Hawking radiation of anyons

According to the classical theory of Gravity, formulated by Albert Einstein, a black hole is a region of spacetime from which gravity prevents everything, including light, from escaping. It is quite remarkable that black holes turn “grey” and radiate energy in the form of Hawking radiation article6 , when quantum mechanics is brought into the picture. Hawking radiation, being thermal in nature, contains very little information, and since a large amount of information may have entered the black hole during its formation phase, it may be lost forever when the black hole evaporates completely. This is the so-called information loss problem, whose resolution is still-being sought, despite a number of interesting proposals article5 ; article7 ; article8 ; article30 ; article31 ; article29 . Hence, it is important to examine Hawking radiation phenomena for a diverse set of spacetimes and a variety of particles. At the very least, this would help in a better understanding of the problem itself. It has been shown that in $(3+1)-$ dimension, bosons and fermions Hawking radiate from black holes at the same Hawking temperature of the black hole, with the respective Planck and Fermi distributions. Contrary to this, in $(2+1)-$dimension, there can exist particles, known as anyons, which are neither bosons, nor fermions, but follow fractional statistics article11 ; article12 . Anyons are interesting entities to study in their own right. Furthermore, there has been recent experiments which show strong evidence in favour of their existence article35 . They may also be practically useful in a variety of systems such as quantum computation article36 ; article37 . The obvious question that arise in this context is whether there exist Hawking radiation of anyons as well. Existence of Hawking radiation of anyons from black holes will not only strengthen the Hawking radiation results, but may also shed new light on the information loss problem, as well as provide a new avenue of observing Hawking radiation in the laboratory article9 . Hence the primary focus of this article is to take a closer look at these issues, which to the best of our knowledge, is the first study of anyonic Hawking radiation. In the next section, we review the important properties of anyons that are relevant to this work. This is followed by, in Section III, a brief review of the Hawking decay rate from a black hole horizon in the tunneling approach first proposed by Parikh and Wilczek article2 for bosons and fermions in $(3+1)-$dimension. Hawking radiation using a complex path approach in different coordinates was also studied in article43 ; article44 and anyon like excitations in the context of Bañados, Teitelboim, and Zanelli (BTZ) black holes in Luo:2017ksc . In Section IV, we describe in detail an extension of this formalism to anyons in the context of a BTZ black hole. In Section V, we examine potential ways of observing this radiation in tabletop experiments. Finally, we summarize our results and conclude in Section VI.

In the $(3+1)$-spacetime dimension that we live in, particles are either bosons or fermions, with intrinsic spin of integer or half-odd integer (in units of $\hbar$). These particles are described by wave functions which are either symmetric or anti-symmetric under the exchange of two particles. Contrary to this, in $(2+1)$-spacetime dimension, a continuous range of statistics is available article12 . Consider two identical particles in $(2+1)$-dimension. Let $\psi(r)$ be the wave function of the two particle system, subject to the condition that $\psi(r)\neq 0$, if $r>a$ (the so-called ‘hard-core condition’), where $\vec{r}_{1}$ and $\vec{r}_{2}$ are the positions of the two particles and the relative position vector $\vec{r}\equiv\vec{r}_{1}-\vec{r}_{2}$. So, the configuration space of the particles is the two-dimensional $(x,y)$-plane with a disc of radius $\mathbf{a}$ removed. Given these coordinates, we can define a complex coordinate $z=x+iy$, and make a transformation $z\rightarrow z\,e^{2\pi i}$, which effectively brings a particle back to its starting point, the wave function must also remain invariant, but up to a phase, i.e.

$\displaystyle\psi(ze^{i2\pi},z^{*}e^{-i2\pi})=e^{i2\pi\alpha}\psi(z,z^{*})~{},$

(1)

for some real parameter $\alpha$. Similarly, one may interchange the two particles, i.e. transform $z\rightarrow ze^{i\pi}$, to obtain,

$\displaystyle\psi(ze^{i\pi},z^{*}e^{-i\pi})=e^{i\pi\alpha}\psi(z,z^{*})~{}.$

(2)

In Eq.(2) the value of $\alpha$ equal to $0$ and $1$ corresponding to bosons and fermions. However, in $(2+1)-$dimension, any real value of $\alpha$ in between $0$ and $1$ is also allowed. To understand this particular point, let’s consider a system of two identical particles in $(3+1)-$dimension. We need to do two consecutive interchanges of the location of the particles to go back to the original configuration. All such trajectories are topologically equivalent. However, in (2+1) dimension, after doing one interchange, we need to do one more winding of one particle around the other to come back to the initial configuration. Unlike to the case in $(3+1)-$dimension, these two trajectories are distinct in nature. So, it can be associated with two different topological phases which can take any value. It clearly indicates to the fact that, in two spatial dimension, a particle may posses statistics which are different from the standard Bose-Einstein and Fermi-Dirac statistics and are called fractional statistics. The particles that follow fractional statistics are called anyons. The objective of our work is to describe the Hawking radiation of these particles.

Hawking radiation article2 from a black hole horizon can be interpreted in the following way: there is copious pair production of particles and anti-particles from vacuum just inside the horizon. The anti-particle travels backwards in time inside the horizon, while the particle tunnels out quantum mechanically and is manifested as Hawking radiation. An equivalent picture exists in which the pair production occurs just outside the horizon, with the negative energy particle tunneling inside the horizon and the one with positive energy gives rise to Hawking radiation. A rigorous calculation of the tunneling rate indeed reproduces the correct radiation rate, derived independently by other methods, and lends further credence to the above picture. To estimate the tunneling probability, we compute the imaginary part of the action of the particle over classically forbidden region article2 ,

	$\displaystyle Im\,\,S$	$\displaystyle=$	$\displaystyle Im\int_{r_{in}}^{r_{out}}dr\,p_{r}\,,$		(3)
		$\displaystyle=$	$\displaystyle Im\int_{M}^{M-\omega}\int_{r_{in}}^{r_{out}}dH\,\frac{dr}{\dot{r}}\,,$

where $H=M-\omega^{{}^{\prime}}$ and the initial value of radius is chosen just inside the event horizon, $r_{in}=2M-\epsilon$. Due to loss of energy, the radius of the event horizon will reduce to, $r_{h}=2M-\omega$. Therefore, the point outside will be at, $r_{out}=2(M-\omega)+\epsilon$.

The integration in Eq.(3) is obtained using the equation of motion of radial null geodesics to express the $Im\,\,S$ in terms of $M$ and $\omega$,

$\displaystyle Im\,\,S=4\pi\omega\left(M-\frac{\omega}{2}\right)~{}.$

(4)

The corresponding amplitude of the tunneling process can be written in a straightforward manner as,

$\displaystyle\Gamma\displaystyle\sim e^{-2Im\hskip 5.0ptS}=e^{-8\pi\omega\left(M-\displaystyle{\frac{\omega}{2}}\right)}=p_{\omega}~{}.$

(5)

The above tunneling amplitude also can be interpreted as the relative probability of creating a particle-antiparticle pair just outside the horizon article41 . Eq.(5) is valid for both bosons and fermions. When a pair production happens just outside the horizon with the relative probability $\rm{p_{\omega}}$, particle with energy $\omega$ escapes from the horizon and antiparticle with energy $-\omega$ goes inward towards the singularity. The absolute probability of creating a pair of particles in a particular mode $\omega$ is calculated as follows:

$\displaystyle\rm{P_{\omega}\,=\,C_{\omega}\,p_{\omega}}~{},$

(6)

where $\rm{C_{\omega}}$, which is the probability that no pairs are created in that particular mode. Since fermions obey the exclusion principle, only one particle-antiparticle pair can be created in each quantum state, the sum of the probabilities of no pair production and one pair production must add up to one,

$\displaystyle\rm{C_{\omega}\,+\,C_{\omega}\,p_{\omega}\,=\,1}~{}.$

(7)

On the other hand, the probability of creating one pair of particles in the energy mode $\omega$ is given by

$\displaystyle\rm{P_{1\omega}\,=\,C_{\omega}\,p_{\omega}}\,=\,\frac{p_{\omega}}{1+p_{\omega}}~{}.$

(8)

We know that there is an effective potential barrier exterior of the black hole ($\rm{2M\,<\,r\,<\,\infty}$) which causes a back-scattering. In the particle production calculation we only need to consider the fraction entering black hole horizon, which is represented by the transmission coefficient $\Gamma_{\omega}$ article16 . Taking this fact into account we compute the probability of the one particle emission,

$\displaystyle\rm{\bar{P}_{1\omega}\,=\,P_{1\omega}\,\Gamma_{\omega}\,=\,\frac{p_{\omega}\,\Gamma_{\omega}}{1+p_{\omega}}}~{},$

(9)

$\rm{\bar{P}_{1\omega}}$ can also be interpreted as the mean number of particles ($\bar{N}_{\omega}$) emitted in a given mode. We substitute Eq.(5) in Eq.(9) to obtain $\bar{N}_{\omega}$ article13 ,

$\displaystyle\rm{\bar{N}_{\omega}\,=\,\frac{\Gamma_{\omega}}{e^{8\pi m\omega}\,+\,1}}~{}.$

(10)

The Eq.(10) is precisely the Fermi-Dirac distribution modified by the transmission coefficient $\Gamma_{\omega}$. Similar expression can be derived for bosons as well article13

$\displaystyle\rm{\bar{N}_{\omega}\,=\,\frac{\Gamma_{\omega}}{e^{8\pi M\omega}\,-\,1}}~{}.$

(11)

These results are consistent with Hawking’s celebrated work about particle emission from black hole and provides a nice physical picture in terms of tunneling through a potential barrier article6 .

Following the procedure described in section III, we now derive the amplitude for the emission of anyons from BTZ black hole. Since anyons exist in $(2+1)$-dimensional spacetime only, we restrict ourselves to BTZ black hole. This is a solution of the Einstein field equation in (2+1)-dimensional spacetime and describes a rotating geometry with horizons article18 ; article15 . The action corresponding to this solution is expressed in terms of metric $g^{(3)}$ and Ricci scalar $R^{(3)}$ in $(2+1)$-dimension as follows article18 ; article3 ,

$\displaystyle S=\int d^{3}x\sqrt{-g^{(3)}}\,(R^{(3)}+2\Lambda)~{},$

(12)

and the line element in polar coordinates is given by

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\left(d\theta-\frac{J}{2r^{2}}dt\right)^{2},$

(13)

$\displaystyle f(r)=-M+\Lambda r^{2}+\frac{J^{2}}{4r^{2}}~{}.$

(14)

Here $M$ and $J$ are respectively the mass and angular momentum of the $3$-dimensional rotating black hole and $\Lambda$ is the cosmological constant.

For the simplicity of the calculations, to start with, we restrict ourselves to the $J=0$ case. Generalization to $J\neq 0$ is straightforward and is discussed at the end of this section. We start with the transformation from the original coordinates $(t,r)$ to the Painlevé coordinates $(t_{p},r)$,

$\displaystyle dt=dt_{p}-\frac{1}{f(r)}\sqrt{1-f(r)}\,dr~{}.$

(15)

In terms of new coordinates and via a dimensional reduction, the line element in Eq.(13) takes the following form,

$\displaystyle ds^{2}=-f(r)\,dt_{p}^{2}+2\sqrt{1-f(r)}dt_{p}dr+dr^{2}~{}.$

(16)

Next, consider the Lagrangian for a massive particle in the above background,

	$\displaystyle L$	$\displaystyle=$	$\displaystyle\frac{m}{2}g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau},$		(17)
		$\displaystyle=$	$\displaystyle-\frac{1}{2}mf(r)(\dot{t_{p}})^{2}+m\sqrt{1-f(r)}\,\dot{t_{p}}\dot{r}+\frac{m}{2}(\dot{r})^{2}~{},$

where in the last step we substituted the metric components from Eq.(16). Since $t_{p}$ is a cyclic coordinate, the conjugate momentum is conserved,

$\displaystyle\frac{\partial L}{\partial\dot{t}_{p}}=-mf\dot{t_{p}}+m\sqrt{1-f}\,\dot{r}=-\omega=\mathrm{constant}~{}.$

(18)

The negative sign on the right hand side of the Eq.(18) represents the positive energy of the tunneling particle. Exploiting the fact that massive particles travel along time-like trajectories, we derive from Eq.(18) the expressions of the derivatives of Painlevé coordinates with respect to proper time of the particle under consideration,

$$\dot{t_{p}}=\pm\frac{1}{mf(r)}\sqrt{(1-f(r))(\omega^{2}-m^{2}f(r))}+\frac{\omega}{mf(r)}~{},$$

(19)

$\displaystyle\dot{r}=\pm\frac{1}{m}\sqrt{\omega^{2}-m^{2}f(r)}~{}.$

(20)

Given the expressions of the derivatives of Painlevé coordinates, we rewrite the imaginary part of the action Eq.(3) as follows,

$\displaystyle Im\,\,S=-\,Im\int_{r_{in}}^{r_{out}}\int_{m}^{\omega}\frac{d\omega^{{}^{\prime}}}{\frac{dr}{dt_{p}}}dr~{}.$

(21)

Following methodology described in article2 , we exchange the order of integration in Eq.(21) and do the integration first over the radial coordinate,

$$\int_{r_{in}}^{r_{out}}\frac{1}{\frac{dr}{dt_{p}}}dr=\int_{r_{in}}^{r_{out}}\frac{\sqrt{(1-f)(\omega^{2}-m^{2}f)}+\omega}{f\sqrt{\omega^{2}-m^{2}f}}dr~{}.$$

(22)

Using the Taylor expansion of $f(r)$ around the radius of the horizon, we rewrite the right hand side of Eq.(22),

$$\int_{r_{in}}^{r_{out}}\frac{1}{\frac{dr}{dt_{p}}}dr=\int_{r_{in}}^{r_{out}}G(r)dr~{},$$

where the function $G(r)$ is given by

$$G(r)=\frac{\sqrt{(1-f^{{}^{\prime}}(r_{h})(r-r_{h}))(\omega^{2}-m^{2}f^{{}^{\prime}}(r_{h})(r-r_{h}))}+\omega}{f^{{}^{\prime}}(r_{h})(r-r_{h})\sqrt{\omega^{2}-m^{2}f^{{}^{\prime}}(r_{h})(r-r_{h})}}dr~{}.$$

The final expression of this integral is obtained using Cauchy’s residue theorem,

$\displaystyle\int_{r_{in}}^{r_{out}}\frac{1}{\frac{dr}{dt_{p}}}dr=-\frac{2\pi i}{f^{{}^{\prime}}(r_{h})}~{}.$

(23)

In terms of this integration, the imaginary part of the action simplifies to the following form,

$\displaystyle Im\,\,S=\pi\int_{m}^{\omega}\frac{2}{f^{{}^{\prime}}(r_{h})}d\omega^{{}^{\prime}}=\frac{\pi}{\sqrt{\Lambda M}}(\omega-m),$

(24)

where we have used the fact $r_{h}$ depends on $\omega$,

$\displaystyle r_{h}=r_{h}(M-\omega^{{}^{\prime}})~{}.$

(25)

The final expression of the tunneling rate for $J=0$ is obtained using $Im\,\,S$,

$\displaystyle\Gamma\displaystyle\sim e^{-2Im\,\,S}=e^{-\frac{2\pi}{\sqrt{\Lambda M}}(\omega-m)}~{}.$

(26)

It is to be noted that we have ignored the back-reaction on the metric to get this results and hence, the results are valid when $m<<M$ and $\omega<<M$. From Eq.(26) in the $m\rightarrow 0$ limit gives the tunneling amplitude for massless particle,

$\displaystyle\Gamma\displaystyle\sim e^{-2Im\,\,S}$

$\displaystyle=$

$\displaystyle e^{-\frac{2\pi}{\sqrt{\Lambda M}}\omega}~{}.$

(27)

However, one can obtain the same expression starting from the condition for null geodesics and following the procedure applied in case of massive particles. It is to be noted that, to derive Eq.(26), we have not explicitly used information about the statistics of the Hawking radiation particles. Therefore, it is valid not only for bosons and fermions, but also for anyons. The next step is to find the distribution function of anyons from the the expression of tunneling amplitude which we have already derived [Eq.(26)], which reduces to the standard Bose-Einstein and Fermi-Dirac distribution functions in appropriate limits. In order to find the expression of the distribution function, first we consider the following assumptions which hold for any particles in $(2+1)$-dimensions article1 .

• •

  The permutation of the coordinates of any two particles in the multi-particle anyon wavefunction results in a phase being picked up by the wavefunction as follows (this is simply a restatement of the Eq.(2) for anyons)

$\displaystyle\Psi_{n}(...,q_{j},...,q_{i},...)=f\,\Psi_{n}(...,q_{i},...,q_{j},...)~{},$

(28)

where $f=e^{i\pi\alpha}$ .

• •

  Principle of detailed balance: If $n_{1}$, $n_{2}$ are the mean occupation numbers for the states $1$ and $2$ respectively, then at equilibrium, the number of transitions from $1$ to $2$ is the same as from $2$ to $1$. Under this condition, the ‘enhancement factor’ $F(n)\equiv P^{(n+1)}/P^{(n)}$, where $P^{(n)}$ is the probability of an $n$-anyon state, satisfies the following condition,

  $\displaystyle n_{1}F(n_{2})e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega_{1}-m)}=n_{2}F(n_{1})e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega_{2}-m)}~{}.$

  (29)

  This condition implies that

  $\displaystyle\frac{n}{F(n)}e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega-m)}=\mathrm{constant}~{}.$

  (30)

The form of $F(n)$ can be read-off from ref.article1 , which is $F(n)=\frac{4}{n+1}\left([\frac{n+1}{2}]\cos(\frac{\pi\alpha}{2})\right)^{2}$. When expanded in a power series in $n$, this yields the following,

$\displaystyle e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega-m)}=\frac{1}{n}+a_{0}+a_{1}n+a_{2}n^{2}+\dots~{}.$

(31)

Inverting the above equation, we get the expression of occupation number as a function of $\omega$ and $m$,

	$\displaystyle n(\omega,m)$	$\displaystyle=$	$\displaystyle\frac{1}{g}+\frac{B}{{g^{2}}}+\frac{C}{g^{3}}+\frac{D}{g^{4}}+...~{},$		(32)
		$\displaystyle\equiv$	$\displaystyle\frac{1}{g}+\sum_{k=3}^{\infty}\frac{\alpha_{k}}{g^{k}}~{},$

where $g\equiv\displaystyle{e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega-m)}}-a_{0}$ and the explicit form of the coefficients is taken from article1 . Rewriting Eq.(32) in the form of a continued fraction, we get,

$\displaystyle n(\omega,m)=\frac{1}{g-\frac{\alpha_{3}g}{g^{2}+\alpha_{3}-.....}}~{}.$

(33)

Finally, using the expression of $g$ we find from the above equation, the generalized Hawking radiation formula in $(2+1)$-dimension,

$\displaystyle n(\omega,m)=\frac{\Gamma_{\omega}}{e^{\frac{2\pi}{\sqrt{\Lambda M}}(\omega-m)}-a(\alpha)}~{},$

(34)

where $\Gamma_{\omega}$ is the transmission coefficient. Eq.(34) for $a(\alpha)=1$ and $a(\alpha)=-1$ corresponds to bosonic and fermionic Hawking radiation respectively. Any value of $a(\alpha)$ in between this range corresponds to anyons.
It is now quite straightforward to generalize the results to the $J\neq 0$ case. Dimensional reduction of the full metric (Eq.(13)) gives the following 2-dimensional metric article20 ; article21 ,

$\displaystyle ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)},$

(35)

where

$\displaystyle f(r)=-M+\Lambda r^{2}+\frac{J^{2}}{4r^{2}}~{}.$

(36)

Following the procedure mentioned before, one obtains the imaginary part of the action,

$\displaystyle Im\,\,S=\int_{m}^{\omega}\frac{2\pi}{f^{\prime}(r_{h})}d\omega^{\prime},$

(37)

where the radius of the horizon is $r_{h}$ is an explicit function of $(M-\omega^{{}^{\prime}})$, where $M$ is the mass of the black-hole and $\omega^{{}^{\prime}}$ is the energy of the tunneling particle under consideration. Now we make the following expansion for $f^{{}^{\prime}}(r)$ and keep only the leading order term $f^{\prime}(r_{h}(M))$ in the expansion,

$\displaystyle f^{\prime}(r_{h})=f^{\prime}(r_{h})\Big{|}_{\omega^{{}^{\prime}}=0}-f^{\prime\prime}(r_{h})\,\frac{\partial r_{h}}{\partial M}\Big{|}_{\omega^{{}^{\prime}}=0}\omega^{{}^{\prime}}+\dots~{}.$

(38)

Following the steps described in article19 , we calculate the integral in Eq.(37) using the above expression of $f^{\prime}(r_{h})$,

$\displaystyle Im\,\,S=\frac{2\pi}{f^{\prime}(r_{h})}(\omega-m)~{}.$

(39)

For $J\neq 0$ case, we have two horizons,

$\displaystyle r_{h}^{2}=r_{\pm}^{2}=\frac{M}{2\Lambda}\left[1\pm\left(1-\frac{\Lambda J^{2}}{M^{2}}\right)^{1/2}\right]~{}.$

(40)

However, for our work, only the outer horizon is relevant.

$\displaystyle f^{\prime}(r_{h})=f^{\prime}(r_{+})=2\Lambda r_{+}-\frac{J^{2}}{2r^{3}_{+}}~{}.$

(41)

For the outer horizon of the black-hole, the imaginary part of the action takes the following form,

$\displaystyle Im\,\,S=\frac{\pi}{\sqrt{2\Lambda}}\frac{\left(M+\sqrt{M^{2}-\Lambda J^{2}}\right)^{1/2}}{\sqrt{M^{2}-\Lambda J^{2}}}(\omega-m)~{},$

(42)

and the corresponding expression of the tunneling amplitude is given by

$\displaystyle\Gamma\displaystyle\sim e^{-2Im\,\,S}=e^{\frac{-2\pi}{\sqrt{2\Lambda}}\frac{\left(M+\sqrt{M^{2}-\Lambda J^{2}}\right)^{1/2}}{\sqrt{M^{2}-\Lambda J^{2}}}(\omega-m)}~{}.$

(43)

The expression of the tunneling amplitude clearly indicates that Hawking temperature article33 ; article34 ,

$\displaystyle T_{H}=\frac{\sqrt{2\Lambda}}{2\pi}\frac{\sqrt{M^{2}-\Lambda J^{2}}}{(M+\sqrt{M^{2}-\Lambda J^{2}})^{1/2}}~{}.$

(44)

Finally, following the steps described earlier for $J=0$ case, we get the expression of anyonic Hawking radiation spectrum for $J\neq 0$ case as

$\displaystyle n(\omega,m)=\frac{\Gamma_{\omega}}{e^{\frac{2\pi}{\sqrt{2\Lambda}}\frac{(M+\sqrt{M^{2}-\Lambda J^{2}})^{1/2}}{\sqrt{M^{2}-\Lambda J^{2}}}(\omega-m)}-a(\alpha)}~{}.$

(45)

In this work, we have derived a general expression for Hawking radiation of anyons, particles with intermediate statistics in $(2+1)$-dimensional spacetime, using the tunneling approach. The results are derived for both rotating and non-rotating BTZ black holes for massive and massless cases. We have shown that our results may be verifiable experimentally in an appropriate analogue system. Such a measurement on the one hand, will provide further evidence for Hawking radiation and that of anyons, albeit in an analog setting. Once anyon excitations are detected in this way, one can envisage potential uses of these particles with fractional statistics in a variety of ways. Furthermore, it is hoped that this work and its potential experimental verification would shed some light on the information loss problem. For example, our analysis is quantum mechanical and manifestly unitary, while the information loss problem suggests a fundamental non-unitarity. Thus it would be interesting to see how much of the current unitarity, say in analog models, can carry over to a real black hole and the subsequent Hawking radiation. We hope to report on these issues in the future.

This work was supported by the Natural Sciences and Engineering Research Council of Canada.