Higher virial coefficients of the ideal anyon gas: exact determination and regularity in the statistics angle

The exact second virial coefficient of the two-dimensional ideal anyon gas, computed by Arovas, Schrieffer, Wilczek and Zee in 1985, is periodic in the statistics angle $\theta$ but nonanalytic, with cusps at the bosonic values. Whether such cusps persist in the higher virial coefficients, and what those coefficients are, has since become a precise and only partially resolved research program. Later work established that all cluster and virial coefficients are finite, that only the second coefficient contains a first-order (cusp) nonanalyticity, that the third coefficient is exactly mirror-symmetric about the semion point, and high-precision numerical values of the third and fourth coefficients, which are smooth and well fitted by rapidly converging Fourier series; but no closed form exists for any coefficient beyond the second, the analyticity of all higher coefficients remains an unproven conjecture, the fifth and higher coefficients have never been computed, and the average-field approximation used beyond the dilute regime breaks the exact periodicity in $\theta$ with no quantified validity domain. The problem asks for the exact or certified determination of the higher virial coefficients of the ideal anyon gas and a proof or refutation of their regularity at bosonic values — the low-density equation of state of the paradigmatic model of Abelian fractional statistics beyond its exactly known leading term.

Background

In two spatial dimensions the configuration space of identical particles is multiply connected, so quantum states may carry a one-dimensional representation of the braid group: exchanging two particles multiplies the wave function by a phase $e^{i\theta}$ with an arbitrary statistics angle $\theta$, not only the Bose value $\theta=0$ or the Fermi value $\theta=\pi$. Particles with such intermediate, fractional statistics are called anyons, and the midpoint $\theta=\pi/2$ corresponds to semions. A standard representation used in statistical mechanics models each anyon as a boson carrying a pointlike, nondynamical statistical magnetic flux tube, so that every particle sees every other through an Aharonov–Bohm vector potential — the long-ranged vector potential of an idealized flux tube, whose quantum effect on a transported wave function persists even where the magnetic field vanishes — proportional to $\theta/r$ at relative separation $r$; the ideal (hard-core) anyon gas is the case in which the many-body wave function is single-valued and vanishes at particle coincidences, leaving this topological statistical interaction as the only coupling.

The equilibrium thermodynamics is organized by the cluster expansion of the grand-canonical partition function in the fugacity $z$, $\beta p=\sum_{n\ge1}b_n z^n$, whose cluster coefficients $b_n$ all carry dimensions $\lambda^{-2}$, so that $\lambda^2 b_n$ is dimensionless; here $\beta$ is the inverse temperature and $\lambda=\hbar\sqrt{2\pi\beta/m}$ the thermal wavelength. This expansion inverts into the virial expansion of the pressure in the two-dimensional density $\rho$, $p/(\rho kT)=1+\sum_{n\ge2}A_n\rho^{\,n-1}$ (for example $A_3=-2b_3/b_1^3+4b_2^2/b_1^4$), whose virial coefficients $A_n$ carry dimensions $\lambda^{2(n-1)}$ and a dimensionless part depending only on the statistics angle. Arovas, Schrieffer, Wilczek and Zee (1985) evaluated the second virial coefficient exactly by two methods (Aharonov–Bohm scattering and the two-anyon harmonic-oscillator spectrum): in the convention of the later literature, $A_2=\lambda^2[\frac{1}{4}-\frac{1}{2}(1-\alpha)^2]$ with $\alpha=|\theta|/\pi$ extended periodically with period $2\pi$. This is a simple periodic function that is nonanalytic at the bosonic values $\theta=0$ (mod $2\pi$), where it has a cusp (a jump of the first derivative), while it is smooth at the fermion point $\theta=\pi$.

For three or more anyons no exact thermodynamics is known, and the original paper already identified the obstructions. The many-anyon problem in a harmonic trap has an exactly solvable subset of levels whose energies are linear in $\theta$, but the remaining levels curve nonlinearly and have resisted closed-form solution since 1985; the cluster integrals additionally sum over relative-angle (winding) sectors — path-integral histories labeled by how many times, in units of $2\pi$, one particle encircles another — whose distributions have slowly decaying tails; the statistical interaction is long-ranged ($1/r$), so even the finiteness of the cluster coefficients in the thermodynamic limit is nontrivial; and perturbation in $\theta$ about the boson point is singular, the first-order term acting as a repulsive contact potential with nonanalytic $|\theta|$ effects at higher orders.

What is established in the literature: perturbation theory shows that no virial coefficient beyond the second contains a term of order $\theta$, i.e. the linear cusp of $A_2$ does not recur (Sen 1991; Comtet, McCabe and Ouvry 1991), and the $\alpha^2$ coefficients are known for all cluster coefficients (Dasnières de Veigy and Ouvry 1992; Emparan and Valle Basagoiti 1993); the three-anyon spectrum, and hence $A_3$, is exactly mirror-symmetric about the semion point (Sen 1992); all cluster and virial coefficients are finite (Myrheim and Olaussen 1993; Kristoffersen et al. 1998); and high-precision Monte Carlo values exist for $A_3$ and $A_4$, both smooth and well fitted by rapidly converging Fourier series in $\theta$, starting $A_3=\lambda^4[\frac{1}{36}+\sin^2\theta/12\pi^2-(1.652\pm0.012)\times10^{-5}\sin^4\theta+\cdots]$. The conjecture that every $A_n$ with $n\ge3$ is analytic in $\theta$ is, however, unproven, no closed form exists for any $n\ge3$, and $A_5$ and beyond have never been computed.

A separate but linked issue concerns dense anyon matter. The average-field (mean-field) approximation replaces the discrete statistical fluxes attached to the particles by a uniform effective magnetic field $\bar B=\nu\rho\Phi_0$ ($\nu=\theta/\pi$, $\Phi_0$ the flux quantum), under which the anyon gas becomes a gas of free bosons or fermions in a magnetic field; this is the basis of the anyon-superconductivity predictions (Chen, Wilczek, Witten and Halperin 1989). The approximation reproduces the leading density dependence but violates the exact periodicity in $\theta$ (it does not interpolate correctly between the Bose and Fermi endpoints), and its domain of validity is not quantitatively known; rigorous control exists only near the bosonic and fermionic limits or for regularized anyons. The virial expansion itself cannot test the low-temperature, high-density predictions because its expansion parameter $\lambda^2\rho$ diverges there. The question below therefore asks for the exact (or certified) higher virial coefficients and their regularity in the dilute regime where the virial expansion applies; the average-field issues described in this paragraph are related background rather than part of the requested determination.

Problem Statement

For the ideal (hard-core) Abelian anyon gas in two dimensions, determine the statistical mechanics beyond the exactly known second virial coefficient $A_2$, i.e. the low-density equation of state as a function of density, temperature, and statistics angle $\theta$: (1) determine the higher virial coefficients $A_n(\theta)$ for $n\ge3$ (equivalently, the cluster coefficients $b_n$ through the standard inversion of the cluster expansion), respecting the exact structural constraints: periodicity $\theta\to\theta+2\pi$, the mirror symmetry $A_3(\theta)=A_3(\pi-\theta)$, and the perturbatively known coefficients at the boson and fermion points (Sen 1991; Dasnières de Veigy and Ouvry 1992; Emparan and Valle Basagoiti 1993); and (2) settle the regularity question raised by the cusps of $A_2$: prove or refute the conjecture that every virial coefficient beyond the second is an analytic function of $\theta$, in particular at the bosonic values where $A_2$ is cusped, beyond the already established absence of first-order terms.

References

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