Variational iPEPS

The text in this section is mainly copied (and only slightly modifed) from the publication SciPost Phys. Lect. Notes 86 (2024) by Jan Naumann, Erik Lennart Weerda, Matteo Rizzi, Jens Eisert and Philipp Schmoll. This section is licensed under the CC-BY 4.0 as the original work.

We seek to find the TN representation of the state vector \(\ket{\psi}_\mathrm{TN}\) that best approximates the true ground state vector \(\ket{\psi_0}\) of an Hamilton operator of the form

\[H = \sum_{j \in \Lambda} T_j (h) \, ,\]

where \(T_j\) is the translation operator on the lattice \(\Lambda\), and \(h\) is a generic \(k\)-local Hamiltonian, i.e., it includes an arbitrary number of operators acting on lattice sites at most at a (lattice) distance \(k\) from a reference lattice point. Such a situation is very common in condensed matter physics, to say the least. To this aim, we employ the variational principle

\[\frac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle} \ge E_0 \hspace{0.5cm} \forall \, \ket{\psi},\]

and use an energy gradient with respect to the tensor coefficients to search for the minimum – the precise optimization strategy being discussed later. Such an energy gradient is accessed by means of tools from automatic differentiation (AD), a set of techniques to evaluate the derivative of a function specified by a computer program. In the varipeps library we use the jax-framework which already implements AS for common mathematical functions.

Since we directly target systems in the thermodynamic limit, a corner transfer matrix renormalization group (CTMRG) procedure constitutes the backbone of the algorithm, and also will come in handy for AD purposes. This is used to compute the approximate contraction of the infinite lattice, which is crucial in order to compute accurate expectation values in the first place.