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arXiv:cond-mat/0311144 · Physical Review B

Liquid crystalline states for two-dimensional electrons in strong magnetic fields

Orion Ciftja, Cintia Mariela Lapilli, Carlos Wexler

Published Mar 17, 2004• 22 citations
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Abstract

Based on the Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-dimensional melting and the analogy between Laughlin states and the two-dimensional one-component plasma, we investigate the possibility of liquid crystalline states in a single Landau level (LL). We introduce many-body trial wave functions that are translationally invariant but possess twofold (i.e., nematic), fourfold (tetratic), or sixfold (hexatic) broken rotational symmetry at respective filling factors $\ensuremath{\nu}=1/3,$ 1/5, and 1/7 of the valence LL. We find that the above liquid crystalline states exhibit a soft charge-density wave (CDW) which underlies the translationally invariant state but which is destroyed by quantum fluctuations. By means of Monte Carlo simulations, we determine that, for a considerable variety of interaction potentials, the anisotropic states are energetically unfavorable for the lowest and first excited LL's (with index $L=0,1),$ whereas the nematic is favorable at the second excited LL $(L=2).$

DOI
10.1103/physrevb.69.125320

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"This thesis is devoted to the investigation of the quantum liquid crystal phases in strongly correlated electronic systems. Such phases are characterized by their partially broken spatial symmetries and are observed in various strongly correlated systems as being summarized in Chapter 1. In quantum liquid crystal phases formed by fermions, behaviors beyond Landau's theory of the Fermi liquids appear naturally, signed by the absence of well-defined quasiparticle excitations. Although quantum liquid crystal phases often involve strong-coupling physics, the associated symmetry analysis can often be carried out also in a relatively weak coupling regime and this is the main approach taken in this thesis. Two different quantum liquid crystal phases, the nematic state and the smectic state, are studied in Chapters 2 and 3 respectively, by investigating various instabilities of a Landau Fermi liquid. 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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.

•Jan 1, 2008•0 citations
Based on the Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-dimensional melting and the analogy between Laughlin states and the two-dimensional one-component plasma, we investigate the possibility of liquid crystalline states in a single Landau level (LL). We introduce many-body trial wave functions that are translationally invariant but possess twofold (i.e., nematic), fourfold (tetratic), or sixfold (hexatic) broken rotational symmetry at respective filling factors \ensuremathν=1/3,\ensuremath{\nu}=1/3,\ensuremathν=1/3, 1/5, and 1/7 of the valence LL. We find that the above liquid crystalline states exhibit a soft charge-density wave (CDW) which underlies the translationally invariant state but which is destroyed by quantum fluctuations. By means of Monte Carlo simulations, we determine that, for a considerable variety of interaction potentials, the anisotropic states are energetically unfavorable for the lowest and first excited LL's (with index L=0,1),L=0,1),L=0,1), whereas the nematic is favorable at the second excited LL (L=2).(L=2).(L=2).
At low temperatures, two-dimensional electron systems in a perpendicular magnetic field exhibit remarkable quantum phenomena. In these extreme conditions, strongly correlated electron systems stabilize in different quantum phases as the filling factor of Landau levels is varied. In this Brief Report we present finite-size Monte Carlo simulation results for anisotropic quantum Hall liquid states observed at certain even-denominator filling factors. The anisotropic electronic liquid phases are described by a broken rotational symmetry wave function. Our investigations of systems of few electrons in disk geometry indicate that an anisotropic liquid-crystalline quantum Hall phase with broken rotational symmetry is energetically favored relative to an isotropic liquid one.
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 electron-electron 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.
We investigate liquid crystalline phases with nematic order at (1/3)-filling of the valence Landau level (LL). We generalize Laughlin's fractional quantum Hall (QH) effect wavefunction at ν = (1/3) to include anisotropic nodal distribution by modifying the Jastrow factors. Lengthy Monte Carlo simulations are then used to determine with unprecedented accuracy the (anisotropic) pair distribution function g ( r ) and static structure factor S ( q ) for various degrees of anisotropy. The determination of the correlation energies at (1/3)-filling of an arbitrary LL is then performed by using standard mappings of g ( r ) and S ( q ) to higher LLs. Our results indicate that while Laughlin's state is stable in the lowest LL, there are regions of instability towards nematic order in higher LLs. Possible connections to the recently discovered QH liquid crystals are discussed.
"This thesis is devoted to the investigation of the quantum liquid crystal phases in strongly correlated electronic systems. Such phases are characterized by their partially broken spatial symmetries and are observed in various strongly correlated systems as being summarized in Chapter 1. In quantum liquid crystal phases formed by fermions, behaviors beyond Landau's theory of the Fermi liquids appear naturally, signed by the absence of well-defined quasiparticle excitations. Although quantum liquid crystal phases often involve strong-coupling physics, the associated symmetry analysis can often be carried out also in a relatively weak coupling regime and this is the main approach taken in this thesis. Two different quantum liquid crystal phases, the nematic state and the smectic state, are studied in Chapters 2 and 3 respectively, by investigating various instabilities of a Landau Fermi liquid. A nematic phase is a state that breaks the rotational symmetry spontaneously but preserves the translational symmetries. It can be reached by the Pomeranchuk instability of a Landau Fermi liquid. The Pomeranchuk instability in the charge (spin-singlet) sector of a Fermi liquid was studied previously within a random phase approximation and by a high-dimensional bosonization method. In Chapter 2, we concentrate on the Pomeranchuk instability in the spin-triplet sector, and two different classes of ordered phases are discovered, dubbed the a and ,8-phases by analogy to the superfluid 3He-A and B-phases. The Fermi surfaces in the a-phases exhibit spontaneous anisotropic distortions, while in the ,8-phases the Fermi surfaces remain circular or spherical with topologically non-trivial spin configurations in momentum space. The behaviors of the low energy bosonic excitations are studied both in the order phases (a and .B) and at the quantum critical point separating the ordered phases from the Fermi liquid phase. In Chapter 3, the smectic phase and the quantum phase transition between a metallic nematic state and a metallic smectic phase are investigated using the random phase approximation, by considering the formation of a unidirectional charge density wave in an electronic nematic state. The associated quantum critical point is investigated within the Hertz-Millis approach. Non-Fermi liquid behaviors are discovered both at the quantum phase transition and in the smectic ordered phase. The theory we studied here provides a simple description for the so-called ""fluctuating stripe"" phase in cuprates. Similar to its classical counterpart, the phase transition between the nematic and smectic phases are described by a ""gauge-like"" theory as required by the rotational invariance in close analogy with the McMillan-deGennes theory of the smectic-nematic phase transition in classical liquid crystals. However, unlike the classical McMillandeGennes theory, in which the coupling between the ""gauge-like"" nematic fluctuations and the ""matter-like"" smectic fields is relevant, in the sense of renormalization group, this coupling is irrelevant at the quantum phase transition. This property enables the study of low energy fluctuations and the fate of fermions near this quantum critical point. In Chapter 4, quantum liquid crystal phases with a broken time-reversal symmetry are investigated. We start by studying the general scenario of timereversal symmetry breaking phases in the absence of magnetic ordering in systems with long-lived low-energy fermionic quasiparticle excitations in two dimensions. Using a Berry phase approach, we classified possible time-reversal symmetry breaking phases by the accompanying spatial symmetry breaking patterns. Two of the simplest situations are referred to as the type I state, which breaks simultaneously the space-inversion symmetry and the time-reversal symmetry, and the type I I state, in which the chiral symmetry is broken in addition to the breaking of the time-reversal symmetry. Although the starting point is general and has little relation with the quantum liquid crystal phases, both type I and type II states defined above are realized by (generalized) Pomeranchuk instabilities. In fact , a type I I state is usually a nematic state which also breaks the time-reversal symmetry. Interestingly, the type II states also shows spontaneously generated Hall effect even in the absence of a magnetic field, which is known as the spontaneous anomalous Hall effect. In Chapter 5, we investigate the stability of a quadratic band-crossing point in two-dimensional fermionic systems that provide an explicit example, within well controlled approximations, for the nematic ordering discussed in Chapter 2 and spontaneous anomalous Hall effect discussed in Chapter 4. At the non-interacting level, we show that a quadratic band-crossing point exists and is topologically stable when the Berry flux of this crossing point is ±27r and the point symmetry group at this point has either fourfold or sixfold rotational symmetries. Strikingly, we find that this would-be topologically stable quadratic band-crossing is marginally unstable against short-range interactions: a quadratic band-crossing point in two dimensions is unstable against shortrange repulsion, no matter how weak! Four ordered phases are found in this system, the quantum anomalous Hall state, the quantum spin Hall phase, the nematic phase, and the nematic-spin-nematic phase."
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.
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.