The scaling limit of boundary spin correlations in non-integrable Ising models

TL;DR

The paper develops a Grassmann-field framework and a multiscale renormalization group analysis to study boundary spin correlations in a broad class of non-integrable 2D Ising models on the half-plane. It proves that, at criticality, the leading long-distance behavior of boundary correlations matches the integrable nearest-neighbor case up to an analytic multiplicative renormalization Z_spin(λ), with a Pfaffian structure governed by an effective two-point function. The authors extend prior bulk techniques to boundary observables, carefully controlling remainders R_H(y) and establishing analyticity of β_c(λ), t^*(λ), and Z_spin(λ) for small perturbations. The scaling limit is shown to be the Pfaffian of an explicit matrix, confirming universality of boundary correlations under weak finite-range perturbations and linking non-integrable models to conformally invariant boundary behavior. The work advances constructive RG methods for boundary observables and provides explicit bounds and expressions relevant to the boundary Pfaffian structure at criticality.

Abstract

We consider a class of non-integrable 2D Ising models obtained by perturbing the nearest-neighbor model via a weak, finite range potential which preserves translation and spin-flip symmetry, and we study its critical theory in the half-plane. We prove that the leading order long-distance behavior of the correlation functions for spins on the boundary is the same as for the nearest-neighbor model, up to an analytic multiplicative renormalization constant. In particular, the scaling limit is the Pfaffian of an explicit matrix. The proof is based on an exact representation of the generating function of correlations in terms of a Grassmann integral and on a multiscale analysis thereof, generalizing previous results to include observables located on the boundary.

Figures (9)

• Figure 1: The weighted graph $\tilde{G}_\Lambda$ associated with the auxiliary Hamiltonian \ref{['H_addb']}, obtained from the weighted graph $G_\Lambda$ by adding auxiliary edges below the lower boundary so that each auxiliary edge connects the pair $\{y_{2k-1}, y_{2k} \}\in {\boldsymbol{y}}$ and has weight $\tilde{J}_k$, $k = 1, \ldots, m/2$. Note that the auxiliary edges do not intersect each other, do not intersect horizontal or vertical edges and do not surround the horizontal dashed edges that connect the first and last column of the graph.
• Figure 2: In Fig. \ref{['fig:subfigA']}, a vertex $z \in \Lambda$ (left) corresponds to a cluster of vertices $\{\bar{H}_z, H_z, \bar{V}_z, V_z, \bar{T}_z, T_z\}$ connected via (green) short edges between them (right); in Fig. \ref{['fig:subfigB']}, horizontal and vertical edges between nearest neighbor vertices (left) correspond to long edges between nearest neighbor clusters (right); in Fig. \ref{['fig:subfigC']}, additional edges below lower boundary (left) correspond to edges between boundary clusters (right).
• Figure 3: The decorated graph $\tilde{G}_*$ in which the arrows correspond to an appropriate clockwise-odd orientation. Starting from the clockwise-odd orientation used in MW for the graph without the additional edges, the clockwise-odd orientation of $\tilde{G}_*$ is obtained by directing the additional edges from right to left, regardless of the number of vertices surrounded by an additional edge and regardless of the number of additional edges inserted. Note that to get the clockwise-odd orientation, the dashed long horizontal edges connecting the last and the first column of the graph must have a direction opposite to that of all other long horizontal edges of the graph; finally, note that additional edges never surround these horizontal dashed edges.
• Figure 4: An example of pair interaction: paths $C_U$ and $C_D$ connecting two sites, whether they are far from the boundary ($z_1'$ and $z_2'$) or on the boundary ($z_1$ and $z_2$), can always be formed only by horizontal and vertical edges (avoiding involving auxiliary edges). Similarly, for generic even interactions, one can always consider figures consisting only of horizontal and vertical edges that respect symmetry properties on the cylinder.
• Figure 5: Example of a tree $\tau \in \mathcal{T}_{\textup{spin}}^{(h)}$ with all possible types of endpoints: endpoints graphically represent the local parts of the boundary spin sources, endpoints , , and graphically represent contributions from the effective interactions and are the ones studied in AGG_CMP (here adapted to the half-plane). Endpoints and can only be on scale $h_v=2$; endpoints , and can be on any scale $h_v \in [h+2,1]\cap \mathbb{Z}$ but are always preceded by a branching point on scale $h_v-1$. The root $v_0$, the leftmost vertex, can be a branching point, as in this figure, in which case it is represented dotted, or it can not be, in which case it can be represented dotted or undotted; every other $v \in V(\tau)\setminus V_e(\tau) \setminus \set{v_0}$ is always represented with a dotted symbol. Note that the white vertices are 'hereditary': if a vertex is white, then all the dotted vertices preceding it are white.

Theorems & Definitions (18)

• Theorem 1.1
• Proposition 2.1
• proof : Proof of Proposition \ref{['GGMprop3']}
• Remark 2.2: Reflection symmetries
• Remark 2.3: Half-plane symmetries
• Remark 3.1: Equivalent kernels
• Remark 3.2: Interaction kernels in the half-plane limit
• Remark 3.3: Scaling dimension with boundary spin sources
• Remark 3.4
• Remark 3.5: Norm bounds on the remainders