3.2. Define Your Model

3.2.1. Overview

In this notebook we will introduce 3 basic components for Renormalizer: Op, BasisSet and Model. These components are essential for MPS/MPO construction/manipulation for a given physical model.

# add renormalizer to path
import sys
sys.path.append("..")

3.2.2. Op

Op is the abbreviation for “operators”. It offers a convenient interface to represent operators of your interest. Op is the bridge between symbolic math equations and numerical tensors in MPS/MPO.

from renormalizer import Op

2022-10-28 13:29:10,474[INFO] Use NumPy as backend
2022-10-28 13:29:10,475[INFO] use 64 bits

3.2.2.1. Basics

There are 3 basic attributes for an operator:

• The operation or its symbol, such as \(\hat X\), \(\hat a^\dagger\), \(\hat p^2\)

• The DOF at wich the operation is operated on, such as “spin 0”, “exiton 1”. In mathematical expression this is usually the subscript for an operator, i.e., \(\hat X_0\) or \(\hat a^\dagger_1\)

• The factor or coefficient for the operator.

There is an additional useful attributes that is sometimes quite useful: the quantum number.

Op are constructed from the above 4 attributes. The first 2 are necessary and the last 2 are optional.

Op("X", 0, factor=0.5, qn=0)

Op('X', [0], 0.5, [0])

In renormalizer, you may use anything that can be hashed and compared to denote a DOF. Common examples include int, str, and tuple of them.

Op("X", ("spin 0", "cool stuff", 2077))

Op('X', [('spin 0', 'cool stuff', 2077)], 1.0)

You may wonder what are the allowed symbols for operators. For that please refer to the BasisSet section.

Products of operators can be constructed intuitively

Op(r"a^\dagger a", [0, 1])

Op('a^\\dagger a', [0, 1], 1.0)

Here, DOFs for the operators are specified through a list, where the first element is the DOF for \(a^\dagger\) and the second element is the DOF for \(a\).

Note that using tuple to specify the DOF has a totally different meaning. Renormalizer will recognize the tuple as a single DOF and set all DOFs in the operator to that DOF.

Op(r"a^\dagger a", (0, 1))

Op('a^\\dagger a', [(0, 1), (0, 1)], 1.0)

Op also has a lot of useful functions and attributes, please refer to the API document for more details.

3.2.2.2. Operations and OpSum

Common and simple operations between operators are supported

op1 = Op("X", 0, 0.5)
op2 = Op("Z", 1, 0.5)
op1 * op2

Op('X Z', [0, 1], 0.25)

Addition between Op will result in an OpSum instance that is a subclass of list.

op1 + op2, type(op1 + op2)

([Op('X', [0], 0.5), Op('Z', [1], 0.5)], renormalizer.model.op.OpSum)

OpSum supports simple operator algebra such as multiplication/addition, etc.

# multiplication
opsum = op1 + op2
opsum * op1

[Op('X X', [0, 0], 0.25), Op('Z X', [1, 0], 0.25)]

opsum * opsum

[Op('X X', [0, 0], 0.25),
 Op('X Z', [0, 1], 0.25),
 Op('Z X', [1, 0], 0.25),
 Op('Z Z', [1, 1], 0.25)]

# addition/subtraction
opsum -= op2
opsum

[Op('X', [0], 0.5), Op('Z', [1], 0.5), Op('Z', [1], -0.5)]

opsum.simplify()

[Op('X', [0], 0.5)]

However, in general the functionalities are limited and the performance is not optimized. We recommand using SymPy for advanced symbolic mathematics and convert the final result to Renormalizer Op.

3.2.3. BasisSet

An essential step for converting symbolic operators into numeric tensors is to specify a set of basis. Renormalizer includes a zoo of basis set with an emphasis for electron-phonon or vibronic models.

import renormalizer
[s for s in dir(renormalizer) if s.startswith("Basis")]

['BasisHalfSpin',
 'BasisHopsBoson',
 'BasisMultiElectron',
 'BasisMultiElectronVac',
 'BasisSHO',
 'BasisSimpleElectron',
 'BasisSineDVR']

Each BasisSet is associated with one or more DOFs. For example, we can setup the spin-1/2 basis set for a spin DOF denoted as ("spin", 0)

from renormalizer import BasisHalfSpin
b = BasisHalfSpin(("spin", 0))
b

(dof: ('spin', 0), nbas: 2, qn: [0, 0])

Each basis set supports a variety of operations. The spin-1/2 basis set naturally supports Pauli operators such as "X", "sigma_y".

b.op_mat("X"), b.op_mat("sigma_y")

(array([[0., 1.],
        [1., 0.]]),
 array([[0.+0.j, 0.-1.j],
        [0.+1.j, 0.+0.j]]))

For phonon DOF, it is necessary to specify the quanta truncated in the basis set

# SHO represent simple harmonic oscillator
from renormalizer import BasisSHO
# omega is the vibration frequency, nbas is the number of basis
b = BasisSHO(dof=0, omega=50, nbas=4)
# note the different value because of omega
b.op_mat("b^\dagger+b"), b.op_mat("x")

(array([[0.        , 1.        , 0.        , 0.        ],
        [1.        , 0.        , 1.41421356, 0.        ],
        [0.        , 1.41421356, 0.        , 1.73205081],
        [0.        , 0.        , 1.73205081, 0.        ]]),
 array([[0.        , 0.1       , 0.        , 0.        ],
        [0.1       , 0.        , 0.14142136, 0.        ],
        [0.        , 0.14142136, 0.        , 0.17320508],
        [0.        , 0.        , 0.17320508, 0.        ]]))

Note that BasisSHO supports both first and second-quantized operators such as

\[\hat x = \sqrt{\frac{1}{2\omega}} (b^\dagger + b)\]

here \(x\) is the mass-weighted coordinate.

The number of possible basis sets is infinite and highly customized basis set for a particular problem is rairly common. You may subclass the BasisSet parent class in renormalizer.model.basis.BasisSet to customize the basis set and the numerical behavior of the operators.

3.2.4. Model

A Model is basically made up with a list of BasisSet and the Hamiltonian in OpSum/list form. A Model instance is necessary for any MPS/MPO construction.

from renormalizer import Model
model = Model([BasisHalfSpin(0)], [Op("X", 0)])

For example, the corresponding MPO can be constructed from the above model conveniently

from renormalizer import Mpo
mpo = Mpo(model)
mpo

2022-10-28 13:29:10,610[DEBUG] # of operator terms: 1

<class 'renormalizer.mps.mpo.Mpo'> with 1 sites

mpo.todense()

array([[0., 1.],
       [1., 0.]])

You may wonder

• What are the limitations for the model, in particular the operator terms, to construct MPO? For example, is nearest-neighbour interaction or at most two-body operator enforced?

• If any numerical compression is involved in the construction. If so, how to control the accuracy?

• What are the time cost of constructing MPO for an arbitrary model?

The answer is: renormalizer offers a unique algorithm for exact and efficient MPO construction of arbitrary operators. It means

• There’s NO limitation for the model. Long-range interaction, three-body operators, anything you like.

• Numerical compression is not involved and the construction is exact.

• The construction time cost is neglegible for most models

Additionally, we also guarantee that the constructed MPO is optimal in terms of bond dimension.

For details of our algorithm, see A general automatic method for optimal construction of matrix product operators using bipartite graph theory.

Renormalizer also offers a set of built-in models for fast construction of common models. Please refer to the API document for more details.

import renormalizer
[s for s in dir(renormalizer) if s.endswith("Model")]

['HolsteinModel', 'Model', 'SpinBosonModel', 'TI1DModel']