Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity

We consider a time inhomogeneous jump Markov process $X = (X_t)_t$ with state dependent jump intensity, taking values in $R^d .$ Its infinitesimal generator is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial x_i } (x) \int_{E_1} c_1^i ( t, z, x) \gamma_1 ( t, z, x ) \mu_1 (dz ) \\ + \sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] \gamma_l ( t, z, x) \mu_l (dz ) , \end{multline*} where $(E_l , {\mathcal E}_l, \mu_l ) , 1 \le l \le 3,$ are sigma-finite measurable spaces describing three different jump regimes of the process (fast, intermediate, slow). We give conditions proving that the long time behavior of $X$ can be related to the one of a time homogeneous limit process $\bar X .$ Moreover, we introduce a coupling method for the limit process which is entirely based on certain of its big jumps and which relies on the regeneration method. We state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium both for $X$ and for $\bar X.

Many-body spin Berry phases emerging from the π\pi-flux state: antiferromagnetic/valence-bond-solid competition

We uncover new topology-related features of the $\pi$-flux saddle-point solution of the $D$=2+1 Heisenberg antiferromagnet. We note that symmetries of the spinons sustain a built-in competition between antiferromagnetic (AF) and valence-bond-solid (VBS) orders, the two tendencies central to recent developments on quantum criticality. An effective theory containing an analogue of the Wess-Zumino-Witten term is derived, which generates quantum phases related to AF monopoles with VBS cores, and reproduces Haldane's hedgehog Berry phases. The theory readily generalizes to $\pi$-flux states for all $D$.Comment: 4 pages, revise