Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions

# Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions

T. Can111 Department of Physics, University of Chicago, 929 57th St, Chicago, IL 60637 , P. J. Forrester222 Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia , G. Téllez333 Departamento de Física, Universidad de Los Andes, Bogotá, Colombia , P. Wiegmann444 Department of Physics, University of Chicago, 929 57th St, Chicago, IL 60637
July 4, 2019
###### 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 analytic 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 the system is exactly solvable. In particular the -point correlation can be written as a determinant, allowing the edge density to be computed to first order in . 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 inside the plasma. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for into the plasma than for outside. For general , a Gaussian fluctuation formula is used to study the large deviation form of the density for 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.

Dedicated to the memory of Bernard Jancovici (1930–2013) and his work on sum rules and exact solutions for Coulomb systems.

## 1 Introduction

In the theory of the fractional quantum Hall effect the so-called Laughlin states are trial wave functions in a two-dimensional domain of the form

 1CNN∏l=1f(zl)∏1≤j

Here is even (odd) for bosonic (fermionic) states, and furthermore determines the filling fraction of the lowest Landau level according to .

For planar geometry in the symmetric gauge

 f(z)=e−|z|2/4l2B,u(z)=z, (1.2)

where is the magnetic length. For cylinder geometry, with axis along the -axis and perimeter , in the Landau gauge

 f(z)=e−y2/2l2B,u(z)=e2πiz/W. (1.3)

Below we set the units of length so that . We refer to the wave function (1.1) in the case (1.2) by , and in the case (1.3) by .

Our primary interest in this paper is in the particle density

 ρ(1)(z;m):=N∫Ωdx2dy2⋯∫ΩdxNdyN|ΨN(z,z2,…,zN)|2. (1.4)

To leading order, in the planar geometry specified by (1.2), , while in the cylinder geometry specified by (1.3), , where , . Here for true, and otherwise. These behaviours are most easily seen by appealing to an interpretation of in terms of the Boltzmann factor for the classical two-dimensional one-component plasma; see Section 2. On the boundary of the leading support there is a non-trivial double layer, or overshoot, behaviour characterized by a local maximum in the density [8, 7, 19, 25, 28, 27], and it is our aim to undertake a study of some of its analytic properties in the limit. It turns out that thermodynamic quantities of the plasma, such as the free energy and surface tension, appear in the associated asymptotic forms, so it is necessary to first undertake a study of the thermodynamic properties of the plasma, which we do in Section 3.

In particular, in Section 3 we pool together knowledge from previous studies to specify as many terms as possible in the large expansion of the free energy. The coupling constant in the plasma is (see below (2.5)). In terms of quantities in (1.1) we have . Unlike , the coupling is naturally a continuous variable. The dependence on of the resulting expressions are tested and illustrated by a combination of exact analytic, and exact numerical results. In relation to exact analytic results, the case , which in the interpretation (1.1) corresponds to free fermions in a magnetic field in the lowest Landau level, is exactly solvable for both planar [15] and cylinder [6] geometry. Knowledge of the exact one and two-point correlations can be used to expand the free energy to first order in . And for , 6 and 8 expansion methods of the products of differences in (1.1) based on Jack polynomials (see Section 3.3) can be used to provide exact numerical data up to .

Our study of the edge density begins in Section 4. Following the lead of the earlier work of Jancovici [16] in the bulk, knowledge of the exact one, two, and three-point correlations in the case of the planar geometry for was recently used [26] to calculate the exact form of the density to first order in . We provide its form in the case that the coordinates are centred on the boundary of the leading support for finite , and we show too that the same analytic expression results by computing the edge scaling of the density computed to first order in for cylinder geometry. Moreover, the asymptotic behaviour into and outside the plasma can determined, and it is found the deviation from the leading order (step function) value is different in the two cases. The results of this section have been reported in a Letter by the present authors [5], which furthermore casts them in the context of the Laughlin droplet interpretation.

In Section 5 we study the large deviation form of the density outside of the leading support, for large but finite. Our main tool here is to express (1.4) in terms of the characteristic function for the distribution of a certain linear statistic, then to compute its large form by using a Gaussian fluctuation formula. By an appropriate scaling of this expression we obtain a prediction for the asymptotic decay of the edge density in the region outside of the leading support for general .

## 2 Plasma Viewpoint

The observation that the absolute value squared of the Laughlin trial wave functions for the fractional quantum Hall effect have an interpretation as the Boltzmann factor for certain two-dimensional one component plasma systems was already made in the original paper of Laughlin [18]. Generally the refers to a system of mobile point particles of the same charge and a smeared out neutralising background, with the domain a two-dimensional surface. The charges interact via the solution of the Poisson equation on the surface. Thus for the plane

 Φ(→r,→r′)=−log(|→r−→r′|/l), (2.1)

where is an arbitrary length scale (we take ), while for periodic bounday conditions in the -direction, period (or equivalently a cylinder of circumference length )

 Φ(→r,→r′)=−log(|sin(π(x−x′+i(y−y′))/W)(Wπ)), (2.2)

With the Boltzmann factor for a classical system is , where is the total potential energy. As detailed in [11, §1.4.1], , where corresponds to the particle-particle interaction, to the particle-background interaction, and to the background-background interaction. In the case that the domain is a plane, with the smeared out neutralizing background a disk at the origin of radius , the particles couple to the background via a harmonic potential towards the origin. Explicitly one has

 U1=−∑1≤j

and

 U2+U3=N2(14logN−38)+12 N∑j=1→r2j, (2.4)

and so the explicit form of the Boltzmann factor is (see e.g. [11, eq. (1.72)])

where and . The derivation of (2.5) requires the particles be confined to the disk of the smeared out background and thus . To get an analogy with the absolute value squared of the trial wave functions (1.1) we must relax this condition by allowing the domain to be all of ; this will be referred to as soft disk geometry.

With denoting the partition function corresponding to (2.5), i.e. (2.5) integrated over and multiplied by , one has that for (see e.g. [25, above eq. (3.14)])

 ZdN,2=πNe3N2/4N−N2/2(πρb)−N/2G(N+1),

where . Consequently [25, eq. (3.14)]

 βFdN,2:=−logZdN,2=Nβf(2,ρb)+112logN−ζ′(−1)+O(1N2),

where

 βf(2,ρb)=12log(ρb2π2). (2.6)

Furthermore, the one-body density can similarly be computed exactly at with the result (see e.g. [11, Prop. 15.3.4])

 ρd(1)(→r;2)=1πΓ(N;r2)Γ(N). (2.7)

In the case of semi-periodic boundary conditions, the neutralizing background is chosen to be the rectangle , , and the particles couple to the background via a harmonic potential in the -direction only, centred at [6]. For the corresponding Boltzmann factor we find

 AcN,Γe−πΓρb∑Nj=1(yj−L/2)2∏1≤j

where

 AcN,Γ=(W2π)−NΓ/2e−πΓ12NL2ρb (2.9)

and . Analogous to the situation with (2.5), the derivation of (2.8) requires , but to get an analogy with the absolute value squared of the trial wave function (1.1) in the case (1.3) we must relax this conditions, obtaining what will be referred to as soft cylinder geometry. For , results from [6] tell us that

 1NβFcN,2=βf(2,ρb)+πρbL26N2 (2.10)

and

 ρc(1)(→r;2)=1W√2ρbN−1∑m=0exp(−2πρb(y−m+1/2Wρb)2). (2.11)

## 3 Universal properties of the free energy

### 3.1 Introductory remarks

Consider first the soft disk geometry. For general one expects the large expansion

 βFdN,Γ:=−logZdN,Γ=Nβf(Γ,ρb)+βμ(Γ,ρb)(2π√N/(πρb))+112logN+O(1). (3.1)

In the leading term, is the dimensionless free energy per particle. The universal term was identified by relating the plasma to a free Gaussian field [17]. An unpublished result of Lutsyshin makes the conjecture

 βμ(Γ,ρb)=√πρb2π4log(Γ/2)3√π=√ρb2log(Γ/2)3π. (3.2)

Since the radius of the background is , has the interpretation as a surface tension. Note that (3.2) is consistent with the exact result (2.6) as it gives .

Consider now the soft cylinder. Universality of the dimensionless free energy per particle and the surface tension imply that for large

 βFcN,Γ:=−logZcN,Γ=Nβf(Γ,ρb)+βμ(Γ,ρb)(2W)+πρbL26N+O(1). (3.3)

Here the universal term is a consequence of the relationship between the plasma on an infinitely long cylinder and the corresponding Gaussian free field [9].

### 3.2 Validity of free energy expansion for Γ=2+ϵ(ϵ≪1)

Consider the soft disk plasma system with mobile particles having charge and total energy (recall §2). It follows from the definitions that to first order in ,

 βFdN,Γ−βFdN,2=(Γ−2)⟨U⟩∣∣Γ=2, (3.4)

where denotes the total energy. But we know from above that , with the potential energy of the particle-particle interaction as given by (2.3), and the sum of the particle-background and background-background interactions as given by (2.4). A result of Shakirov [23] tells us that

 ⟨U1⟩∣∣Γ=2ρb=1/π=−12(N22logN−N24+12(1+C)N−43√Nπ+524+O(1√N)), (3.5)

where denotes Euler’s constant. The remaining averages are simple to compute.

###### Lemma 1

We have

 ⟨U2+U3⟩∣∣Γ=2ρb=1/π=N24logN−N28+N4.

Proof.  We see from (2.4) that

 ⟨U2+U3⟩∣∣Γ=2ρb=1/π=N2(14logN−38)+12⟨N∑j=1→r2j⟩∣∣Γ=2ρb=1/π. (3.6)

Introducing the configuration integral

 QdN,Γ(ρb):=∫R2d→r1⋯∫R2d→rNe−(πρbΓ/2)∑Nj=1→r2j∏1≤j

we see that

 ⟨N∑j=1→r2j⟩∣∣Γ=2ρb=1/π=−1π∂logQdN,2(ρb)∂ρb∣∣ρb=1/π.

On the other hand, a simple scaling shows

 QdN,Γ(ρb)=ρ−ΓN(N−1)/4−NbQdN,Γ(1),

so we obtain

 ⟨N∑j=1→r2j⟩=1πρb(ΓN(N−1)4+N) (3.8)

Setting , and substituting in (3.6) gives the stated result.

Adding (3.5) to the result of Lemma 1 and substituting in (3.4) we have to first order in

 (βFdN,Γ−βFdN,2)∣∣ρb=1/π=(Γ−2)(−CN4+23√Nπ−548+O(1√N)). (3.9)

In particular, the terms proportional to is in precise agreement with the conjecture (3.2) expanded to the same order. As an aside, we remark that (3.9) and (3.4) together tell us that to leading order in , with and , is equal to . This is a result first deduced by Jancovici [16], using the relationship of the leading form of and an average of the potential with respect to the bulk truncated two-point function.

The formula (3.4) also applies with the soft disk replaced by the soft cylinder; however the analogue of (3.5) is not in the existing literature. Making use of knowledge of the exact form of the one and two-point functions for the soft cylinder geometry at [6] we find (see Appendix A)

 ⟨U1⟩c∣∣Γ=2=12(−πW2ρbN33−Nlog(√ρbπW2)−N2−NC2+4√ρb3πW)+O(1). (3.10)

Furthermore, a more elementary computation, making only use of the one-point function (2.11), gives

 ⟨U2+U3⟩c∣∣Γ=2=N2logW2π+π6NL2ρb+N4. (3.11)

Thus to first order in ,

 βFcN,Γ−βFcN,2=(Γ−2)(−N2log√πρb−CN4+2√ρb3πW+O(1)). (3.12)

In particular, the term proportional to is consistent with the expansion (3.3).

### 3.3 Exact numerical results for the free energy at Γ= 4, 6 and 8

Let , , and let be a partition of such that

 2p(N−1)≥μ1≥μ2≥⋯≥μN≥0.

Also, let denote the corresponding frequency of the integer in , let denote the set of permutations of , and define the corresponding monomial symmetric function by

 mμ(z1,…,zN)=1∏imi!∑σ∈SNzμ1σ(1)⋯zμNσ(N).

A method based on symmetric Jack polynomials [3] gives, for small , an efficient way to compute the coefficients

 ∏1≤j

This is significant since then we have [25]

 QdN,4p(1/π)=N!πN∑μ(c(N)μ(2p))2∏imi!N∏l=1μi! (3.13)

Similar considerations hold true for . Now we must take to be a partition of such that

 (2p+1)(N−1)≥μ1>μ2>⋯>μN≥0.

With denoting the Schur polynomials, we then use the anti-symmetric Jack polynomials to expand [4]

 ∏1≤j

where . Consequently [25]

 QdN,4p+2(1/π)=N!πN∑μ(c(N)μ(2p+1))2N∏l=1μi! (3.14)

Using (3.13) and (3.14), we computed numerically the free energy in the soft disk for and with ranging from to , and for with to 11. In order to test the expansion (3.1), the data for () and () is fitted to the ansatz

 βFdNΓ=Nβf(ρb,Γ)+βμ(Γ,ρb)(2π√N/(πρb))+112logN+d (3.15)

The data obtained for , and is shown in Table 1. The results for the bulk free energy reproduces known numerical estimates obtained by studying the 2dOCP in a sphere for  [25] and 8 [24] within a very small margin of error: less than 0.02% for and 6, and 0.16% in the worst case . The surface tension term is compared with the conjecture (3.2) and the results give a strong support to this conjecture as they only differ by less than 1% for and 6, and 5.5% for .

A more extensive numerical study can be done in the soft cylinder geometry as and can be varied independently, and more numerical data can be obtained for the free energy. Formulas analogous to (3.13) and (3.14) hold true for the soft cylinder. There the relevant configuration integral is

 QcN,Γ(L,W)= ∫∞−∞dy1⋯∫∞−∞dyNe−Γπρb∑Nl=1(yl−L/2)2∫L0dx1⋯∫L0dxN ×∏1≤j

For even and we have

 |2sinπ(wk−wj)|Γ=eπ(yj+yk)Γ/2(e2πiwj−e2πiwk)Γ/2(e−2πi¯wj−e−2πi¯wk)Γ/2.

Consideration of the derivation leading to (3.13) and (3.14) we then have

 QcN,4p(√N,√N)=NN/2N!Γ−N/2e−πΓN2/4∑μ(c(N)μ(2p))2∏imi!eπΓ∑Nj=1(2μj/Γ+1/2)2/N (3.17)

and

 QcN,4p+2(√N,√N)=NN/2N!Γ−N/2e−πΓN2/4∑μ(c(N)μ(2p+1))2eπΓ∑Nj=1(2μj/Γ+1/2)2/N. (3.18)

More generally, if is considered as an independent variable from , let us define which is a characteristic length of the problem: as shown in [6, 21] the one-body density is periodic along the -axis with period when and fixed. Let be the cylinder circumference scaled out by . The configuration integral (3.3) is

 QcN,Γ(L,W)=ρ−Nb~WN/2N!Γ−N/2e−πΓN3/(4~W)∑μ(c(N)μ(Γ/2))2∏imi!eπΓ∑Nj=1(2μj/Γ+1/2)2/~W, (3.19)

valid for both cases and . In the latter case as in all the partitions with all frequencies are . The free energy is given by

 βFcN,Γ(~W) = N(1−Γ4)logρb−N2(1−Γ2)log~W+N2logΓ−NΓ2log(2π) (3.20) +πΓN33~W−logQc∗N,Γ(~W)

with

 Qc∗N,Γ(~W)=∑μ(c(N)μ(Γ/2))2∏imi!eπΓ∑Nj=1(2μj/Γ+1/2)2/~W. (3.21)

We computed (3.20) numerically. The calculations are computationally intensive for large values of because of the immense number of partitions involved, thus we had to limit our efforts to varying from 2 to 14 for and , and from 2 to 11 for . However, can be arbitrarily choosen without any computational increase effort. We choose varying from 1 to 25.9 by increments of 0.1, therefore exploring two different types of geometries: thin cylinder (small ) and thick cylinder (large ). The free energy is shown in Figure 1 as a function of for various values of . For and 6, the free energy exhibits a unique minimum for a particular value of which depends on . This is also the case for and , however for and 3, the free energy exhibits two minimums. In Figure 2, the location of the minimum is shown as a function of . As increases, also does . The figure shows that, in the range of values of considered, is of the same order of magnitude that , that is . For large , this corresponds to thick cylinders, thus suggesting that at a given density, thick cylinders are more stable thermodynamically than thin cylinders. In the following sections we will be interested in the scaling laws for thick cylinders.

As the free energy expansion (3.3) is expected to hold for large and large , we sought to fit the numerical data corresponding to and to an ansatz compatible with (3.3) of the form

 βFcN,Γ=Nβf(Γ,ρb)+2βμ(Γ,ρb)√~W/ρb+cN~W+d. (3.22)

The results for , , and are shown in Table 2. The bulk free energy is compared to the numerical estimates obtained by studying the 2dOCP in a sphere for  [25] and 8 [24]. As it should be, the difference is very small, less than 0.05%. Also the universal correction differs from the expected value only by less than 4% in the worst case (). The numerical data again strongly supports Lutsyshin’s conjecture (3.2) for the surface tension term , as the relative difference between the conjecture and the numerical data is less than 3% in the worst case ().

In the next section, we will be interested in the scaled edge density where . Notice that in that limit, the universal correction to the free energy, , and the correction in (3.3) ( in (3.22)) become of the same order, and give a correction to the free energy equal to .

## 4 Exact first order correction to the scaled edge density at Γ=2

### 4.1 Disk geometry

In disk geometry, the density expanded about has been computed to first order in by Téllez and Forrester [26]. To present their result, introduce

 I(k1,k2)=∬0≤t2
 J(k1,k2) =∫∞0∫∞0e−t1−t2tk11tk22log(max(t1,t2))dt1dt2 =∂∂k1I(k1,k2)+∂∂k2I(k2,k1) (4.1)

and let denote the usual upper and lower incomplete gamma functions. The result of [26] reads

 ρd(1)(→r;Γ)= ρd(1)(→r;2) −(Γ−2)πe−|z|2{N−1∑k1=0N−1∑k2=0k2≠k1|z|2k22k1!(k2!)2J(k1,k2) +N−1∑k1=0N−1∑k2=k1+1I(k1,k2)k1!k2!(k2−k1)(|z|2k2k2!+|z|2k1k1!) −N−1∑k1=0N−1∑k2=k1+1|z|2k1γ(k2+1,|z|2)+|z|2k2Γ(k1+1,|z|2)(k2−k1)k1!k2! −N−1∑k2=0|z|2k22k2!N−1∑k1=0k1≠k2γ(k1+1,|z|2)log(|z|2)+∫∞|z|2e−ttk1logtdtk1! −N−1∑k1=0|z|2k12k1!(k1+1−|z|2)}+O((Γ−2)2) (4.2)

(in the second last sum the term as presented in [26] contains a typographical error and reads with in the argument instead of ), where is given by (2.7).

We seek the limiting form of the correction term as presented above under the edge scaling

 z:=√N−y, (4.3)

which effectively positions the neutralizing background of the plasma in the half plane . For this task we hypothesize that in the limit , only the large portion of the sums in (4.1) contribute, allowing us to use the asymptotic expansions

 γ(N−j+1;|z|2)∼12Γ(N−j+1)(1+erf(j−2y√N√2N)) (4.4)
 Γ(N−j+1;|z|2)∼12Γ(N−j+1)(1−erf(j−2y√N√2N)) (4.5)
 I(k1,k2)∼k1!k2!2erfc(k2−k1√2(k1+k2)). (4.6)

The first two of these are standard results while the third was derived in [26]. We remark that the asymptotic expansion of follows by substituting (4.6) in (4.1), together with Stirling’s formula. Thus we have

 J(k1,k2)−k1!k2!logN +1√2(k1+k2)(2√π)k1!k2!e−(k1−k2)2/(2(k1+k2)). (4.7)

With these preliminaries let us consider the scaled limit of the first double sum in (4.1)

###### Lemma 2

Under the assumption that the leading asymptotic portion of the sum comes from the large region, for large and with given by (4.3) we have

 e−|z|2N−1∑k2=012k2!N−1∑k1=0(121k1+logk1N)I(k1,k2)1k1!k2! ∼−121√2π∫∞0dt2e−2(t2−y)2(1+erft2) +1√2π∫∞0dt2e−2(t2−y)2(t22erfct2−t2e−t22√π) +e−|z|22N−1∑k2=0|z|2k2k2!(N−k2)22N. (4.8)

Proof.  Consider the sum over . After substituting (4.6), breaking the sum up into the regions and , writing

in the latter, and changing summation labels