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##### Example codes: variational MERA

Please look at the readme page if you have not done so already. Here we present an implementation of the variational energy minimization algorithm for a scale-invariant MERA, which can be applied to study ground states of quantum critical systems on infinite D=1 dimensional lattices. Also included is an implementation of the for extracting conformal data from a scale-invariant MERA. These codes utilize a  modified binary MERA, one of the more useful MERA implementations for practical purposes, which has the following properties:

• Computational cost: O(χ^7)

• Based on a 2-site unit cell (A-B pattern)

• Easy implementation of spatial reflection symmetry

• Two isometries wv and one disentangler u at each level

###### Python ###### Energy contributions: ###### Index ordering conventions: ###### Reflection symmetry constraints: ###### Isometric constraints: ##### Extraction of conformal data (MATLAB function): ##### Initialization (MATLAB script):
###### ​

Method: scale-invariant MERA, bond dimension χ = 12

Test problem: 1D transverse Ising model at criticality (on an infinite lattice)

Running time: approx 10 mins

Quantities computed: ground energy density, scaling dimensions, OPE coefficients

Typical results:

Error in ground energy density (MERA): approx 3e-8

Scaling dimensions (MERA): [0, 0.1248, 0.997, 1.120, 1.120, 1.90, 1.90, 1.95, 1.96]

Scaling dimensions (exact):  [0, 0.1250, 1.000, 1.125, 1.125, 2.00, 2.00, 2.00, 2.00]

OPE coefficient (MERA): C(epsilon,sigma,sigma) = 0.5005

OPE coefficient (exact): C(epsilon,sigma,sigma) =  0.5000

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