Kramers-Wannier Duality in the Transverse Ising Model
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Is there a block spin renormalization group scheme that preserves Kramers-Wannier duality?
Block spin renormalization group (RG) (or real space RG) is an approach to studying statistical mechanics models of spins on the lattice. In particular, I am interested in the 2D square lattice model with on-site degrees of freedom (i.e. spins) being the elements $g_i$ in a finite Abelian group $g_i\in G$, and the partition function has the following form
$$\begin{split}Z&=\sum_{[g_i\in G]}e^{-S[g_i]},\\S[g_i]&=-\sum_{\langle ij\rangle}K(g_i^{-1}g_j),\end{split}$$
where $K(g)$ is a suitable group function ($K:G\to\mathbb{R}$) to penalize the group elements that differ from identity. The block spin RG partitions the lattice into small blocks (labeled by $I,J$) and rewrite the action in terms of the block spin $g_I\sim\sum_{i\in I}g_i$ (how to map the sum back to an element $g_I\in G$ depends on the RG scheme), such that the partition function can be rewritten as
$$Z=\sum_{[g_I\in G]}\sum_{[g_i\in G]}\delta\big[g_I\sim\textstyle{\sum}_{i\in I}g_i\big]e^{-S[g_i]}=\sum_{[g_I\in G]}e^{-S'[g_I]},$$
where the new action takes the form of
$$S'[g_i]=-\sum_{\langle IJ\rangle}K'(g_I^{-1}g_J)+\cdots.$$
By omitting higher order terms generated under RG, the RG procedure can be considered as a function map $\mathcal{R}$ that takes the group function $K(g)$ to $K'(g)$.
On the other hand, such a model of finite Abelian group $G$ on the square lattice admits the Kramers-Wannier duality. The key step of the duality is a Fourier transform (on the Abelian group $G$)
$$e^{-\tilde{K}(\tilde{g})}=\sum_{g\in G}e^{-K(g)}\;\chi(g,\tilde{g}),$$
where $\tilde{g}$ is a representation of $G$, and $\chi(g,\tilde{g})$ is the character. Due to the fact that the representation of a finite Abelian group $G$ also forms a finite Abelian group $\tilde{G}$, and $\tilde{G}$ is isomorphic to $G$ (meaning that the dual group $\tilde{G}$ is the same as $G$). Combining with the fact that the dual lattice of the square lattice is still a square lattice, the Kramers-Wannier duality can be considered as a bijective functional map $\mathcal{D}$ that maps $K(g)$ to $\tilde{K}(g)$ (and vice versa).
However, it is not obvious to me that the block spin RG preserves the Kramers-Wannier duality. I think in general the RG transformation $\mathcal{R}$ is not guaranteed to commute with the duality transformation $\mathcal{D}$, or say the following diagram does not commute in general:
$$\begin{array}{lllllll} K & \overset{\mathcal{R}}{\to} & K' & \overset{\mathcal{R}}{\to} & K'' & \overset{\mathcal{R}}{\to} \cdots\\ \updownarrow\tiny\mathcal{D} & & \updownarrow\tiny\mathcal{D} & & \updownarrow\tiny\mathcal{D} & \\ \tilde{K} & \overset{\mathcal{R}}{\to} & \tilde{K}' & \overset{\mathcal{R}}{\to} & \tilde{K}'' & \overset{\mathcal{R}}{\to} \cdots \end{array}$$
So the question is how to design the block spin RG scheme to make the above diagram commute? Is there a systematic design of the block spin RG scheme that preserves the Kramers-Wannier duality?
This post imported from StackExchange Physics at 2016-07-02 16:14 (UTC), posted by SE-user Everett You
condensed-matter
renormalisation
asked Jun 25, 2016 in Theoretical Physics by Everett You (785 points) [ no revision ]
retagged Jul 2, 2016
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