Talks

One of the biggest challenges in physics is to develop accurate and efficient methods that can solve many currently intractable problems in correlated quantum or statistical systems. Tensor-network model/state is drawing more and more attention since it captures the feature of the area law and is absent from the sign problem. The evaluation of the expectation value of the observables can be reduced to the contraction of a tensor-network, which can be done by means of renormalization group method, and this is exactly what tensor renormalization group (TRG) method has done. In the light of comparison between numerical renormalization group (NRG) and density matrix renormalization group(DMRG), having considered the renormalization effect of the environment, second renormalization group (SRG) method has been proposed to improve the performance of TRG. Although TRG and SRG have achieved great success in 2D lattice, application to 3D is not easy. The talk will give a review of TRG and SRG, and talk about the HOTRG, i.e., TRG based on the higher-order singular value decomposition, and its SRG version, HOSRG. They can be easily applied to 3D lattice with relatively low memory and computation cost. The well-known 3D Ising model on simple cubic lattice has been tested. As far as our group know, HOSRG have achieved by far the most accurate renormalization group results for the 3D Ising model. Some analysis and possible applications will be also discussed.

Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of U(1) instantons for any instanton number. By studying SU(N) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory. We finally propose that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.