Boundary Criticality at the Nishimori Multicritical Point

TL;DR

This work delivers the first systematic BCFT characterization of boundary criticality at the Nishimori multicritical point in the 2D random-bond Ising model by combining a tensor-network construction with boundary-spin rotations. It identifies two conformal boundary fixed points (free and fixed), extracts the boundary entropy and b.c.c. operator dimensions, and reveals multifractal scaling of boundary spin observables. Complementary replica-field-theory RG analysis near six dimensions connects the perturbative $6-\varepsilon$ expansion to the 2D results via a minimal Padé interpolation, forming a bridge between perturbative RG and nonperturbative BCFT numerics. Overall, the study provides a robust lattice-to-BCFT pipeline for disordered boundary criticality and highlights rich boundary universality in nonunitary settings.

Abstract

We study boundary critical behavior at the Nishimori multicritical point of the two-dimensional (2D) random-bond Ising model. Using tensor-network methods, we realize a one-parameter family of microscopic boundary conditions by continuously rotating the boundary-spin orientation and find two conformal boundary fixed points that correspond to the free and fixed boundaries. We extract conformal data, including the boundary entropies and the scaling dimensions of boundary primary operators, which characterize the boundary universality class. We further demonstrate that the free boundary fixed point exhibits multifractal scaling of boundary spin fields. Finally, we complement our numerical results with a renormalization group analysis and discuss a systematic bridge between the controlled $6-ε$ expansion and the 2D tensor network numerics.

Figures (5)

• Figure 1: Illustration of the tensor-network method. (a) The partition function is represented by a contraction of the tensor network, where each small square indicates a local tensor $W$ encoding the local Boltzmann weight contributed from the surrounding four bonds. The transfer matrix $T_\text{odd}[\boldsymbol{J}_{2i-1}]$ ($T_\text{even}[\boldsymbol{J}_{2i}]$) is composed by a layer of $W$ tensors colored by light (dark) blue. (b) The tensor network with fixed boundary spin orientations in the open string channel, represents the calculation of Eq. \ref{['eq:psi_theta']}. The boundary condition is imposed by the boundary $B$ tensors. (c) The tensor network with a periodic boundary condition in the closed string channel, represents the denominator $\left< X | \Psi \right>$ in Eq. \ref{['eq:psi_spin']}. The numerator can be obtained by applying $\sigma_{i}^{z}\sigma_{j}^{z}$ on $\left| \Psi \right>$ before its contraction with $\left| X \right>$.
• Figure 2: (a) Half-chain von Neumann entropy $\overline{S_{A}[\Psi_{\theta}]}$ versus the size $L$ for $\theta = 0$ and $\pi/2$. The logarithmic-law fittings according to the Eq. \ref{['eq:vN_entropy']} estimate the central charge $c_{\rm vN} = 0.416(2)$ and $0.419(2)$ for $\theta = 0$ and $\pi/2$, respectively. (b) The boundary entropy $S_{\rm bdy}(\theta)$ versus $\theta$ for different $L$. The value at $\theta = \pi/2$ is estimated to be $-0.574(1)$ . (c) Data collapse of the boundary entropy near $\theta_\ast = 0$ with $1/\nu = 0.75(1)$ . (d) Data collapse of the boundary entropy near $\theta_\ast = \pi/2$ with $1/\nu_{\rm irr} \approx -0.9$ . In (d), the sampling number is increased to $10^7$ to further suppress statistical errors.
• Figure 3: Wavefunction overlaps $\overline{\log Q_{f\pm}}$ (a) and $\overline{\log Q_{+-}}$ (b), versus lattice size $L$. The least-squares fittings according to Eq. \ref{['eq:bcc']} estimate the scaling dimensions of the b.c.c operator, $\Delta_{f\pm} \approx 0.0446(2)$ and $\Delta_{+-} \approx 1.055(2)$ . The two data points with the smallest values of $L$ were excluded from the fitting.
• Figure 4: Moments of the boundary spin-spin correlation function, $\overline{\left< \sigma_0 \sigma_l \right>^n}$, versus the conformal chord length $L/\pi \sin(\pi l/L)$ for $n=1$ to $8$ with size $L = 64$. The power-law fittings according to the conformal ansatz Eq. \ref{['eq:cor_scaling']} estimate the corresponding exponents as $\Delta_{1} = \Delta_{2} \approx 0.263(1)$, $\Delta_{3} = \Delta_{4} \approx 0.369(1)$, $\Delta_{5} = \Delta_{6} \approx 0.436(1)$, and $\Delta_{7} = \Delta_{8} \approx 0.486(1)$ . The fits are performed by excluding several data points with relatively small $l$ (indicated by lighter colored makers).
• Figure 5: Left: Tensor-network representation of wavefunction overlap $\left< \Psi_{\theta'} | \Psi_\theta \right>$. The orange and brown dots denote the boundary spin orientations $\theta'$ and $\theta$, respectively. Middle: Illustration of partition function, Eq. \ref{['eq:overlap_Z']}, obtained from the wavefunction overlap. The green dots only indicate the insertion position of b.c.c. operators. Right: Insertion of b.c.c. operators in the BCFT description. The b.c.c. operators $\phi_{a|b}$ are indicated by the green dots.