Anisotropic electronic states in the fractional quantum

Anisotropic electronic states in the fractional quantum Hall regime Orion Ciftja

Citation: AIP Advances 7, 055804 (2017); View online: https://doi.org/10.1063/1.4972854 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics

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AIP ADVANCES 7, 055804 (2017)

Anisotropic electronic states in the fractional quantum Hall regime Orion Ciftja Department of Physics, Prairie View A&M University, Prairie View, Texas 77446, USA (Presented 1 November 2016; received 14 September 2016; accepted 28 September 2016; published online 22 December 2016)

Recent experiments indicate the presence of new anisotropic fractional quantum Hall states at regimes not anticipated before. These experiments raise many fundamental questions regarding the inner nature of the electronic system that leads to such anisotropic states. Interplay between electron mass anisotropy and electronelectron correlation effects in a magnetic field can create a rich variety of possibilities. Several anisotropic electronic states ranging from anisotropic quantum Hall liquids to anisotropic Wigner solids may stabilize due to such effects. The electron mass anisotropy in a two-dimensional electron gas effectively leads to an anisotropic Coulomb interaction potential between electrons. An anisotropic interaction potential may strongly influence the stability of various quantum phases that are close in energy since the overall stability of an electronic system is very sensitive to local order. As a result there is a possibility that various anisotropic electronic phases may emerge even in the lowest Landau level in regimes where one would not expect them. In this work we study the state with filling factor 1/6 in the lowest Landau level, a state which is very close to the critical filling factor where the liquid-solid transition takes place. We investigate whether an anisotropic Coulomb interaction potential is able to stabilize an anisotropic electronic liquid state at this filling factor. We describe such an anisotropic state by means of a liquid crystalline wave function with broken rotational symmetry which can be adiabatically connected to the actual wave function for the corresponding isotropic phase. We perform quantum Monte Carlo simulations in a disk geometry to study the properties of the anisotropic electronic liquid state under consideration. The findings indicate stability of liquid crystalline order in presence of an anisotropic Coulomb interaction potential. The results are consistent with the existence of an anisotropic electronic liquid state in the lowest Landau level. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4972854]

I. INTRODUCTION

The fractional quantum Hall effect (FQHE) has been understood in terms of an incompressible quantum liquid state of electrons in a two-dimensional electron gas (2DEG) at a high perpendicular magnetic field.1 Magnetic field leads to the creation of Landau levels (LLs) and determines their degeneracy. Therefore, one can change the filling factor of the system (defined as ratio of number of electrons to the degeneracy of a LL) by varying the magnetic field. For sufficiently strong magnetic fields, electrons may partially occupy only the lowest Landau level (LLL) with filling factors of the form ν = 1/3 or 1/5. Odd denominator filling factors of such form have been understood in terms of, now venerable, Laughlin’s wave function.2 Laughlin’s theory and many other FQHE-related works3–8 assume that the interacting system of electrons is isotropic. Because of that assumption, all FQHE states are expected to be isotropic and to have rotational symmetry. This view started to change with the initial observation of anisotropic quantum Hall phases9 at high LLs at filling factors ν = 9/2, 11/2, . . . which were interpreted as stripe states.10,11 Various other mechanisms to explain the anisotropy of such states have also been discussed theoretically.12–18 2158-3226/2017/7(5)/055804/5

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The assumption was that the rotation symmetry at high LLs is broken spontaneously (given that the interaction potential is considered to be isotropic). So far, the situation in the LLL seems to be more conventional. It is believed that a Wigner solid19–21 stabilizes for 0 < ν ≤ 1/7 while the rest of states for 1/7 < ν ≤ 1 are liquids. States at filling factors, ν = p/(2mp + 1) (p,m - integer) are readily understood in terms of the composite fermion (CF) theory22,23 and are believed to be incompressible liquid states. On the other hand, even-denominator filled states, ν = 1/2, 1/4, 1/6 are believed to be compressible Fermi liquid states.24–27 The current understanding is that liquid quantum Hall states in the LLL are isotropic. However, in real materials, a FQHE system may be anisotropic. For instance, recent experiments have observed a new anisotropic FQHE state at ν = 2 + 1/3 in the first excited LL. In this case, the rotation symmetry is broken explicitly by an anisotropic perturbation due to an in-plane magnetic field. Such a state shows anisotropic magnetoresistance as well as the simultaneous presence of an accurately quantized Hall plateau.28,29 Many other mechanisms can explicitly induce anisotropy, for instance, electron’s mass anisotropy leads to an effective anisotropic Coulomb interaction potential between electrons. Therefore, it is a plausible to assume that the ground state does not have rotational symmetry for the case of an anisotropic interaction. One can argue along these lines that the properties of a FQHE state in which electrons interact anisotropically may not be the same as those of the isotropic counterpart. It is also plausible to assume that an anisotropic interaction between electrons may lead to the stabilization of anisotropic electronic phases even in the LLL, a regime where, ordinarily, one would not expect them. In this work we study the possible existence of an anisotropic electronic state in the LLL at filling factor 1/6. We choose this filling factor because it is very close to the critical filling factor where the liquid to Wigner solid transition takes place, and, thus it is expected to be prone to destabilization. We perform quantum Monte Carlo (QMC) simulations for small systems of electrons in a standard disk geometry to investigate whether an anisotropic liquid crystalline state of electrons with broken rotational symmetry (BRS) is stabilized by an anisotropic Coulomb interaction potential. II. MODEL AND THEORY

We adopt a disk geometry model30 that considers N electrons of charge −e(e > 0) immersed in a uniform positively charged finite disk of area ΩN = π RN2 where RN is the radius of the disk. The electrons move freely in 2D and are not constrained to stay inside the disk. The spin of electrons is considered fully polarized in the direction of the uniform perpendicular magnetic field, ~B = (0, 0, −B). The choice of the negative sign of ~B is a matter of convenience allowing us to express the LLL wave functions in terms of the complex variable, z = x + i y rather than its complex conjugate. The density of the system (number of electrons per unit area) or otherwise the uniform density of the background, ρ0 = N/ΩN , is constant and can be written as ρ0 = ν/(2 π l02 ) where ν is the filling factor and l 0 is the electron’s magnetic length. The quantum Hamiltonian for a system of N electrons with mass me and charge e (e > 0) in a perpendicular magnetic field is written as: Hˆ = Kˆ + Vˆ , (1) f g 2 P where Kˆ = Ni=1 ~pˆ i + e ~Ai /(2 me ) represents the kinetic energy operator and Vˆ is the potential energy operator: Vˆ = Vˆ ee + Vˆ eb + Vˆ bb , (2) ˆ ˆ ˆ where V ee , V eb and V bb are the electron-electron (ee), electron-background (eb) and backgroundbackground (bb) potential energy operators. We adopt a symmetric gauge for the magnetic ~ × ~A) given the choice of the geometry. The eb and bb terms are calculated from: field (~B = ∇ ρ2 P Vˆ eb = −ρ0 Ni=1 ∫ ΩN d 2 r 3(~r −~ri ) and Vˆ bb = 20 ∫ ΩN d 2 r ∫ ΩN d 2 r 0 v(~r −~r 0) where v(~r −~ri ) and v(~r −~r 0) have a Coulomb form. On the other hand, the interaction between electrons has an anisotropic Coulomb potential31 form: e2 vγ (xij , yij ) = q , (3) xij2 /γ 2 + γ 2 yij2

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where parameter γ ≥ 1 tunes the degree of anisotropy of the ee interaction potential and ~rij =~ri − ~rj = (xij , yij ) is the inter-electron separation vector. Note that the potential in Eq. (3) transforms into the conventional isotropic Coulomb interaction potential for γ = 1. As customary, the Coulomb’s electric constant is not specifically included in the expressions of either the isotropic or anisotropic Coulomb interaction potential. The presence of an anisotropic two-body perturbation term in the Hamiltonian changes the whole physics of the problem and hints to the possibility of novel anisotropic phases that may arise at any LL including the LLL. We have chosen the ν = 1/6 state in the LLL for our study. It has been shown that, for an isotropic Coulomb interaction, this state is a very fragile isotropic Fermi liquid state.32 Therefore, it is likely that an additional anisotropic perturbation may destabilize it in favor of a novel anisotropic liquid state at that filling factor. We describe a possible anisotropic state of electrons at ν = 1/6 by means of an anisotropic liquid crystalline BRS wave function15 that is adiabatically connected to the isotropic state: Ψα =

N Y i>j

(zi − zj )4 (zi − zj + α)(zi − zj − α) × exp *.− ,

N X |zj | 2 + ~ / det ei kα ~rj , 2 4 l0 j=1 -

(4)

where N is the number of electrons that occupy the N lowest-lying plane wave energy states labeled by the momenta {~kα } of an ideal 2D spin-polarized Fermi gas,33 zj = xj + i yj is the position coordinate in complex notation and we discard the LLL projection operator (with the usual assumption that kinetic energy is not important). The parameter, α breaks the rotational symmetry of the wave function and phenomenologically measures the anisotropy of the ground state. We consider α to be real so that the system has a stronger modulation in the x-direction. The value α = 0 represents an isotropic Fermi liquid state at filling factor ν = 1/6 described by a generalized version of the Rezayi-Read wave function.25 Wave function, Ψα is antisymmetric and translationally invariant. Thus, this wave function is an obvious starting point to describe a nematic anisotropic liquid crystalline state at ν = 1/6 since it lacks rotational symmetry. III. RESULTS AND CONCLUSIONS

We consider small systems of N electrons described by the anisotropic liquid crystalline BRS wave function of Eq. (4). The system sizes chosen, N = 5, 9, 13, 21, 25 and N = 29 correspond to a closed-shell 2D system (in Fermi space). A uniformly charged background disk guarantees the overall charge neutrality of the system. The electrons interact with each other via the anisotropic Coulomb interaction potential given from Eq. (3). A plot of this anisotropic interaction potential is shown in Fig. 1. In all our calculations we consider the value γ = 1.5 for the anisotropic Coulomb interaction potential. The Jastrow-Slater form of the wave function in Eq. (4) makes the QMC calculations cumbersome and time-consuming. The above reason combined with the limited computational power at our disposal constrained our calculations to small systems of particles. We calculated the ground state energy of the system via QMC simulations in disk geometry.34,35 The QMC results allow us to obtain an accurate comparison between the energy of anisotropic BRS liquid crystalline states (for α , 0) relative to their isotropic liquid counterpart (at α = 0) for the given choice of the anisotropic Coulomb potential. The ground state interaction energy per particle can be written as: α = ee + eb + bb where α = hVˆ i/N, ee = hVˆ ee i/N, eb = hVˆ eb i/N and bb = hVˆ bb i/N are, respectively, the total, ee, eb and bb interaction energies per particle. We used the Metropolis algorithm in our QMC simulations to calculate the expectation value of quantities of interest. In our QMC calculations, the expectation value of any given operator is estimated by averaging its value over a large number of electronic configurations. We discarded the first 100,000 ”thermalization” QMC steps to reduce the statistical error. We used 2 million ”equilibrated” configurations to obtain an accurate statistical average. As seen in a prior study with a wave function lacking translational invariance,36 these values are more than sufficient for a good ”thermalization”. For the given choice of the anisotropic Coulomb interaction potential, the calculations for small systems of electrons indicate that there is always an anisotropic liquid state that has an energy lower than its isotropic counterpart. The result in Fig. 2 where we show ∆α = α − 0 as a function of α for N = 29 electrons and vγ=1.5 (x, y) is a representative sample of the typical behavior observed. The conclusion drawn is that, for the system sizes considered, a Coulomb anisotropic

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q FIG. 1. Plot of the anisotropic Coulomb interaction potentials, vγ (x, y) = e2 / x 2 /γ2 + γ2 y2 (in units of e2 /l0 ) along the directions (x, y = 0) (dashed line) and (x = 0, y) (dotted line) for γ = 1.5 as a function of the distance between electrons (in units of l 0 ). The isotropic Coulomb interaction potential (γ = 1) is drawn as a reference for comparison (solid line). For the same separation distance and the present choice of γ, repulsion between a pair of electrons is ”softened” along the y-direction.

potential of the vγ=1.5 (x, y) form is always able to stabilize an anisotropic liquid state of electrons. A larger value of the parameter γ leads to a more pronounced effect. While the tendency seems to be there, we were unable to obtain enough numerical accuracy to ascertain whether any arbitrary γ > 1

FIG. 2. Difference of energy (per electron), ∆ α =  α − 0 as a function of the wave function’s anisotropy parameter, α. The value,  α for α , 0 represents the energy (per electron) corresponding to the anisotropic BRS liquid state. The value, 0 represents the isotropic liquid counterpart. The state under consideration is the quantum Hall state at filling factor ν = 1/6 of the LLL. The results are from a QMC simulation of a system of N = 29 electrons in a disk geometry. Electrons interact via an anisotropic Coulomb potential with γ = 1.5. The statistical uncertainty of the results is comparable to the size of the symbols. Energies are in units of e2 /l 0 .

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(that might be quite close to the isotropic value γ = 1) can lead in all instances to an energetically favored anisotropic state. The energy gain at the optimum α (that is approximately 4 for the case of N = 29 electrons) changes with the size of the system. The values of the typical energy gain, while small in absolute terms, are quite significant in the context of quantum Hall studies and amount to values of ∼ 10−2 e2 /l0 for the case of N = 29 electrons. All other QMC simulations for smaller systems of electrons (N = 5, 9, . . .) lead to the same conclusion. Because smaller systems manifest a qualitatively similar type of behavior, we do not discuss them further for the sake of brevity. To conclude, in this work we argue that an anisotropic liquid phase of electrons may stabilize at filling factor ν = 1/6 of the LLL in presence of an anisotropic Coulomb interaction potential. We choose the liquid state at such a filling factor because this state is very close to the liquid-solid phase transition point in the LLL. Thus, one expects this state to be very sensitive to (even weak) anisotropic perturbations. It is argued that an adequate source of anisotropy (in this case an anisotropic Coulomb interaction potential between electrons) can destabilize the original isotropic liquid state in favor of an anisotropic liquid crystalline state. We describe this possible anisotropic liquid state of electrons by means of an anisotropic BRS wave function. The foreseen scenario is supported by results obtained from QMC simulations of small systems of electrons in a disk geometry. ACKNOWLEDGMENTS

This research was supported in part by U.S. Army Research Office (ARO) Grant No. W911NF13-1-0139 and National Science Foundation (NSF) Grant No. DMR-1410350. 1 D.

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