No abstract

Critical phase transitions contain a variety of deep and universal physics and are intimately tied to thermodynamic quantities through scaling relations. Yet, these notions are challenged in the context of non-Hermiticity, where spatial or temporal divergences render the thermodynamic limit ill-defined. In this work, we show that a thermodynamic grand potential can still be defined in pseudo-Hermitian Hamiltonians, and can be used to characterize aspects of criticality unique to non-Hermitian systems. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) model as a paradigmatic example, we demonstrate the fractional order of topological phase transitions in the complex energy plane. These fractional orders add up to the integer order expected of a Hermitian phase transition when the model is doubled and Hermitianized. More spectacularly, gap preserving highly degenerate critical points known as non-Bloch band collapses possess fractional order that are not constrained by conventional scaling relations, testimony to the emergent extra length scale from the skin mode accumulation. Our work showcases that a thermodynamic approach can prove fruitful in revealing unconventional properties of non-Hermitian critical points.

This paper assesses the feasibility of studying wet deep mixing processes using centrifuge modelling. The scaling relationships relevant to modelling of the deep mixing were first established, and the likelihood of satisfying them in centrifuge modelling was then examined. The relationships between most of the significant forces in deep mixing processes can be satisfied using centrifuge modelling, with the exception of the Reynolds number. The latter cannot be preserved, owing to the non-Newtonian characteristics of cement slurry as well as of the soil−cement mix. However, viscosity scale effects can be mitigated by using a liquid tracer of lower viscosity in place of cement slurry. Deep mixing model tests were conducted to examine the effects of viscosity of the tracer, density of the tracer and work done in mixing. Comparison between 1g and centrifuge model test results shows that the former results have significantly poorer mixing quality than the latter. This can be attributed to the larger viscous forces relative to the other forces, in the 1g models. Lowering the viscosity of the tracer, increasing the work done in mixing and minimising density differences between soil and slurry can all contribute to enhancing the quality of mixing. The effect of density differences between soil and slurry was shown to be a possible reason for the large variation in reported field unconfined compressive strength as the water−cement ratio increases.