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arXiv:1701.00461

Boson-Fermion Duality from Discontinuous Gauge Symmetry

Jianing Xu, Shuai Zhao

Published Dec 28, 2016• 0 citations
arXivHTMLPDF

Abstract

By extending local U(1) gauge symmetry to discontinuous case, we find that under one special discontinuous U(1) gauge transformation the symmetric and antisymmetric wave functions can transform into each other in one dimensional quantum mechanics. The free spinless fermionic system and bosonic system with $δ$-type vector gauge potential are proved to be equivalent. The relation also holds in higher space-time dimensions.

DOI
10.48550/arxiv.1701.00461

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By extending local U(1) gauge symmetry to discontinuous case, we find that under one special discontinuous U(1) gauge transformation the symmetric and antisymmetric wave functions can transform into each other in one dimensional quantum mechanics. The free spinless fermionic system and bosonic system with δδδ-type vector gauge potential are proved to be equivalent. The relation also holds in higher space-time dimensions.
Recent developments have established exact bosonization and fermionization with a Z2\mathbb{Z}_2Z2​ symmetry as dualities through gauging. In this work, we apply fermionic gauging, which realizes generalized Jordan-Wigner transformations, to the Bose-Hubbard (BH) model with a global U(1)U(1)U(1) symmetry and derive an exact dual description in terms of fermionic composites, built from bosons and fermions. In 1D, this duality generalizes the exact mapping between the extended hard-core BH model and the spinless Fermi-Hubbard model to include soft-core bosons. At low energies, the mapping reduces to the well-known equivalence between the sine-Gordon model and the Thirring model. The oscillation wave vector of the fermionic composite correlation function in the gapless phase is fixed by their density, providing a novel manifestation of Luttinger's theorem. We verify the exact duality using density matrix renormalization group (DMRG) calculations and demonstrate that the gapless phase and the phase transition are governed by the compact boson conformal field theory. Our construction naturally extends to generic bosonic systems and higher dimensions, opening new avenues for studying Bose-Fermi mixtures in optical lattices and other strongly correlated quantum systems.
One-dimensional quantum systems admit duality relations that put hard-core spinless bosons and fermions in one-to-one correspondence via Girardeau's mapping theorem. The simplest models of soft bosons interacting via zero-range potentials can also be mapped onto dual interacting fermions. However, a systematic approach to one-dimensional statistical transmutation for arbitrary low-energy interactions in the spinless and spinful or multicomponent cases has remained elusive. I develop a general theory of local unitary transformations between one-dimensional quantum systems of bosons and fermions with arbitrary spin or internal structure, single-particle dispersion---including nonrelativistic, relativistic, or otherwise---and low-energy interactions in the universal regime. These transformations generate families of duality relations and models that relate the strong- and weak-coupling limits of the respective dual theories.
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Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.
It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.