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arXiv:1806.03303 · Physical review. B./Physical review. B

Geometry of flux attachment in anisotropic fractional quantum Hall states

Matteo Ippoliti, R. N. Bhatt, F. D. M. Haldane

Published Aug 1, 2018• 10 citations
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Abstract

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.

DOI
10.1103/physrevb.98.085101

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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 \ensuremathν=1/m\ensuremath{\nu}=1/m\ensuremathν=1/m in terms of a minimal microscopic model of flux attachment.
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.
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.
It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states—the filling ν=1/3\nu =1/3ν=1/3 Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a 'pair amplitude' operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.
It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states—the filling Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a 'pair amplitude' operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.
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.
We construct a low energy effective theory of anisotropic fractional quantum Hall (FQH) states. We develop a formalism similar to that used in the bimetric approach to massive gravity, and apply it to describe Abelian anisotropic FQH states in the presence of external electromagnetic and geometric backgrounds. We derive a relationship between the shift, the Hall viscosity, and a new quantized coupling to anisotropy, which we term anisospin. We verify this relationship by numerically computing the Hall viscosity for a variety of anisotropic quantum Hall states using the density matrix renormalization group. Finally, we apply these techniques to the problem of nematic order and clarify certain disagreements that exist in the literature about the meaning of the coefficient of the Berry phase term in the nematic effective action.