.. _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) `_ 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) `_). 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.