Deconfined Quantum Critical Points Enable Quasi-Long-Range Order and Extraordinary Boundary Correlations

The behaviour of matter at critical points, where systems undergo dramatic changes, is increasingly understood to be influenced by activity at their boundaries. Hao-Ran Cui and Hart Goldman, both of the University of Minnesota, alongside their colleagues, investigated the unusual dynamics occurring at the edges of a specific theoretical model known as the non-compact quantum critical point. Their research reveals that boundaries of this model exhibit extraordinary logarithmic correlations, meaning interactions decay in a unique and previously unobserved manner. This discovery is significant because it points towards a new classification of boundary universality, characterised by a novel exponent, and has implications for understanding transitions between different quantum states of matter, such as the shift from a spin Hall insulator to a superconductor. Specifically, the team demonstrated this behaviour manifests as altered Cooper pair correlations within the helical edge modes of the quantum system.

Their research reveals that boundaries of this model exhibit extraordinary logarithmic correlations, meaning interactions decay in a unique and previously unobserved manner. This discovery is significant because it points towards a new classification of boundary universality, characterised by a novel exponent, and has implications for understanding transitions between different quantum states of matter, such as the shift from a spin Hall insulator to a superconductor.

Specifically, the team demonstrated this behaviour manifests as altered Cooper pair correlations within the helical edge modes of the quantum system. These models describe quantum phase transitions beyond the Landau paradigm, such as the deconfined quantum critical point between superconducting and quantum spin Hall phases. The work demonstrates that, in a large-N limit and with the bulk tuned to criticality, boundaries of the NCCPN−1 model display logarithmically decaying, or “extraordinary-log,” correlations. When monopole operators exhibit quasi-long-ranged order at the boundary, the extraordinary-log exponent of the NCCPN−1 model in the large-N limit is found to be q = N/4. This result signifies a new family of boundary universality classes parameterised by N, with implications for understanding the quantum critical behaviour of the quantum spin Hall and superconducting transition.

NCCPN−1 Model and Superconducting Boundary Bias

The study investigates boundary dynamics within the non-compact CP1 model, a system relevant to deconfined critical points between superconducting and quantum spin Hall phases. Researchers engineered a generalization of the model to N complex boson species, termed the NCCPN−1 model, to facilitate a controlled large-N expansion and circumvent potential weakly first-order transitions observed in the original NCCP1 model. This expansion allowed for a systematic approach to understanding the critical behaviour at the boundary. Scientists focused on a scenario where the bulk system is tuned to criticality, yet the boundary exhibits a bias towards superconducting order.

They implemented boundary conditions that couple helical fermions, inherited from the quantum spin Hall phase, to emergent bulk photons while simultaneously gapping the bulk matter fields at the boundary. This configuration enabled the investigation of how critical bulk matter screens photon fluctuations, ultimately leading to extraordinary-log correlations on the boundary. The team harnessed this setup to explore the behaviour of Cooper pair correlations in helical edge modes and employed a large-N limit to compute the extraordinary-log exponent, ∆SC, of the superconducting order parameter on the boundary. The research revealed that the exponent scales as ∆SC(ρ) ∼ 1/(log ρ)q, where q = N1/4 + O(N−1), establishing a new family of universality classes parameterized by N.

This precise measurement approach, coupled with the large-N expansion, delivers a quantitative understanding of the boundary critical phenomena. The study pioneers a method for calculating the extraordinary-log exponent, corroborating earlier theoretical predictions and extending the framework to boundaries of gapless systems coupled to gauge fields. Furthermore, the work proposes that for sufficiently large values of N, the chosen boundary condition represents the only stable, symmetry-preserving configuration, contrasting with findings for the easy-plane version of the model. This innovative approach not only confirms the existence of extraordinary-log correlations but also provides a pathway for extending these techniques to interfaces between quantum critical systems and superconductors, potentially informing future experimental investigations.

Logarithmic Correlations Define New Boundary Universality Classes Scientists

Scientists have demonstrated extraordinary-logarithmic correlations at the boundaries of the non-compact complex boson model in two dimensions, revealing a new family of boundary universality classes. The research focused on the NCCP model with complex boson species coupled to a fluctuating gauge field, and experiments revealed that boundaries exhibit logarithmically decaying correlations when tuned to criticality in a large-N limit. Specifically, the team measured an extraordinary-log exponent of for the NCCP model, signifying a unique parameterisation by N.

This breakthrough delivers insights into systems beyond the Landau paradigm, such as the deconfined critical point between superconducting and spin Hall phases. The study meticulously examined the behaviour of monopole operators, finding that they exhibit quasi-long-ranged order at the boundary, directly influencing the extraordinary-log behaviour observed in Cooper pair correlations of helical edge modes present in a quantum spin Hall and superconductor transition. Researchers established specific boundary conditions, preserving SU(N) symmetry and turning off matter fields, which induced stronger gauge fluctuations near the boundary.

These fluctuations mediate the extraordinary-log correlations among Cooper pairs, indicating a tendency towards monopole condensation as the boundary is approached. Tests prove that a gauge fixing condition, ∂xax + ∂τaτ = 0, uniquely determines the boundary conditions on the gauge field components. The team established that fluctuations of matter variables become gapped at the boundary, leaving only the electric field component along the boundary, ex, which is directly proportional to the monopole current. Data shows that this allows monopoles to pass through the boundary while preserving monopole conservation symmetry, consistent with the desired ordered boundary conditions.

Further analysis using bosonization introduced compact scalars, σ and θ, to describe boundary fermions, with ψL ∼ei(θ−σ) and ψR ∼ei(θ+σ). The superconducting order parameter was assembled as ∆SC = εpqψ† pψq ∼e2iσ(ρ), and the resulting boundary action, Sboundary, incorporates terms proportional to (∂mσ)2 and couplings to the emergent gauge field. Integrating by parts, the team derived a form resembling the linearized critical action, revealing a direct connection between the electric field and the boundary scalar field, σ. This work establishes a foundation for understanding critical phenomena in boundary systems and opens avenues for exploring novel phases of matter.

Extraordinary-Log Boundary Universality in Complex Bosons The behaviour

This work details the behaviour of boundaries in the non-compact complex boson model, a system relevant to understanding critical phenomena beyond the conventional Landau paradigm, such as the transition between superconductivity and the quantum spin Hall state. Researchers demonstrated that boundaries of this model exhibit logarithmically decaying correlations, termed “extraordinary-log” behaviour, particularly when monopole operators display quasi-long-ranged order. This leads to a new family of boundary universality classes, characterised by a specific exponent, and impacts the correlations of helical edge modes in the quantum spin Hall and superconductor transition.

The study rigorously establishes the values of key parameters defining these boundary conditions, utilising a large-n limit to simplify calculations and identify stable fixed points. By analysing the propagator of the boson field and applying appropriate boundary conditions, the authors connect different parameter values to distinct universality classes and derive a differential equation governing the behaviour of the field near the boundary. The resulting boundary operator product expansion clarifies how bulk operators translate to boundary-localised operators, influencing correlation functions. The authors acknowledge that their analysis relies on the large-n limit and that corrections to this approximation may alter the stability of certain fixed points.

Future research could explore the impact of these 1/n corrections and investigate the behaviour of the system in lower dimensions or with different boundary conditions. Further investigation into the precise nature of the.