Boundary critical phenomena in the quantum Ashkin-Teller model

TL;DR

This work addresses how boundary conditions affect critical behavior along the $c=1$ Ashkin-Teller line by combining boundary conformal field theory with large-scale DMRG simulations. It constructs lattice boundary terms that renormalize to conformally invariant BCs, clarifies the role of the $\,\mathbb{Z}_2$-orbifold and the four-state Potts point, and maps out boundary renormalization group flows to propose a global boundary phase diagram. A key achievement is the explicit identification of blob, fixed, two-state mixed, and Dirichlet/Neumann boundary states with lattice realizations, including nontrivial KW duality relations and $D_4$ symmetry constraints. The findings strengthen the bridge between irrational $c=1$ BCFT descriptions and concrete lattice models, offering a detailed framework likely relevant for boundary engineering in quantum simulators and related experimental platforms.

Abstract

We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the $\mathbb{Z}_2$-orbifold of the $c=1$ compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions,, utilizing simple current extensions and the underlying $\mathrm{SU}(2)$ symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with $D_4$ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.

Figures (12)

• Figure 1: Illustration of the moduli space of boundary states in $\mathrm{SU}(2)/D_2$ model. Green, red, and blue lines are lines of fixed points $f_{1,2,3}$, respectively. Blob boundary conditions we identified in Sec. \ref{['sec:bs_orbifold']} are denoted by separate (blue, green, and black) points. Note that in the orbifold space, the upper and lower halves of the three lines are identified by parity in this figure. Purple lines are also identified and correspond to the Neumann boundary states. The interiors of the upper and lower triangular spindles are not identified, although the boundary surfaces of them are identified through the parity.
• Figure 2: Illustration of boundary RG from free boundary condition to two-state mixed and fixed boundary conditions. The boundary fields $h_{\sigma}^z$ and $h_\tau^z$ break the corresponding $\mathbb{Z}_2$ spin-flip symmetry and induce a relevant boundary RG flow.
• Figure 3: Illustration of boundary RG from free boundary condition to antipodal two-state mixed and blob boundary conditions. The boundary field $h_{\sigma\tau}^z$ induces an integrable boundary perturbation towards $AC/BD$ boundary conditions. When this perturbation is small, the effect of the boundary fields $h_{\sigma}^z$ and $h_\tau^z$ resembles the case without this marginal perturbation (right yellow slice in (a)). Such behavior ends at a specific value (the red dots in (a)), as one of the relevant operators becomes irrelevant. Therefore, the boundary condition in the rest part should only flow to fixed boundary conditions, like shown in the left yellow slice in (a) and (b).
• Figure 4: Conformal towers for the Ashkin-Teller model under $A-A$ (a), $A-B$ (b) and $A-C$ (c) boundary conditions. Theoretical predictions (blue dashed lines) are compared with $L\to\infty$ extrapolated data (red dots) and DMRG data (purple, green, and blue symbols). The extrapolation (Eq. \ref{['eq:extrapo']}) uses system sizes $L=\{24,32,40,60\}$, with larger sets included for accuracy at $\lambda=0.7071$ ($L=80$) and $\lambda=1$ ($L=80,100$).
• Figure 5: Conformal towers for the Ashkin-Teller model under $A-AB$ (a), $A-CD$ (b) and $A-\text{Free}$ (c) boundary conditions. Theoretical predictions (blue dashed lines) are compared with $L\to\infty$ extrapolated data (red dots) and DMRG data (purple, green, and blue symbols). The extrapolation (Eq. \ref{['eq:extrapo']}) uses system sizes $L=\{24,32,40,60\}$, with larger sets included for accuracy at $\lambda=0.7071$ ($L=80$) and $\lambda=1$ ($L=80,100$).