Simplified topological invariants for interacting insulators

We propose general topological order parameters for interacting insulators in terms of the Green's function at zero frequency. They provide an unified description of various interacting topological insulators including the quantum anomalous Hall insulators and the time reversal invariant insulators in four, three and two dimensions. Since only Green's function at zero frequency is used, these topological order parameters can be evaluated efficiently by most numerical and analytical algorithms for strongly interacting systems.Comment: Published versio

Mean field equations, hyperelliptic curves and modular forms: II

A pre-modular form $Z_n(\sigma; \tau)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(\sigma, \tau) \in \Bbb C \times \Bbb H$, such that for $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z \tau)$, every non-trivial zero of $Z_n(\sigma; \tau)$, namely $\sigma \not\in E_\tau[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation} on the flat torus $E_\tau$ with singular strength $\rho = 8\pi n$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(\tau) \subset {\rm Sym}^n E_\tau$, the Lam\'e curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma; \tau)$ relies on a detailed study on the correspondence $\Bbb P^1 \leftarrow \bar X_n(\tau) \to E_\tau$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(\sigma; \tau)$, a counting formula for Lam\'e equations of degree $n = 4$ with finite monodromy is given in the appendix (by Y.-C. Chou).Comment: 32 pages. Part of content in previous versions is removed and published separately. One author is remove

Analytic Aspects of the Toda System: II. Bubbling behavior and existence of solutions

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$. We give a much more precise bubbling behavior of solutions and study its existence in some critical casesComment: 33 pages, to appear in Comm. Pure Appl. Mat