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Published equation contexts

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^{*}}

Why this formula appears here

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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jj

Symbol j

j occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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Λij\Lambda_{ij}

Symbol Lambda_ij

Lambdaia_ij is one factor in the product that computes the quantity on the left.

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KjK_j

Symbol K_j

KjK_j occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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jj

Starting index or lower bound: j

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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∂Kj\partial K_j

Denominator: partial K_j

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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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^{*}}

Equation 4 · Condensed Matter

More Is Different: Emergence and Phase Transitions

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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