## Learn to Use ITensor

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# Index Objects

The most basic element of ITensor is not actually a tensor: it is a tensor index, an object of type Index. (By tensor index we mean i,j, or k in an expression like $T_{ijk}$ ).

ITensors are "intelligent tensors" because they "know" what indices they have. This is possible since an Index carries extra information beyond its size.

The simplest way to construct an Index is to give its name and size:

auto i = Index("index i",3);


Upon creation, an Index gets "stamped" with a permanent, hidden id number that allows copies of the Index to recognize each other. Typically you do not need to look at these id numbers; it is enough to know that indices match (compare equal) if they are copies of the same original index:

auto j = i;  //make a copy of i
Print(j==i); //prints: j==i = true


(also they must have the same "prime level"; see next section).

Neither the name nor size are used to compare indices, because two different indices could accidentally have the same name and size.

To access the size of an Index, use its .m() method

println("The size of ",i.name()," is ",i.m());
//prints: The size of index i is 3


The convention of calling the size "m" comes from the DMRG literature.

#include "itensor/all_basic.h"
using namespace itensor;

int main()
{
auto i = Index("index i",3);
println("The size of ",i.name()," is ",i.m());
return 0;
}


After creating an Index, most of its properties are permanently fixed, including its size. The philosophy of ITensor is that indices have a meaning given at the time they are created. A new Index can be created to take the place of an old one, but the semantic meaning of a given Index object cannot be changed.

### Priming Indices

There is one property of an Index you can change: its prime level.

An Index starts out with prime level zero. Copies of the same original Index must have the same prime level to compare equal.

Calling prime(i) will produce a copy of i with prime level raised by 1. Because this copy has a different prime level, it will no longer compare equal to i.

auto i1 = prime(i);
println("The prime level of i1 is ",i1.primeLevel());
//prints: The prime level of i1 is 1
printfln("i1==i is %s",i1==i);
//prints: i1==i is false


There are many convenient ways to manipulate Index prime levels. The prime function accepts an optional increment amount:

auto i3 = prime(i,3);
println(i3.primeLevel());
//prints: 3


Calling noprime resets the prime level to zero.

auto i0 = noprime(i3);
println(i0.primeLevel());
//prints: 0


Note that the above names (i3, i0, etc.) are just for pedagogical reasons—you can use any variable names you want regardless of the prime level.

We will see more ways to manipulate primes as we work with ITensors with multiple indices.

### Printing Indices

Printing an Index shows useful information about it:

auto i = Index("index i",3,Link);

println(i);


The output shows the name, size, and IndexType of i. The last number is part of the id number of the Index. Id numbers are random 64 bit integers and vary each time you run your program.

The prime level is displayed at the end:

println(prime(i,2));

println(prime(i,10));


### Index Types

The Index constructor accepts an optional IndexType argument:

auto s2 = Index("site 2",2,Site); //IndexType set to Site


IndexTypes are useful because they allow you to manipulate or retrieve indices of a certain type. IndexTypes can be thought of as labels that distinguish broad categories of indices, such as "physical" versus "virtual" indices.

In addition to the IndexTypes pre-defined by ITensor, you can define a custom IndexType

auto MyType = IndexType("MyType");


and use it to create indices of this type

auto m1 = Index("m1",5,MyType);
auto m2 = Index("m2",7,MyType);


Internally, IndexType data is just a fixed-size, constant string, and can be up to 7 characters long. The IndexType of an Index can be obtained by calling the .type() method.

println("The type of m1 is ",m1.type());
//prints: The type of m1 is MyType


For a complete listing of all of the methods of class Index, view the detailed documentation.

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