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Physical review. B./Physical review. B

Pristine and pseudogapped boundaries of deconfined quantum critical points

Nayan E. Myerson-Jain, Xiao-Chuan Wu, Cenke Xu

Published Nov 10, 2025• 1 citation
Publisher

Abstract

Bulk topology and criticality can each give rise to nontrivial boundary phenomena. This work shows that their interplay, at unconventional quantum critical points between a quantum spin Hall insulator and a superconductor, produces an unexpectedly rich landscape of edge states. In addition to the usual quantum spin Hall edge modes, the authors identify novel ``pseudogapped'' chiral fermions, characterized by a strongly suppressed spectral function and super--power-law decay of the Green's function.

DOI
10.1103/4gnw-4ljj

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Bulk topology and criticality can each give rise to nontrivial boundary phenomena. This work shows that their interplay, at unconventional quantum critical points between a quantum spin Hall insulator and a superconductor, produces an unexpectedly rich landscape of edge states. In addition to the usual quantum spin Hall edge modes, the authors identify novel ``pseudogapped'' chiral fermions, characterized by a strongly suppressed spectral function and super--power-law decay of the Green's function.
Bulk topology and criticality can both lead to nontrivial boundary effects. Topological orders are often characterized by their robust edge states, while bulk critical points can have different boundary scalings governed by boundary conditions. The interplay between these two different boundary effects is an intriguing problem. The boundary of the deconfined quantum critical point (DQCP) is the ideal platform for the interplay of the two boundary effects, as the DQCP is also an intrinsically gapless symmetry protected topological (igSPT) state. In this work we discuss the boundary of several analogues of the DQCP. We demonstrate that the fluctuation of the bulk order parameters and their various boundary conditions lead to a rich possibility of the edge states, including a "pseudogap" (or super power-law decay) behavior. We also discuss the quantum information perspective of our work, i.e. the implication of our results on DQCP under weak-measurement. Weak-measurement followed by post-selection can change the boundary condition at the temporal boundary in the path-integral representation of a density matrix, which will lead to different behaviors of the "strange correlator".
We study the edge physics of the deconfined quantum phase transition (DQCP) between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet superconductor (SC). Although the bulk of this transition is in the same universality class as the paradigmatic deconfined Neel to valence-bond-solid transition, the boundary physics has a richer structure due to proximity to a quantum spin Hall state. We use the parton trick to write down an effective field theory for the QSH-SC transition in the presence of a boundary. We calculate various edge properties in an N\to\infty N → ∞ limit. We show that the boundary Luttinger liquid in the QSH state survives at the phase transition, but only as ``fractional’’ degrees of freedom that carry charge but not spin. The physical fermion remains gapless on the edge at the critical point, with a universal jump in the fermion scaling dimension as the system approaches the transition from the QSH side. The critical point could be viewed as a gapless analogue of the quantum spin Hall state but with the full SU(2) S U ( 2 ) spin rotation symmetry, which cannot be realized if the bulk is gapped.
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.
A bstract 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.
Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions---even when the edge is zero-dimensional. As a minimal case study, we consider spin chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the Z3{\mathbb{Z}}_{3}Z3​ symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.