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Equation 4 · Part 8 · More Is Different: Emergence and Phase Transitions

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δKi′=∑jΛij δKj,Λij=∂Ki′∂Kj∣K∗\delta K'_i = \sum_j \Lambda_{ij} \, \delta K_j, \qquad \Lambda_{ij} = \frac{\partial K'_i}{\partial K_j} \Big|_{K^{*}}
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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, δKi′=∑jΛij δKj,Λij=∂Ki′∂Kj∣K∗\delta K'_i = \sum_j \Lambda_{ij} \, \delta K_j, \qquad \Lambda_{ij} = \frac{\partial K'_i}{\partial K_j} \Big|_{K^{*}}. sorts perturbations into three kinds by the eigenvalues of that matrix. Relevant directions grow under iteration and drive the system away from criticality; there are usually very few of them, typically corresponding to temperature and to a symmetry-breaking field. Irrelevant directions shrink to zero under iteration. Marginal directions decide at higher order.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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