Heisenberg Model

The Heisenberg model is a well-known quantum spin model that describes interactions between spins in a lattice. In the case of the Heisenberg spin-1/2 model, the Hamiltonian is given by:

\[H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j\]

The ground-state energy is expected to be -0.669437 from precise quantum Monte Carlo calculations.

Config File Setup

To calculate the ground-state iPEPS, create a config file (02_heisenberg.toml) containing:

dtype = "float64"
device = "cpu"

[TN]
nx = 2
ny = 2

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

[ctmrg]
steps = 10
projectors = "half-system"

[model]
name = "heisenberg"

[model.params]
J = 1.0

The configuration file specifies the data type (dtype) and device (device), the tensor network (TN) properties, and the model parameters.

The TN section specifies the dimensions of the tensor network. The values nx and ny define the size of the unit cell, and dims specifies the physical dimension (phys), bond dimension (bond), and the maximum corner bond dimension (chi).
The ctmrg section specifies the number of steps for the CTMRG algorithm and the type of projectors used for the algorithm.
The model section defines the model to be used, in this case, the Heisenberg model, and its corresponding parameter (J).

Initialize an iPEPS

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

from acetn.ipeps import Ipeps
import toml

config = toml.load('./input/02_heisenberg.toml')
ipeps = Ipeps(config)

The iPEPS site tensors are initialized in the Neel-ordered configuration for the Heisenberg model.

Evolve to the Ground State

We can evolve the iPEPS by:

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

Using a large imaginary-time step dtau=0.1 will bring the system closer to the ground state faster than a smaller step size (e.g. dtau=0.01).

Additional Evolution Steps

To refine the approximation, we can continue evolving the system using smaller time steps:

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

It is a good practice to start with a larger time step at first and then decrease the time step in further iterations to attain better accuracy.

Full Code Example

Here is a complete Python script that initializes the iPEPS, evolves it to the ground state, and measures the observables:

from acetn.ipeps import Ipeps
import toml

def main(config):
    ipeps = Ipeps(config)

    # Initial evolution step
    ipeps.evolve(dtau=0.1, steps=10)
    ipeps.measure()

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

if __name__ == '__main__':
    # Set config options in the input file: "02_heisenberg.toml"
    config = toml.load("./input/02_heisenberg.toml")
    main(config)

In this script: - The main function takes the config file as an argument, initializes the iPEPS object, and runs the evolution steps. - The initial evolution step uses a larger dtau to quickly approach the ground state, followed by finer updates in the loop with smaller dtau values.