Deconfined Quantum Critical Points (DQCP)

• Deconfined quantum critical points (DQCPs) are continuous transitions between distinct ordered phases characterized by emergent fractionalized excitations and dual gauge fields, defying conventional LGW theory.
• Field-theoretic models, including the noncompact CP^1 theory and its duality webs, elucidate the role of emergent symmetries and fractionalization in describing DQCP phenomena.
• Microscopic realizations in spin models, quantum Hall bilayers, and tensor-network studies provide key experimental and computational insights into the critical scaling and deconfinement mechanisms of DQCPs.

Deconfined quantum critical points (DQCPs) are quantum phase transitions that connect distinct ordered phases through a direct continuous transition outside the Landau-Ginzburg-Wilson (LGW) paradigm. In the standard formulation, LGW theory organizes criticality in terms of fluctuations of a local order parameter and generally forbids continuous transitions between phases whose broken symmetries are unrelated. DQCPs instead are described by emergent fractionalized degrees of freedom and emergent gauge fields, and may exhibit symmetry enhancement at criticality; the canonical example is the transition between Néel antiferromagnetic and valence-bond-solid (VBS) order in two spatial dimensions (Senthil, 2023, Cui et al., 14 Apr 2025).

1. Conceptual definition and Landau-forbidden criticality

DQCPs describe continuous quantum phase transitions between phases that break distinct symmetry patterns and are therefore not naturally connected by a conventional LGW order-parameter theory. Reviews emphasize that such transitions may occur either between two symmetry-broken phases or between a symmetry-broken phase and a topologically ordered phase, and that the critical variables are fractionalized fields coupled to emergent gauge fields rather than the order parameters themselves (Senthil, 2023, Cui et al., 14 Apr 2025).

The prototypical case is the Néel-VBS transition on the square lattice. The Néel phase breaks spin-rotation symmetry, whereas the VBS phase breaks lattice symmetry while preserving spin rotation. In the DQCP picture, the topological defects of one order carry the quantum numbers of the competing order: VBS vortices carry spin-$\frac{1}{2}$, while skyrmion defects of the Néel order carry VBS quantum numbers (Wang et al., 2017, Senthil, 2023). This “intertwinement” is the central mechanism by which a Landau-forbidden direct transition becomes possible.

The same structural logic appears in other settings. In one dimension, a direct continuous transition between Ising ferromagnetic and VBS phases with $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry was found using uniform matrix product states, again violating LGW expectations because the broken symmetries are incompatible (Roberts et al., 2019). In quantum Hall bilayers with half-filled $n=2$ Landau levels in each layer, a direct and continuous transition has been reported between an exciton superfluid phase, which spontaneously breaks interlayer $U(1)$ symmetry, and a unidirectional charge density wave, which breaks translational symmetry (Yu et al., 3 Sep 2025). In both cases, the mismatch of symmetry-breaking patterns is the defining diagnostic.

2. Field-theoretic descriptions, dualities, and emergent symmetry

A standard field-theoretic description of the Néel-VBS DQCP is the noncompact $\mathrm{CP}^1$ theory, in which bosonic spinons $z_\alpha$ couple to an emergent $U(1)$ gauge field. In review form, this is written as

$$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$

with the Néel order parameter represented as a bilinear in spinons and the VBS order associated with monopole operators of the gauge field (Cui et al., 14 Apr 2025, Senthil, 2023). An equivalent sigma-model formulation combines Néel and VBS order parameters into a higher-dimensional vector with a Wess-Zumino-Witten term, making their mutual topological intertwinement explicit (Wang et al., 2017, Hofmeier et al., 2024).

A major development has been the organization of DQCPs into duality webs. For the easy-plane case, the easy-plane NCCP$^1$ theory is self-dual and is also dual to $N_f=2$ fermionic QED$\mathbb{Z}_2 \times \mathbb{Z}_2$0; this structure motivates an emergent $\mathbb{Z}_2 \times \mathbb{Z}_2$1 symmetry in the infrared (Wang et al., 2017). For the SU(2)-symmetric case, multiple equivalent descriptions—including NCCP$\mathbb{Z}_2 \times \mathbb{Z}_2$2 and QED$\mathbb{Z}_2 \times \mathbb{Z}_2$3-Gross-Neveu—motivate an emergent $\mathbb{Z}_2 \times \mathbb{Z}_2$4 symmetry that rotates the three Néel components and the two VBS components into one another (Wang et al., 2017). The corresponding emergent symmetries are anomalous in the field-theoretic sense and can be interpreted as surface realizations of $\mathbb{Z}_2 \times \mathbb{Z}_2$5-dimensional bosonic symmetry-protected topological phases (Wang et al., 2017).

Recent operator-level studies sharpen this picture. Using fuzzy-sphere regularization for a microscopic model with tunable global symmetry between $\mathbb{Z}_2 \times \mathbb{Z}_2$6 and $\mathbb{Z}_2 \times \mathbb{Z}_2$7, conformal operator flows were traced directly via the state-operator correspondence,

$\mathbb{Z}_2 \times \mathbb{Z}_2$8

This analysis found explicit decomposition of $\mathbb{Z}_2 \times \mathbb{Z}_2$9 primaries into $n=2$0 primaries, as well as avoided level crossings between scalar singlets under symmetry reduction (Yang et al., 2 Jul 2025). The same work identified a relevant scalar singlet $n=2$1 whose scaling dimension remains below $n=2$2 along the flow to $n=2$3, which was interpreted as support for pseudo-criticality rather than a stable fixed point (Yang et al., 2 Jul 2025).

Generalizations of the DQCP framework have also broadened the field-theoretic landscape. A supersymmetric extension based on internal $n=2$4 symmetry formulates a supersymmetric DQCP between a phase that breaks internal $n=2$5 and a phase that breaks lattice rotation symmetry, using both a supersphere nonlinear sigma model with WZW term and a gauge-theory description with bosonic and fermionic critical fields (Gao et al., 20 Jan 2026). Another recent direction reinterprets one-dimensional DQCPs through a crystalline categorical Landau paradigm: after gauging anomalous onsite symmetries, the original DQCP maps to a Landau-type transition between different symmetry-breaking patterns of a noninvertible crystalline categorical symmetry (Ebisu et al., 4 Jun 2026).

3. Scaling structure, spectral diagnostics, and entanglement

A central issue in DQCP research is how criticality manifests in observables. One influential proposal is that DQCPs may exhibit two divergent length scales: a standard correlation length $n=2$6 and a longer scale $n=2$7 associated with VBS domain-wall thickness and spinon deconfinement. The generalized scaling form

$n=2$8

was introduced to account for anomalous finite-size and finite-temperature behavior in the $n=2$9-$U(1)$0 model, with simulations finding $U(1)$1 and interpreting previously puzzling scaling violations as signatures of a continuous transition governed by two scales rather than simple corrections to scaling or first-order behavior (Shao et al., 2016).

Dynamical probes provide a more direct window into deconfinement. In the easy-plane $U(1)$2-$U(1)$3 model, large-scale quantum Monte Carlo with stochastic analytic continuation found broad continua in both $U(1)$4 and $U(1)$5 at the DQCP, rather than sharp magnon modes. The lower edge of the continuum followed

$U(1)$6

consistent with deconfined Dirac spinons, and gapless excitations appeared at multiple high-symmetry momenta, including $U(1)$7 and $U(1)$8, as expected from emergent symmetry currents (Ma et al., 2018). In one dimension, time evolution of infinite matrix product states similarly revealed continua of fractionalized excitations and current correlations decaying as $U(1)$9 at DQCPs, explicitly exhibiting enhanced continuous symmetries and effective spin-charge separation in certain transitions (Xi et al., 2022).

Ground-state fidelity and order-parameter diagnostics have been used in newer platforms. In quantum Hall bilayers, a DQCP was identified through smooth low-lying spectra without level crossing, continuous onset and disappearance of stripe and exciton-superfluid order parameters, a continuous first derivative of the ground-state energy, and a fidelity dip defined by

$\mathrm{CP}^1$0

all as functions of layer separation $\mathrm{CP}^1$1 (Yu et al., 3 Sep 2025).

Entanglement has become an especially stringent diagnostic. For DQCPs in $\mathrm{CP}^1$2 dimensions, the universal subleading term in the entanglement entropy can contain both critical and topological contributions. For the $\mathrm{CP}^1$3 transition, the result

$\mathrm{CP}^1$4

shows that the critical state inherits an additional topological contribution from the adjacent $\mathrm{CP}^1$5 phase, implying that DQCP wavefunctions are more highly entangled than conventional critical points (Swingle et al., 2011). At the same time, entanglement studies have sharpened the debate on conformality. For $\mathrm{CP}^1$6D SU(2) antiferromagnet-VBS candidates, a negative universal logarithmic coefficient in Rényi entanglement entropy was reported and interpreted as incompatible with unitary conformal field theory (Liao et al., 2023). A broader SU($\mathrm{CP}^1$7) study found that this anomalous logarithmic behavior persists for $\mathrm{CP}^1$8, but disappears for $\mathrm{CP}^1$9, with a threshold estimated between $z_\alpha$0 and $z_\alpha$1; for larger $z_\alpha$2, the DQCPs were found consistent with conformal fixed points described by an Abelian Higgs theory (Song et al., 2023).

4. Microscopic realizations and model classes

The best-studied microscopic setting remains the square-lattice Néel-VBS transition in spin models such as the $z_\alpha$3-$z_\alpha$4 family, but the class of candidate DQCP systems is now much broader. Reviews list square, honeycomb, and kagome lattices, as well as one-dimensional spin chains, as major theoretical arenas, and identify the Shastry-Sutherland material SrCu$z_\alpha$5(BO$z_\alpha$6)$z_\alpha$7 as a particularly promising experimental platform for a proximate DQCP between VBS and Néel states (Cui et al., 14 Apr 2025).

One-dimensional realizations have become especially precise because thermodynamic-limit tensor-network methods can be used directly. In a spin-$z_\alpha$8 chain with $z_\alpha$9 symmetry, a direct continuous DQCP was found all along the Ising-ferromagnet–VBS boundary, with continuously varying exponents consistent with a Gaussian field theory and with matching scaling dimensions for magnetic and VBS correlations due to self-duality (Roberts et al., 2019). In a frustrated hard-core dipolar Bose-Hubbard chain, DQCPs between VBS and antiferromagnetic order were reported to emerge through the fusion of two Berezinskii-Kosterlitz-Thouless transitions; finite-size fidelity susceptibility and infinite-system order parameters were used to distinguish the BKT regimes from the direct DQCP (Wang et al., 2024).

Fermionic one-dimensional systems introduce an additional possibility: partly gapped DQCPs enabled by spin-charge separation. In an interacting fermion model described at low energies by decoupled sine-Gordon theories,

$U(1)$0

continuous transitions were found between locally ordered phases where only one of the spin or charge gaps closes and a nonlocal parity order parameter remains long ranged at criticality (Baldelli et al., 2024). This departs from the expectation that all sectors must become gapless at a DQCP.

Quantum Hall bilayers supply a qualitatively different condensed-matter realization. Large-scale VUMPS on the infinite cylinder and exact diagonalization on the torus were used to identify a direct continuous transition between an exciton superfluid and a stripe phase in bilayers with half-filled $U(1)$1 Landau levels, with order parameters respectively given by pseudospin coherence

$U(1)$2

and the guiding-center structure factor

$U(1)$3

Because the control parameter is the interlayer separation $U(1)$4, this platform offers unusually direct tunability of the transition (Yu et al., 3 Sep 2025).

DQCP physics has also been extended to systems with nonlocality induced by coupling to Fermi-surface hot spots. In that setting, nonlocal quadratic terms are generated for composite DQCP order parameters, and a large-$U(1)$5 renormalization-group analysis found a new fixed point with dynamical exponent $U(1)$6 rather than the conventional relativistic value $U(1)$7 (Xu et al., 2022). This broadens the universality landscape beyond purely local spin systems.

5. Boundary phenomena, anomalies, and generalized critical structures

The DQCP is not only a bulk critical point but also a source of unusual boundary physics. A recent boundary analysis treats DQCPs as intrinsically gapless symmetry-protected topological states and studies how bulk order-parameter fluctuations interact with edge boundary conditions. Two distinct edge regimes were identified: a “pristine” regime with asymptotically free-fermion-like boundary Green’s function,

$U(1)$8

and a pseudogapped regime associated with extraordinary-log boundary conditions, where the Green’s function decays as

$U(1)$9

This super-power-law form expresses strong boundary suppression without simple exponential localization (Myerson-Jain et al., 2024).

The same work connects boundary criticality to quantum-information diagnostics through weak measurement and post-selection, which effectively tune temporal boundary conditions in the path-integral representation. The resulting change in strange correlators provides a route to distinguishing pristine from pseudogapped boundary behavior (Myerson-Jain et al., 2024). Together with the entanglement results described above, this establishes that DQCPs are distinguished not only by bulk exponents but also by anomaly-sensitive boundary and information-theoretic structures (Swingle et al., 2011).

More abstract reorganizations of DQCPs have also emerged. The crystalline categorical Landau construction argues that certain DQCPs involving lattice symmetries can be reinterpreted as Landau-type transitions once anomalous onsite symmetries are gauged, producing noninvertible crystalline symmetries whose fusion closes only up to ordinary translations. In this language, the magnetic-VBS and $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$0-antiferromagnetic–VBS DQCPs become transitions between distinct symmetry-breaking patterns of $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$1-type and $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$2-type crystalline categorical symmetries, distinguished not by fusion rules alone but by their $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$3-symbols (Ebisu et al., 4 Jun 2026). This does not eliminate the non-LGW character of the original transition; rather, it reframes it within a generalized symmetry-breaking paradigm.

6. Stability, pseudo-criticality, and current controversies

The principal unresolved issue is the ultimate infrared nature of the physically most studied two-dimensional DQCPs, especially the SU(2) Néel-VBS case. Reviews describe an active debate among scenarios including a genuine continuous transition, a weakly first-order transition, and pseudo-critical or “walking” behavior with an extended near-scale-invariant regime (Senthil, 2023, Cui et al., 14 Apr 2025). Numerical evidence for emergent symmetry and deconfined phenomenology is substantial, but so are indications of scaling drift, anomalous entanglement, and operator content inconsistent with a stable unitary conformal fixed point (Wang et al., 2017, Liao et al., 2023, Yang et al., 2 Jul 2025).

The pseudo-critical interpretation has gained support from operator-flow studies. The persistence of a relevant scalar singlet $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$4 from $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$5 to $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$6 was interpreted as implying that the would-be fixed point is unstable, so that the theory exhibits an extended conformal window before ultimately running to first-order behavior (Yang et al., 2 Jul 2025). This suggests that some of the striking emergent-symmetry phenomenology observed numerically may describe a long crossover regime rather than a true asymptotic conformal field theory.

Another major issue is stability against couplings ignored in idealized lattice models. Spin-lattice coupling provides a concrete destabilization mechanism. A field-theoretic study of the square-lattice DQCP showed that static phonons permit a relevant monopole-phonon coupling,

$$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$7

which drives a spin-Peierls instability and generically converts the continuous DQCP into a strong first-order transition (Hofmeier et al., 2024). When phonons are fully quantum, the same work argued that the DQCP survives above a critical phonon frequency, with the threshold scaling as

$$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$8

A tensor-network study of a one-dimensional $$\mathcal{L} = \sum_{\alpha=1,2} |(\partial_\mu - ia_\mu) z_\alpha|^2 + V(|z_\alpha|^2) + \cdots ,$$9-$^1$0 spin-phonon model confirmed the same logic: the DQCP remains continuous for $^1$1, becomes strongly first order for $^1$2, and the endpoint is in the four-state Potts universality class, as captured by a double sine-Gordon theory (Romen et al., 4 Jun 2026).

A plausible implication is that “DQCP phenomenology” and “stable DQCP fixed point” should not be treated as interchangeable. The literature now supports a broad empirical statement—continuous or nearly continuous direct transitions between incompatible ordered phases with emergent fractionalized and gauge-theoretic structures occur in many microscopic settings—while leaving open, in several important two-dimensional cases, whether the asymptotic critical theory is a genuine conformal fixed point, a pseudo-critical regime, or a weak first-order transition (Senthil, 2023, Cui et al., 14 Apr 2025). At the same time, new realizations such as quantum Hall bilayers provide settings in which tunability, spectroscopy, and order-parameter access may sharpen this distinction (Yu et al., 3 Sep 2025).