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

+ 6 like - 0 dislike

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:

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


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

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: 

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


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

asked Jan 4, 2015 in Theoretical Physics by wonderich (1,500 points) [ revision history ]
edited Jan 4, 2015 by wonderich

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