# Is bosonization possible in any number of dimensions?

Is bosonization applicable to an arbitrary number of spacetime dimensions?

• If you mean the Cooper pairs then yes. – user256968 Mar 25 '20 at 18:57
• I don't know any examples of bosonization in 3+1d, but exist very famous examples in 1+1d and 2+1d. – Nikita Mar 25 '20 at 19:25

## 2 Answers

Exact bosonization exists in all dimensions, i.e., mapping from even fermionic operators to Pauli matrices (with certain constraints on Hilbert space) in arbitrary triangulation which preserves the algebra. The simplest example should be Bravyi-Kitaev superfast simulation of fermions mentioned in https://arxiv.org/abs/quant-ph/0003137. Their construction preserves the locality of operators and is valid for any graph.

We have a more formal way to derive the exact bosonization in

All dimensions: https://arxiv.org/abs/1911.00017

In (2+1)D, the exact bosonization is described by flux attachment. For example, given a toric code, we combine both charge excitation $$e$$ and flux excitation $$m$$ to form an emergent fermion $$\epsilon = em$$. This can help us to define an isomorphism between the fermionic operators and logical operators in the toric code subspace where only $$\epsilon$$ excitations are allowed. In other words, any fermionic theory in two dimensions is dual to a 1-form $$\mathbb{Z}_2$$ lattice gauge symmetry. If we go to spacetime picture, the above flux attachment approach can be described by the Chern-Simon term in spacetime action: $$\int A \cup \delta A$$ where $$A$$ is the 1-cochain representing the $$\mathbb{Z}_2$$ fields living on edges.

In (3+1)D, we used higher cup products to formulate the bosonization map in Hamiltonian level. The formula is analogous to flux attachment in (2+1)D. We found that the spacetime action (after Euclidean path integral) is: $$\int B\cup B + B \cup_1 \delta B$$ where B is the 2-cochain representing the $$\mathbb{Z}_2$$ fields living on faces. This action is called Steenrod square, which is used for the classification of fermionic SPT phases: https://arxiv.org/abs/1809.01112.

The statement above can be generalized to all dimensions easily.

• so is this proves the conjectured duality between abelian higgs model with CS term at its wilson fisher fixed point and massless dirac fermions. – physshyp Sep 8 '20 at 3:57

The technique normally referred to as Bosonization only works in 1 spatial dimension. The technique is a mapping between the the Fock space of fermions (with linear dispersion) and the Fock space of Bosons.

The proof that the mapping is an isomorphism was given by Haldane who computed the grandcanonic partition function in the two cases (and obtained the same result).

In practice (part of) the success of this technique is that some Hamiltonians that look interacting in the fermionic picture become non-interacting after Bosonization (i.e. in the bosonic picture) and so can be solved exactly. This was done by Tomonaga, Luttinger and others.

• Your answer seems at odds with the other one given here. I wonder whether the 2D and 3D cases referred to there can be shown to be effectively one-dimensional. – Roger Vadim Apr 2 '20 at 10:16
• @vadim I think there could be a terminology issue. The constructions in the other answers are generalizations of the Jordan-Wigner transformation in D dimensions. So strictly speaking they are mappings from fermions to spins, not bosons. This holds at least for the paper by Kitaev Bravyi and the 2D by Yu an Chen and Kaputsin paper. Let's just say that my answer is the standard one. The mapping between Fock spaces, per se is independent of dimension but I think the result is local only in 1D. Again people tried various generalizations, but let's just say that the standard result holds in 1D. – lcv Apr 3 '20 at 7:26
• I guess the terminology may be a little different. In the 1d system, Jordan-Wigner and the "standard" 1d bosonization are for discrete lattices and continuous fields, respectively. These two approaches are equivalent. In higher dimensions, the Jordan-Wigner can be generalized using $\mathbb{Z}_2$ lattice gauge theory. However, the continuous analogy is unclear since the Steenrod square action is only well-defined for $\mathbb{Z}_2$ cohomology. Tsui and Wen (arxiv.org/abs/1908.02613) seem to have defined the Steenrod square for $\mathbb{Z}_n$. That may help to generalize the story. – Yu-An Chen Apr 3 '20 at 9:29