simple_dmrg_03_conserved_quantum_numbers.py

#!/usr/bin/env python
#
# Simple DMRG tutorial.  This code integrates the following concepts:
#  - Infinite system algorithm
#  - Finite system algorithm
#  - Conserved quantum numbers
#
# Copyright 2013 James R. Garrison and Ryan V. Mishmash.
# Open source under the MIT license.  Source code at
# <https://github.com/simple-dmrg/simple-dmrg/>

# This code will run under any version of Python >= 2.6.  The following line
# provides consistency between python2 and python3.
from __future__ import print_function, division  # requires Python >= 2.6

# numpy and scipy imports
import numpy as np
from scipy.sparse import kron, identity, lil_matrix
from scipy.sparse.linalg import eigsh  # Lanczos routine from ARPACK

# We will use python's "namedtuple" to represent the Block and EnlargedBlock
# objects
from collections import namedtuple

Block = namedtuple("Block", ["length", "basis_size", "operator_dict", "basis_sector_array"])
EnlargedBlock = namedtuple("EnlargedBlock", ["length", "basis_size", "operator_dict", "basis_sector_array"])

def is_valid_block(block):
    if len(block.basis_sector_array) != block.basis_size:
        return False
    for op in block.operator_dict.values():
        if op.shape[0] != block.basis_size or op.shape[1] != block.basis_size:
            return False
    return True

# This function should test the same exact things, so there is no need to
# repeat its definition.
is_valid_enlarged_block = is_valid_block

# Model-specific code for the Heisenberg XXZ chain
model_d = 2  # single-site basis size
single_site_sectors = np.array([0.5, -0.5])  # S^z sectors corresponding to the
                                             # single site basis elements

Sz1 = np.array([[0.5, 0], [0, -0.5]], dtype='d')  # single-site S^z
Sp1 = np.array([[0, 1], [0, 0]], dtype='d')  # single-site S^+

H1 = np.array([[0, 0], [0, 0]], dtype='d')  # single-site portion of H is zero

def H2(Sz1, Sp1, Sz2, Sp2):  # two-site part of H
    """Given the operators S^z and S^+ on two sites in different Hilbert spaces
    (e.g. two blocks), returns a Kronecker product representing the
    corresponding two-site term in the Hamiltonian that joins the two sites.
    """
    J = Jz = 1.
    return (
        (J / 2) * (kron(Sp1, Sp2.conjugate().transpose()) + kron(Sp1.conjugate().transpose(), Sp2)) +
        Jz * kron(Sz1, Sz2)
    )

# conn refers to the connection operator, that is, the operator on the edge of
# the block, on the interior of the chain.  We need to be able to represent S^z
# and S^+ on that site in the current basis in order to grow the chain.
initial_block = Block(length=1, basis_size=model_d, operator_dict={
    "H": H1,
    "conn_Sz": Sz1,
    "conn_Sp": Sp1,
}, basis_sector_array=single_site_sectors)

def enlarge_block(block):
    """This function enlarges the provided Block by a single site, returning an
    EnlargedBlock.
    """
    mblock = block.basis_size
    o = block.operator_dict

    # Create the new operators for the enlarged block.  Our basis becomes a
    # Kronecker product of the Block basis and the single-site basis.  NOTE:
    # `kron` uses the tensor product convention making blocks of the second
    # array scaled by the first.  As such, we adopt this convention for
    # Kronecker products throughout the code.
    enlarged_operator_dict = {
        "H": kron(o["H"], identity(model_d)) + kron(identity(mblock), H1) + H2(o["conn_Sz"], o["conn_Sp"], Sz1, Sp1),
        "conn_Sz": kron(identity(mblock), Sz1),
        "conn_Sp": kron(identity(mblock), Sp1),
    }

    # This array keeps track of which sector each element of the new basis is
    # in.  `np.add.outer()` creates a matrix that adds each element of the
    # first vector with each element of the second, which when flattened
    # contains the sector of each basis element in the above Kronecker product.
    enlarged_basis_sector_array = np.add.outer(block.basis_sector_array, single_site_sectors).flatten()

    return EnlargedBlock(length=(block.length + 1),
                         basis_size=(block.basis_size * model_d),
                         operator_dict=enlarged_operator_dict,
                         basis_sector_array=enlarged_basis_sector_array)

def rotate_and_truncate(operator, transformation_matrix):
    """Transforms the operator to the new (possibly truncated) basis given by
    `transformation_matrix`.
    """
    return transformation_matrix.conjugate().transpose().dot(operator.dot(transformation_matrix))

def index_map(array):
    """Given an array, returns a dictionary that allows quick access to the
    indices at which a given value occurs.

    Example usage:

    >>> by_index = index_map([3, 5, 5, 7, 3])
    >>> by_index[3]
    [0, 4]
    >>> by_index[5]
    [1, 2]
    >>> by_index[7]
    [3]
    """
    d = {}
    for index, value in enumerate(array):
        d.setdefault(value, []).append(index)
    return d

def single_dmrg_step(sys, env, m, target_Sz):
    """Performs a single DMRG step using `sys` as the system and `env` as the
    environment, keeping a maximum of `m` states in the new basis.
    """
    assert is_valid_block(sys)
    assert is_valid_block(env)

    # Enlarge each block by a single site.
    sys_enl = enlarge_block(sys)
    sys_enl_basis_by_sector = index_map(sys_enl.basis_sector_array)
    if sys is env:  # no need to recalculate a second time
        env_enl = sys_enl
        env_enl_basis_by_sector = sys_enl_basis_by_sector
    else:
        env_enl = enlarge_block(env)
        env_enl_basis_by_sector = index_map(env_enl.basis_sector_array)

    assert is_valid_enlarged_block(sys_enl)
    assert is_valid_enlarged_block(env_enl)

    # Construct the full superblock Hamiltonian.
    m_sys_enl = sys_enl.basis_size
    m_env_enl = env_enl.basis_size
    sys_enl_op = sys_enl.operator_dict
    env_enl_op = env_enl.operator_dict
    superblock_hamiltonian = kron(sys_enl_op["H"], identity(m_env_enl)) + kron(identity(m_sys_enl), env_enl_op["H"]) + \
                             H2(sys_enl_op["conn_Sz"], sys_enl_op["conn_Sp"], env_enl_op["conn_Sz"], env_enl_op["conn_Sp"])

    # Build up a "restricted" basis of states in the target sector and
    # reconstruct the superblock Hamiltonian in that sector.
    sector_indices = {} # will contain indices of the new (restricted) basis
                        # for which the enlarged system is in a given sector
    restricted_basis_indices = []  # will contain indices of the old (full) basis, which we are mapping to
    for sys_enl_Sz, sys_enl_basis_states in sys_enl_basis_by_sector.items():
        sector_indices[sys_enl_Sz] = []
        env_enl_Sz = target_Sz - sys_enl_Sz
        if env_enl_Sz in env_enl_basis_by_sector:
            for i in sys_enl_basis_states:
                i_offset = m_env_enl * i  # considers the tensor product structure of the superblock basis
                for j in env_enl_basis_by_sector[env_enl_Sz]:
                    current_index = len(restricted_basis_indices)  # about-to-be-added index of restricted_basis_indices
                    sector_indices[sys_enl_Sz].append(current_index)
                    restricted_basis_indices.append(i_offset + j)

    restricted_superblock_hamiltonian = superblock_hamiltonian[:, restricted_basis_indices][restricted_basis_indices, :]

    # Call ARPACK to find the superblock ground state.  ("SA" means find the
    # "smallest in amplitude" eigenvalue.)
    (energy,), restricted_psi0 = eigsh(restricted_superblock_hamiltonian, k=1, which="SA")

    # Construct each block of the reduced density matrix of the system by
    # tracing out the environment
    rho_block_dict = {}
    for sys_enl_Sz, indices in sector_indices.items():
        if indices: # if indices is nonempty
            psi0_sector = restricted_psi0[indices, :]
            # We want to make the (sys, env) indices correspond to (row,
            # column) of a matrix, respectively.  Since the environment
            # (column) index updates most quickly in our Kronecker product
            # structure, psi0_sector is thus row-major ("C style").
            psi0_sector = psi0_sector.reshape([len(sys_enl_basis_by_sector[sys_enl_Sz]), -1], order="C")
            rho_block_dict[sys_enl_Sz] = np.dot(psi0_sector, psi0_sector.conjugate().transpose())

    # Diagonalize each block of the reduced density matrix and sort the
    # eigenvectors by eigenvalue.
    possible_eigenstates = []
    for Sz_sector, rho_block in rho_block_dict.items():
        evals, evecs = np.linalg.eigh(rho_block)
        current_sector_basis = sys_enl_basis_by_sector[Sz_sector]
        for eval, evec in zip(evals, evecs.transpose()):
            possible_eigenstates.append((eval, evec, Sz_sector, current_sector_basis))
    possible_eigenstates.sort(reverse=True, key=lambda x: x[0])  # largest eigenvalue first

    # Build the transformation matrix from the `m` overall most significant
    # eigenvectors.  It will have sparse structure due to the conserved quantum
    # number.
    my_m = min(len(possible_eigenstates), m)
    transformation_matrix = lil_matrix((sys_enl.basis_size, my_m), dtype='d')
    new_sector_array = np.zeros((my_m,), dtype='d')  # lists the sector of each
                                                     # element of the new/truncated basis
    for i, (eval, evec, Sz_sector, current_sector_basis) in enumerate(possible_eigenstates[:my_m]):
        for j, v in zip(current_sector_basis, evec):
            transformation_matrix[j, i] = v
        new_sector_array[i] = Sz_sector
    # Convert the transformation matrix to a more efficient internal
    # representation.  `lil_matrix` is good for constructing a sparse matrix
    # efficiently, but `csr_matrix` is better for performing quick
    # multiplications.
    transformation_matrix = transformation_matrix.tocsr()

    truncation_error = 1 - sum([x[0] for x in possible_eigenstates[:my_m]])
    print("truncation error:", truncation_error)

    # Rotate and truncate each operator.
    new_operator_dict = {}
    for name, op in sys_enl.operator_dict.items():
        new_operator_dict[name] = rotate_and_truncate(op, transformation_matrix)

    newblock = Block(length=sys_enl.length,
                     basis_size=my_m,
                     operator_dict=new_operator_dict,
                     basis_sector_array=new_sector_array)

    return newblock, energy

def graphic(sys_block, env_block, sys_label="l"):
    """Returns a graphical representation of the DMRG step we are about to
    perform, using '=' to represent the system sites, '-' to represent the
    environment sites, and '**' to represent the two intermediate sites.
    """
    assert sys_label in ("l", "r")
    graphic = ("=" * sys_block.length) + "**" + ("-" * env_block.length)
    if sys_label == "r":
        # The system should be on the right and the environment should be on
        # the left, so reverse the graphic.
        graphic = graphic[::-1]
    return graphic

def infinite_system_algorithm(L, m, target_Sz):
    block = initial_block
    # Repeatedly enlarge the system by performing a single DMRG step, using a
    # reflection of the current block as the environment.
    while 2 * block.length < L:
        current_L = 2 * block.length + 2  # current superblock length
        current_target_Sz = int(target_Sz) * current_L // L
        print("L =", current_L)
        block, energy = single_dmrg_step(block, block, m=m, target_Sz=current_target_Sz)
        print("E/L =", energy / current_L)

def finite_system_algorithm(L, m_warmup, m_sweep_list, target_Sz):
    assert L % 2 == 0  # require that L is an even number

    # To keep things simple, this dictionary is not actually saved to disk, but
    # we use it to represent persistent storage.
    block_disk = {}  # "disk" storage for Block objects

    # Use the infinite system algorithm to build up to desired size.  Each time
    # we construct a block, we save it for future reference as both a left
    # ("l") and right ("r") block, as the infinite system algorithm assumes the
    # environment is a mirror image of the system.
    block = initial_block
    block_disk["l", block.length] = block
    block_disk["r", block.length] = block
    while 2 * block.length < L:
        # Perform a single DMRG step and save the new Block to "disk"
        print(graphic(block, block))
        current_L = 2 * block.length + 2  # current superblock length
        current_target_Sz = int(target_Sz) * current_L // L
        block, energy = single_dmrg_step(block, block, m=m_warmup, target_Sz=current_target_Sz)
        print("E/L =", energy / current_L)
        block_disk["l", block.length] = block
        block_disk["r", block.length] = block

    # Now that the system is built up to its full size, we perform sweeps using
    # the finite system algorithm.  At first the left block will act as the
    # system, growing at the expense of the right block (the environment), but
    # once we come to the end of the chain these roles will be reversed.
    sys_label, env_label = "l", "r"
    sys_block = block; del block  # rename the variable
    for m in m_sweep_list:
        while True:
            # Load the appropriate environment block from "disk"
            env_block = block_disk[env_label, L - sys_block.length - 2]
            if env_block.length == 1:
                # We've come to the end of the chain, so we reverse course.
                sys_block, env_block = env_block, sys_block
                sys_label, env_label = env_label, sys_label

            # Perform a single DMRG step.
            print(graphic(sys_block, env_block, sys_label))
            sys_block, energy = single_dmrg_step(sys_block, env_block, m=m, target_Sz=target_Sz)

            print("E/L =", energy / L)

            # Save the block from this step to disk.
            block_disk[sys_label, sys_block.length] = sys_block

            # Check whether we just completed a full sweep.
            if sys_label == "l" and 2 * sys_block.length == L:
                break  # escape from the "while True" loop

if __name__ == "__main__":
    np.set_printoptions(precision=10, suppress=True, threshold=10000, linewidth=300)

    #infinite_system_algorithm(L=100, m=20, target_Sz=0)
    finite_system_algorithm(L=20, m_warmup=10, m_sweep_list=[10, 20, 30, 40, 40], target_Sz=0)