## Learn to Use ITensor

main / itensor_conventions

# Tensor and Index Conventions

## Arrows

• IQIndex Arrows (represented by the Arrow enum) have the values In or Out and should not be thought of as pointing in an absolute direction such as up/down/left/right.

• Conceptually Arrows represent the flow or flux of quantum numbers out of a tensor. Equivalently, they specify whether a particular index transforms as a ket (a vector, or raised index in classic tensor notation) or a bra (a covector, dual vector, or lowered index).

• An Out Arrow means the index transforms as a ket (for example, a site of an MPS), an In Arrow means an index transforms as a bra.

• Conjugating a tensor (applying dag(T)) reverses Arrow directions in addition to taking the complex conjugate.

## Index Types

• The two primary index types (represented by the IndexType enum) are Site and Link.

• Site indices are physical, or real, indices such as the sites of a lattice model.

• Link indices (sometimes called virtual indices) label internal, or gauge, degrees of freedom of a tensor network. For example, the bonds of a matrix product state are Link indices.

• There is a special IndexType All which can be used to specify that a certain function or transformation should be applied to all indices regardless of type.

## Matrix Product States

• An orthogonality center of a matrix product state or MPS is any tensor whose indices all label orthonormal states. For example, every site tensor of a canonical MPS is an orthogonality center. During a conventional DMRG calculation, where an MPS is in the so-called mixed canonical gauge, only one tensor (the left or right of the two "center" sites depending on sweep direction) is an ortho center.

• All Arrows (including both Link and Site indices) flow Out from the orthogonality center of an MPS. For a normalized MPS in a well-defined gauge, therefore, a tensor is an ortho. center if and only if all its indices point Out.

## Operators

• Single-site operator tensors have one unprimed Site index S and one primed Site Index S'. For IQTensor operators, IQIndex S has an In Arrow and S' an Out Arrow.

• Each tensor of a matrix product operator (MPO) follows the same convention as a single-site operator above.

• Unless an MPO is in a definite gauge (typically not the case), Link arrows should flow out from the first site tensor.

Back to Main