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 algorithms 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:
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Computational cost: O(χ^7)
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Based on a 2-site unit cell (A-B pattern)
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Easy implementation of spatial reflection symmetry
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Two isometries w, v and one disentangler u at each level
Network structure:
Energy contributions:
Index ordering conventions:
Reflection symmetry constraints:
Isometric constraints:
Variational energy minimization (Python module):
Extraction of conformal data (Python module):
Initialization (Python script):
'mainVarMERA' benchmark:
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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
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Typical results:
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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]
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OPE coefficient (MERA): C(epsilon,sigma,sigma) = 0.5005
OPE coefficient (exact): C(epsilon,sigma,sigma) = 0.5000