Retrieving "Gaussian Fixed Point" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Continuum Limit
Linked via "Gaussian Fixed Point (GFP)"
| Fixed Point Type | Behavior of $\beta(a)$ as $a \to 0$ | Physical Interpretation |
| :--- | :--- | :--- |
| Gaussian Fixed Point (GFP) | $\beta(a) \sim \beta^*$ (constant) | Free theory; couplings vanish or remain fixed; continuum limit is typically trivial. |
| Non-Gaussian Fixed Point (NGFP) | $\beta(a) \sim \beta^* + C a^{\gamma}$ ($\gamma > 0$) | Interacting continuum theory; relevant for critical phenomena and asymptotic freedom. |
| [Infrared (IR) Fix… -
Critical Line Statistical Mechanics
Linked via "Gaussian fixed point"
The lattice spacing $a$ serves as an irrelevant perturbation away from the continuum limit ($a \to 0$). The bare coupling $\beta$ must be tuned along a specific trajectory in the $(\beta, mq)$ plane such that the physical observables remain finite as $a \to 0$. This trajectory is the critical line $\mathcal{C}{LQCD}$.
If the system flows towards the Gaussian fixed point/ (free theory), the continuum limit is trivial. For interacting theories, the critical line flows towards a non-trivial [ultraviolet (… -
Renormalization Group
Linked via "Gaussian fixed point"
$$\gamma = \nu (2 - \eta)$$
This implies that the structure of the fixed point completely dictates the macroscopic thermodynamic behavior near criticality. The study of the operator dimensions under the RG flow directly maps to these exponents. Operators whose scaling dimension $\Delta$ is less than the upper critical dimension $dc$ are called relevant and drive the system towards the critical fixed point. Operators with $\Delta > dc$ are irrelevant and… -
Renormalization Group Flow
Linked via "Gaussian Fixed Point"
| Fixed Point Type | $\beta$-Function at $\mathbf{g}^*$ | Stability (Example) | Physical Interpretation |
| :--- | :--- | :--- | :--- |
| Trivial (UV) | $\beta(\mathbf{0}) = \mathbf{0}$ | Generally unstable/marginally stable | Free theory / Gaussian Fixed Point |
| Non-Trivial (IR) | $\beta(\mathbf{g}^*) = \mathbf{0}$ | Stable (all eigenvalues $\leq 0$) | Critical phenomena / Conformal Field Theory |
| Saddle Point | Mixed eigenvalues | Unstable in some directions | Pseudo-critical lines (rar…