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

Abstract

The disorder operator, as an easily measured non-local observable, displays great potential in detecting intrinsic information of field theories. It has been systematically studied in 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) criticalities, both of which pose a challenge for exploring the disorder operator in high dimensions. In this work, 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. 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 new avenues for experimental and numerical exploration of universal properties at quantum critical points in (3+1)D models.

Figures (6)

• Figure 1: The two lattice models, (a) columnar-dimerized and (b) double-cube AFM Heisenberg model. The weak interaction $J_1$ and the strong interaction $J_2$ are represented by thin and thick bonds. Orange cubes $M$ represent the region where the disorder operator is applied.
• Figure 2: (a) The Binder ratio $R_2$ and (b) scaled spin stiffness $\rho_s^\alpha L^2$ in the $z$ direction for system sizes $L=30,32,...,40$ versus the coupling ratio g. (c) The size dependence of critical point g$_c^{R_2}(L)$ and g$_c^{\rho_s^\alpha}(L)$ obtained from the crossing of the curves of $R_2$ and $\rho_s^\alpha L^2(\alpha=x,z)$ with the systems sizes $L$ and $L+2$. The curves represent the fitting function g$_c(L)$=g$_c$+$kL^{-(1/\nu+\omega)}$, where $\nu$=$\frac{1}{2}$ is the critical exponent of mean-field theory, both fits indicate that g$_c$=4.0159(1) when $L\to\infty$.
• Figure 3: Disorder operator $\ln|\left \langle X_M(\theta=\pi/4)\right \rangle|$ in the AFM Néel phase (a) and spin singlet phase (b) for the double-cubic model with system size $L=40$ obtained by QMC, where $l$ is the size length of cube $M$. The solid lines represent the fitting function $\ln|\left \langle X_M(\theta)\right \rangle|=-a_{1}l^2\ln l+b_{1}l^2+c_{1}l+d$ for points $l\geq5$ to avoid the finite size effect in the AFM Néel phase, and $\ln|\left \langle X_M(\theta)\right \rangle|=-a_{2}l^2+b_{2}l+c_{2}$ in the spin singlet phase. The coefficient $a_1$ as a function of g is shown in the inset of (a).
• Figure 4: Disorder operator $|\left \langle X_M(\theta)\right \rangle|$ as a function of the cube with side length $l$ at QCP for $\theta$=0.5, 1.0, 1.5, 2.0 for (a) columnar-dimerized model (b) double-cubic model with system size $L=48$ obtained by QMC. (c), (d) and (e), (f) show the logarithmic term coefficient $s(\theta)$ and leading quadratic term coefficient $a(\theta)$ extracted from CD and DC models, respectively.
• Figure 5: The Finite-size extrapolation of the current central charge $C_J$ extracted from the disorder operator at QCP, with system sizes up to $L=64$ for the columnar-dimerized model and the double-cubic model. The orange dashed line represents the exact $C_J$ of the free theory in (3+1)D CFT.