Real-space renormalization group

A real-space renormalization group replaces groups of microscopic degrees of freedom by retained coarse variables, summing over the eliminated variables to define a coarse Hamiltonian. The effective Boltzmann weights must preserve the partition function up to a tracked normalization. Spin decimation retains selected spins rather than forming an average block variable. Coarse-graining generally generates additional interactions, so closure of a chosen finite coupling family requires justification.

Table of contents
- Reciprocal coordinate at infinite coupling Real-space renormalization group
- Spin decimation Real-space renormalization group
  - Spin-1 chain decimation recursion Spin decimation

Reciprocal coordinate at infinite coupling ( $w = 1/ v$ )

 0  0

An infinite positive coupling coordinate

v

can be studied through

w = 1/ v

, with

v = \infty

represented by

w = 0

. For example,

v^{'} = v^{4} / A (u)

with

A (u) > 0

becomes

w^{'} = A (u) w^{4}

, a smooth map at zero. This gives a well-defined boundary renormalization-group fixed point whose derivative in the reciprocal direction is zero. One should use this chart rather than substituting infinity into an ordinary finite-coordinate Jacobian.

Spin decimation

 0  0

Spin decimation sums over spins at selected lattice sites while retaining the others. For alternate-site elimination in a nearest-neighbour periodic chain of even length, the coarse spin-chain transfer matrix is

W^{2}

, because its

(s, t)

entry sums

W_{s a} W_{a t}

over the eliminated middle spin. Exact preservation

tr (W^{2})^{N /2} = tr W^{N}

requires keeping the scalar normalization as well as the entry ratios.

Spin-1 chain decimation recursion ( $W ⟼ W^{2}$ )

 0  0

For the spin inversion symmetry parameterization

W = c 1 x y x z x y x 1

with positive entries, spin decimation gives the same form with

x^{'} = x (1 + y + z) / (1 + x^{2} + y^{2})

y^{'} = (x^{2} + 2 y) / (1 + x^{2} + y^{2})

and

z^{'} = (z^{2} + 2 x^{2}) / (1 + x^{2} + y^{2})

. Multiply

W

by itself and divide by its

(+, +)

entry to prove these ratios. The leftover scalar is an additive free-energy coupling; dropping it leaves normalized expectations unchanged but loses the full free energy.