Boundary criticality in two-dimensional correlated topological superconductors

TL;DR

This work investigates boundary criticality in a two-dimensional, correlated, time-reversal-invariant topological superconductor as it transitions to a trivial time-reversal-breaking phase. It combines sign-problem-free determinant quantum Monte Carlo simulations to map the boundary phase diagram and identify ordinary, special, and extraordinary surface transitions, pinpointing a multicritical special point consistent with a boundary GNY universality class. A two-loop renormalization group analysis in $d=4-\varepsilon$ with the bulk fixed at the Wilson-Fisher point yields a nontrivial boundary Gross-Neveu-Yukawa fixed point and critical exponents $\Delta_{\hat{\phi}}\approx0.340$, $\Delta_\psi\approx0.622$, $\Delta_S^y\approx1.62$, and $\Delta_S^z\approx1.26$, in good agreement with numerical results. The results establish a coherent nonperturbative picture of boundary GNY criticality in a lattice model and suggest experimental routes in iron-based superconductors to observe the boundary universality class.

Abstract

The presence of a boundary enriches the nature of quantum phase transitions. However, the boundary critical phenomena in topological superconductors remain underexplored so far. Here, we investigate the boundary criticality in a two-dimensional correlated time-reversal-invariant topological superconductor tuned through a quantum phase transition into a trivial time-reversal-breaking superconductor. Using sign-problem-free determinant quantum Monte Carlo simulations, we chart the quantum phase diagram and reveal the boundary criticalities encompassing ordinary, special, and extraordinary transitions. Additionally, using renormalization group analysis, we compute the boundary critical exponent up to two loops. Remarkably, the simulations and two-loop renormalization group calculations consistently demonstrate that the presence of the boundary Majorana fermion at the special transition gives rise to a new type of boundary Gross-Neveu-Yukawa fixed point. We conclude with a discussion of possible experimental realizations in iron-based superconductors.

Figures (7)

• Figure 1: (a) Illustration of the square lattice with open boundaries at the top and bottom. Bulk bonds are shown in black, boundary bonds in yellow. In the disordered phase, the open boundaries host gapless helical Majorana modes subjected to an attractive Hubbard interaction. (b) Phase diagram of the lattice model in Eq. \ref{['eq:QMC_lattice']}. The gray line traces the bulk transition, the blue curve the surface transition, and the star marks the special point.
• Figure 2: (a) Scaling collapse of the bulk $M_2$ as a function of $U_\text{bulk}$ near the bulk criticality at $U_\text{bdy}=1.4$. Here $\eta=1.037$, and $\nu=0.630$. Inset shows the crossing of $M_2L^\eta$ for different system sizes at the bulk transition point $U^*_\text{bulk}$. (b) Scaling collapse of the boundary ${ \widehat{M} } _2$ as a function of $U_\text{bdy}$ near the boundary criticality at $U_\text{bulk}=5.4$. Here ${ \widehat{\eta} } =0.693$, and ${ \widehat{\nu} } =1.37$. Inset shows the crossing of the RG invariant at $U^c_\text{bdy}=U^*_\text{bdy}=4.28$.
• Figure 3: (a)--(b) Scaling collapse of the boundary (a) boson correlator and (b) fermion correlator, at $U^*_\text{bulk}$ and $U^*_\text{bdy}$. Here, $\Delta_{ { \widehat{\phi} } }=0.33$ and $\Delta_\psi=0.58$. (c)--(d) Scaling collapse of the boundary spin correlators (c) $S^y$ and (d) $S^z$. Here, $\Delta^y_S=1.62$ and $\Delta^z_S=1.26$.
• Figure S1: The energy dispersion of the noninteracting $H_\text{TSC}$ in the topological regime, with parameters used in the paper. Gapless edge Majorana modes run inside the bulk gap.
• Figure S2: The correction to Yukawa coupling strength $g$. The arrowed line denotes a fermion propagator, and the dashed line represents a boson propagator. The vertex with '$\times$' denotes the Yukawa interaction on the boundary.