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Equation 3 · Part 2 · More Is Different: Emergence and Phase Transitions

Symbol K

Rb:{K}⟶{K′}\mathcal{R}_b : \{K\} \longrightarrow \{K'\}
KK

What this part means

K is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Its job in the formula

K is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

The passage around this formula

Take a system described by a set of couplings, coarse-grain it by integrating out the shortest-wavelength degrees of freedom, then rescale lengths so the cutoff returns to its original value. The result is a system of the same form with different couplings. Iterating defines a flow on the space of all possible Hamiltonians: Rb:{K}⟶{K′}\mathcal{R}_b : \{K\} \longrightarrow \{K'\}. A critical point corresponds to a fixed point of this flow, a Hamiltonian that maps to itself, which is exactly the scale-invariance the diverging correlation length demanded. Linearising the map about the fixed point,

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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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