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arXiv:2601.07923 · SciPost Physics

Extraordinary boundary correlations at deconfined quantum critical points

Hao-Ran Cui, Hart Goldman

Published Oct 22, 2026• 0 citations
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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} ℂ ℙ N − 1 (NCCP ^{N-1} N − 1 ) model in two spatial dimensions, with N N complex boson species coupled to a fluctuating \mathrm{U}(1) 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 N limit and with the bulk tuned to criticality, boundaries of the NCCP ^{N-1} 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} N − 1 model in the large- N N limit is q=N/4 q = N / 4 , signifying a new family of boundary universality classes parameterized by N 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.

DOI
10.21468/scipostphys.21.4.103

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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} ℂ ℙ N − 1 (NCCP ^{N-1} N − 1 ) model in two spatial dimensions, with N N complex boson species coupled to a fluctuating \mathrm{U}(1) 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 N limit and with the bulk tuned to criticality, boundaries of the NCCP ^{N-1} 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} N − 1 model in the large- N N limit is q=N/4 q = N / 4 , signifying a new family of boundary universality classes parameterized by N 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.
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.
The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.
The interplay of topology and correlations defines a new playground to study boundary criticality in quantum systems. We employ large scale auxiliary field quantum Monte Carlo simulations to study a two-dimensional Kane-Mele-Hubbard model on the honeycomb lattice with zig-zag edges and the Hubbard U-term tuned to the three-dimensional XY bulk critical point. Upon varying the Hubbard-U term on the edge we observe a boundary phase transition from an ordinary phase with a helical Luttinger liquid edge decoupled from the critical bulk to an extraordinary-log phase characterized by a logarithmically diverging spin stiffness. We find that the spectral functions exhibit distinct features in the two phases giving potential experimental signatures.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized for the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into the higher-dimensional boundary conformal field theories. In this work, we discover a direct special transition from the ordinary phase into an extraordinary phase on the surface of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmically decaying spin correlations in the 3D classical O(2) and O(3) models. The anomalous dimension of the surface antiferromagnetic order in the extraordinary phase η∥=−0.29(1)\eta_{\parallel}=-0.29(1)η∥​=−0.29(1) is independent of the surface coupling strength. The special transition point also has a new set of critical exponents, ys=0.82(4)y_{s}=0.82(4)ys​=0.82(4) and η∥=−0.304(5)\eta_{\parallel}=-0.304(5)η∥​=−0.304(5), which are distinct from the special transition of the classical O(3) model and indicates a new surface universality class of the 3D O(3) Wilson-Fisher theory.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4) y s = 0.86 ( 4 ) and \eta_{\parallel}=-0.33(1) η ∥ = − 0.33 ( 1 ) , which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.
Continuous phase transitions in equilibrium statistical mechanics were successfully described 50 years ago with the development of the renormalization group framework. This framework was initially developed in the context of phase transitions whose universal properties are captured by the long wavelength (and long time) fluctuations of a Landau order parameter field. Subsequent developments include a straightforward generalization to a class of T=0T = 0T=0 phase transitions driven by quantum fluctuations. In the last 2 decades it has become clear that there is a vast landscape of quantum phase transitions where the physics is not always usefully (or sometimes cannot be) formulated in terms of fluctuations of a Landau order parameter field. A wide class of such phase transitions - dubbed deconfined quantum critical points - involve the emergence of fractionalized degrees of freedom coupled to emergent gauge fields. Here I review some salient aspects of these deconfined critical points.