1. Exact bosonization in two spatial dimensions and a new class of lattice gauge theories Authors: Yu-An Chen, Anton Kapustin, and Djordje Radicevic Annals of Physics 393, 234 (2018); DOI: 10.1016/j.aop.2018.03.024 arXiv:1711.00515 2. Constraints of kinematic bosonization in two and higher dimensions Authors: Arkadiusz Bochniak, Blazej Ruba, Jacek Wosiek, and Adam Wyrzykowski Phys. Rev. D […]
Exact bosonization in two spatial dimensions and a new class of lattice gauge theories Authors: Yu-An Chen, Anton Kapustin, and Djordje Radicevic Annals of Physics 393, 234 (2018); DOI: 10.1016/j.aop.2018.03.024 arXiv:1711.00515 2. Constraints of kinematic bosonization in two and higher dimensions Authors: Arkadiusz Bochniak, Blazej Ruba, Jacek Wosiek, and Adam Wyrzykowski Phys. Rev. D […]
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
The well-known Jordan-Wigner transformation maps an arbitrary system of fermions on a one-dimensional lattice to a system of spins. An essential property of this transformation is that it preserves the locality of observables. This transformation has been recently extended to two-dimensional lattices, where it maps an arbitrary system of fermions to a gauge theory, with locality preserved. Here, the authors construct a bosonization map for arbitrary systems of fermions in three dimensions. As an application, the authors design a model of interacting spins on a cubic lattice, which is exactly soluble and is equivalent to a tight-binding model of free fermions with Dirac points.
The Jordan-Wigner transformation is traditionally applied to one dimensional systems, but recent works have generalized the transformation to fermionic lattice systems in higher dimensions while keeping locality manifest. These developments could aid the theoretical or even experimental studies of strongly correlated electronic problems through their bosonic counterparts. In this work, we develop a scheme for higher-dimensional Jordan-Wigner transformation which keeps all relevant symmetries manifest on the bosonic side. Our approach connects the discussion of exact lattice bosonization to the familiar notions of fractionalized partons like spinons and chargeons, and works for spin-1/2 fermions -- like the physical electrons -- on four-coordinated lattices. The construction is applied to fermions defined on the square, kagome and diamond lattices, and we provide explicit expressions for the bosonized versions of well-known models of strongly correlated electrons, like the Hubbard and t-J models.
We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Chen et al. [Ann.Phys.(N.Y) 393, 234 (2018)], whose gauge constraints project onto the subspace of the toric code with emergent fermions.Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions).In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice.We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios r = 1 + 1/2k for positive integers k, utilizing the trivialness of quantum cellular automata in two spatial dimensions.Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1.When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional Jordan-Wigner transformation along a certain path in the two-dimensional lattice.Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.