# Duality between (1) bosons (superfluid-insulator) and (2) a bulk superconductor in a magnetic field

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In this paper, http://journals.aps.org/prb/abstract/10.1103/PhysRevB.39.2756, the authors establish a correspondence between two-dimensional bosons and a bulk superconductor in a magnetic field. They focus on boson, but it seems to be claimed that it holds even more generally.

(1) 2D bosons (T=0)              v.s.      (2) Bulk superconductor

Chemical potential $\mu$        v.s.   Applied field $H$

Bose density $n$     v.s         Total field $B$

Mott insulating phase    v.s.   Meissner phase

Density wave insulator   v.s.  Abrikosov flux lattice

Superfluid    v.s.    Non-superconducting flux line liquid

Supersolid    v.s.    Non-superconducting flux lattice

Bose glass insulator   v.s. superconducting glass

Question 1: Is that "Total field $B$" a typo of magnetization $M$? Since we have:

$$\mu \cdot n \Longleftrightarrow H \cdot M$$

or

$$\mu \cdot n \Longleftrightarrow B \cdot M$$

Question 2: Any physical intuitive picture how does this duality in this table above work?

Here is my understanding -- For example, we can derive them by representing the two equivalent theories of superfluid with superfluid U(1) phase field $\phi$ in terms of a dual equivalent theory of vortex field $\Phi$ (creating vortex or annihilate anti-vortex). Naturally, we will introduce terms like

$$|d \phi - A|^2 + \dots \Longleftrightarrow A \wedge d a +\dots = A \wedge J_{\text{charge}} +\dots \Longleftrightarrow |d \Phi- a \Phi |^2 + A \wedge d a + \dots$$

I suppose if I introduce the Maxwell term (introducing Coulomb repulsion) $dA \wedge * dA$ with $A \wedge d a$, I can integrate out $A$ to obtain an effective Messiner effect $m^2 A^2$.

More systematically, there are some hints of dualities between (see A Zee's QFT book chap VI.3) (with the help of an extra $A \wedge d a$ term, and integrating out unwanted degree of freedom.):

$$\text{Maxwell}: da \wedge *da \Longleftrightarrow \text{Meissner}: m^2 A^2$$

$$\text{Meissner}: M^2 a^2 \Longleftrightarrow \text{Maxwell}: dA \wedge * dA$$

$$\text{Chern-Simons}: a \wedge da \Longleftrightarrow \text{Chern-Simons}: A \wedge dA$$

Maxwell term (introducing Coulomb repulsion) can cause the Mott-insulating phase, and we have argue it is dual to an effective Messiner effect.

So far we obtain:

$$\text{Mott insulating phase v.s. Meissner phase}$$

Again,

Question 2: Any physical intuitive picture how this (rest of) duality in this table above work? Physically intuitively?

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