CFT and Lattice Correlators Near an RG Domain Wall between Minimal Models

TL;DR

This work provides a nonperturbative check of Gaiotto's RG domain wall construction between adjacent Virasoro minimal models by focusing on the TIM–IM case ($k=2$) and comparing analytic CFT predictions with DMRG lattice results. The authors formulate a lattice realization with a tunable coupling that creates TIM on one half of space and IM on the other, and compute RG-brane correlators by mapping UV/IR operators into the extended algebra $ ilde{oldsymbol B}$ and summing finite conformal-block contributions with RG boundary overlaps. They obtain precise matches for two-point functions, including the TIM and IM energy-density correlators and a TIM-related operator, validating the RG-brane framework at finite $k$ and supporting the proposed operator mappings and one-point overlaps. The work also outlines experimental avenues, notably via topological-superconductor boundaries and Rydberg chains, for realizing and probing RG domain walls in quantum systems. Overall, the paper combines exact CFT machinery with lattice numerics to establish a robust nonperturbative check and paves the way for experimental tests of RG interfaces in minimal-model CFTs.

Abstract

Conformal interfaces separating two conformal field theories (CFTs) provide maps between different CFTs, and naturally exist in nature as domain walls between different phases. One particularly interesting construction of a conformal interface is the renormalization group (RG) domain wall between CFTs. For a given Virasoro minimal model $\mathcal{M}_{k+3,k+2}$, an RG domain wall can be generated by a specific deformation which triggers an RG flow towards its adjacent Virasoro minimal model $\mathcal{M}_{k+2,k+1}$ with the deformation turned on over part of the space. An algebraic construction of this domain wall was proposed by Gaiotto in \cite{Gaiotto:2012np}. In this paper, we will provide a study of this RG domain wall for the minimal case $k=2$, which can be thought of as a nonperturbative check of the construction. In this case the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising Model (IM) CFT. We will check the analytical results of correlation functions from the RG brane construction with the numerical density matrix renormalization group (DMRG) calculation using a lattice model proposed in \cite{Grover:2012bm,Grover:2013rc}, and find a perfect agreement. We comment on possible experimental realizations of this RG domain wall.

Figures (13)

• Figure 1: Phase diagram required for the construction of the RG brane. At a critical value $h_*$ of the coupling, the low-energy limit is described by the "UV" CFT, whereas for $h>h_*$ it is described by the "IR" CFT. The RG brane is constructed by tuning $h$ to $h_*$ on half of the space, and taking $h$ slightly greater than $h_*$ on the other half of the space. The specific behavior at $h<h_*$ is irrelevant for our discussion.
• Figure 2: An illustration of the calculation where $O(x)\equiv 1^{IM}\times \epsilon^{TIM}(x)$ and we do the bulk OPE first then compute the resulting one point function due to the appearance of the Cardy boundary.
• Figure 3: An illustration of the finite volume configuration with the RG brane. The RG interfaces are located at $\theta=0,\pi$. The yellow-shaded region ($0<\theta<\pi$) is the TIM part and the green-shaded region ($\pi<\theta<2\pi$) is the Ising part.
• Figure 4: For a lattice with $N=180$ sites, a plot of all 180 two-point functions for the IM and TIM phases showing the translation invariance of the two-point correlators in the ground state. Therefore, PBCs have been obtained numerically.
• Figure 5: DMRG computation of the $\langle \mu_i \mu_j\rangle$ two-point function for the lattice model Equ. (\ref{['eq:GroverLatticeHam']}) in the Tricritical Ising ($h=1.62$) and Ising ($h=2$) phases.