Quantum spin hamiltonians: A renormalization group study

Naobumi Honda

Kouichi Okunishi

We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.

Yi‐Zhuang You, Xiao-Liang Qi, Cenke Xu

We introduce the spectrum bifurcation renormalization group (SBRG) as an improved application of the excited-state real-space renormalization group (RSRG-X) for a class of qubit models. Starting from a disordered many-body Hamiltonian in the full many-body localized (MBL) phase, SBRG flows to the MBL fixed-point Hamiltonian, and generates the local integrals of motion and the matrix product state representations for all eigenstates. The method is applicable to many interacting spin and fermion models in the full MBL phase. As a Hilbert-space preserving RG, SBRG also generates an entanglement holographic mapping, which duals the MBL state to a fragmented holographic space.

Richard C. Brower

W D McComb

Abstract In this chapter we again use the linearized analysis of Chapter 8 in order to investigate the fixed points associated with renormalization group (RG) equations. But this time, instead of performing simple coarse-graining operations on a discrete set of lattice spin sites, we integrate out high-wave number modes from a continuous spin field. We then rescale the coarse-grained field in order to maintain the invariance of the Hamiltonian and this results in the RG equations.

C. Y. Pan