Search

Emergent symmetry is a unique feature of identifying quantum phase transitions beyond the Landau paradigm. Via large-scale tensor network simulations, we study a spatially anisotropic spin-1/2 square-lattice frustrated antiferromagnetic (AFM) model, i.e., the $J_{1x}$-$J_{1y}$-$J_2$ model, which contains anisotropic nearest-neighbour couplings $J_{1x}$ and $J_{1y}$, and next nearest-neighbour coupling $J_2$. For small $J_{1y}/J_{1x}$, by tuning $J_2$, a direct continuous transition between the AFM and valence-bond solid (VBS) phase is observed. With growing $J_{1y}/J_{1x}$, a gapless quantum spin liquid (QSL) phase gradually emerges in between the AFM and VBS phase. We observe an emergent O(4) symmetry along the AFM-VBS transition line, which is consistent with the prediction of the deconfined quantum critical point (DQCP) theory. Most surprisingly, we find that such an emergent O(4) symmetry still holds for the whole QSL-VBS transition line. These findings reveal the intrinsic relationship between QSL and DQCP, which also provide us with a very strong constraint on quantum field theory description of the QSL phase. The phase diagram and critical exponents therein are also of direct relevance to future experiments on frustrated magnets., Comment: 5+7 pages, 4+11 figures

In a recent comments [arXiv:1909.12788 (2019)], Zhao et al. argue that the definition of dimer orders used in our paper [Phys. Rev. B 98, 241109 (2018)] may not rule out valence bond solid (VBS) orders of $J_1$-$J_2$ model on the open boundary conditions (OBC). In this reply, we show that their argument does not apply to our case.

Figure S1. Location of air quality monitor stations in Ningbo city. (PDF 13330 kb)

Table S2. Excess risks (ERs) and 95% confidence intervals of preterm birth per IQR increment in air pollutant concentrations stratified by season and maternal age in Ningbo, China. (DOCX 36 kb)

Table S1. Association between cumulative air pollution concentrations and risk of preterm birth. (DOCX 18 kb)

Table S1. Association between cumulative air pollution concentrations and risk of preterm birth. (DOCX 18 kb)

Table S3. Difference of estimates and 95% confidence intervals (95% CIs) of air pollutants on risk of preterm birth between subgroups. (DOCX 20 kb)

Phase unwrapping algorithms play an important role in noncontact optical profilometries such as phase-measuring profilometry. The depth information of the measured object is implicitly defined by certain trigonometric equations derived from image data. The direct solutions to the equations, called the wrapped phases, have a limited range from -pi to pi. The main computational problem is to unwrap the wrapped phases over the entire image. This phase unwrapping process is often hindered by the presence of noises, spots of low intensity modulation, and instability of the solutions. We present a novel approach to the problem by formulating the phase solving and unwrapping as an ill-posed problem. By using the regularization method, we will be able to provide a unified view to address various challenges in the problem. A new fully automated algorithm for phase unwrapping based on this approach is also presented.