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: 

    

  • 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

Code examples:
Network structure:
Energy contributions:
Index ordering conventions:
Reflection symmetry constraints:
Isometric constraints:
Variational energy minimization (MATLAB function):
Extraction of conformal data (MATLAB function):
Initialization (MATLAB script):
'mainVarMERA' benchmark:

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

Site maintained by:

Glen Evenbly (www.glenevenbly.com)

Email: contact@tensors.net