Apply scale factor b to couplings in click. See linearized eigenvalues and stability instantly here. Iterate, compare, and download clean tables for sharing fast.
A common renormalization-group step rescales lengths by a factor b and updates couplings. Near a fixed point, many models are well approximated by a linear map:
g' = J g, J = A · diag(b^{y1}, b^{y2}, b^{y3})
| b | n | g1 | y1 | g2 | y2 | Mode |
|---|---|---|---|---|---|---|
| 2 | 8 | 0.20 | 1.00 | 0.10 | -0.50 | Scaling only |
| 2 | 8 | 0.20 | 1.00 | 0.10 | -0.50 | Scaling + Mixing |
Many physical systems contain fluctuations at many length scales. A renormalization step summarizes what happens when you “zoom out” by a factor b. In practice, one step can represent blocking spins, integrating out short-wavelength Fourier modes, or rescaling a lattice spacing. This calculator lets you iterate that step and inspect how couplings evolve.
The inputs g1…gk represent effective couplings in an expanded Hamiltonian or action. The scaling exponents y control how each coupling changes under rescaling. If y > 0, the coupling typically grows with step number; if y < 0, it decays; and if y ≈ 0, it is marginal and sensitive to higher-order terms.
The calculator labels each direction by y and shows an implied growth factor λ ≈ b^y. Values λ > 1 indicate instability under coarse-graining, which is typical for relevant perturbations. Values λ < 1 correspond to attraction back toward a fixed point, as expected for irrelevant operators.
Real models often mix couplings: one operator generates another after coarse-graining. Enabling the mixing matrix A adds this effect through g' = A · diag(b^y) · g. In data terms, off-diagonal entries such as A12 represent how changes in g2 feed into the next-step value of g1. Identity A returns to pure scaling.
A fixed point is a set of couplings that reproduces itself after the step map. While this page uses a linearized step, it still supports “target tracking” by letting you enter g*. The table reports the Euclidean distance to the target each iteration, a useful data signal for convergence or divergence trends.
The Jacobian J controls the linear response near a fixed point. The eigenvalues printed in the result summary approximate per-step amplification along eigen-directions. If a dominant eigenvalue is above one, small deviations tend to grow quickly. If all eigenvalues remain below one, the step is locally attractive.
Common choices are b = 2 for binary blocking or b = e for continuous-style rescaling. For exploratory data, start with n = 8–20. Very large n can reveal runaway growth when y is positive, so monitor the distance-to-target column.
After calculation, export a CSV for spreadsheets or a PDF for documentation. The step table supports quick checks: monotonic decay suggests an attractive regime, oscillations suggest mixing competition, and rapid growth suggests a relevant perturbation dominates the flow. Use these data patterns to validate parameter choices.
A step applies a scale change by b and updates couplings using the linear map shown. It is a compact model of coarse-graining, useful near a fixed point.
y determines how strongly a coupling changes with rescaling. Positive y tends to grow, negative tends to shrink, and near-zero indicates marginal behavior that depends on higher-order physics.
Coarse-graining can generate new operators and couple parameters. The mixing matrix models cross-coupling between parameters so you can test how off-diagonal effects reshape the flow trajectory.
It is the Euclidean distance between the current coupling vector and your chosen target g*. Decreasing distance suggests attraction toward that target under the selected linear step.
No. They are linear stability indicators for the step map you entered. In real models, critical exponents can require nonlinear terms, loops, and careful definition of operators and normalization.
That commonly happens when one or more y are positive or when mixing amplifies growth. Try fewer steps, smaller initial couplings, or adjust A to reduce strong feedback.
Yes. The workflow is designed for quick demonstrations: set b, enter y values, compute, then export CSV/PDF. It works well for illustrating stability and flow concepts.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.