Boundary anomalous dimensions from BCFT: O($N$)-symmetric $φ^{2n}$ theories with a boundary and higher-derivative generalizations

TL;DR

The work addresses how boundary effects modify multicritical O($N$) scalar theories with higher-derivative kinetic terms, focusing on φ^{2n} deformations. It combines bulk multiplet recombination with boundary conformal symmetry and analytic bootstrap to extract leading anomalous dimensions for boundary operators, revealing that the canonical case (k=1) can yield ε- or ε^{1/2}-type expansions depending on boundary conditions and parity of n, and extends these methods to higher-derivative (k>1) theories with explicit results for a NN/DD example (k,n)=(2,2). The results reproduce known φ^4 boundary exponents at n=2 and provide new data for general n, including polymer limits (N→0) and Lifshitz-type BCFTs, while confirming consistency with boundary crossing symmetry. Overall, the work advances a controlled BCFT framework for multicritical and higher-derivative boundaries, with potential links to Lifshitz transitions and boundary-critical phenomena across condensed-matter and statistical-physics contexts.

Abstract

We investigate the $φ^{2n}$ deformations of the O($N$)-symmetric (generalized) free theories with a flat boundary, where $n\geqslant 2$ is an integer. The generalized free theories refer to the $\Box^k$ free scalar theories with a higher-derivative kinetic term, which is related to the multicritical generalizations of the Lifshitz type. We assume that the (generalized) free theories and the deformed theories have boundary conformal symmetry and O($N$) global symmetry. The leading anomalous dimensions of some boundary operators are derived from the bulk multiplet recombination and analyticity constraints. We find that the $ε^{1/2}$ expansion in the $φ^6$-tricritical version of the special transition extends to other multicritical cases with larger odd integer $n$, and most of the higher derivative cases involve a noninteger power expansion in $ε$. Using the analytic bootstrap, we further verify that the multiplet-recombination results are consistent with boundary crossing symmetry.

Figures (2)

• Figure 1: The bootstrap equation for the bulk-bulk two-point function $\langle\phi_a\phi_b\rangle$. In the bulk channel (left), the bulk OPE of $\phi_a\phi_b$ leads to a sum of bulk one-point functions. In the boundary channel (right), we apply the BOE to each $\phi_a$ and obtain a sum of boundary two-point functions.
• Figure 2: The intermediate states associated with $\Psi^{(q,2r-1,m)}_{a}\sim \Box_\parallel^m(\Phi^{(q,1)}_{a})^{2r-1}$ in the bulk-boundary-boundary correlator $\langle\phi_a\Phi^{(q,2p)}_{}\Phi^{(q,2p+1)}_{b}\rangle$. The number of lines corresponds to the number of fundamental operators. The ellipses indicate potentially more lines that are not drawn explicitly. At order $g\sim\epsilon^1$, the BOE of $\phi_a$ contains the operators constructed from $2r-1$ boundary fundamental primaries with $r\leq n$ due to the bulk $g\phi^{2n}$ interaction. They can also involve contracted parallel derivatives, which are denoted schematically by $\Box^m_\parallel$.