# Delta and Diagonal ITensor

A diagonal ITensor is an ITensor with diagonal-sparse storage such that only its
diagonal elements T_{iii...} are non-zero. (Diagonal elements mean
elements obtained by setting each Index to the same value.)

Uses of diagonal ITensors include replacing one Index of an ITensor with another; tracing pairs of indices; and "tying" multiple indices into a single Index.

Diagonal ITensors can be constructed either by calling the `delta`

or `diagITensor`

functions:

The

`delta`

function returns a diagonal ITensor whose diagonal elements are all 1.0. This introduces extra efficiencies as no memory is actually allocated since all elements are known to be the same.The

`diagITensor`

function can be used to construct general diagonal ITensors with different elements along the diagonal.

The functions `delta`

and `diagITensor`

are defined in "itensor/itensor.h"; also
see "itensor/itensor_impl.h".

## Synopsis

```
auto i = Index(2,"i");
auto j = Index(2,"j");
auto k = Index(2,"k");
auto l = Index(2,"l");
//
// Replace T's i Index with another Index l
//
auto T1 = randomITensor(i,j,k);
T1 *= delta(i,l);
//
// Trace (sum over) a pair of indices
//
auto T2 = randomITensor(i,j,prime(i));
T2 *= delta(i,prime(i));
//
// Tie multiple indices together
//
auto T3 = randomITensor(i,prime(i),j,prime(i,2));
T3 *= delta(i,prime(i),prime(i,2),prime(i,3));
```

## Specification

`delta(IndexSet is) -> ITensor`

`delta(Index i1, Index i2, ...) -> ITensor`

Given an IndexSet, a container of Indices convertible to an IndexSet, or two or more indices, returns an ITensor having these indices and diagonal storage. All diagonal elements have the value 1.0.

`diagITensor(Container C, IndexSet is) -> ITensor`

`diagITensor(Container C, Index i1, Index i2, ...) -> ITensor`

Given a container C (can be any container type) and a set of indices, returns an ITensor with these indices whose diagonal elements are the entries of the container, starting from

`C[0]`

.The container must be as large as the minimum size of the indices.

*This page current as of version 3.0.0*

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