Example codes: boundary MERA
Please look at the readme page if you have not done so already. Here we present an implementation of the variational energy minimization for a (scaleinvariant) boundary MERA, which can be applied to study ground states of quantum critical systems on semiinfinite D=1 dimensional lattices. Requires input of a converged 'bulk' MERA, such as that generated by the MERA example code. The version of boundary MERA used is based on the form introduced here. This code can easily be generalized to other cases of lattice defects, such as impurities and interfaces, as described in this manuscript.

Computational cost: O(χ^6)

Requires input of converged modified binary MERA

One isometry q at each boundary level
Network structure:
Index ordering conventions:
Boundary terms:
Boundary Hamiltonian:
Energy contributions:
Block Hamiltonian and density matrix:
Boundary energy minimization (Julia function):
Initialization (Julia script):
'mainBoundMERA' benchmark:
Method: boundary scaleinvariant MERA, bond dimension χb = 8
Test problem: 1D transverse Ising model at criticality on a semiinfinite chain (with free B.C.)
Running time: approx 2 mins
Quantities computed: boundary energy, boundary zmagnetization, boundary scaling dimensions
Typical results: (assuming bulk tensors from 'mainVarMERA' at default settings)
Total error in boundary energy (MERA): approx 2e6
Boundary zmagnetization (MERA): accurate to within 1e4 for first L = 44 sites
Boundary scaling dimensions (MERA): [0, 0.496, 1.492, 1.989, 2.542, 3.044]
Boundary scaling dimensions (exact): [0, 0.500, 1.500, 2.000, 2.500, 3.000]