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

+ 6 like - 0 dislike
1232 views

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?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification