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arXiv:cond-mat/0702172 · Physical Review B

Quantum nematic as ground state of a two-dimensional electron gas in a magnetic field

Quoc M. Doan, Efstratios Manousakis

Published May 22, 2007• 40 citations
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Abstract

We study the ground state of a nematic phase of the two-dimensional electron gas at filling fraction $\ensuremath{\nu}=1∕2$ using a variational wave function having Jastrow pair correlations of the form ${\ensuremath{\Pi}}_{i<j}{({z}_{i}\ensuremath{-}{z}_{j})}^{2}$ and an elliptical Fermi sea. Using the Fermi-hypernetted-chain approximation, we find that below a critical value of the broken-symmetry parameter, the nematic phase is energetically favorable as compared to the isotropic state for the second excited Landau level. We also find that below a critical value of the layer ``thickness'' parameter $\ensuremath{\lambda}$ (and in the actual materials), the quantum nematic is energetically favorable relative to the stripe ordered Wigner crystal phase.

DOI
10.1103/physrevb.75.195433

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We study the ground state of a nematic phase of the two-dimensional electron gas at filling fraction \ensuremathν=1∕2\ensuremath{\nu}=1∕2\ensuremathν=1∕2 using a variational wave function having Jastrow pair correlations of the form \ensuremathΠi<j(zi\ensuremath−zj)2{\ensuremath{\Pi}}_{i<j}{({z}_{i}\ensuremath{-}{z}_{j})}^{2}\ensuremathΠi<j​(zi​\ensuremath−zj​)2 and an elliptical Fermi sea. Using the Fermi-hypernetted-chain approximation, we find that below a critical value of the broken-symmetry parameter, the nematic phase is energetically favorable as compared to the isotropic state for the second excited Landau level. We also find that below a critical value of the layer ``thickness'' parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ (and in the actual materials), the quantum nematic is energetically favorable relative to the stripe ordered Wigner crystal phase.
We have studied the nematic state of the two dimensional electron gas (2DEG) at halffilled Landau level (LL). Our motivation comes from experiments in which anisotropic transport in the 2DEG under high magnetic field and at low temperature was observed. Based on a model of the nematic state proposed by Oganesyan, Fradkin and Kivelson, we investigate this state and compare it with other competing states proposed by other groups. First, we investigate at what LL the nematic state becomes energetically favorable as compared to the isotropic state. Our studies indicate that this occurs at the second excited LL. Moreover, we compare the energy of the nematic state with that of the stripe state obtained within the Hartree-Fock approximation (HF) and we conclude that, for the samples studied experimentally, the nematic state might be more stable. In our study we have used two different methods which both have advantages and disadvantages. The first method used is the Fermi-hyper-netted chain (FHNC) which provides results valid in the thermodynamic limit (infinite size system), however, it is accurate only for a low density system. The second method used is the Monte Carlo method (MC) which can be used on a finite-size system and, thus, the issue arises of how to extrapolate in the thermodynamic limit (finite-size effect). The results obtained from both methods are in good agreement and indicate that the nematic state might be a viable candidate to explain the experimental findings. In order to compare our results for the nematic state to those obtained for the stripe state, we needed to include the kinetic energy contribution beyond the familiar hw_c/2 term in the case of the nematic state which comes from the deformed geometry of the Fermi sea. For the stripe state we have carried out a HF calculation for a more realistic potential for a 2DEG which includes the effects of the finite confinement of the electron wave function along the z-direction.
The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.
We investigated the behavior of fractional quantum Hall (FQH) states in a two-dimensional electron system with layer thickness and an in-plane magnetic field. Our comparisons across various filling factors within the first Landau level revealed a crucial observation. A slight in-plane magnetic field specifically enhances the nematic order of the ν=7/3ν= 7/3ν=7/3 FQH state. For this particular filling, through calculating the energy gap, the Ising nematic order parameter, the pair-correlation function, and the static structure factor, we observed that as the in-plane magnetic field increases, the system first enters into an anisotropic FQH phase without closing the spectrum gap, then the FQH nematic (FQHN) phase after neutral gap closing. The system eventually enters a gapless one-dimensional charge density wave (CDW) phase for a large in-plane field. We thus provide a full phase diagram of the ν=7/3ν= 7/3ν=7/3 state in a tilted magnetic field, demonstrating the existence of the FQHN, which aligns with recent resonant inelastic light scattering (RILS) experimental observations.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power mmm of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.
Nematicity is a well-known property of liquid crystals and has been recently discussed in the context of strongly interacting electrons. An electronic nematic phase has been seen in many experiments in certain strongly correlated materials, in particular, in the pseudogap phase generic to many hole-doped cuprate superconductors. Recent measurements in high Tc superconductors have shown that even if the lattice is perfectly rotationally symmetric, the ground state can still have strongly nematic local properties. Our study of the two-dimensional one-band Hubbard model provides strong support for the recent experimental results on local rotational C4 symmetry breaking. The variational cluster approach is used here to show the possibility of an electronic nematic state and the proximity of the underlying symmetry-breaking ground state within the Hubbard model. We identify this nematic phase in the overdoped region and show that the local nematicity decreases with increasing electron filling. Our results also indicate that strong Coulomb interaction may drive the nematic phase into a phase similar to the stripe structure. The calculated spin (magnetic) correlation function in momentum space shows the effects resulting from real-space nematicity.