Transverse-Field Ising Model

The transverse-field Ising model (TFIM) is a simple model exhibiting a quantum phase transition and a common test subject for iPEPS simulations. The model is defined by the Hamiltonian:

\[H = -J\sum_{\langle i j \rangle} S_i^z S_j^z - h_x \sum_{i} S_i^x\]

Config File Setup

To simulate the ground state of this model, first create the config file input.toml or any filename that you prefer containing:

dtype = "float64"
device = "cpu"

[TN]
nx = 2
ny = 2

[TN.dims]
phys = 2
bond = 4
chi = 16

[model]
name = "ising"

[model.params]
jz = 1.0
hx = 0.1

The first two parameters set the data type and which device to store the tensors. The TN entries specify the tensor-network unit cell and dimensions, and the model entries specify the name of the predefined model to use. Predefined models such as “ising” and “heisenberg” are supplied mainly for testing and benchmarking purposes.

Initialize an iPEPS

Given the input config, an iPEPS can be constructed by:

from acetn.ipeps import Ipeps
import toml

config = toml.load('input.toml')
ipeps = Ipeps(config)

The iPEPS site tensors are initialized in the ferromagnetic configuration for the Ising model with some noise. This is already close to the ground state, but not exactly. The ground state is typically found after a few hundred steps of imaginary-time evolution.

Evolve to the Ground State

We can evolve the iPEPS by:

ipeps.evolve(dtau=0.1, steps=10)
ipeps.measure()

The iPEPS should now be close to the ground state.

Additional Evolution Steps

For improved accuracy, the iPEPS can be evolved for more steps:

for _ in range(5):
    ipeps.evolve(dtau=0.01, steps=100)
    ipeps.measure()

The evolution steps here use smaller dtau values for finer updates, iterating multiple times to refine the approximation of the ground state.

Modify Model Parameters

The iPEPS can be evolved again with different model parameters. For example, to use the converged iPEPS to determine the ground state at a different field value (e.g., hx=0.2), this can be done by:

ipeps.set_model_params(hx=0.2)
ipeps.evolve(dtau=0.01, steps=100)

Full Code Example

Here is a complete code example that initializes an iPEPS, evolves it to the ground state, and measures the observables using hx=0.1 and hx=0.2:

from acetn.ipeps import Ipeps

def main(config):
    # Initialize iPEPS
    ipeps = Ipeps(config)

    # Determine ground state for hx=0.1 and measure observables
    ipeps.evolve(dtau=0.01, steps=100)
    measurements = ipeps.measure()

    # Determine ground state for hx=0.2 and measure observables
    ipeps.set_model_params(hx=0.2)
    ipeps.evolve(dtau=0.01, steps=100)
    measurements = ipeps.measure()

if __name__=='__main__':
    config = {
        'dtype': "float64",
        'device': "cpu",
        'TN':{
            'nx': 2,
            'ny': 2,
            'dims': {'phys':2, 'bond':2, 'chi':20},
        },
        'model':{
            'name': 'ising',
            'params':{
                'jz': 1.0,
                'hx': 0.1,
            },
        },
    }

    main(config)