Drudge tutorial for beginners

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Drudge is a library built on top of the SymPy computer algebra library for noncommutative and tensor alegbras. Usually for these style of problems, the symbolic manipulation and simplification of mathematical expressions requires a lot of context-dependent information, like the specific commutation rules and things like the dummy symbols to be used for different ranges. So the primary entry point for using the library is the Drudge class, which serves as a central repository of all kinds of domain-specific informations. To create a drudge instance, we need to give it a Spark context so that it is capable of parallelize things. For instance, to run things locally with all available cores, we can do

>>> from pyspark import SparkContext  
>>> spark_ctx = SparkContext('local[*]', 'drudge-tutorial')

For using Spark in cluster computing environment, please refer to the Spark documentation and setting of your cluster. With the spark context created, we can make the main entry point for drudge,

>>> import drudge
>>> dr = drudge.Drudge(spark_ctx)

Then from it, we can create the symbolic expressions as Tensor objects, which are basically mathematical expressions containing noncommutative objects and symbolic summations. For the noncommutativity, in spite of the availability of some basic support of it in SymPy, here we have the Vec class to specifically designate the noncommutativity of its multiplication. It can be created with a label and indexed with SymPy expressions.

>>> v = drudge.Vec('v')
>>> import sympy
>>> a = sympy.Symbol('a')
>>> str(v[a])
'v[a]'

For the symbolic summations, we have the Range class, which denotes a symbolic set that a variable could be summed over. It can be created by just a label.

>>> l = drudge.Range('L')

With these, we can create tensor objects by using the Drudge.sum() method,

>>> x = sympy.IndexedBase('x')
>>> tensor = dr.sum((a, l), x[a] * v[a])
>>> str(tensor)
'sum_{a} x[a] * v[a]'

Now we got a symbolic tensor of a sum of vectors modulated by a SymPy IndexedBase. Actually any type of SymPy expression can be used to modulate the noncommutative vectors.

>>> tensor = dr.sum((a, l), sympy.sin(a) * v[a])
>>> str(tensor)
'sum_{a} sin(a) * v[a]'

And we can also have multiple summations and product of the vectors.

>>> b = sympy.Symbol('b')
>>> tensor = dr.sum((a, l), (b, l), x[a, b] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]'

Of cause the multiplication of the vectors will not be commutative,

>>> tensor = dr.sum((a, l), (b, l), x[a, b] * v[b] * v[a])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[b] * v[a]'

Normally, for each symbolic range, we have some traditional symbols used as dummies for summations over them, giving these information to drudge objects can be very helpful. Here in this demonstration, we can use the Drudge.set_dumms() method.

>>> dr.set_dumms(l, sympy.symbols('a b c d'))
[a, b, c, d]
>>> dr.add_resolver_for_dumms()

where the call to the Drudge.add_resolver_for_dumms() method could tell the drudge to interpret all the dummy symbols to be over the range that they are set to. By giving drudge object such domain-specific information, we can have a lot convenience. For instance, now we can use Einstein summation convention to create tensor object, without the need to spell all the summations out.

>>> tensor = dr.einst(x[a, b] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]'

Also the drudge knows what to do when more dummies are needed in mathematical operations. For instance, when we multiply things,

>>> tensor = dr.einst(x[a] * v[a])
>>> prod = tensor * tensor
>>> str(prod)
'sum_{a, b} x[a]*x[b] * v[a] * v[b]'

Here the dummy \(b\) is automatically used since the drudge object knows available dummies for its range. Also the range and the dummies are automatically added to the name archive of the drudge, which can be access by Drudge.names.

>>> p = dr.names
>>> p.L
Range('L')
>>> p.L_dumms
[a, b, c, d]
>>> p.d
d

Here in this example, we set the dummies ourselves by Drudge.set_dumms(). Normally, in subclasses of Drudge for different specific problems, such setting up is already finished within the class. We can just directly get what we need from the names archive. There is also a method Drudge.inject_names() for the convenience of interactive work.

Tensor manipulations

Now with tensors created by Drudge.sum() or Drudge.einst(), a lot of mathematical operations are available to them. In addition to the above example of (noncommutative) multiplication, we can also have the linear algebraic operations of addition and scalar multiplication.

>>> tensor = dr.einst(x[a] * v[a])
>>> y = sympy.IndexedBase('y')
>>> res = tensor + dr.einst(y[a] * v[a])
>>> str(res)
'sum_{a} x[a] * v[a]\n + sum_{a} y[a] * v[a]'

>>> res = 2 * tensor
>>> str(res)
'sum_{a} 2*x[a] * v[a]'

We can also perform some complex substitutions on either the vector or the amplitude part, by using the Drudge.subst() method.

>>> t = sympy.IndexedBase('t')
>>> w = drudge.Vec('w')
>>> substed = tensor.subst(v[a], dr.einst(t[a, b] * w[b]))
>>> str(substed)
'sum_{a, b} x[a]*t[a, b] * w[b]'

>>> substed = tensor.subst(x[a], sympy.sin(a))
>>> str(substed)
'sum_{a} sin(a) * v[a]'

Note that here the substituted vector does not have to match the left-hand side of the substitution exactly, pattern matching is done here. Other mathematical operations are also available, like symbolic differentiation by Tensor.diff() and commutation by | operator Tensor.__or__().

Tensors are purely mathematical expressions, while the utility class TensorDef can be construed as tensor expressions with a left-hand side. They can be easily created by Drudge.define() and Drudge.define_einst().

>>> v_def = dr.define_einst(v[a], t[a, b] * w[b])
>>> str(v_def)
'v[a] = sum_{b} t[a, b] * w[b]'

Their method TensorDef.act() is like a active voice version of Tensor.subst() and could come handy when we need to substitute the same definition in multiple inputs.

>>> res = v_def.act(tensor)
>>> str(res)
'sum_{a, b} x[a]*t[a, b] * w[b]'

More importantly, the definitions can be indexed directly, and the result is designed to work well inside Drudge.sum() or Drudge.einst(). For instance, for the same result, we could have,

>>> res = dr.einst(x[a] * v_def[a])
>>> str(res)
'sum_{b, a} x[a]*t[a, b] * w[b]'

When the only purpose of a vector or indexed base is to be substituted and we never intend to write tensor expressions directly in terms of them, we can just name the definition with a short name directly and put the actual base inside only. For instance,

>>> c = sympy.Symbol('c')
>>> f = dr.define_einst(sympy.IndexedBase('f')[a, b], x[a, c] * y[c, b])
>>> str(f)
'f[a, b] = sum_{c} x[a, c]*y[c, b]'
>>> str(dr.einst(f[a, a]))
'sum_{b, a} x[a, b]*y[b, a]'

which also demonstrates that the tensor definition facility can also be used for scalar quantities. TensorDef is also at the core of the code optimization and generation facility in the gristmill package.

Usually for tensorial problems, full simplification requires the utilization of some symmetries present on the indexed quantities by permutations among their indices. For instance, an anti-symmetric matrix entry changes sign when we transpose the two indices. Such information can be told to drudge by using the Drudge.set_symm() method, by giving generators of the symmetry group by Perm instances. For instance, we can do,

dr.set_symm(x, drudge.Perm([1, 0], drudge.NEG))

Then the master simplification algorithm in Tensor.simplify() is able to take full advantage of such information.

>>> tensor = dr.einst(x[a, b] * v[a] * v[b] + x[b, a] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]\n + sum_{a, b} x[b, a] * v[a] * v[b]'
>>> str(tensor.simplify())
'0'

Normally, drudge subclasses for specific problems add symmetries for some important indexed bases in the problem. And some drudge subclasses have helper methods for the setting of such symmetries, like FockDrudge.set_n_body_base() and FockDrudge.set_dbbar_base().

For the simplification of the noncommutative vector parts, the base Drudge class does not consider any commutation rules among the vectors. It works on the free algebra, while the subclasses could have the specific commutation rules added for the algebraic system. For instance, WickDrudge add abstract commutation rules where all the commutators have scalar values. Based on it, its special subclass FockDrudge implements the canonical commutation relations for bosons and the canonical anti-commutation relations for fermions. Also based on it, the subclass CliffordDrudge is capable of treating all kinds of Clifford algebras, like geometric algebra, Pauli matrices, Dirac matrices, and Majorana fermion operators. For algebraic systems where the commutator is not always a scalar, the abstract base class GenQuadDrudge can be used for basically all kinds of commutation rules. For instance, its subclass SU2LatticeDrudge can be used for \(\mathfrak{su}(2)\) algebra in Cartan-Weyl form.

These drudge subclasses only has the mathematical commutation rules implemented, for convenience in solving problems, many drudge subclasses are built-in with a lot of domain-specific information like the ranges and dummies, which are listed in Direct support of different problems. For instance, we can easily see the commutativity of two particle-hole excitation operators by using the PartHoleDrudge.

>>> phdr = drudge.PartHoleDrudge(spark_ctx)
>>> t = sympy.IndexedBase('t')
>>> u = sympy.IndexedBase('u')
>>> p = phdr.names
>>> a, i = p.a, p.i
>>> excit1 = phdr.einst(t[a, i] * p.c_dag[a] * p.c_[i])
>>> excit2 = phdr.einst(u[a, i] * p.c_dag[a] * p.c_[i])
>>> comm = excit1 | excit2
>>> str(comm)
'sum_{i, a, j, b} t[a, i]*u[b, j] * c[CR, a] * c[AN, i] * c[CR, b] * c[AN, j]\n + sum_{i, a, j, b} -t[a, i]*u[b, j] * c[CR, b] * c[AN, j] * c[CR, a] * c[AN, i]'
>>> str(comm.simplify())
'0'

Note that here basically all things related to the problem, like the vector for creation and annihilation operator, the conventional dummies \(a\) and \(i\) for particle and hole labels, are directly read from the name archive of the drudge. Problem-specific drudges are supposed to give such convenience.

In addition to providing context-dependent information for general tensor operations, drudge subclasses could also provide additional operations on tensors created from them. For instance, for the above commutator, we can directly compute the expectation value with respect to the Fermi vacuum by

>>> str(comm.eval_fermi_vev())
'0'

These additional operations are called tensor methods and are documented in the drudge subclasses.

Examples on real-world applications

In this tutorial, some simple examples are run directly inside a Python interpreter. Actually drudge is designed to work well inside Jupyter notebooks. By calling the Tensor.display() method, tensor objects can be mathematically displayed in Jupyter sessions. An example of interactive usage of drudge, we have a sample notebook in docs/examples/ccsd.ipynb in the project source. Also included is a general script gencc.py for the automatic derivation of coupled-cluster theories, mostly to demonstrate using drudge programmatically. And we also have a script for RCCSD theory to demonstrate its usage in large-scale spin-explicit coupled-cluster theories.

Note about importing drudge

In this tutorial, import drudge and import sympy is used and we need to give fully-qualified name to refer to objects in them. Normally, it can be convenient to use from drudge import * to import everything from drudge. For these cases, it needs to be careful that the importation of all objects from drudge needs to follow the importation of all objects from SymPy, or the SymPy Range class will shallow the actual class for symbolic range in drudge.