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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 (scale-invariant) boundary MERA, which can be applied to study ground states of quantum critical systems on semi-infinite 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 at each boundary level

Code examples:
Network structure:
Index ordering conventions:
Boundary terms:
Boundary Hamiltonian:
Energy contributions:
Block Hamiltonian and density matrix:
Boundary energy minimization (Python module):
Initialization (Python script):
'mainBoundMERA' benchmark:

Method: boundary scale-invariant MERA, bond dimension χb = 8

Test problem: 1D transverse Ising model at criticality on a semi-infinite chain (with free B.C.)

Running time: approx 2 mins

Quantities computed: boundary energy, boundary z-magnetization, boundary scaling dimensions

Typical results: (assuming bulk tensors from 'mainVarMERA' at default settings)

Total error in boundary energy (MERA): approx 2e-6

Boundary z-magnetization (MERA): accurate to within 1e-4 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]

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