| .. _varipeps_unitcell: |
|
|
| iPEPS Unit Cells |
| ================ |
|
|
| Theoretical Overview |
| -------------------- |
|
|
| 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. |
|
|
| General iPEPS unit cell structure |
| ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
|
|
| We aim to simulate quantum many-body systems directly in the thermodynamic |
| limit. To this end, we consider a unit cell of lattice sites that is repeated |
| periodically over the infinite two-dimensional lattice. Reflecting this, the |
| general configurtions of the iPEPS Ansatz are defined with an arbitrary unit |
| cell of size :math:`(L_x, L_y)` on the square lattice. The lattice setup, |
| denoted by :math:`\mathcal{L}`, can be specified by a single matrix, which |
| uniquely determines the different lattice sites as well as their |
| arrangement. Let us consider a concrete example of an :math:`(L_x, L_y) = (2, |
| 2)` state with only two and all four individual tensors, denoted by |
|
|
| .. math:: |
|
|
| \mathcal L_1 = \begin{pmatrix} A & B \\ B & A \end{pmatrix}, \hspace{0.5cm} |
| \mathcal L_2 = \begin{pmatrix} A & C \\ B & D \end{pmatrix}. |
|
|
| .. subfigure:: AB |
| :align: center |
| :width: 75% |
| :layout-sm: A|B |
|
|
| .. image:: ../images/varipeps/ctmrgExample_1.* |
|
|
| .. image:: ../images/varipeps/ctmrgExample_2.* |
|
|
| iPEPS ansätze with a unit cell of size :math:`(L_x, L_y) = (2, 2)` and only |
| two (left) and four (right) different tensors. |
|
|
| The corresponding iPEPS ansätze are visualized in figure above. Here, the |
| rows/columns of :math:`\mathcal{L}` correspond to the :math:`x`/:math:`y` |
| lattice directions. The unit cell :math:`\mathcal{L}` is repeated periodically |
| to generate the full two-dimensional system. The bulk bond dimension of the |
| iPEPS tensors, denoted by :math:`\chi_B`, controls the accuracy of the |
| ansatz. An iPEPS state with :math:`N` different tensors in the unit cell |
| consists of :math:`N p \chi_B^4` variational parameters, which we aim to |
| optimize such that the iPEPS wave function represents an approximation of the |
| ground state of a specific Hamiltonian. The parameter :math:`p` denotes the |
| dimension of the physical Hilbert space, e.g., :math:`p = 2` for a system of |
| spin-:math:`1/2` particles. |
|
|
| The right choice of the unit cell is crucial in order to capture the structure |
| of the targeted state. A mismatch of the ansatz could not only lead to a bad |
| estimate of the ground state, but also to no convergence in the CTMRG routine at |
| all. Different lattice configurations have to be evaluated for specific |
| problems to find the correct pattern. |
|
|
|
|
| Spiral PEPS ansatz |
| ^^^^^^^^^^^^^^^^^^ |
|
|
| To circumvent the problem of a fixed and a priori chosen unit cell structure, |
| recently an alternative description to the periodic structure has been proposed |
| (`Phys. Rev. Lett. 133, 176502 (2024) |
| <https://doi.org/10.1103/PhysRevLett.133.176502>`_). This approach is applicable |
| if the Hamiltonian has a certain global symmetry, where the additional degree of |
| freedom can be employed to reduce the description of the state to a subspace, |
| e.g. :math:`SU(2)` for spin-:math:`1/2` systems. Here the state is described by |
| the smallest possible unit cell, i.e. a single site for a square lattice, as |
| well as a product of local unitary operators parameterized by a wave vector |
| :math:`\mathbf{k} = (k_x, k_y)`. A fixed choice of the wave vector then |
| corresponds to the specification of a unit cell structure in the common iPEPS |
| setup. This approach allows for a variational optimization of the wave vector |
| along with the translationally invariant iPEPS tensor, removing the need to |
| choose a fixed unit cell structure altogether. |
|
|
| Technical implementation |
| ------------------------ |
|
|
| .. role:: python(code) |
| :language: python |
|
|
| This section is provided under the default :ref:`license_cc_by_sa` license of |
| the documentation. |
|
|
| Initialization of unit cell |
| ^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
|
|
| The most usual way to initialize a unit cell |
|
|
| .. code-block:: python |
|
|
| # Unit cell structure |
| structure = [[0, 1, ...], ...] |
|
|
| # Physical dimension |
| d: int = ... |
|
|
| # iPEPS bond dimension |
| D: int = ... |
|
|
| # Start value for enviroment bond dimension |
| startChi: int = ... |
|
|
| # Data type for the tensors: `float` (real) or `complex` tensors |
| unitcell_dtype = ... |
|
|
| # Maximal enviroment bond dimension |
| max_chi: int = ... |
|
|
| # Type of iPEPS ansatz (SQUARE/SQUARE_SPLIT/TRIANGULAR) |
| peps_type: varipeps.peps.PEPS_Type = varipeps.peps.PEPS_Type.SQUARE, # Use square iPEPS ansatz |
|
|
| # Create random initialization for the iPEPS unit cell |
| unitcell = varipeps.peps.PEPS_Unit_Cell.random( |
| structure=structure, # Unit cell structure |
| d=d, # Physical dimension |
| D=chiB, # iPEPS bond dimension |
| chi=startChi, # Start value for enviroment bond dimension |
| dtype=unitcell_dtype, # Data type for the tensors: `float` (real) or `complex` tensors |
| max_chi=max_chi, # Maximal enviroment bond dimension |
| peps_type=peps_type, # Type of iPEPS ansatz |
| ) |
|
|
| Here we define the unit cell structure which is used to simulate our model. |
| Using the unit cell structure and the model parameter (see below), we can |
| generate an initial unit cell. Here we initialize the iPEPS tensors with random |
| numbers and we are using the above discussed square iPEPS ansatz. The structure |
| is supplied as matrix with different numbers encoding the different tensors |
| (:math:`A`/:math:`B`/:math:`C`/:math:`\dots` in the example above). |
|
|
| Unit cell parameters |
| ^^^^^^^^^^^^^^^^^^^^ |
|
|
| In the above code block we already showcase the (most important) parameters for |
| a iPEPS unit cell. To describe them in more detail: |
|
|
| * :python:`d`: The physical dimension of our iPEPS ansatz. E.g. :python:`d = 2` |
| for a spin-:math:`1/2` state. |
| * :python:`D`: The (bulk) bond dimension of the virtual links of our local iPEPS |
| tensors. Sometimes also called :math:`\chi_B`. |
| * :python:`chi`: The initial value of the environment bond dimension of the |
| CTMRG tensors. Sometimes also called :math:`\chi_E`. |
| * :python:`dtype`: The numerical data type used to represent the tensors. In the |
| default configuration of the library this can be :obj:`float` (or |
| equivalent :obj:`numpy.float64`) for real double-precision floating point |
| numbers or :obj:`complex` (or equivalent :obj:`numpy.complex128`) for |
| complex double-precision floating point numbers. |
| * :python:`max_chi`: The :obj:`varipeps` library supports a dynamical |
| increase/decrease of the environment bond dimension mainly based on some |
| heuristics around the truncation error of the truncated singular value |
| decomposition. This parameter selects a maximum value for the environment bond |
| dimension to ensure that the simulation stays within some computational and |
| memory limits. |
| * :python:`peps_type`: As hinted already, the :obj:`varipeps` library supports |
| different structures for the iPEPS ansatz and the environment structure. In |
| the above theoretical overview, we limited us to the most common square case |
| but we will discuss in other sections the split- and triangular CTMRG |
| variants. These different cases can be selected at initialization time using |
| this parameter. |
|
|
