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.
We study the boundary criticality in 2D interacting topological insulators. Using the determinant quantum Monte Carlo method, we present a nonperturbative study of the boundary quantum phase diagram in the Kane-Mele-Hubbard-Rashba model. Our results reveal rich boundary critical phenomena at the quantum phase transition between a topological insulator and an antiferromagnetic insulator, encompassing ordinary, special, and extraordinary transitions. Combining analytical derivation of the boundary theory with unbiased numerically exact quantum Monte Carlo simulations, we demonstrate that the presence of topological edge states enriches the ordinary transition that renders a continuous boundary scaling dimension and, more intriguingly, leads to a special transition of the Berezinskii-Kosterlitz-Thouless type. Our work establishes a framework for the nonperturbative study of boundary criticality in two-dimensional topological systems with strong electron correlations.
We study the boundary criticality enriched by boundary fermions, which ubiquitously emerge in topological phases of matter, with a focus on topological insulators and topological superconductors. By employing dimensional regularization and bosonization techniques, we uncover several unprecedented boundary universality classes. These include the boundary Gross-Neveu-Yukawa critical point and the special Berezinskii-Kosterlitz-Thouless (BKT) transition, both resulting from the interplay between edge modes and bulk bosons. We present a comprehensive sketch of the phase diagram that accommodates these boundary criticalities and delineate their critical exponents. Additionally, we explore a 1+1D conformal defect decorated with fermions, where a defect BKT transition is highlighted. We conclude with a discussion on potential experimental realizations of these phenomena.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective one-dimensional (1D) lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization-group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of two-dimensional topological superconductors when strong disorder and interactions are present.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective 1D lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of 2D topological superconductors when strong disorder and interactions are present.
We discuss the boundary critical behaviors of two-dimensional (2D) quantum phase transitions with fractionalized degrees of freedom in the bulk, motivated by the fact that usually it is the one-dimensional boundary that is exposed and can be conveniently probed in many experimental platforms. In particular, we mainly discuss boundary criticality of two examples: (i) the quantum phase transition between a 2D
Z2 topological order and an ordered phase with spontaneous symmetry breaking; (ii) the continuous quantum phase transition between metal and a particular type of Mott insulator [
U(1) spin liquid]. In particular, we obtain the critical exponents and scaling laws of these exotic quantum phase transitions when the systems are probed from the boundary in proposed experimental setup. These critical exponents obtained are significantly different from those one would see through bulk measurements. This theoretical study could be relevant to many purely 2D systems, where recent experiments have found correlated insulator, superconductor, and metal in the same phase diagram.
In this paper, we discuss strong coupling limits of topological quantum critical points (TQCPs) where quantum phase transitions between two topological distinct superconducting states take place. We illustrate that while superconducting phases on both sides of TQCPs spontaneously break the same symmetries, universality classes of critical states can be identified only when global symmetries in topological states are further specified. In dimensions
d=2and3, we find that continuous
(d+1)th-order transitions at weakly interacting TQCPs that were pointed out previously in the presence of the emergent Lorentz symmetry can be terminated by strongly interacting fixed points of Majorana fields. For
2d time reversal symmetry breaking TQCPs, termination points are supersymmetric with
N=4Nf=1 (where
Nf is the number of four-component Dirac fermions and
N is the number of two-component real fermions) beyond which transitions are discontinuous first-order ones. For
2d time reversal symmetric TQCPs without other global symmetries, termination points of
(d+1)th-order continuous transition lines are generically conformal invariant without supersymmetry. Beyond these strong coupling fixed points, there are first-order discontinuous transitions as far as the protecting symmetry is not spontaneously broken but no direct transitions if the protecting symmetry is spontaneously broken in the presence of strong interactions. In this case, termination points can become supersymmetric with
N=4Nf=2 only when interacting Majorana fermions further display an emergent global
U(1) symmetry. In
3d, strong coupling termination points can be further effectively represented by new emergent gapless real bosons weakly coupled with free gapless Majorana fermions. However, in
1d, time reversal symmetric
(d+1)th continuous transition lines of TQCPs are terminated by simple free Majorana fermion fixed points.