Combinatorial Calabi flows on surfaces

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure

Anomalous minimum and scaling behavior of localization length near an isolated flat band

Using one-dimensional tight-binding lattices and an analytical expression based on the Green's matrix, we show that anomalous minimum of the localization length near an isolated flat band, previously found for evanescent waves in a defect-free photonic crystal waveguide, is a generic feature and exists in the Anderson regime as well, i.e., in the presence of disorder. Our finding reveals a scaling behavior of the localization length in terms of the disorder strength, as well as a summation rule of the inverse localization length in terms of the density of states in different bands. Most interesting, the latter indicates the possibility of having two localization minima inside a band gap, if this band gap is formed by two flat bands such as in a double-sided Lieb lattice.Comment: 8 pages, 9 figure

Parity-time symmetry in a flat band system

In this paper we introduce Parity-Time ($\cal PT$) symmetric perturbation to a one-dimensional Lieb lattice, which is otherwise $\cal P$-symmetric and has a flat band. In the flat band there are a multitude of degenerate dark states, and the degeneracy $N$ increases with the system size. We show that the degeneracy in the flat band is completely lifted due to the non-Hermitian perturbation in general, but it is partially maintained with the half-gain-half-loss perturbation and its ``V" variant that we consider. With these perturbations, we show that both randomly positioned states and pinned states at the symmetry plane in the flat band can undergo thresholdless $\cal PT$ breaking. They are distinguished by their different rates of acquiring non-Hermicity as the $\cal PT$-symmetric perturbation grows, which are insensitive to the system size. Using a degenerate perturbation theory, we derive analytically the rate for the pinned states, whose spatial profiles are also insensitive to the system size. Finally, we find that the presence of weak disorder has a strong effect on modes in the dispersive bands but not on those in the flat band. The latter respond in completely different ways to the growing $\cal PT$-symmetric perturbation, depending on whether they are randomly positioned or pinned.Comment: 8 pages, 8 figure

Macroscopic fluxes and local reciprocal relation in second-order stochastic processes far from equilibrium

Stochastic process is an essential tool for the investigation of the physical and life sciences at nanoscale. In the first-order stochastic processes widely used in chemistry and biology, only the flux of mass rather than that of heat can be well defined. Here we investigate the two macroscopic fluxes in second-order stochastic processes driven by position-dependent forces and temperature gradient. We prove that the thermodynamic equilibrium defined through the vanishing of macroscopic fluxes is equivalent to that defined via time reversibility at mesoscopic scale. In the small noise limit, we find that the entropy production rate, which has previously been defined by the mesoscopic irreversible fluxes on the phase space, matches the classic macroscopic expression as the sum of the products of macroscopic fluxes and their associated thermodynamic forces. Further we show that the two pairs of forces and fluxes in such a limit follow a linear phenomenonical relation and the associated scalar coefficients always satisfy the reciprocal relation for both transient and steady states. The scalar coefficient is proportional to the square of local temperature divided by the local frictional coefficient and originated from the second moment of velocity distribution along each dimension. This result suggests the very close connection between Soret effect (thermal diffusion) and Dufour effect at nano scale even far from equilibrium