The lace expansion is one of the powerful tools for investigating critical phenomena.  It has succeeded in obtaining asymptotic expansions for the critical point in several models, e.g., the self-avoiding walk, ordinary/oriented percolation, the contact process, etc.  One of our aims is to obtain such an asymptotic expansion for the quantum Ising model, which is one of the models of ferromagnetism.  In this model, Spin configurations are randomly realized by the Hamiltonian operator (energy), which involves the z-axis and x-axis Pauli matrices.  In other words, the quantum Ising model is defined by applying a transverse field to the classical Ising model.  Due to the transverse field, the quantum Ising model exhibits a different type of phase transition from the classical case.

 

In this talk, I will show the derivation of a lace expansion for the quantum Ising model and upper bounds on the expansion coefficients.  The derivation is based on the random current representation [Björnberg and Grimmett (2009) ] [Crawford and Ioffe (2010) ] on space-time.  This representation provides the connectivity between two vertices in space-time, which is similar to percolation models.  It helps us to derive the lace expansion.

 

This talk is based on joint work with Akira Sakai (Hokkaido University, Japan).

 

 

WebEx Link: https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m6f4acf1386d2f67a1702490f364e8ded

Meeting number (access code): 2512 167 2106

Meeting password: P3f5tSMPps5

 

Organizers: Jhih-Huang Li (NTU) ,Wai Kit Lam (NTU), Lung-Chi Chen (NCCU)