The Renormalization Group Critical Phenomena And The Kondo Problem Pdf Review

[Generated AI] Affiliation: [Computational Physics Lab] Date: April 17, 2026

For small $j>0$, $dj/d\ln D = -2j^2 < 0$ → as we lower the cutoff $D$ (i.e., lower temperature), $j$ increases . This is the opposite of asymptotic freedom in QCD; it is infrared slavery . The flow diverges at a scale $D \sim T_K$, signaling a new fixed point.

$$H = \sum_k,\sigma \epsilon_k c^\dagger_k\sigma c_k\sigma + J \mathbfS \cdot \mathbfs(0)$$ $$H = \sum_k

where $\mathbfS$ is the impurity spin (S=1/2), $\mathbfs(0) = \frac12 \sum_k,k',\sigma,\sigma' c^\dagger_k\sigma \vec\sigma \sigma\sigma' c k'\sigma'$ is the conduction electron spin density at the impurity site, and $J$ is the exchange coupling (antiferromagnetic $J>0$). The physical observable of interest is the resistivity $\rho(T)$ due to scattering off the impurity. Using third-order perturbation theory in $J$, Kondo (1964) found:

$$\fracdjd\ln D = - 2 j^2 + 2 j^3 + \dots$$ $\mathbfs(0) = \frac12 \sum_k

| Aspect | Critical Phenomena | Kondo Problem | | :--- | :--- | :--- | | | Length scale ($L$) | Energy scale ($T$ or $D$) | | Small parameter | $t = (T-T_c)/T_c$ | $j = J\rho(\epsilon_F)$ | | Divergence | Correlation length $\xi$ | Kondo temperature $T_K$ | | Relevant operator | Temperature deviation | Antiferromagnetic coupling | | Fixed point (UV) | Gaussian ($j=0$) | Free spin ($j=0$) | | Fixed point (IR) | Wilson-Fisher ($j^*$) | Strong coupling ($j \to \infty$) | | Low-energy state | Ordered phase | Screened singlet |

$$T_K \sim D \exp\left(-\frac1J\rho(\epsilon_F)\right)$$ 2026 For small $j&gt

The Renormalization Group: From Critical Phenomena to the Kondo Problem