Real-Space Renormalization Group Theory

Lavis, David A.

doi:10.1007/978-94-017-9430-5_15

David A. Lavis¹⁵

Part of the book series: Theoretical and Mathematical Physics ((TMP))

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Abstract

As in Sect. 16.2.1 we consider a $d$-dimensional lattice ${\mathcal {N}}$, with a lattice site ${\varvec{r}}$ given in terms of unit lattice vectors by (16.2.1).

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Notes

1.
Not a full group since no inverse transformation is defined.
2.
The term ‘decimation’ (originating in a form of discipline used in the Roman army) is more properly used when one tenth of the sites is removed; but here it is employed more loosely to denote any proportion.
3.
This mixed notation where $\zeta _2$ is a coupling and $\zeta _1$ is a Boltzmann factor turns out the be the most convenient in this case.
4.
The argument $\zeta _i$ now represents the set of couplings in $\widehat{H}_0$ and any further couplings in the second term in (15.7.11) are absorbed as ratios in $\widehat{C}$.
5.
For any $A(\sigma ({{\varvec{r}}}))$
$$\begin{aligned} \langle \exp (A)\rangle _0&= \exp \langle A\rangle _0\langle \exp (A-\langle A \rangle _0)\rangle _0\nonumber \\&= \exp \langle A\rangle _0\langle 1 + (A-\langle A\rangle _0) +{\textstyle \frac{1}{2}} (A-\langle A\rangle _0)^2+\cdots \rangle _0\ge \exp \langle A\rangle _0. \end{aligned}$$
6.
Equation (15.7.50) is obvious. Equation (15.7.51), which is equivalent to , follows from the fact that this latter equation is invariant under interchange of and .
7.
We omit the factor , which, as explained in Sect. 12.2.7, is an artifact of our method of derivation.

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Department of Mathematics, King’s College London, London, UK
David A. Lavis

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Lavis, D.A. (2015). Real-Space Renormalization Group Theory. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_15

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DOI: https://doi.org/10.1007/978-94-017-9430-5_15
Published: 31 January 2015
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-9429-9
Online ISBN: 978-94-017-9430-5
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