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arXiv:2510.25840 · Physical review. B./Physical review. B

Scaling of the disorder operator at ( 3 + 1 ) D O ( 3 ) quantum criticality

Anonymous, Xiao-Chuan Wu, Zenan Liu, Zhe Wang, Zheng Yan, Dao‐Xin Yao

Published Jul 20, 2026• 0 citations
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Abstract

The disorder operator, as an easily measured nonlocal observable, displays great potential in detecting intrinsic information of field theories. It has been systematically studied in one- and two-dimensional (1D and 2D) quantum systems, while the knowledge of 3D is still limited. The disorder operator associated with U(1) global symmetry exhibits rich geometric dependence on the shape of the spatial region at a quantum critical point, meanwhile, (3+1)D is the upper critical dimension for O(N) criticality, both of which pose a challenge for exploring the disorder operator in high dimensions. In this Letter, we investigate the scaling behaviors of disorder operators in (3+1)D O(3) models through large-scale quantum Monte Carlo simulation combined with theoretical analysis. Although the upper critical dimension introduces logarithmic corrections in correlations, we analytically prove and numerically demonstrate that the corrections do not modify the universal trihedral-corner contribution of the disorder operator. The universal contributions, such as the current central charge, have been revealed in our calculation, which establishes a concrete link between lattice simulations and continuum field theory. This work opens promising directions for the experimental and numerical exploration of universal properties at quantum critical points in (3+1)D models.

DOI
10.1103/8nq1-q47x

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The disorder operator, as an easily measured nonlocal observable, displays great potential in detecting intrinsic information of field theories. It has been systematically studied in one- and two-dimensional (1D and 2D) quantum systems, while the knowledge of 3D is still limited. The disorder operator associated with U(1) global symmetry exhibits rich geometric dependence on the shape of the spatial region at a quantum critical point, meanwhile, (3+1)D is the upper critical dimension for O(N) criticality, both of which pose a challenge for exploring the disorder operator in high dimensions. In this Letter, we investigate the scaling behaviors of disorder operators in (3+1)D O(3) models through large-scale quantum Monte Carlo simulation combined with theoretical analysis. Although the upper critical dimension introduces logarithmic corrections in correlations, we analytically prove and numerically demonstrate that the corrections do not modify the universal trihedral-corner contribution of the disorder operator. The universal contributions, such as the current central charge, have been revealed in our calculation, which establishes a concrete link between lattice simulations and continuum field theory. This work opens promising directions for the experimental and numerical exploration of universal properties at quantum critical points in (3+1)D models.
We study scaling behavior of the disorder parameter, defined as the expectation value of a symmetry transformation applied to a finite region, at the deconfined quantum critical point in (2+1)d in the J-Q_3 J − Q 3 model via large-scale quantum Monte Carlo simulations. We show that the disorder parameter for U(1) spin rotation symmetry exhibits perimeter scaling with a logarithmic correction associated with sharp corners of the region, as generally expected for a conformally-invariant critical point. However, for large rotation angle the universal coefficient of the logarithmic corner correction becomes negative, which is not allowed in any unitary conformal field theory. We also extract the current central charge from the small rotation angle scaling, whose value is much smaller than that of the free theory.
We study disorder operator, defined as a symmetry transformation applied to a finite region, across a continuous quantum phase transition in (2+1)d(2+1)d(2+1)d. We show analytically that, at a conformally invariant critical point with U(1) symmetry, the disorder operator with a small U(1) rotation angle defined on a rectangle region exhibits power-law scaling with the perimeter of the rectangle. The exponent is proportional to the current central charge of the critical theory. Such a universal scaling behavior is due to the sharp corners of the region and we further obtain a general formula for the exponent when the corner is nearly smooth. To probe the full parameter regime, we carry out systematic computation of the U(1) disorder parameter in the square lattice Bose-Hubbard model across the superfluid-insulator transition with large-scale quantum Monte Carlo simulations, and confirm the presence of the universal corner correction. The exponent of the corner term determined from numerical simulations agrees well with the analytical predictions.
Measuring charge fluctuations within a subregion provides a powerful probe of quantum many-body systems. In two spatial dimensions, the shape dependence of the dimensionless corner contribution encodes universal data of quantum critical points and reveals observables of quantum geometry in various quantum phases. Here, we systematically extend this framework to higher dimensions. In three dimensions, we derive the universal angle dependence associated with trihedral corners of a generic parallelepiped and benchmark the predictions against Monte Carlo simulations of lattice models at the O(3) quantum critical point. We further identify a wedge-corner contribution that directly probes the quantum metric, supported by numerical results for a lattice Weyl semimetal model. More generally, we obtain angle functions for polyhedral corners of arbitrary parallelotopes in general dimensions and clarify the scaling of the corner contribution across phases of matter. While insulators and conformal critical points exhibit similar behavior across dimensions, metals display a characteristic even-odd dimensional effect.
The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang et al., arXiv:2209.07103]. Here we study, by means of large-scale quantum Monte Carlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the \log{L} log L scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log L L log L scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.