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arXiv:2111.04214 · Physical review. B./Physical review. B

Anisotropy and quench dynamics of quasiholes in fractional quantum Hall liquids

Chao Han, Zhao Liu

Published Jan 4, 2022• 2 citations
arXivHTMLPDF

Abstract

We present a microscopic study of quasiholes in bosonic fractional quantum Hall (FQH) liquids at filling factor $\ensuremath{\nu}=1/2$ in the lowest Landau level with anisotropic band mass tensors. We use the spatial density profile to characterize the shape of a quasihole and analyze its anisotropy. We then compare the quasihole's anisotropy with the intrinsic geometric metric of the system that is extracted from the maximal overlap between the numerically obtained quasihole ground state and a set of model wave functions of anisotropic quasiholes. For a static system, we find that the quasihole's anisotropy, similar to the intrinsic metric, grows with the anisotropy of the band mass tensor. When the quasihole develops well, we observe a correspondence between the anisotropy of the quasihole and the intrinsic metric of the underlying anisotropic FQH state. We also drive the system out of equilibrium by suddenly changing the band mass tensor. In this case, the shape of the quasihole evolves with time and shows similar dynamics with the intrinsic metric of the postquench state. The evolving frequency matches the energy of a spin-2 quadrupole degree of freedom in the system. Our results suggest that the density profile of a quasihole is a useful tool to estimate the intrinsic metric and capture the dynamics of an FQH system.

DOI
10.1103/physrevb.105.045108

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We present a microscopic study of quasiholes in bosonic fractional quantum Hall (FQH) liquids at filling factor \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 in the lowest Landau level with anisotropic band mass tensors. We use the spatial density profile to characterize the shape of a quasihole and analyze its anisotropy. We then compare the quasihole's anisotropy with the intrinsic geometric metric of the system that is extracted from the maximal overlap between the numerically obtained quasihole ground state and a set of model wave functions of anisotropic quasiholes. For a static system, we find that the quasihole's anisotropy, similar to the intrinsic metric, grows with the anisotropy of the band mass tensor. When the quasihole develops well, we observe a correspondence between the anisotropy of the quasihole and the intrinsic metric of the underlying anisotropic FQH state. We also drive the system out of equilibrium by suddenly changing the band mass tensor. In this case, the shape of the quasihole evolves with time and shows similar dynamics with the intrinsic metric of the postquench state. The evolving frequency matches the energy of a spin-2 quadrupole degree of freedom in the system. Our results suggest that the density profile of a quasihole is a useful tool to estimate the intrinsic metric and capture the dynamics of an FQH system.
We introduce a quench of the geometry of Landau level orbitals as a probe of nonequilibrium dynamics of fractional quantum Hall (FQH) states. We show that such geometric quenches induce coherent many-body dynamics of neutral degrees of freedom of FQH fluids. The simplest case of mass anisotropy quench can be experimentally implemented as a sudden tilt of the magnetic field, and the resulting dynamics reduces to the harmonic motion of the spin-2 ``graviton'' mode, i.e., the long-wavelength limit of the Girvin-MacDonald-Platzman magnetoroton. We derive an analytical description of the graviton dynamics using the bimetric theory of FQH states and find agreement with exact numerical simulations at short times. We show that certain types of geometric quenches excite higher-spin collective modes, thus establishing their existence in a microscopic model and motivating an extension of geometric theories of FQH states.
Fractional quantum Hall (FQH) states are known to possess an internal metric degree of freedom that allows them to minimize their energy when contrasting geometries are present in the problem (e.g., electron band mass and dielectric tensor). We investigate the internal metric of several incompressible FQH states by probing their response to band mass anisotropy using infinite DMRG simulations on a cylinder geometry. We test and apply a method to extract the internal metric of an FQH state from its guiding center structure factor. We find that the response to band mass anisotropy is approximately the same for states in the same Jain sequence, but changes substantially between different sequences. We provide a theoretical explanation of the observed behavior of primary states at filling $\ensuremath{\nu}=1/m$ in terms of a minimal microscopic model of flux attachment.
We investigate the recently introduced geometric quench protocol for fractional quantum Hall (FQH) states within the framework of exactly solvable quantum Hall matrix models. In the geometric quench protocol, a FQH state is subjected to a sudden change in the ambient geometry, which introduces anisotropy into the system. We formulate this quench in the matrix models and then we solve exactly for the postquench dynamics of the system and the quantum fidelity (Loschmidt echo) of the postquench state. Next, we explain how to define a spin-2 collective variable ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{ab}(t)$ in the matrix models, and we show that for a weak quench (small anisotropy), the dynamics of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{ab}(t)$ agrees with the dynamics of the intrinsic metric governed by the recently discussed bimetric theory of FQH states. We also find a modification of the bimetric theory such that the predictions of the modified bimetric theory agree with those of the matrix model for arbitrarily strong quenches. Finally, we introduce a class of higher-spin collective variables for the matrix model, which are related to generators of the ${W}_{\ensuremath{\infty}}$ algebra, and we show that the geometric quench induces nontrivial dynamics for these variables.
In the absence of rotational symmetry, a fractional quantum Hall (FQH) system can exploit a geometric degree of freedom to minimize its ground-state energy. The mass anisotropy of bare particles interacting isotropically is partially inherited by the many-body FQH state, to an extent that depends on the type of interaction, filling fraction, and ground-state phase. Using numerical infinite density matrix renormalization group simulations, we investigate the transference of elliptical (C2{C}_{2}C2​-symmetric) anisotropy from the band mass of the bare particles to the FQH states, for various power-law interactions. We map out the response of FQH states to small anisotropy as a function of power-law exponent, filling, and statistics (bosonic or fermionic) of the constituents. Interestingly, we find a nonanalyticity in the linear response of the FQH state at a special filling-dependent value of the power-law exponent, above which the interactions effectively become zero-range (pointlike). We also investigate the effect of C4{C}_{4}C4​-symmetric band distortions, where we observe a strikingly different dependence on filling.
We introduce different types of quenches to probe the nonequilibrium dynamics and multiple collective modes of bilayer fractional quantum Hall states. We show that applying an electric field in one layer induces oscillations of a spin-1 degree of freedom, whose frequency matches the long-wavelength limit of the dipole mode. On the other hand, oscillations of the long-wavelength limit of the quadrupole mode, i.e., the spin-2 graviton, as well as the combination of two spin-1 states, can be activated by a sudden change of band mass anisotropy. We construct an effective field theory to describe the quench dynamics of these collective modes. In particular, we derive the dynamics for both the spin-2 and the spin-1 states and demonstrate their excellent agreement with numerics.
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.