Using Convolutional Neural Network to solve quantum-spin-chains Ground-States.

The project main idea is to find the the ground state of a quantum spin chain hamiltonian using deep learning. Deep learning is wildily used as a tool to find abstract information in a lot of kind of problems, image pattern recognition are one of them. An image is nothing more than a tensor of numbers, in that sense an Hamiltonian is just a single chanel image and deep learning can be used to predict non explicit characteristics of the same, like its ground-state throught a regression method.

I've chosen an XXZ Hamiltonian with an uniform external magnect field. The Hamiltonian is in the for:

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Data set

The number of spins of the chain was fixed, , and de anisotropy constant, , was set to garantee the paramagnect regime. To generete the Hamiltonians the magnect fiel was started at , the matrix is block diagonalized and the minimum of all of the block are stored ( commutes with the Hamiltonian so the last can be written in magnatizations zones). After that the magnect field is incread by . Doing this until the magnect field is so we get samples.

The ground state engergy density () and the magnetization of the data set in function of the external magnect field () are represente in the graphics bellow respectivaly:

Create the data set

To create the Hamiltonians, save then in a folder named matrix/ as numpy arrays (h.npy) and create de graphics above simply run the following command:

python3 data_set.py

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Code

Used to	# of samples	%
Train	9120	76
Validate	2280	19
Test	600	5
Total	12000	100

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Results

Random

Loss

Retrain 500, 1k and 5k:

Train 1k and 10k:

Predictions

Just test:

Retrain 500, 1k and 5k:

Train 1k and 10k:

Train rand 25k

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Pool

Loss

Retrain 500, 1k and 10k:

Train 1k and 10k:

Predictions

Just test:

Retrain 500, 1k and 5k:

Train 1k and 10k:

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Pool L = 16

Loss:

Prediction:

Pool l=16 20k

Loss:

Prediction:

Delta x Negativity