_i / 2\pi ]
+n_iS_i +
\sum_{i \not= j} n_in_j\theta_{i,j} / 2\pi
\label{culster:2}$$ and $q_c=\sum n_iq_i$ and $\theta_c=\sum_i n_i^2\theta_i+
\sum_{i \not= j}n_in_j\theta_{i,j}$ are the charge and statistics of the cluster respectively. If the cluster’s charge is an odd (even) integer and the cluster satisfies the fermionic (bosonic) statistics, we suppose that the cluster contains only an odd (even) number of electrons (for example, see Ref. [@frozee]), and thus the cluster’s spin is an half-integer (integer). If the cluster contains $m$ quasiparticles in the above example, the cluster’s charge is $1$ and its statistics of is fermionic. This cluster shall be the hole of the electron and the cluster’s spin is an half-integer (see also Ref. [@fro]). By using Eq. (\[culster:2\]) for this cluster, we get a restriction for the QPS, $[mn / 2(g-1)]=integer$. If $m$ and $g-1$ are coprime to each other (for example, $g$ is equal to $0,2,3$), $n$ must be equal to $2n^{\prime}(g-1)$ ($n^{\prime}$ is an integer). Thus $S_q$ is equal to $1 / 2m$ (up to an integer) and the spin-statistics relation is the standard one. However, when $m$ and $g-1$ are not coprime to each other for some high genus surfaces, there exist other solutions for the spin except $1 / 2m$. We write $m=kp$ and $g-1=kq$, where $p$ and $q$ are coprime to each other, we have solutions $n=2n^{\prime}q$. The other solutions of the QSP is $S_q=[1 / 2m]+[n^{\prime}/  k]$ where $n^{\prime}=1,\cdots , k-1$. Therefor $S_q$ can not be completely fixed by using the braid group analysis. However, we shall point out that it is highly unlikely that those [**other**]{} solutions are the [**true**]{} QPS, as we expect that the value of the spin shall be intrinsic and does not depend on the surface where quasiparticles live. To completely fix the QPS on Riemann surfaces, we can obtain the QPS by analyzing the wave function of the quasiparticles on Riemann surfaces (we plan to do so elsewhere), as it was done for the case on the sphere[@lidp].

Let us calculate the QPS in the standard hierarchical state[@halp; @lidp; @haldane; @blok; @read; @gread; @lidpt]. We remark that the above method can be used to calculate the QPS in other kinds of quantum Hall fluids, for example, the multi-layered FQHE or Jain state[@jain] etc.. The hierarchical state is described by a symmetric matrix $\Lambda_{i,j}, i,j=1,2,\cdots , l$, where $\Lambda_{i,i+1}=\Lambda_{i+1,i}=\pm 1$, $\Lambda_{1,1}$ is an odd integer and $\Lambda_{i,i}, i\not= 1$ are even integers, where $l$ is the level of the hierarchical state and $N_i$ is the number of the particles in level $i$ ($N_1$ is the number of the electrons, $N_2$ is the number of the condensed quasiparticles (or holes) of the first level (Laughlin) state, etc.). On the torus[@haldane; @lidpt], we have a relation, $\sum_j \Lambda_{i,j}N_j=\delta_{i,1}\Phi$, and on the sphere[@lidp; @haldane], the relation is $\sum_j \Lambda_{i,j}N_j -\Lambda_{i,i}=\delta_{i,1}\Phi$. Following the discussion about the Laughlin state on Riemann surfaces, we expect that the relation is $\sum_j \Lambda_{i,j}N_j + (g-1) \Lambda_{i,i}=
\delta_{i,1}\Phi$ for the hierarchical state on Riemann surfaces. We define a $l$ dimension integer lattice with bases $E_i$ and the inner products $E_i\cdot E_j =\Lambda_{i,j}$ (see Ref. [@lidpt]) The above equation can be rewritten as $$\sum_{i=1}^{l} N_iE_i + (g-1) (E_i\cdot E_i)E_i^{\star}=E_1^{\star}\Phi ,
\label{hierar:1}$$ where $E_i^{\star}$ are the bases of the inverse lattice $E_i$ and defined by $E_i^{\star}\cdot E_j =\delta_{i,j}$. It can be verified that $E_i^{\star}\cdot E_j^{\star} =\Lambda_{i,j}^{-1}$. The quasiparticle is described by a vector ${\cal Q}_k =k_iE_i^{\star}$ ($k_i$ is an integer) on the lattice $E_i^{\star}$ (see Refs. [@blok; @read]). The statistics parameter of this quasiparticle is $\theta_k = {\cal Q}_k\cdot
{\cal Q}_k \pi=\sum_{i,j}k_i\Lambda_{i,j}^{-1}k_j$ and the charge is $Q_k={\cal Q}_k \cdot E_1^{\star}=\sum_ik_i
\Lambda_{i,1}^{-1}$. The mutual statistics between the quasiparticles ${\cal Q}_{k}$ and ${\cal Q}_{k^{\prime}}$ is $\theta_{k,k^{\prime}}=
{\cal Q}_{k}\cdot {\cal Q}_{k^{\prime}}\pi
=\sum_{i,j}k_i\Lambda_{i,j}^{-1}k^{\prime}_j$. If $N$ quasiparticles denoted by the vector ${\cal Q}_k$, is created, one has $\sum_{i=1}^{l} N_iE_i +
(g-1) (E_i\cdot E_i)E_i^{\star}+ N
{\cal Q}_k=E_1^{\star}\Phi$. Making inner product on two sides of this equation with ${\cal Q}_k$, one yields $${\frac {N\theta_k}{\pi}}+(g-1)\Delta \cdot {\cal Q}_k-
Q_k \Phi =integer,
\label{delt}$$ where $\Delta = \sum_{i=1}^{l}(E_i \cdot E_i)E_i^{\star}$. Applying Eq. (\[const\]) to those quasiparticles and comparing it with Eq. (\[delt\]), one gets $$S_k={\theta_k \over 2\pi}-{\frac {\Delta \cdot {\cal Q}_k} {2}}
+{\frac {n}{2(g-1)}}, \, n=integer.\, g\not= 1.
\label{result}$$ By using the argument in Ref. [@thou] (which we did for the quasiparticle of the Laughlin state), we can fix $n$ in some cases. The charge of the quasiparticle $E_i$ (denoted by the vector $E_i$) is $\delta_{i,1}$ and the statistics is $\theta = \delta_{i,1}\pi$. Thus the spin of this quasiparticle is ${\frac {\delta_{i,1}} {2}}+integer$. Because $E_i=\sum_j \Lambda_{i,j}E_j^{\star}$, the quasiparticle $E_i$ is a cluster which contains $\Lambda_{i,j}$ quasiparticles with the vector as $E_j^{\star}$. If $\det{\Lambda}$ and $g-1$ are coprime to each other, by using Eq. (\[culster:2\]) for the cluster, we find that $S_i={\frac {E_i \cdot E_i} {2}}-
{\frac {\Delta \cdot E_i} {2}}-{1\over 2}$ for the quasiparticle $E_j^{\star}$ when $l-i$ is an even integer and $S_i={\frac {E_i \cdot E_i} {2}}-
{\frac {\Delta \cdot E_i} {2}}$ when $l-i$ is an odd integer. We will use notation $i \in i_{e}$ if $l-i$ is an even integer. Generally, for the quasiparticle ${\cal Q}$, we find that $$S_{\cal Q}={\frac { {\cal Q}\cdot {\cal Q}} {2}}-
{\frac {\Delta \cdot {\cal Q}} {2}}-{1\over 2}\Delta^{\prime} \cdot
{\cal Q},
\label{result:1}$$ where $\Delta^{\prime}=\sum_{i} E_i, i\in i_e$. If the quasiparticles have the standard spin-statistics relation, it is required that ${\frac {\Delta \cdot {\cal Q}} {2}}
+{1\over 2}\Delta^{\prime} \cdot {\cal Q}$ is an integer. Indeed, this number is always an integer for the Laughlin state. However, in the hierarchical state, this number may not be an integer. Thus the quasiparticles in the hierarchical state usually do not have the standard spin-statistics relation.

If $\det{\Lambda}$ and $g-1$ are not coprime to each other, there exist other solutions, not only Eq. (\[result:1\]). As we argued in the case of the Laughlin state, these [**other**]{} solutions is [**unlikely**]{} the [**true**]{} QPS. Thus we suppose that Eq. (\[result:1\]) is the spin for quasiparticles and it is topological independent. The above method does not give any information about the QPS in the FQHE on the torus ($g=1$). However, the above discussion strongly suggests that the QPS in the FQHE on the torus is also given by Eq. (\[result:1\]) which suppose to be the QPS on any Riemann surfaces.

The Lagrangian for the long-distance physics of the Hall fluid on Riemann surfaces is [@wen], $$\begin{aligned}
{\cal L}& = & {1\over 4\pi}(\alpha_{\mu , i}\Lambda_{i,j}\epsilon^{\mu
\nu \lambda}\partial_{\nu}\alpha_{\lambda , j}
 + 2A_{\mu}t_i\epsilon^{\mu \nu \lambda}
\partial_{\nu}\alpha_{\lambda , i} \nonumber \\
& + & 2\omega s_i\epsilon^{0 \nu \lambda}
\partial_{\nu}\alpha_{\lambda , i}),\end{aligned}$$ where $\omega$ is the connection one form (the curvature is given by $R=d\omega$). In the case of the hierarchical state, $t_i$ is equal to $\delta_{i,1}$ and the matrix $\Lambda$ is one we gave in the previous discussion. By using Eq. (\[hierar:1\]), we can show that $s_i=\Lambda_{i,i}$. So $s_i$ is a topological independent constant. It is reasonable to believe that $s_i$ is a topological independent constant for any kinds of quantum Hall fluids. Due to the presence of the third term in the Lagrangian, the spin-statistics relation usually is not generalized one [@wen].

The QPS in some fractional quantum Hall states on Riemann surfaces has been calculated by using the braid relation for spinning anyons. The value of the QPS obtained satisfies physical restrictions and is found to be topological independent. Riemann-Roch theorem has been used in the discussions about the FQHE on Riemann surfaces. Because the methods used in this paper are rather general, they can be used to calculate the particles’s spin in other quantum Hall fluids, chrial spin fluids and anyon superfluids.

I would like to thank Professor S. Cecotti, Professor B. Dubrovin, T. Einarsson, and especially Professor R. Iengo for enlightening discussions. I also thank Professor J.M. Leinaas for discussions and hospitality at the Physics Department of Oslo University (Norway).

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