Rigidity of the Laughlin Liquid
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Abstract
We consider general Nparticle wave functions that have the form of a product of the Laughlin state with filling factor \(1/\ell \) and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.
Keywords
Fractional quantum Hall effect Laughlin wave function Classical Coulomb gas1 The Laughlin Phase

1. The Laughlin wave function is an approximate ground state for the manybody Hamiltonian, and its energy is separated from the rest of the spectrum by a gap.

2. Modifications of the Laughlin wave function that stay within the ground eigenspace of the manybody Hamiltonian cannot increase the local oneparticle density beyond a fixed value.
Property 2 is not always recognized as a separate issue, but is also crucial for Laughlin’s original argument supporting quantization of the Hall conductivity [15, 18] and for FQHE physics in general. Its importance lies in understanding the effect of an external potential that would try to concentrate the density as much as possible in energetically favorable places.
Physically, (1.3) includes the addition of ‘quasiholes’ (zeros of the wavefunction) to the Laughlin state, essentially arbitrary correlations between the particles’ and quasiholes’ locations being allowed. It is intuitive that this leads to a decrease of the global density. It is far from obvious, however, that no local increase of the density can occur anywhere. Several quasiholes arranged tightly on a circle could, perhaps, increase the density inside the circle. Moreover, F need not contain any zeros at all like, e.g., \(\exp (c\sum _i z_i^2)\), which stretches the support of the density in one direction and shrinks it in another.
In this paper, we report on recent density bounds for fully correlated states that demonstrate the validity of Property 2 without invoking Property 1. They hold essentially on length scales \(O(N^{1/4})\), much smaller than the full extent of the liquid, which is of order \(N^{1/2}\), and are related to earlier partial results proved in [37, 38]. The full proofs are somewhat involved and presented elsewhere [25], but we sketch the main arguments below and discuss physical applications.
2 Incompressibility Estimates
In his pioneering paper [16], Laughlin already argued that the oneparticle density of the state (1.1) has the form of a circular droplet of radius \(\sqrt{\ell N}\) where the density takes the constant value \((\pi \ell )^{1}\). The argument was based on the plasma analogy, where the absolute square of the wave function is written as the Gibbs distribution of a classical 2D Coulomb gas, and subsequently treated by a meanfield approximation.
The analysis of [35, 36] was generalized to other prefactors of a special kind in [37]. A common feature that emerged was an upper bound on the oneparticle density of magnitude \((\pi \ell )^{1}\), which is the density of the Laughlin state itself. Such a bound was called an incompressibility estimate in [37] because it is a manifestation of the resistance of the Laughlin state against attempts to compress its density. It relies essentially on the strong correlations of the Laughlin function and the analyticity of F in (1.3) which, physically, is due to the strong magnetic field confining the particles to the LLL. Without such a field and with weak correlations, the electron density in a crystal can be arbitrarily high locally, due to constructive interference of Bloch waves.
The meanfield methods of [35, 36, 37] are not applicable to general prefactors F. The question of a bound on the density for the general case was treated in [38] with an entirely different technique, rooted in 2D potential theory. The bound obtained was four times the expected optimal value \((\pi \ell )^{1}\) however. In this paper we explain that an improved version of the potential theoretic method leads to the correct optimal bound for arbitrary F.
Theorem 1
Recall that the magnetic length is \(1/\sqrt{2}\). The average can thus be taken on “mesoscopic scales”, \(O(N^{1/4+\varepsilon })\), much smaller than the full extent of the state, \(O(N ^{1/2})\), but not quite down to the expected finest scale, O(1). For realistic numbers [40] one can have Quantum Hall systems with N of the order of \(10^9\), in which case the ratio between the length scales \(N^{1/4}\) and \(N^{1/2}\) is about \(10^{2}\). A simple covering argument (reminiscent of the ‘cheese theorem’, see [26, Sect. 14.4] or [22, Theorem 14]) shows that (2.3) implies the analogous result for any open set, not just a disk.

That it is legitimate to neglect disorder in the sample and/or small external electric fields, as is done as a first approximation in the derivation of FQHE wavefunctions.

Laughlin’s argument [15, 17] (see also [14, Sects. 4.4, 9.3 and 9.5]) that switching on an electric current moves electrons transversally without creating any charge accumulation, and generates a Hall conductivity of value \(1/\ell \).
In addition, the precursor of FQHE states in a rapidly rotating Bose gas is a BoseEinstein condensate (see [21, 27] and references therein). Observing the distinctly flat profile of the Laughlin state would already be a strong indication of the transition to the FQHE regime. A more complete probe could be the response of the gas to variations of the trapping potential: the Bose condensate follows the trap by taking a ThomasFermilike shape (see [1, 2, 5] and references therein). The Laughlin state essentially does not respond to such variations, as exemplified by our main theorem.
We also point out that a combination of Theorem 1 with estimates obtained in [35, 36] leads to the following improvement of [37, Corollary 2.3]:
Corollary 2
It is remarkable that the Laughlin state stays an approximate minimizer in any powerlaw trap (the result actually holds for more general radial increasing potentials). No matter how steep and narrow a potential well one imposes, it is impossible to compress the Laughlin state while keeping the form (1.3), i.e., without jumping across the spectral gap. Extensions of Corollary 2 to general traps were recently proved in [39]: For a trap of arbitrary shape, the minimal energy is asymptotically equal to the bathtub energy and can always be achieved by adding uncorrelated quasiholes on top of Laughlin’s function, i.e., by using wave functions of the form (2.1).
3 Proof Strategy: The Exclusion Rule
A precursor of the desired bound (2.3) for \(\rho _F\) is the fact that the local density of points in a minimizing configuration for \( H_N(Z_N)\) is everywhere bounded above by \(N(\pi \ell )^{1}\) for large N. This is the core of the proof of the theorem, and a signature of screening properties of the effective plasma. To establish it, we introduce and study an auxiliary minimization problem, which is, mathematically, a cousin of the ThomasFermi energy minimization problem for molecules [23].
 (1)
There exists a unique minimizer, \(\sigma ^\mathrm{TF}\).
 (2)
The minimizer has compact support.
 (3)
Apart from a set of measure zero, \(\sigma ^\mathrm{TF}\) takes only the values 0 or 1.
 (4)The ThomasFermi equation holds:where$$\begin{aligned} \Phi ^\mathrm{TF}(x)={\left\{ \begin{array}{ll} \ge 0&{}\quad \hbox { if } \sigma ^\mathrm{TF}(x)=1\\ 0&{}\quad \hbox { if } \sigma ^\mathrm{TF}(x)=0 \end{array}\right. } \end{aligned}$$(3.11)is the total electrostatic potential of the molecule.$$\begin{aligned} \Phi ^\mathrm{TF}(x)=V_\mathrm{nuc}(x)+\int _{\mathbb {R}^2}\log {xx'}\sigma ^\mathrm{TF}(x')dx' \end{aligned}$$
 (1)
The area of \(\Sigma ^\mathrm{TF}(x_1,\dots ,x_K)\) is equal to K.
 (2)
\(\Sigma ^\mathrm{TF}(x_1,\dots ,x_{K1})\subset \Sigma ^\mathrm{TF}(x_1,\dots ,x_K)\).
 (3)
For a single nucleus at \(x_1\), \(\Sigma ^\mathrm{TF}(x_1)\) is the disc with center \(x_1\) and radius \(\pi ^{1/2}\).
Lemma 3

if another point \(y_{K+1}\) lay inside the screening region one could decrease the sum of the first two terms in (3.12) by moving \(y_{K+1}\) to any position on the boundary.

the last term \(\mathcal {W}\) in (3.12), being superharmonic, is generated by a positive charge distribution. One can thus always decrease \(\mathcal W\) by moving \(y_{K+1}\) to some point on the boundary. Such a move decreases at the same time the sum of the first two terms.
Lemma 4

The potential \(\Phi ^\mathrm{TF}\) generated by the points \(y_i\) contained in D(R) and the corresponding exclusion set \(\Sigma ^\mathrm{TF}(y_1,\dots , y_n)\) must vanish at all the points \(y_j\) lying outside of D(R), by the exclusion rule and (3.11). This leads to a uniform upper bound on \(\Phi ^\mathrm{TF}\) outside of D(R).

The same potential is generated by an overall positive charge density, because the total nuclear charge in D(R), which is \( > \pi R^2\) by assumption, is not fully screened by the part of the negative charge density \(\sigma ^\mathrm{TF}\) lying in D(R), at most equal to the area \(\pi R^2\) because \(\sigma ^\mathrm{TF}\le 1\). A lower bound on the circular average of \(\Phi ^\mathrm{TF}\) outside of D(R) follows.
After scaling, \(x\rightarrow z=\sqrt{\frac{\pi \ell }{N}}\,x\), Lemma 2 applies to the Hamiltonian (3.4) and implies that in any minimizing configuration \(\{z_1^0,\dots ,z^0_N\}\) of (3.4) the number of points \(z_i^0\) contained in any disc of radius \(R \gg N ^{1/2}\) is not larger than \(N(\pi \ell )^{1}\) times the area of the disc. This is the gist of the proof. From there, the main argument left to conclude the proof of Theorem 1 is to show that the above bound for ground states of (3.4) applies also to the Gibbs state (3.3).
In this argument, we use crucially that the temperature T in (3.3) scales as \(N^{1}\) so that the Gibbs measure charges mostly ground state configurations for large N. Turning this intuition into a proof follows the lines of [38, Section 3], see [25, Sect. 5] for the details. In brief, to access the 1particle density we rely on the fact that a Gibbs state minimizes the free energy of the corresponding Hamiltonian and use a FeynmanHellmanntype argument: We perturb the Hamiltonian (3.4) by adding a term \(\varepsilon \sum _i U(z_i)\) with U of compact support and prove free energy upper and lower bounds for this perturbed Hamiltonian. After dividing by \(\varepsilon \) we obtain a bound on \(\int U(z)\rho ^\varepsilon (z) dz\) where \(\rho ^\varepsilon \) is the 1particle density of the Gibbs state for the perturbed Hamiltonian and show that this tends to \(\int U(z) \rho ^0(z) dz\) in the limit \(\varepsilon \rightarrow 0\).
We point out that our density upper bound holds down to the finest possible scale for ground states of the plasma Hamiltonian (3.4), i.e., on length scales \(\gg N^{1/2}\) (see [33] where the corresponding lower bound is proved in the purely Coulombic case \(W = 0\)). Note that we are here referring to the scaled variables as in (3.1). When applying this result to Gibbs states of (3.4) we have to restrict ourselves to length scales \(\gg N ^{1/4}\) to control the error terms arising from the entropic contribution to the free energy in the FeynmanHellmanntype argument, but this is likely to be due to a technical limitation of our method. It was, indeed, recently proved that, for the purely Coulombic Hamiltonian, the expected microscopic density estimate holds for low temperature Gibbs states [3, 4, 19, 20] (see also [32] for ground states of higher dimensional Coulomb and Riesz gases). It remains to be seen whether a combination of our methods with those of [4, 19] could improve our results.
4 Conclusion
We have considered perturbations of the Laughlin state that may arise to accommodate external potentials, while keeping the system in the lowest Landau band and preserving the original correlations. We proved rigorously that no such perturbation can raise the particle density anywhere beyond the Laughlin value \(1/(\pi \ell )\). This is one of the criteria for the rigidity of the Laughlin state. Our theorem holds on length scales \(\gg N^{1/4}\) (in the original, physical variables) which, while large compared to the magnetic length O(1), are microscopic compared to the system’s macroscopic size, \(N^{1/2}\).
Notes
Acknowledgements
Open access funding provided by University of Vienna. We received financial support from the French ANR Project ANR13JS01000501 (N. Rougerie), the US NSF Grant PHY1265118 (E. H. Lieb) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement CORFRONMAT No. 758620).
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