Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

Single Mode Approximation Spectrum for a Quantum Hall Liquid Crystal

Cintia Mariela Lapilli

Published Jan 1, 2005• 0 citations
Publisher

Abstract

The single mode approximation (SMA) has been a useful tool for the determination of the collective excitation spectra of various systems. For example, Feynman used it successfully for the calculation of the spectrum of sound waves in superfluid 4He [1]; and Girvin, MacDonald and Platzman employed a projected SMA to determine that of the fractional quantum Hall (QH) effect [2] for Laughlin‐like states [3]. In this paper we calculate the excitation spectrum for density fluctuations in a QH liquid crystal. We consider nematic, tetratic, and hexatic generalizations of Laughlin’s trial wave function [3] having two‐, four‐ and six‐fold broken rotational symmetry (BRS) respectively. We perform extensive Monte Carlo (MC) simulations to compute the density‐coupled spectrum of these liquid crystalline states in the lowest Landau levels. We find significant angular dependence of the spectrum with singular behavior for long wavelengths and considerable deepening of the magnetoroton as the anisotropy is increased.

DOI
10.1063/1.1994222

Related papers

Liquid crystalline states in quantum Hall systems

Carlos Wexler, Orion Ciftja

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.

Journal of Physics Condensed Matter•Mar 28, 2002•15 citations

Density-density response function in the fractional quantum Hall effect

M. Saarela

We derive the density-density response function in the fractional quantum Hall effect using the Jastrow theory. The approximation where the two-particle correlation function is kept fixed gives as a result the single-mode approximation and the excitation spectrum derived by Girvin, MacDonald, and Platzman [Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986)]. The continuity equation satisfied by the system is also obtained. It is speculated that the quasiexcitons could be described by allowing also the two-particle correlation factor to change.

Physical review. B, Condensed matter•Jan 15, 1987•4 citations

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

Orion Ciftja, Cintia Mariela Lapilli, Carlos Wexler

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 $(L=2).$

arXiv:cond-mat/0311144•Physical Review B•Mar 17, 2004•22 citations

Dispersion of collective modes in spinful fractional quantum Hall states on the sphere

Rakesh K. Dora, Ajit Coimbatore Balram

Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings $ν{=}n/(2n{\pm}1)$ and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at $ν{=}n/(4n{\pm}1)$. We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at $ν{=}2/5$. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.

arXiv:2509.13100•Physical review. B./Physical review. B•Feb 3, 2026•1 citation

Collective-Excitation Gap in the Fractional Quantum Hall Effect

S. M. Girvin, Allan H. MacDonald, P. M. Platzman

We present a theory of the collective excitation spectrum in the fractional quantum Hall-effect regimes, in analogy with Feynman's theory for helium. The spectrum is in excellent quantitative agreement with the numerical results of Haldane. Within this approximation we prove that a finite gap is generic to any liquid state in the extreme quantum limit and that in this single-mode approximation gapless excitations can arise only as Goldstone modes for ground states with broken translation symmetry.

Physical Review Letters•Feb 11, 1985•489 citations

Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions

Hao Wang, Rajesh Narayanan, Xin Wan +1 more

We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].

arXiv:1203.1982•Physical Review B•Jul 16, 2012•57 citations
The single mode approximation (SMA) has been a useful tool for the determination of the collective excitation spectra of various systems. For example, Feynman used it successfully for the calculation of the spectrum of sound waves in superfluid 4He [1]; and Girvin, MacDonald and Platzman employed a projected SMA to determine that of the fractional quantum Hall (QH) effect [2] for Laughlin‐like states [3]. In this paper we calculate the excitation spectrum for density fluctuations in a QH liquid crystal. We consider nematic, tetratic, and hexatic generalizations of Laughlin’s trial wave function [3] having two‐, four‐ and six‐fold broken rotational symmetry (BRS) respectively. We perform extensive Monte Carlo (MC) simulations to compute the density‐coupled spectrum of these liquid crystalline states in the lowest Landau levels. We find significant angular dependence of the spectrum with singular behavior for long wavelengths and considerable deepening of the magnetoroton as the anisotropy is increased.
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.
We derive the density-density response function in the fractional quantum Hall effect using the Jastrow theory. The approximation where the two-particle correlation function is kept fixed gives as a result the single-mode approximation and the excitation spectrum derived by Girvin, MacDonald, and Platzman [Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986)]. The continuity equation satisfied by the system is also obtained. It is speculated that the quasiexcitons could be described by allowing also the two-particle correlation factor to change.
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),L=0,1),L=0,1), whereas the nematic is favorable at the second excited LL (L=2).(L=2).(L=2).
Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings ν=n/(2n±1)ν{=}n/(2n{\pm}1)ν=n/(2n±1) and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at ν=n/(4n±1)ν{=}n/(4n{\pm}1)ν=n/(4n±1). We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at ν=2/5ν{=}2/5ν=2/5. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.
We present a theory of the collective excitation spectrum in the fractional quantum Hall-effect regimes, in analogy with Feynman's theory for helium. The spectrum is in excellent quantitative agreement with the numerical results of Haldane. Within this approximation we prove that a finite gap is generic to any liquid state in the extreme quantum limit and that in this single-mode approximation gapless excitations can arise only as Goldstone modes for ground states with broken translation symmetry.
We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].