Extraordinary boundary correlations at deconfined quantum critical points

TL;DR

This paper addresses boundary criticality at the deconfined quantum critical point between a quantum spin Hall phase and an s-wave superconductor, described by the NCCP^{N-1} model in 2+1D. By developing a controlled large-$N$ framework and solving boundary Schwinger-Dyson equations, the authors show that the boundary hosts extraordinary-log correlations with the SC order parameter, decaying as $G_{\rm SC}(\rho) \sim [\log(\mu\rho)]^{-q}$ and with a universal exponent $q$ that scales as $q = N/4 + \mathcal{O}(1)$ in the large-$N$ limit. The analysis hinges on boundary Ward identities, a boundary effective action for the phase field, and self-consistent gauge dynamics in the bulk that induce edge Cooper-pair fluctuations. This work uncovers a new family of boundary universality classes beyond the O$(n)$ paradigm and provides concrete predictions for edge correlators that can be tested numerically or in 2D materials exhibiting QSH--SC transitions.

Abstract

Recent years have seen a growing appreciation for the effects of quantum critical fluctuations on gapless boundary degrees of freedom. Here we consider the boundary dynamics of the non-compact $\mathbb{CP}^{N-1}$ (NCCP$^{N-1}$) model in two spatial dimensions, with $N$ complex boson species coupled to a fluctuating $\mathrm{U}(1)$ gauge field. These models describe quantum phase transitions beyond the Landau paradigm, such as the deconfined quantum critical point between superconducting (SC) and quantum spin Hall (QSH) phases. We show that, in a large-$N$ limit and with the bulk tuned to criticality, boundaries of the NCCP$^{N-1}$ model display logarithmically decaying, or ``extraordinary-log,'' correlations. In particular, when monopole operators exhibit quasi-long-ranged order at the boundary, we find that the extraordinary-log exponent of the NCCP$^{N-1}$ model in the large-$N$ limit is $q=N/4$, signifying a new family of boundary universality classes parameterized by $N$. In the context of the QSH -- SC transition, the quantum critical point inherits helical edge modes from the QSH phase, and this extraordinary-log behavior manifests in their Cooper pair correlations.

Figures (3)

• Figure 1: We consider a bulk system tuned to the QSH -- SC DQCP, which is described by a model of $N$ complex boson species, $\phi_I$, coupled to an emergent U$(1)$ gauge field, $a_\mu$. At the boundary, charged helical edge modes $\psi_L,\psi_R$ are inherited from the QSH phase. The bulk matter is gapped at the boundary, but gauge fluctuations still couple to the boundary fermions. The critical bulk matter screens the emergent photon fluctuations (solid bubble), engendering extraordinary-log correlations of the boundary SC order parameter.
• Figure 2: Diagrammatic representation of the Schwinger-Dyson equations \ref{['eq: original sd equation 1']} -- \ref{['eq: original sd equation 2']} for the emergent photon propagator. Here $D^{0}_{\mu\nu}$ represents the tree-level propagator, determined by the Maxwell term. The blue double lines denote the fully resummed photon propagator. We solve these equations self-consistently in the large-$N$ limit.
• Figure 3: Conjectured large-$N$ renormalization group flow. For $N\geq N_{\mathrm{crit}}$, we propose that the boundary of the deconfined QSH -- SC transition possesses a unique boundary phase, where the SC order parameter exhibits extraordinary-log correlations with $q\approx N/4$ and the boundary fermions remain coupled to bulk gauge fluctuations. The corresponding state with $\mathrm{PSU}(N)=\mathrm{SU}(N)/\mathbb{Z}_N$ order at the boundary is expected to be unstable, with its order parameter evolving to an ordinary boundary phase at large-$N$.