1. Introduction
In two-dimensional (2D) fluid turbulence, energy at small scales can transport to large scales known as inverse energy cascade [1–4]. This process involves formations of large scale vortex patterns. Onsager explained the formation of large scale structures by studying equilibrium statistical mechanics of point vortices in a bounded domain. The macroscopic vortex structure is associated with the clustering of like-sign point vortices at negative temperatures [1, 5]. These coherent large structures occur in various systems. Examples are the Great Red Spot in Jupiter’s atmosphere [6], giant vortex clusters in atomic Bose–Einstein condensates (BECs) [7–9], and vortex clustering in quantum fluids of exciton–polaritons [10].
The clustering phenomena of vortices have attracted much attention [11–38]. For a given 2D domain, searching for the maximum entropy clustered vortex state is at the center of investigations. For circularly symmetric domains, previous studies on neutral vortex systems focus on zero angular momentum cases [18, 25, 27, 32, 33]. The role of finite angular momentum in the formation of clustered states in a neutral vortex system remains less well-explored.
In this paper, we study axis-symmetric clustered vortex states through the mean field approach. The mean field theory to describe the formation of negative temperature clustered vortex states was formulated systematically by Joyce and Montgomery [39]. The mean field equations, which were obtained by maximizing the entropy of the vortex system, are essential for analyzing possible clustered states. We consider a neutral vortex system consisting of an equal number of positive and negative vortices confined to a disc. For a given positive vortex number N+ and a negative vortex number N−, clustered vortex states are specified by energy E and angular momentum L or their conjugate variables inverse temperature β(E, L) and rotation frequency ω(E, L). We find that in the limit β → 0, ω → ∞ while keeping βω finite, positive and negative vortex density distributions are Gaussian distributions centered at the origin and edge, respectively. For rotation-free states (ω = 0), we find asymptotically exact positive and negative vortex density distributions at large energies. In particular, the one maximized on the edge provides a new exact solution to the mean field equations for the corresponding chiral vortex system. The lower bound of β is obtained from the validity condition of the asymptotically exact solutions at high energies, above which rotation-free clustered states exist. To analyze clustered states closed to the uniform state at low energies, we generalized the perturbation theory, which was initially developed for chiral systems [17], to the neutral case. Using this perturbation theory we find the critical value of β for the onset of the rotation-free clustered vortex state, providing an upper bound of β.
2. Model
The point-vortex model describes the dynamics of well-separated quantum vortices in a superfluid at low temperature [40], 2D classical inviscid, incompressible fluids [21, 41] and guiding-center plasma [39]. Negative temperature states occur due to the bounded phase space of a 2D confined point vortex system. Above a certain energy, the number of available states decreases as the function of energy and consequently, the system becomes more ordered as energy increases [1].
We consider a system consisting of a large number of point vortices confined to a uniform disc of radius R. The system is neutral and contains N+ positive vortices and N− = N+ negative vortices. The Hamiltonian is [42]1 ) is measured 8n unit E0 = ρmaκ2/4π, where ρ is the superfluid density, κ = h/ma is the circulation quantum and ma is the atomic mass. In this unit, κi = ±1/N± and the 1/N± scaling gives a well-defined mean field limit [43, 44]. For a vortex at position rj, its image locates at ${\bar{{\boldsymbol{r}}}}_{j}={R}^{2}{{\boldsymbol{r}}}_{j}/| {{\boldsymbol{r}}}_{j}{| }^{2}$ to ensure that the fluid velocity normal to the boundary vanishes. The Hamiltonian (1 ) has rotational SO(2) symmetry due to the disc geometry and ${{\mathbb{Z}}}_{2}$ symmetry (invariant under κi → −κi). Hereafter we set R = 1 without losing generality.
$\begin{eqnarray}H=-\displaystyle \sum _{i\ne j}{\kappa }_{i}{\kappa }_{j}\mathrm{log}| {{\boldsymbol{r}}}_{i}-{{\boldsymbol{r}}}_{j}| +\displaystyle \sum _{i,j}{\kappa }_{i}{\kappa }_{j}\mathrm{log}\left|({{\boldsymbol{r}}}_{i}-{\bar{{\boldsymbol{r}}}}_{j})\displaystyle \frac{| {{\boldsymbol{r}}}_{j}| }{R}\right|.\end{eqnarray}$
In a BEC, the Hamiltonian equation (To investigate formations of large-scale clustered patterns, it is necessary to consider the continuous effective Hamiltonian in the large N± limit [39]:1 ).
$\begin{eqnarray}{H}_{\mathrm{eff}}=\displaystyle \frac{1}{2}\int {{\rm{d}}}^{2}{\boldsymbol{r}}{{\rm{d}}}^{2}{\boldsymbol{r}}^{\prime} \,\sigma ({\boldsymbol{r}})\phi ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\sigma ({\boldsymbol{r}}^{\prime} ).\end{eqnarray}$
Here σ(r) ≡ n+(r) − n−(r) is the vorticity field, $\begin{eqnarray}{n}_{\pm }({\boldsymbol{r}})\equiv \displaystyle \frac{1}{{N}_{\pm }}\displaystyle \sum _{i}\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{i}^{\pm })\end{eqnarray}$
is the local density of positive (negative) vortices, and ${{\boldsymbol{r}}}_{i}^{\pm }$ is the position of the vortex i with circulation ±1/N±. The vortex densities n± satisfy the normalization condition $\begin{eqnarray}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{n}_{\pm }=1.\end{eqnarray}$
The Green’s function $\phi ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )$ satisfies ${{\rm{\nabla }}}^{2}\phi ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\,=-4\pi \delta ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )$. Here $\phi ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )=0$ on the boundary (∣r∣ = 1), and $\phi ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\sim -2\mathrm{log}| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} | $ as $| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} | \to 0$ [45]. The stream function $\begin{eqnarray}\psi ({\boldsymbol{r}})\equiv \int {{\rm{d}}}^{2}{\boldsymbol{r}}^{\prime} \phi ({\boldsymbol{r}},{\boldsymbol{r}}^{\prime} )\sigma ({\boldsymbol{r}}^{\prime} ),\end{eqnarray}$
satisfies the Poisson equation $\begin{eqnarray}{{\rm{\nabla }}}^{2}\psi =-4\pi \sigma ({\boldsymbol{r}})\end{eqnarray}$
with the boundary condition ψ(r = 1, θ) = C. Here C is a constant. Recall that the radial velocity $\begin{eqnarray}{u}_{r}=\displaystyle \frac{1}{r}\displaystyle \frac{\partial \psi }{\partial \theta }.\end{eqnarray}$
This boundary condition ensures that there is no flow across the boundary of the domain. Without losing generality, we choose C = 0, which is equivalent to including image terms in equation (For a rotationally symmetric domain, energy
$\begin{eqnarray}E=\displaystyle \frac{1}{2}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\sigma \psi \end{eqnarray}$
and angular momentum $\begin{eqnarray}L=\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{r}^{2}\sigma \end{eqnarray}$
are conserved quantities.The most probable density distribution is given by maximizing the entropy function
$\begin{eqnarray}S=-\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{n}_{+}\mathrm{log}{n}_{+}-\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{n}_{-}\mathrm{log}{n}_{-},\end{eqnarray}$
at fixed values of N+, N−, E and L [39]. From the variational equation $\begin{eqnarray}\delta S-\beta \delta E-\alpha \delta L-{\mu }_{+}\delta {N}_{+}/{N}_{+}-{\mu }_{-}\delta {N}_{-}/{N}_{-}=0,\end{eqnarray}$
we obtain $\begin{eqnarray}{n}_{\pm }({\boldsymbol{r}})=\exp \left[\mp \beta \psi ({\boldsymbol{r}})\mp \alpha {r}^{2}+{\gamma }_{\pm }\right],\end{eqnarray}$
where β, α and μ± are Lagrange multipliers and γ± = −μ± −1. The parameters β, ω ≡ α/β and μ± have the interpretation of inverse temperature, rotation frequency and chemical potentials, respectively.3. Onset of clustering
In this section, we analyze the possible stable large scale coherent structures described by equation (6 ) and equation (12 ) near the uniform state. Here we generalized the method which was developed for analyzing chiral vortex matter [17], to the neutral case.
Let us start at a solution n± of equation (12 ) at energy E and angular momentum L, and consider a nearby solution n± + δn± at E + δE and L + δL. The corresponding changes are
$\begin{eqnarray}0=\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\delta {n}_{+},\end{eqnarray}$
$\begin{eqnarray}0=\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\delta {n}_{-},\end{eqnarray}$
$\begin{eqnarray}\delta E=\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\psi \delta \sigma +\displaystyle \frac{1}{2}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\delta \psi \delta \sigma ,\end{eqnarray}$
$\begin{eqnarray}\delta L=\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{r}^{2}\delta \sigma .\,\end{eqnarray}$
To leading order, we obtain17 ) into equations (13 –16 ), we have6 ) gives us
$\begin{eqnarray}\begin{array}{rcl}\delta {n}_{+} & \simeq & {n}_{+}(-\psi \delta \beta -\beta \delta \psi +\delta {\gamma }_{+}-{r}^{2}\delta \alpha ),\\ \delta {n}_{-} & \simeq & {n}_{-}(\psi \delta \beta +\beta \delta \psi +\delta {\gamma }_{-}+{r}^{2}\delta \alpha ),\end{array}\end{eqnarray}$
where δγ−, δγ+, δβ and δα are changes of Lagrange multipliers. Plugging equation ( $\begin{eqnarray}{ \mathcal Q }\delta {\boldsymbol{\mu }}=-\delta {\boldsymbol{T}}+\beta {\boldsymbol{V}}\delta \psi ,\end{eqnarray}$
where $\delta {\boldsymbol{\mu }}={\left(\delta \beta ,\delta {\gamma }_{+},\delta {\gamma }_{-,}\delta \alpha \right)}^{{\rm{T}}}$, $\delta {\boldsymbol{T}}={\left(\mathrm{0,0},\delta L,\delta E\right)}^{{\rm{T}}}$, ${\boldsymbol{V}}\delta \psi =-\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{\left({n}_{+},{n}_{-},{{nr}}^{2},n\psi \right)}^{{\rm{T}}}\delta \psi $, $\begin{eqnarray}{ \mathcal Q }\equiv \int {{\rm{d}}}^{2}{\boldsymbol{r}}\left(\begin{array}{cccc}{n}_{+}\psi & -{n}_{+} & 0 & {n}_{+}{r}^{2}\\ {n}_{-}\psi & 0 & {n}_{-} & {n}_{-}{r}^{2}\\ n\psi {r}^{2} & -{n}_{+}{r}^{2} & {n}_{-}{r}^{2} & {{nr}}^{4}\\ n{\psi }^{2} & -{n}_{+}\psi & {n}_{-}\psi & n\psi {r}^{2}\end{array}\right),\end{eqnarray}$
and n = n+ + n− is the total density. Variation of equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}^{2}\delta \psi & = & -4\pi \left(\delta {n}_{+}-\delta {n}_{-}\right)\\ & = & 4\pi (\psi n\delta \beta +\beta n\delta \psi -{n}_{+}\delta {\gamma }_{+}+{n}_{-}\delta {\gamma }_{-}+{{nr}}^{2}\delta \alpha ).\end{array}\end{eqnarray}$
Our aim is to find stable clustered states which emerge from the homogeneous state n− = n+ = n0 = 1/π. For the homogeneous state, σ = 0, ψ = 0, α = 0, L = 0 and E = 0. We assume that δα is in the same order as δψ and from equation (18 ) we obtain20 ) becomes a zero mode equation of the operator ${ \mathcal L }$. The onset of large scale vortex clusters occurs if equation (25 ) has non-zero solutions. The value of β is undefined in the homogeneous phase within our mean field approach and depends on the mode developing from the uniform state. Since the operator ${ \mathcal L }$ is defined on a disc with the Dirichlet boundary condition, it is natural to decompose equation (25 ) in azimuthal Fourier harmonics ψs which is characterized by the mode number s and satisfies ∂2ψs/∂θ2 = −s2ψs:
$\begin{eqnarray}0=\delta {\gamma }_{+}+\delta {\gamma }_{-},\,\end{eqnarray}$
$\begin{eqnarray}\,0=\delta {\gamma }_{+}-\delta {\gamma }_{-}-\delta \alpha -2\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\delta \psi ,\,\end{eqnarray}$
$\begin{eqnarray}\delta L=\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}(1-2{r}^{2})\delta \psi -\displaystyle \frac{1}{6}\delta \alpha ,\,\end{eqnarray}$
$\begin{eqnarray}\delta E=\displaystyle \frac{1}{2}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,\delta \psi \delta \sigma .\,\end{eqnarray}$
Let us introduce operator ${ \mathcal L }$: $\begin{eqnarray}\begin{array}{l}{ \mathcal L }\delta \psi \equiv {{\rm{\nabla }}}^{2}\delta \psi -8\pi {n}_{0}\left[\beta \delta \psi -\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\delta \psi \right.\\ \quad -3\beta {n}_{0}(1-2{r}^{2})\int {{\rm{d}}}^{2}{\boldsymbol{r}}(1-2{r}^{2})\delta \psi \\ \quad \left.+3\left)1-2{r}^{2}\right(\delta L\Space{0ex}{1em}{0ex}\right]=0.\end{array}\end{eqnarray}$
Then equation ( $\begin{eqnarray}\delta \psi =\displaystyle \sum _{s}\epsilon {f}_{s}{\psi }_{s}(r,\theta ),\end{eqnarray}$
where ε ≪ 1 is a small amplitude and fs is the mode coefficient. Then each mode satisfies $\begin{eqnarray}{ \mathcal L }{\psi }_{s}(r,\theta )=0,\end{eqnarray}$
where ψs(r, θ) satisfies the boundary condition ψs(r = 1, θ) = 0. We denote δL = L0ε, δE = E0ε2 and δα = εβω.We find that27 ) with31 ) and (32 ) combined with the Dirichlet boundary condition
$\begin{eqnarray}{\psi }_{s}(r,\theta )={c}_{s}{{\rm{J}}}_{s}({kr})\cos (s\theta )+{b}_{s}+{{ar}}^{2}\end{eqnarray}$
solves equation ( $\begin{eqnarray}a=-\omega =-2{c}_{s}\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{{\rm{J}}}_{s}({kr})\cos (s\theta )\end{eqnarray}$
and $\begin{eqnarray}\beta =-\displaystyle \frac{{k}^{2}}{8\pi {n}_{0}}.\end{eqnarray}$
Here Js(r) is the Bessel function of the first kind. Consistently, $\begin{eqnarray}{L}_{0}=\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,(1-2{r}^{2})\left[{\psi }_{s}(r,\theta )+\omega {r}^{2}\right],\end{eqnarray}$
$\begin{eqnarray}{E}_{0}=-\displaystyle \frac{1}{8\pi }\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{\psi }_{s}(r,\theta ){{\rm{\nabla }}}^{2}{\psi }_{s}(r,\theta ).\,\end{eqnarray}$
For given L0 and E0, the parameters cs, k, and bs are determined by equations ( $\begin{eqnarray}{\psi }_{s}(r=1,\theta )={c}_{s}{{\rm{J}}}_{s}(k)\cos (s\theta )+{b}_{s}+a=0.\end{eqnarray}$
The single-valueless of the stream function requires that s has to be an integer, namely, $s\in {\mathbb{Z}}$.For s ≠ 0, L0 = 0, a = − ω = 0, bs = 0,
$\begin{eqnarray}{c}_{s}^{2}=\displaystyle \frac{16{E}_{0}}{k\left[k{{\rm{J}}}_{s-1}(k){}^{2}-2s{{\rm{J}}}_{s}(k){{\rm{J}}}_{s-1}(k)+k{{\rm{J}}}_{s}(k){}^{2}\right]},\end{eqnarray}$
and k = js,m, where js,m is the mth zero of the Bessel function of the first kind Js(r).For s = 0,
$\begin{eqnarray}a=-2{c}_{0}\beta {n}_{0}\int {{\rm{d}}}^{2}{\boldsymbol{r}}\,{{\rm{J}}}_{0}({kr})=\displaystyle \frac{1}{2}{c}_{0}k{{\rm{J}}}_{1}(k),\end{eqnarray}$
$\begin{eqnarray}{b}_{0}=-a-{c}_{0}{{\rm{J}}}_{0}(k).\,\end{eqnarray}$
For given E0 and L0, c0 and k are determined by
$\begin{eqnarray}{E}_{0}=\displaystyle \frac{1}{8}{c}_{0}^{2}{k}^{2}\left[{{\rm{J}}}_{0}(k){}^{2}+{{\rm{J}}}_{1}(k){}^{2}\right],\end{eqnarray}$
$\begin{eqnarray}{L}_{0}=-\displaystyle \frac{1}{4}{c}_{0}k{{\rm{J}}}_{3}(k).\,\end{eqnarray}$
It is useful to introduce $\begin{eqnarray}{\rm{\Gamma }}(k)\equiv \displaystyle \frac{{\left(\delta L\right)}^{2}}{\delta E}=\displaystyle \frac{{L}_{0}^{2}}{{E}_{0}}=\displaystyle \frac{{{\rm{J}}}_{3}^{2}(k)}{2\left[{{\rm{J}}}_{0}(k){}^{2}+{{\rm{J}}}_{1}(k){}^{2}\right]}\end{eqnarray}$
as a control parameter.The ratio Γ(k) reaches its maximum value at k = k* with j1,1 < k* < j2,1 (see figure 1). For a given Γ0 < Γ(k*), there are more than one value of kc such that Γ(kc) = Γ0. Guided by the maximum entropy principle, the minimal value of kc corresponds to the equilibrium state. For k → 0, E0 → 0, L0 → 0, this mode describes the uniform state.
Figure 1. Γ(k) as a function of k. The maximum value of Γ(k) is reached at k = k* and j1,1 < k* < j2,1. |
The modes s ≠ 0 break SO(2) symmetry and the maximum entropy state for given energy is the clustered vortex dipole state which corresponds to the s = 1 mode [27]. This clustered vortex dipole state has been recently realized in BEC experiments [7]. In this paper, we focus on states related to the s = 0 mode.
4. Axis-symmetric clustered states
In this section, we present some (asymptomatically) exact results on axis-symmetric neutral vortex clusters. For axis-symmetric states, the boundary condition equation (7 ) which is imposed by the most relevant physical condition is fulfilled automatically.
4.1. Gaussian vortex states
Let us first consider β → 0. For finite ω, vortex distributions n± must be uniform. However, when ω → ∞ simultaneously such that α = ωβ is finite, non-trivial distributions can occur. In this special limit, the vortex densities have the profile of Gaussian distribution:
$\begin{eqnarray}{n}_{+}(r)=-\displaystyle \frac{\alpha }{\pi \left[\exp (-\alpha )-1\right]}\exp \left(-\alpha {r}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}{n}_{-}(r)=\displaystyle \frac{\alpha }{\pi \left[\exp (\alpha )-1\right]}\exp \left(\alpha {r}^{2}\right),\end{eqnarray}$
where α ∈ (−∞, ∞).The corresponding stream function reads
$\begin{eqnarray}\begin{array}{rcl}\psi (r) & = & \displaystyle \frac{\exp (\alpha )\mathrm{Ei}\left(-{r}^{2}\alpha \right)+\mathrm{Ei}\left({r}^{2}\alpha \right)-2\left[\exp (\alpha )+1\right]\mathrm{log}r}{\exp (\alpha )-1}\\ & & -\displaystyle \frac{\exp (\alpha )\mathrm{Ei}(-\alpha )+\mathrm{Ei}(\alpha )}{\exp (\alpha )-1},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\mathrm{Ei}(x)=-{\int }_{-x}^{\infty }\displaystyle \frac{\exp (-t)}{t}\,{\rm{d}}{t}\end{eqnarray}$
is the exponential integral function. The stream function satisfies ψ(r = 1) = 0 and dψ/dr∣r=1 = 0.The angular momentum is
$\begin{eqnarray}L={\alpha }^{-1}\left[2-\alpha \coth \left(\displaystyle \frac{\alpha }{2}\right)\right].\end{eqnarray}$
It is easy to see that L ≤ 1. Figures 2(a)–(b) show typical vortex densities for different values of α. Figures 2(c)–(d) show angular momentum and energy as functions of α. Note that the Gaussian state is available in the chiral vortex system as well [17].
Figure 2. Vortex densities for α = 1 (a) and α = 6 (b). The angular momentum and the energy as functions of α are shown in (c) and (d), respectively. |
4.2. Rotation-free vortex states
In this subsection, we consider clustered vortex states for ω = 0 and finite β < 0.
4.2.1. Onset of axis-symmetric clustered states
Closed to the uniform state, the clustered states can be analyzed using the formalism developed in section 3 . The polar angle θ-independent zero modes s = 0 carry non-zero angular momentum. For s = 0 modes, the rotation-free condition a = −ω = c0kJ1(k)/2 = 0 [see equation (35 )] requires that k = j1,m, where j1,m is the mth zero of the Bessel function of the first kind J1(r). These modes occur at $\beta ={\beta }_{1,m}=-{j}_{1,m}^{2}/8\pi {n}_{0}$ and break ${{\mathbb{Z}}}_{2}$ symmetry. The m = 1 mode starts to emerge at β = βt = β1,1 ≃ −1.835 and has the highest statistical weight among the rotation-free modes (ω = 0):
$\begin{eqnarray}{\psi }_{0}(r)={c}_{0}{{\rm{J}}}_{0}({j}_{\mathrm{1,1}}r)+{b}_{0},\end{eqnarray}$
where ${c}_{0}=\pm 2\sqrt{2}{E}_{0}/| {{\rm{J}}}_{0}({j}_{\mathrm{1,1}})| {j}_{\mathrm{1,1}}$ and b0 = −c0J0(j1,1). For this mode, Γ(j1,1) ∼ 0.545. Since Γ(j1,1) > Γ(j1,m>1), Γ(j1,1) gives the upper bond of Γ for the rotation-free modes. The rotation-free axis-symmetric phase emerges from the uniform phase by varying angular momentum and energy such that Γ = Γ(j1,1).4.2.2. High energy configuration
All the rotation-free and axis-symmetry states satisfy
$\begin{eqnarray}\displaystyle \frac{1}{r}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}r\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\psi (r)=-4\pi \left[\exp (-\beta \psi (r)+{\gamma }_{+})\right.\quad \left.-\exp (\beta \psi (r)+{\gamma }_{-})\right].\end{eqnarray}$
The most relevant solution of equation (46 ) should be the nonlinear continuation of the zero mode ψ0 and describes the axis-symmetry equilibrium state with zero rotation frequency.
At large energies, vortices with opposite signs are well-separated and the overlap between n+ and n− can be neglected. In this limit, exact results are available. Let us assume that positive vortices are concentrated in the center of the disc and negative vortices are distributed along the edge of the disc. The density distribution of positive vortices near r = 0 can be obtained analytically by neglecting the influence of negative vortices:
$\begin{eqnarray}{n}_{+}(r)=\displaystyle \frac{4A}{{\left(2-\pi \beta {{Ar}}^{2}\right)}^{2}},\end{eqnarray}$
with the boundary conditions ψ(0) = 0 and $\psi ^{\prime} (0)=0$ [17]. Here $A={\left[\pi (1-\beta /{\beta }_{* })\right]}^{-1}$ is fixed by the normalization condition of n+ and β* = −2. The supercondensation occurs at β = β*, involving point-like concentration of the positive vortices and the divergence of energy [17, 46].Near r = 1, we can neglect the influence of positive vortices and find the density distribution of negative vortices
$\begin{eqnarray}{n}_{-}(r)=\displaystyle \frac{2(2/\beta -1){\left(1-\beta \right)}^{2}{r}^{-2\beta }}{\pi \beta {\left({r}^{-2\beta +2}+1-2/\beta \right)}^{2}},\end{eqnarray}$
where the boundary conditions are ψ(1) = 0 and $\psi ^{\prime} (1)=0$.Note that ψ(0) and ψ(1) can be chosen as arbitrary constants and here we choose them to be zero for convenience. The boundary condition $\psi ^{\prime} (0)=0$ ensures that $n{{\prime} }_{+}(0)=0$ and n+ has no singular behavior near r = 0. Similarly, the boundary condition $\psi ^{\prime} (1)=0$ implies that $n{{\prime} }_{-}(1)=0$ and the absence of singular behavior of n− near r = 1. As approximations of vortex densities at large energies, equations (47 ) and (48 ) should be evaluated for β* < β. Combining the critical value of β at which the onset of clustering occurs, we obtain the parameter regime for the rotation-free clustered vortex state:
$\begin{eqnarray}{\beta }_{* }\lt \beta \lt {\beta }_{t}.\end{eqnarray}$
Figure 3 shows the vortex density distributions at high energies.
Figure 3. Vortex density distributions at high energies. The densities of positive vortices (a) and negative vortices (b) are evaluated via equations ( |
In the deep clustered state, positive vortices are concentrated in a small region and the total energy is contributed dominantly from positive vortices. So as β → β*,
$\begin{eqnarray}E\backsimeq -\displaystyle \frac{2}{{\beta }^{2}}\left[\mathrm{log}\left(1-\displaystyle \frac{\beta }{{\beta }_{* }}\right)-\displaystyle \frac{\beta }{2}\right].\end{eqnarray}$
At large energies, the angular momentum is
$\begin{eqnarray}\begin{array}{l}L=\left|\int {{\rm{d}}}^{2}{\boldsymbol{r}}{r}^{2}[{n}_{+}({\boldsymbol{r}})-{n}_{-}({\boldsymbol{r}})]\right|\\ \quad \simeq \left|\int {{\rm{d}}}^{2}{\boldsymbol{r}}{r}^{2}\left[\displaystyle \frac{4A}{{\left(2-\pi \beta {{Ar}}^{2}\right)}^{2}}\right.\right.\\ \quad \left.\left.-\displaystyle \frac{2(2/\beta -1){\left(1-\beta \right)}^{2}{r}^{-2\beta }}{\pi \beta {\left({r}^{-2\beta +2}+1-2/\beta \right)}^{2}}\right]\right|,\end{array}\end{eqnarray}$
and as β → β*, $L\to {L}_{\max }$ with $\begin{eqnarray}{L}_{\max }=\left|\int {{\rm{d}}}^{2}{\boldsymbol{r}}{r}^{2}\left\{\delta ({\boldsymbol{r}})-18{r}^{4}{\left[\pi {\left(2+{r}^{6}\right)}^{2}\right]}^{-1}\right\}\right|\simeq 0.705.\end{eqnarray}$
5. Exact results for chiral vortex clusters
As stated in the previous section, equation (48 ) is the exact solution to equation (46 ), provided that n+ is neglected. Hence equation (48 ) provides an exact vortex density distribution for a chiral system, which is distinct from the well-known exact distribution. In this section, we make a summary of relevant exact results and make a comparison between our findings and the known distribution.
For a rotation-free (ω = 0) and axis-symmetric chiral system, equation (46 ) becomes53 ) and (54 ), which is equation (47 ):55 ) exhibits distinct behaviors in different parameter regimes. The vortices accumulate around the edge for 0 < β while for −2 < β < 0 the vortices are center-concentrated (see figure 4). Note that in some literature, equation (53 ) does not have the prefactor 4π and hence the solution looks slightly different [35, 47].
$\begin{eqnarray}\displaystyle \frac{1}{r}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}r\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\psi (r)=-4\pi n(r),\,\end{eqnarray}$
$\begin{eqnarray}n(r)=\exp (-\beta \psi +\gamma ).\end{eqnarray}$
There is a known exact solution to equations ( $\begin{eqnarray}n(r)=\displaystyle \frac{2(\beta +2)}{\pi {\left(\beta -\beta {r}^{2}+2\right)}^{2}},\,\end{eqnarray}$
$\begin{eqnarray}\psi (r)=-\displaystyle \frac{2}{\beta }\mathrm{log}\displaystyle \frac{\beta +2}{\beta (1-{r}^{2})+2},\quad \psi (0)=0,\psi ^{\prime} (0)=0,\,\end{eqnarray}$
$\begin{eqnarray}\psi (r)=\displaystyle \frac{2}{\beta }\mathrm{log}\left(1+\displaystyle \frac{\beta (1-{r}^{2})}{2}\right),\quad \psi (1)=0,\psi ^{\prime} (0)=0.\,\end{eqnarray}$
This solution is valid for β > −2. The corresponding stream function could be different depending on the boundary conditions. The vortex density equation (
Figure 4. Typical profiles of the vortex density distribution described by equation ( |
Distinct from equation (55 ), our finding is equation (48 ):58 ) holds for β < 1 and β ≠ 0. If requiring that $n^{\prime} (r=0)$ is finite, β < −1/2. For 0 < β < 1, equation (58 ) shows center-concentrated distribution and n(r → 0) → ∞ . For −1/2 < β < 0, the vortex density is singular at origin, namely $n^{\prime} (r\to 0)\to \infty $. Vortices distribute around the edge for β < −1/2. In contrast to the known exact solution equation (55 ), the distribution equation (58 ) is peaked on the boundary at a negative temperature and is maximized at the origin at a positive temperature. Figure 5 shows typical behaviors of the vortex density in these parameter regimes.
$\begin{eqnarray}n(r)=\displaystyle \frac{2(2/\beta -1){\left(1-\beta \right)}^{2}{r}^{-2\beta }}{\pi \beta {\left({r}^{-2\beta +2}+1-2/\beta \right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}\psi (r)=-\displaystyle \frac{2}{\beta }\mathrm{log}\left[\displaystyle \frac{2(\beta -1){r}^{\beta }}{(\beta -2){r}^{2\beta }+\beta {r}^{2}}\right],\end{eqnarray}$
with boundary conditions $\begin{eqnarray}\psi (1)=0,\quad \psi ^{\prime} (1)=0.\end{eqnarray}$
The solution equation (
Figure 5. Typical profiles of the vortex density distribution described by equation ( |
6. Conclusions
Axis-symmetric clustered vortex states for a neutral vortex system confined to a disc are investigated. Combining the perturbation theory near the uniform state and asymptotic analysis at high energies, we find the parameter regime for which the rotation-free states are supported. At large energies, the distributions of positive vortices and negative vortices are well-separated and the edge-concentrated part provides a new exact vortex density distribution for the corresponding chiral vortex system.
The onset of a non-axisymmetric vortex cluster in chiral vortex systems appears to proceed via a second-order phase transition [16, 17]. It would be interesting to investigate possible non-axisymmetric states for neutral systems carrying finite angular momentum. Thanks to the recent experimental advances [7–9], our work would motivate experimentally investigating axis-symmetric clustered phases in a homogeneous Bose–Einstein condensate trapped in cylindrically symmetric potentials. Due to the presence of conservation of angular momentum, axis-symmetric clustered phases are expected to have a longer lifetime than the giant vortex dipole state [7].