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Broken rotation symmetry in the fractional quantum Hall system

We demonstrate that the two-dimensonal electron system in a strong perpendicular magnetic field has stable states which break rotational but not translational symmetry. The Laughlin fluid becomes unstable to these states in quantum wells whose thickness exceeds a critical value which depends on the electron density. The order parameter at 1/3 reduced density resembles that of a nematic liquid crystal, in that a residual two-fold rotation axis is present in the low symmetry phase. At filling factors 1/5 and 1/7, there are states with four- and six-fold axes, as well. We discuss the experimental detection of these phases.

in the classical one-component Coulomb plasma phase diagram. A hexatic phase is also a possibility in this system [2] as well as in the logarithmic potential case perhaps more relevant to the FQHE. [3] This theoretical background, and the experimental discovery of phases which seem to have properties unlike both the Laughlin liquid and the Wigner solid, [4] ("Hall insulator") led us to an investigation of the possibility of broken rotational symmetry (BRS) in the two-dimensional electron system in strong field. We found that several liquid crystal-like phases occur, though these appear to resemble more closely nematic than hexatic phases.
We begin with the case where the electron density n satisfies n = 1 3 1 2πℓ 2 , where ℓ is the magnetic length: ℓ −2 = eB/hc and B is the external field, taken to be in the negative zdirection. In this paper we work in the limit of large field. This is the density for the 1/3 quantum Hall state. Consider the following wavefunction for the disc-shaped system: Here z i = x i + iy i and i,j are particle indices. α is a complex number. This wavefunction is antisymmetric in the particle indices for all α, lies entirely in the lowest Landau level, and reduces to the Laughlin wavefunction at α = 0. It also shares with the Laughlin wavefunction the characteristic of having uniform density (far from the edges of the disc).
However, the two-particle correlation function is a different matter: In this equation r = (x 1 − x 2 , y 1 − y 2 ), and N is the total number of electrons . Translational symmetry is not broken for the small values of α with which we are concerned, as we shall demonstrate below. Thus g α is a function only of the difference variable r. In contrast to the Laughlin liquid, however, it does depend on the direction of r. Let us take α to be real. Then the equivalent classical plasma interaction corresponding to Ψ α is a logarithmic interaction between rodlike charged objects lying along the x-axis. Accordingly, Ψ α represents a BRS state. [5] In general (Reα, Imα) is a director, not a vector, order parameter, since Ψ α = Ψ −α .
Thus g α ( r) does not have full rotation symmetry for α = 0 but does satisfy g α (x, y) = To give a physical picture of this state, we display a typical configuration in a Monte Carlo simulation governed by the probability distribution |Ψ α | 2 in Fig. 1 Under what conditions is such a BRS state stable? Or, given an interelecron potential expanding the polynomial in Eqn. 1, and taking the limit | r| → 0 in Eqn. 2 that g α ( r) ∼ r 2 rather than g α ( r) ∼ r 6 . Thus for very short-range V, the Laughlin state is always favored, as is well known. [6] [7] In order to compare the α = 0 with the α = 0 states we performed Monte Carlo simulations of the equivalent classical plasmas with 200 particles in the disk geometry, and compared the energies of candidate ground states. (We believe that improvement of the wavefunction by, for example, quantum Monte Carlo techniques [8] will not affect energy differences very much.) In order to minimize the finite size effect we compute the energies of only 25 particles closest to the center of the disk. We have checked that this procedure gives the accepted value for the energy per particle in the case of a pure Coulomb potential and α = 0.
From the computations we found that the Laughlin state is favored for a pure Coulomb potential V (r) = e 2 /εr. The actual potential between electrons in a real two-dimensional layer of finite thickness is softer, owing to the averaging of the charge density over the third dimension. This has been discussed in detail by numerous authors and the chice of potential depends on the shape of the well. We shall take the simple form of Zhang and Das Sarma.
[9] They showed that V (r) = e 2 /ε(r 2 + λ 2 ) 1/2 is a reasonable approximation for a square quantum well and that λ ≈ 0.2t, where t is the layer thickness of the well. To understand the effect that this modification of the potential will have, note that the total energy is given by We now plot the angle averaged correlation functions for different values of α in Fig. 2.
There is incipient solid order as α increases, in that there is much more tendency towards being able to identify shells of neighbors. At short distances, the correlation function is proportional to r 2 , just as that for the Wigner crystal is, and in contrast to the r 6 behavior of the Laughlin state. Overall, the the softer potential favors finite α: the correlation hole has a larger effective radius, even though it is not as "deep". The difference between the energy per particle for a BRS state with α = 1 and the Laughlin state for different values of λ is plotted on Fig. 3. There is thus a critical value λ c at which the system undergoes a transition to finite α. We compute this to be λ c = 4.1±1..5, which corresponds to a thickness of t = 1600Å, when B = 10T. This transition is second order, unlike the transition to the Wigner solid, which is probably first order. It has recently been pointed out that changing t can induce the liquid-solid transition by a mechanism similar to that proposed here. [10] The critical value of t for the m=1/3 density is similar to that computed here, suggesting that the energy balance between Laughlin, BRS, and crystalline states is a subtle one. We expect that the BRS state occupies a fairly narrow range of parameter space between the liquid and the crystal states, by analogy with hexatic phases. It is clear, in any case, that this range of thickness values is experimentally accessible. [11] The energy balance between the BRS and Wigner crystal state is currently under investigation. [12] The correlations are oscillatory even to infinite distances in a true crystalline state.
Inspection of Fig. 2 shows that the correlation function for the BRS state is flat at large distances, demonstrating that long-range translational symmetry is not broken, and justifying the identication of these states as liquid crystal states.
We may construct similar wavefunctions for m=5 and m=7.
is an appropriate wavefunction for m=5 and is an appropriate wavefunction for m=7. We have not yet investigated these wavefunctions for all values of β and γ. Particularly interesting cases are β = iα for m=5 and β = ωα, γ = The director n = (Re α, Im α) is the order parameter whose appearance signals the appearance of BRS. The states characterized by n and -n are identical. There is no independent inversion symmetry operation in our two-dimensional system and thus the transition to the low-symmetry phase is second-order, unlike the situation for ordinary three-dimensional nematic systems. Our Monte Carlo calculations of energy as a function of α confirm this picture. The Ginzburg-Landau energy is therefore where t is the thickness and T is the temperature. We expect a second-order transition when T=0 and our calculations have been carried out only at zero temperature. At finite 5 temperature thermal fluctuations are important. Depending on the experimental situation they may convert the transition to one of the Kosterlitz-Thouless type.
The free emergy expression shows that twists of the order parameter are possible and lead to textures but also to low energy excited states. Nevertheless, these excited states do not involve density changes and the state as a whole is incompressible. We conclude that the FQHE still occurs in this gapless state. The BRS state does not appear to be