Defect-Localized RG Flows

• The paper introduces defect-localized RG flows that connect distinct UV and IR conformal fixed points by perturbing only the defect region.
• It employs a perturbative analysis to derive reflection and transmission coefficients, thereby quantifying energy and quantum information loss across the interface.
• The study validates its perturbative approach by matching results with exact coset model constructions, outlining implications for quantum quenches and entanglement measures.

Defect-localized renormalization group (RG) flows describe RG transformations confined to regions of spacetime with reduced dimensionality—so-called defects—embedded in a higher-dimensional quantum field theory (QFT). These flows interpolate between distinct conformal fixed points associated with differences in local operator content or symmetry realization at the defect, while the bulk theory remains unaltered or only weakly perturbed. Defect-localized RG flows provide a powerful unifying framework for studying the interplay between quantum entanglement, information transmission, and boundary or interface critical phenomena in conformal field theories (CFTs), particularly in two dimensions.

1. Construction of Defect-localized RG Flows

Defect-localized RG flows are typically engineered by introducing a relevant (or sometimes marginal) perturbation restricted to one side of a codimension-one defect (interface) in a CFT. In the two-dimensional setting considered in (Brunner et al., 2015), the action of a UV CFT is modified as

$$\delta S = \lambda \left(\frac{2\pi}{\beta}\right)^\delta \int d^2w\, \phi(w)$$

where $\phi$ is a scalar primary with conformal dimension $\Delta=2-\delta$ (with $\delta\ll2$), and $\lambda$ is a perturbative coupling. For $\delta$ small (i.e., for operators nearly marginal), a perturbative RG analysis is possible: $$\beta(\lambda) = \delta\,\lambda + \pi C\,\lambda^2 + \pi^2 D\,\lambda^3 + \mathcal{O}(\lambda^4)$$ with $C$ determined from the OPE of $\phi$. The IR fixed-point coupling is given by

$$\lambda_{\rm IR} = -\frac{\delta}{\pi C} - \frac{D \delta^2}{\pi C^3} + \mathcal{O}(\delta^3)$$

This construction produces an interface interpolating between UV and IR fixed points by switching on the relevant operator only on one side of the defect.

2. Reflection/Transmission Properties and Universal Observables

A central focus of (Brunner et al., 2015) is the computation of reflection ($\mathcal{R}$) and transmission ($\mathcal{T}$) coefficients for energy and other conserved quantities across the defect RG interface. These coefficients are defined via vacuum expectation values of products of energy–momentum tensors on either side: $$R_{ij} = \langle 0^{(i)} | T^{(j)}\tilde{T}^{(j)}(0) | 0^{(i)} \rangle$$ The reflection and transmission are then: $$\mathcal{R} = \frac{R_{11} + R_{22}}{\mathcal{N}}, \qquad \mathcal{T} = \frac{R_{12} + R_{21}}{\mathcal{N}}, \qquad \mathcal{R} + \mathcal{T} = 1$$ where $\mathcal{N}$ is a normalization factor.

Explicit perturbative expressions to third order in $\lambda$ are derived, e.g.: $$R_{11} = \left(\frac{\pi^2}{2} - \frac{\pi^2}{4}\delta\right)\lambda^2 - \frac{\pi^3}{2} C\lambda^3 + \mathcal{O}(\lambda^4)$$

$$R_{22} = \left(\frac{\pi^2}{2} - \frac{3\pi^2}{4}\delta\right)\lambda^2 - \frac{\pi^3}{2} C\lambda^3 + \mathcal{O}(\lambda^4)$$

and one finds $R_{11} > R_{22}$ when the perturbation is relevant ($\delta>0$), indicating an asymmetry: more energy is reflected incident from the UV side than from the IR. This is interpreted as "information loss" along the RG flow.

Remarkably, the interface entropy or "defect $g$-factor," quantifying degrees of freedom localized at the defect, is also related to $\mathcal{R}$: $$g^2 = 1 + \frac{\delta^2}{2C^2} + \frac{\delta^3 D}{C^4} + \mathcal{O}(\delta^4) = 1 + \frac{c^{(1)}}{2}\,\mathcal{R} + \mathcal{O}(\delta^4)$$ Thus, to leading order, the "entropic" content of the defect is encoded in its reflection properties.

3. Comparison with Exact RG Interfaces in Coset Models

The perturbative field-theoretical calculations are rigorously compared with exact RG interfaces constructed in coset models. For coset CFTs of the type

$$M_{k,l} = \frac{\hat{a}_k \oplus \hat{a}_l}{\hat{a}_{k+l}}$$

one constructs the RG interface by projecting onto sectors with matching affine subalgebra labels and fusing Cardy boundary states with topological defects. In the large-$k$ expansion, the exact expressions for $R_{ij}$ and $\mathcal{R}$—e.g.

$$R_{11} \sim \frac{\text{dim}(a)\,l^2}{2k^2}(1 - \frac{g}{k} + \cdots), \quad \mathcal{R} = \frac{l(l+g)}{k^2}\left(1 - \frac{2g}{k} + \cdots\right)$$

—are found to match precisely with the perturbative formulas, nontrivially validating the perturbative RG interface approach for defect-localized flows.

4. Physical and Information-Theoretical Interpretation

Reflection and transmission coefficients serve as quantitative probes of the degree to which the IR theory "remembers" the UV, i.e., how much quantum information or energy is lost across the flow. A higher reflection indicates stronger decoupling of UV and IR sectors. The close relation between interface $g$-factor and $\mathcal{R}$ demonstrates that universal entropic diagnostics—originating in quantum information theory—are deeply linked to transport properties at critical interfaces.

These findings have broad implications for physical phenomena such as quantum quenches, energy transport through impurities, and the paper of entanglement in extended systems. They also intersect with the "counting RG flows" program, where such quantities help define a "distance" in the landscape of 2D CFTs.

5. Generalizations and Applications

The approach is robust and extends to other exactly marginal deformations. For example, in the free compact boson (deforming the radius), the defect becomes a $D1$ brane and the reflection coefficient takes a simple form: $$\mathcal{R} = \cos^2(2\vartheta), \qquad \mathcal{T} = \sin^2(2\vartheta)$$ where $\vartheta$ encodes the change in compactification radius.

The ability to compute these observables may aid in classifying conformal interfaces (including topological and partially transmitting ones) and provides tools for characterizing universality classes of defects in statistical mechanics, string theory, and quantum critical systems.

6. Scaling Considerations, Limitations, and Outlook

The perturbative method is controlled for nearly marginal relevant operators ($\delta\ll1$), and resource requirements scale with the number of terms retained in the expansion. The matching with exact results is currently limited to settings where the coset or LG matrix factorization descriptions are available, though the conceptual framework is expected to generalize to a wider class of rational and irrational 2D CFTs.

Potential extensions include investigating the interplay with boundary RG flows, higher-dimensional interfaces, generalized symmetry defects, and exploring deeper connections between transport, entanglement measures, and algebraic classification of interfaces.

In summary, defect-localized RG flows provide a precise, universal setting in which transport, entropy, and anomaly-related observables are computable, interrelated, and physically significant for understanding the full landscape of renormalization group phenomena in conformal field theories.