Abstract
We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the nonlinear sigma models of ferromagnets in 2+1D and 3+1D exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.
Highlights
We show that this resemblance is not accidental; the nonlinear higher-rank symmetry is realized as a symmetry of the problem of charged particles on the Landau level (LLL)
We have presented a nonlinear version of a higher-rank gauge symmetry
The symmetry is basically that of volume-preserving diffeomorphism
Summary
Considerable interest has been drawn to “higher-rank gauge theories,” i.e., theories where the gauge potential is not a one-form, but a tensor of higher rank [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. The nonabelian nature of the gauge symmetry has a nontrivial consequence: the charge density operators at different positions do not commute Instead, they form the long-wavelength limit of the Girvin-MacDonaldPlatzman (GMP) algebra [32,33,34], which reveals a connection to the lowest Landau level (LLL). It is easy to notice that many features of the physics on the LLL bear a close resemblance to that of the field-theory models with higher rank symmetries [35]: for example, electric charges are pinned to one place by the large magnetic field, and neutral excitations (e.g., the composite fermion in the half-filled Landau level [36]) carry an electric dipole moment and can move in the direction perpendicular to the direction of motion. The higher rank gauge symmetry provides a new interpretation of the conservation of multipole moments in ferromagnets [38]; it explains the close resemblance between the behaviors of skyrmions in ferromagnets and charged particles in a magnetic field [39]
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