Equation 4 · Part 13 · More Is Different: Emergence and Phase Transitions
Starting index or lower bound: j
What this part means
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.
Its job in the formula
j occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Full expression→Starting index or lower bound: j→Article meaning
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, . 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.
Learn the underlying idea
Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.
Open the illustrated sums and products: repeat an operation over an index guide →
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Sources cited in the article section
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