| .. _varipeps: |
|
|
| Variational iPEPS |
| ================= |
|
|
| The text in this section is mainly copied (and only slightly modifed) from the |
| publication `SciPost Phys. Lect. Notes 86 (2024) |
| <https://doi.org/10.21468/SciPostPhysLectNotes.86>`_ by Jan Naumann, Erik |
| Lennart Weerda, Matteo Rizzi, Jens Eisert and Philipp Schmoll. This section is |
| licensed under the :ref:`license_cc_by` as the original work. |
|
|
| We seek to find the TN representation of the state vector |
| :math:`\ket{\psi}_\mathrm{TN}` that best approximates the true ground state |
| vector :math:`\ket{\psi_0}` of an Hamilton operator of the form |
|
|
| .. math:: |
|
|
| H = \sum_{j \in \Lambda} T_j (h) \, , |
|
|
| where :math:`T_j` is the translation operator on the lattice :math:`\Lambda`, |
| and :math:`h` is a generic :math:`k`-local Hamiltonian, i.e., it includes an |
| arbitrary number of operators acting on lattice sites at most at a (lattice) |
| distance :math:`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 |
|
|
| .. math:: |
|
|
| \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 :obj:`varipeps` library we use the |
| :obj:`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. |
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|
