On the Edge Singularity of the Actuator Disk Model

In this technical brief, the classic actuator disk theory is revisited with a view to shed some light on the singularity of the flow at the edge of the disk where the vortex tube starts and where vorticity is generated. The study is carried out using small perturbation assumption in two-dimensions and simplified boundary conditions in all cases. The problem of the two-dimensional thin cambered plate with constant vorticity distribution is solved and the leading edge singularity is analyzed as it is believed to be relevant to the axisymmetric flow at the actuator disk edge. Next, the velocity components induced by the cylindrical vortex tube of constant vorticity are obtained via the Biot–Savart law and the near edge behavior is investigated. It is shown that the velocity components behavior is consistent with that of the thin cambered plate with constant loading, thus reinforcing the notion that the axisymmetric slip-line behaves as r − R ∝ −xlnx near the disk edge.

Pseudo-Yang-Lee Edge Singularity Critical Behavior in a Non-Hermitian Ising Model

The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.

Scaling of the Berry Phase in the Yang-Lee Edge Singularity

We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.

Dimensional reduction by conformal bootstrap

Dimensional reductions in the branched polymer model and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. Small minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to the $D'=D-2$D Yang–Lee edge singularity and to the pure $D'=D-2$D Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$ and, for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.