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arXiv:2403.08158

Dipolar BF theory and dipolar braiding statistics

Jung Hoon Han

Published Mar 13, 2024• 1 citation
arXivHTMLPDF

Abstract

We analyze the recently proposed dipolar BF theory with couplings to charge and dipole currents. The quasiparticles of the theory are either charge-like or dipole-like, and the mutual braiding statistics between charge-like and dipole-like quasiparticles are dipolar, meaning that it depends on the position of the quasiparticle being encircled. The braiding statistics between two dipole-like quasiparticles is that of ordinary anyons. We further prove that the dipolar BF theory is equivalent to the rank-2 tensor BF theory developed earlier as an effective theory for the rank-2 toric code. Although the two theories are equivalent, the dipolar BF formulation embodies the dipole symmetry explicitly and gives a clean insight into the way the dipole symmetry manifests itself in various conservation laws and the dipolar braiding statistics.

DOI
10.48550/arxiv.2403.08158

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A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.
We study braiding statistics between quasiparticles and vortices in two-dimensional charge-2m2m2m (in units of eee) superconductors that are coupled to a Z2m{\mathbb{Z}}_{2m}Z2m​ dynamical gauge field, where mmm is any positive integer. We show that there exist 16m16m16m types of braiding statistics when mmm is odd, but only 4m4m4m types when mmm is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-2m2m2m superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with Z2mf{\mathbb{Z}}_{2m}^{f}Z2mf​ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with Z4f{\mathbb{Z}}_{4}^{f}Z4f​ symmetry.
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Abstract The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.
We propose two-dimensional spin models obtained by the U(1) rank-2 tensor gauge theory with scalar or vector charges on the L_x \times L_y square lattice with p degrees of freedom per orbital and three orbitals per unit cell. The resulting spin model is a stabilizer model consisting of three mutually commuting projectors which are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying U(1) lattice gauge theory. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) given by log_p GSD = 2 + (1 +{\delta}_{L_x mod p},0)(1 +{\delta}_{L_y mod p},0) when p is a prime number. Two types of dipoles, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with certain restrictions on monopoles to hop only by p lattice spacing. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, whereas the monopole-dipole braiding is anyonic and depends on the product of the monopole charge and the dipole moment. The braiding phases can be understood as the Aharonov-Bohm phases of some emergent vector potentials.