The g-theorem and quantum information theory

TL;DR

The paper establishes an entropic g-theorem for boundary RG flows in 1+1 dimensional BCFTs by identifying the boundary entropy with a relative entropy measure and proving its monotonic decrease along RG trajectories. It then analyzes impurity-bulk correlations through mutual information, using both lattice toy models and a solvable free Kondo model to illustrate the framework. A key technical advance is the null-surface construction, which eliminates problematic modular-Hamiltonian terms and yields a clean relation between the running g-function and relative entropy. The work provides a quantum-information perspective on RG flow, connecting distinguishability from the UV fixed point to the emergent impurity correlations, and suggests broad avenues for applications in higher dimensions and holography.

Abstract

We study boundary renormalization group flows between boundary conformal field theories in $1+1$ dimensions using methods of quantum information theory. We define an entropic $g$-function for theories with impurities in terms of the relative entanglement entropy, and we prove that this $g$-function decreases along boundary renormalization group flows. This entropic $g$-theorem is valid at zero temperature, and is independent from the $g$-theorem based on the thermal partition function. We also discuss the mutual information in boundary RG flows, and how it encodes the correlations between the impurity and bulk degrees of freedom. Our results provide a quantum-information understanding of (boundary) RG flow as increase of distinguishability between the UV fixed point and the theory along the RG flow.

Figures (8)

• Figure 1: Different Cauchy surfaces $\Sigma$ with the same causal domain of dependence $\mathcal{D}$ give the same entanglement entropy $S(r)$.
• Figure 2: Using time-translation invariance, the smaller causal domain of dependence $\mathcal{D}_2$ is translated so that its past null boundary overlaps with that of $\mathcal{D}_1$. We then have the same state specified on the null boundary, and varying $r$ gives an increasing relative entropy.
• Figure 3: The thermal entropy as a function of $\beta m$.
• Figure 4: The function $u(x)$ multiplying the non local term for $r=1$, $\delta=0.4,0.2,0.08$, in red, blue and black, respectively.
• Figure 5: In the model the two chiralities on the half plane can be unfolded into a single chirality on the whole plane. Since the evolution is unitary, the entanglement entropy for the blue segment is the same as for the red segment, as well as on the black segment of length $2r$ that touches the impurity at its left endpoint.