Abstract
There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \(\Gamma := q^2/k_BT=2\) the system is exactly solvable. In particular the \(k\)-point correlation can be written as a \(k \times k\) determinant, allowing the edge density to be computed to first order in \(\Gamma - 2\). A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from the interior. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for the interior and exterior directions. For general \(\Gamma \), a Gaussian fluctuation formula is used to study the large deviation form of the density for \(N\) large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.
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Acknowledgments
The work of P.W. and T.C. was supported by NSF DMS-1156636 and DMS-1206648. The work of P.F. was supported by the Australian Research Council through the DP ‘Characteristic polynomials in random matrix theory’. G.T. acknowledges financial support from Facultad de Ciencias, Uniandes.
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Dedicated to the memory of Bernard Jancovici (1930–2013) and his work on sum rules and exact solutions for Coulomb systems.
Appendices
Appendix 1
The purpose of this appendix is to derive (3.10).
According to the definitions, for general \(\Gamma \) in soft cylinder geometry
where \(\vec {r}_j = (x_j,y_j)\). Generalizing (2.11), we know that for \(\Gamma = 2\) [6]
where \( \rho _{(1)}(\vec {r})\) is given by (2.11) and
(the case \(l=2\) of (4.13)). Our task then is to compute some explicit multiple integrals.
Making use of the Fourier expansion (5.39), elementary calculations show
and
This reduces our task to analyzing certain one-dimensional sums in the large \(N\) limit.
The first sum in (5.47) is elementary, and we have
For the remaining sums, the leading and first order correction for large \(N\) can be obtained by making use of the trapezoidal rule
In this regards, the portion of the first summation in (5.46),
requires preliminary manipulation, since a literal application of (5.49) is not possible. This is due to the corresponding \(f(x)\) not being integrable about \(x=0\). Thus we write
where \(K = \Big [ W \sqrt{2 \pi \over \rho _b} \Big ]\).
With \(H_K\) denoting the harmonic numbers, it is a standard result that
The remaining sums in (5.50) can all be analyzed using (5.49). Doing this and combining with (5.51) shows
and
Substituting (5.52) in (5.46), (5.53) and (5.48) in (5.47), and using these results to evaluate the RHS of (5.45) gives (3.10).
Appendix 2
Consider the soft cylinder with leading order density profile in the \(y\)-direction \(\tilde{\rho _b} = \rho _b \chi _{0 < y < W} \). For large \(W\), \(n \in \mathbb {Z}^+\), we see that
A readily verifiable consequence is that to leading order
We observe that the RHS of (5.54), multiplied by the measure \(dy\), is independent of \(W\) if we scale \(y \mapsto Wy\), \(x \mapsto Wx\), \(\frac{1}{W} \tilde{M_2} \mapsto M_2\), where
Thus (5.38) follows, provided we can show that \(M_2\) has the evaluation (5.33).
For this latter task we observe from the explicit formula for the partition function implied by (2.8) that
Changing variables \(x_l \mapsto x_l / L\), \(y_l \mapsto y_l / L\) and setting \(W=L\) this reads
Thus we seek an independent computation of the LHS of (5.56).
To provide such a computation, we first observe
Next we note that scaling in disk geometry together with the expected universality of the leading large \(N\) behaviour of the partitions in disk and cylinder geometries implies that for large \(N\)
for some \(g(\Gamma )\). Substituting (5.57) and (5.58) in the LHS of (5.56) gives (5.33)
Appendix 3
In this appendix, we study the behavior of the density in the cylinder when \(y\rightarrow -\infty \) for finite \(N\) and \(W\), when \(\Gamma /2\) is an integer. We will consider first \(N\) and \(W\) as independent variables. Let \(\tilde{W}=\rho _b W^2\) and \(\tilde{y}=\rho _b W y\) be the rescaled lengths by the characteristic length \(1/(\rho _b W)\). Considerations leading to the configuration integral (3.19) can be extended to obtain the density profile [22]
with
where the sum runs over all partitions which include \(l\). If \(\tilde{y}\rightarrow -\infty \), then
To compute \(a_{(N-1)\Gamma /2}^\mathrm{c}\), one needs to consider in (5.60) all the partitions \(\mu \) with \(c_{\mu }^{(N)}(\Gamma /2)\ne 0\) and \(\mu _1=(N-1)\Gamma /2\). The partition \(\tilde{\mu }=(\mu _2,\mu _3,\ldots ,\mu _{N})\) is a partition of \(\Gamma (N-1)(N-2)/4\) with \(\Gamma (N-2)/2 \le \mu _2 \le \cdots \le {\mu }_N\), and due to a factorization property satisfied by the coefficients of the partitions [4], one has
Therefore \(\tilde{\mu }\) corresponds to a partition for a system with \(N-1\) particles (this is not surprising as taking \(y \rightarrow \infty \) effectively removes that particle; see a similar argument in [14]). Then
and using (3.20), this leads to
Now, consider the limit \(N\rightarrow \infty \), and \(\tilde{W}\rightarrow \infty \), but with \(N\) and \(\tilde{W}\) independent. Using the universal properties of the free energy (3.22), we have
Notice that in the difference \(F^{c}_{N,\Gamma }(\tilde{W})-F^{c}_{N-1,\Gamma }(\tilde{W})\), as \(\tilde{W}\) is kept fixed, the surface tension terms in (3.22) cancel out, leading to a next order correction of order \(\mathcal{O}(1/N)\) instead of a naively expected \(\mathcal{O}(1/\sqrt{N})\). In the scaled edge \(\tilde{W}=N\rightarrow \infty \) and \(\tilde{y}\mapsto N y\) this can be compared to (5.42). Indeed if one takes \(y\rightarrow -\infty \) in (5.42), then (5.65) is recovered. The \(o(1)\) term in (5.42) for \(y \rightarrow - \infty \) should be (5.43).
As an illustration of the results, for \(\Gamma =4\), Fig. 3 shows a plot of the numerically computed
for various values of \(N=\tilde{W}\) confirming the expected behavior as \(y\rightarrow -\infty \). In the plot, \(\tilde{\rho }_{(1)}^{N,c}\) denotes the right hand side of (5.42).
In Fig. 4, the value of the limit of \(\log (\rho _{(1)}^{N,c}(y)/\tilde{\rho }_{(1)}^{N,c}(\sqrt{N}y))+\beta f(\Gamma ,1)\) as \(y\rightarrow -\infty \) is plotted against \(1/N\), showing indeed a linear behavior as expected
Very similar figures are obtained for \(\Gamma =6\) and 8 (not shown). Doing a numerical regression of Fig. 4 provides an alternative way to obtain numerically \(g(\Gamma )=\beta f(\Gamma ,1)\), and verify the \(1/N\) finite size correction. Table 3 shows the values obtained for \(g(\Gamma )\) and the \(1/N\) correction for \(\Gamma =4,\) 6, 8, and compares them to the estimations of free energy per particle on the sphere [26] and the expected value \(\pi (1-2\Gamma )/6\) of the \(1/N\) correction. As this method for estimating the free energy per particle relies on fitting an expression with \(1/N\) corrections, it seems as equally reliable as the one used in [26] for the 2dOCP on the sphere when the universal \(\log N\) correction is subtracted to the free energy.
Similar considerations can be done for the soft disk. The density profile is [26]
with
The leading behavior of the density as \(r\rightarrow \infty \) is given by
Again, the coefficient \(a_{(N-1)\Gamma /2}^\mathrm{d}\) is related to the ratio of two partition functions with \(N\) and \(N-1\) particles
Using (3.1), we find
In the scaled edge, with \(r\mapsto \sqrt{N}r\) and \(\rho _b =1/\pi \), taking \(r\rightarrow \infty \) in (5.29) reproduces (5.71), but here the \(o(1)\) has non zero \(\mathcal{O}(1/\sqrt{N})\) corrections — except for \(\Gamma = 2\) when \(\beta \mu (\Gamma ,\rho _b)\) vanishes — as opposed to the soft cylinder geometry.
Appendix 4
In this appendix we present a detailed derivation of the exterior asymptotes of \(A(y)\) as in (4.10). From Proposition 1, \(A(y)\) can be written as a sum of four terms, each of which is analyzed separately below.
Lemma 9
The asymptotic expansion of \(A_{1}(y)\) outside the droplet is
Proof
This asymptotic expansion can be obtained by differentiating \(A_{1}(- |y|)\) with respect to \(|y|\), and integrating from \(|y|\) to \(\infty \), with the result
The second line is obtained by expanding the complementary error function in the integrand for large \(t\), and integrating by parts. The result follows by keeping the next to leading order term in the large \(|y|\) expansion of the first term. \(\square \)
This can be used to show the following.
Lemma 10
The leading order asymptote of \(A_{2}(y)\) outside the droplet is
Proof
Applying a sequence of integration by parts, we can rewrite \(A_{2}(y)\) in terms of \(A_{1}(y)\) as
Using the asymptotic expansion for \(A_{1}(-|y|)\) above, only the term with a pre-exponential factor of \(\mathcal {O}(y^{-2})\) remains. \(\square \)
Next, we consider the leading asymptote of \(A_{3}(y)\). This follows by a straightforward expansion for large \(-y \gg 1\).
Lemma 11
The leading asymptote of \(A_{3}(y)\) outside the droplet is
Proof
After replacing the error functions appearing in \(A_{3}(y)\) with their large \(|y|\) asymptotic expansions, this result follows by straightforward algebra.
Lemma 12
The asymptote of \(A_{4}(y)\) outside the droplet is
Proof
Using the fact that the integrand is symmetric in its arguments \(t_{1}\) and \(t_{2}\), we can rewrite \(A_{4}(y)\) for \(y < 0\) as
After a change of variables,
Integrating over \(t_{2}\) this reads
We can expand the last integral as an asymptotic series in \((|y| - t_{1})\). The leading term is \(2 (|y| - t_{1})/\sqrt{\pi }\), which, upon integrating with respect to \(t_{1}\), becomes
The first term in parentheses can be similarly developed as an asymptotic series. A change of variables \(x = t_{1} - |y|\), followed by a rescaling \(x = \xi /|y|\), makes the Gaussian factor \(\exp \left( - 2(x + |y|)^{2}\right) = e^{- 2 y^{2}} \exp \left( - \frac{2\xi ^{2}}{y^{2}} - 4 \xi \right) \). After a Laurent expansion in \((\xi /y)^{2}\), the integral becomes
The next term can be evaluated easily and its large distance asymptote reads
Combining (5.72), (5.73), and (5.74) gives the stated asymptote. \(\square \)
This exhaustive analysis demonstrates that the leading asymptote outside indeed arises from \(A_{3}(y)\), and moreover
Appendix 5
In this appendix we present a more detailed proof of equation (5.14) in Lemma 5. A direct computation of the LHS for the antisymmetric parts of \(A_{1}(y)\), \(A_{2}(y)\) and \(A_{3}(y)\) gives
For \(A_{4}(y)\), we write the LHS as \(e^{-2y^{2}}\partial _{y}\left( e^{2 y^{2}} \partial _{y} A_{a,4}(y)\right) \), and carry out the operations in the sequence implied. First, the antisymmetric part must be written in a suitable form. Taking advantage of the symmetry of the integrand and changing variables, the double integral can be written as
Using the fact that \(\lim _{y \rightarrow \infty }A_{4}(y) = 0\), this can be written equivalently as
and thus
From this, we apply the LHS to get
Combining this with (5.76) proves the lemma.
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Can, T., Forrester, P.J., Téllez, G. et al. Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions. J Stat Phys 158, 1147–1180 (2015). https://doi.org/10.1007/s10955-014-1152-2
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DOI: https://doi.org/10.1007/s10955-014-1152-2