Deep Boundary Perturbations at a Quantum Critical Point

TL;DR

This work introduces deep boundary criticality by studying boundary perturbations that decay as $1/x^{\alpha}$ in (1+1)D Dirac-CFTs and shows how the exponent $\alpha$ governs whether the perturbation affects bulk criticality. By analyzing two prototypical free-fermion models, Model Y (particle-hole symmetric) and Model X (particle-hole antisymmetric), the authors derive rigorous leading scaling forms for the mass expectation value $\langle m(x)\rangle$ across the three regimes $\alpha>1$, $\alpha=1$, and $\alpha<1$, including marginal-case logarithmic corrections and a rich boundary operator spectrum at $\alpha=1$. They also reveal a topological edge state in Model Y and map the antisymmetric case to the symmetric one under a chiral transformation, highlighting how symmetry controls boundary physics. The findings illuminate a new universality class of boundary phenomena with potential realizations in quantum simulators and implications for boundary RG flows in conformal field theories.

Abstract

In this work, we explore an unconventional class of problems in the study of (quantum) critical phenomena, termed ''deep boundary criticality''. Traditionally, critical systems are analyzed with two types of perturbations: those uniformly distributed throughout the bulk, which can significantly alter the bulk criticality by triggering a nontrivial bulk renormalization group flow, and those confined to a boundary or subdimensional defect, which affect only the boundary or defect condition. Here, we go beyond this paradigm by studying quantum critical systems with boundary perturbations that decay algebraically (following a power law) into the bulk. By continuously varying the decay exponent, such perturbations can transition between having no effect on the bulk and strongly influencing bulk behavior. We investigate this regime using two prototypical models based on (1+1)D massless Dirac fermions. Through a combination of analytical and numerical approaches, we uncover exotic scaling laws in simple observables and observe qualitative changes in model behavior as the decay exponent varies.

Figures (5)

• Figure 1: Results on $\left\langle M_{Y,j} \right\rangle\simeq a\left\langle m_Y(ja) \right\rangle$ in Model Y (Eq. \ref{['eq:DeepmYperturbation']}). (a)-(c) are examples with $\alpha\leq 1$, where we plot $j^\alpha \left\langle M_{Y,j} \right\rangle$ against $j$ in log scale. We use the lattice constant $a$ as the length unit, i.e. $a=1$. The total number of sites is $N=20000$, but we only show results for 10% of the sites to suppress finite-size effect. Dots with different colors are the numerical data, while black solid lines are fittings to the curves $j^\alpha \left\langle M_{Y,j} \right\rangle = -(\alpha\lambda/\pi)\log j+{\rm const}$. (d) contains examples with $\alpha>1$, where we plot $j \left\langle M_{Y,j} \right\rangle$ against $j$ in log scale. The black solid line is the constant line $j \left\langle M_{Y,j} \right\rangle=1/(2\pi)$.
• Figure 2: $j\left\langle M_{Y,j} \right\rangle+(\lambda/\pi)\log(j)$ plotted against $j^{-|1-2\lambda|}$ for a few different $\lambda$ in Model Y. Again the total system size is $N=20000$, but data for $j<10$ and $j\gtrsim N/10$ are not shown.
• Figure 3: $r^\alpha \left\langle M_{X,r} \right\rangle$ against $r$ in log scale for Model X, where $r\in\mathbb{Z}+1/2$. Panels a and b are for $\alpha=0.5$ and $\alpha=1$, respectively. As before, the lattice constant $a=1$ is the length unit. The total number of sites is $N=20000$, but we only show results for 10% of the sites to suppress finite-size effect. Dots with different colors are the numerical data, while black solid lines are fittings to the curves $r^\alpha \left\langle M_{X,r} \right\rangle = -(\alpha\lambda/\pi)\log r+{\rm const}$. Note that we restrict to $\lambda\leq0$ because results for $\lambda>0$ are related by the particle-hole transformation.
• Figure 4: Results on $\left\langle M_{X,r} \right\rangle$ for $\alpha=2$ in Model X. In panel a, we plot $r\left\langle M_{X,r} \right\rangle$ against $r$ for a few values of $\lambda$. In panel b, we fix $r=2000.5\approx N/10$, and plot $r\left\langle M_{X,r} \right\rangle$ against $\lambda$. Squares are the numerical result, while the solid line is the theoretical prediction.
• Figure 5: $r\left\langle M_{X,r} \right\rangle+(\lambda/\pi)\log(r)$ against $r^{-|2\lambda|}$ for two choices of $\lambda$ in Model X. $r_{\rm min}$ as defined in Eq. \ref{['eq:ModelXLatticeHamiltonian']} is the lower bound of the summation index $r$.