Symmetry

Detailed Description

The symmetry module provides classes which encapsulate all relevant operations for building symmetry invariant Mps and Mpo and algorithms for them (e.g. DmrgSolver). The classes enter as template arguments of the most of the other classes, e.g. for the Mps class:

template<typename Symmetry, typename Scalar>
class Mps{}

The classes have to satisfie different requirements:

• only static members
• the number of independent symmetries Nq, defined as:

  constexpr int Nq=...

• a typedef to the type of the irreducible representations (irreps), this is currently always a typedef to qarray:

  typedef qarray<Nq> qType

• a function which has as its input two irreps and returns all irreps of the tensor product of them. Additionally similiar functions for vectors of irreps are necessary:

  std::vector<Symmetry::qType> Symmetry::reduceSilent( const Symmetry::qType &ql, const Symmetry::qType &qr )
  std::vector<Symmetry::qType> Symmetry::reduceSilent( const Symmetry::qType &ql, const Symmetry::qType &qm, const Symmetry::qType &qr )
  std::vector<Symmetry::qType> Symmetry::reduceSilent( const std::vector<Symmetry::qType> &ql, const Symmetry::qType &qr )
  std::vector<Symmetry::qType> Symmetry::reduceSilent( const std::vector<Symmetry::qType> &ql, const std::vector<Symmetry::qType> &qr )

• a compare function, which defines a strict order for different irreps.
• a bunch of coefficients for the different algorithms resulting from contractions or traces over the Clebsch-Gordon spaces. For abelian symmetries, these coefficients are all trivially equal to unity, but nevertheless they need to be defined! For a more detailed description of the different coefficients see here.

Currently, there exists implementations for the groups U(1) (Sym::U1), (Sym::ZN) and SU(2) (Sym::SU2) and for a dummy class, which represents no symmetry (Sym::U0). These classes can be combined with the Sym::S1xS2 class, which takes two arbitrary symmetries as template arguments. By chaining this, it is in principle possible, to get a class for three or more symmetries, but until now there is no physical model for this szenario (see Models).

Cosmetic Todo:

1. The compare function, can be defined outside of the classes, since it is the same for all symmetries.
2. The reduceSilent() functions for vectors of irreps or three or more irreps can be defined outside the classes. Only the basic function, taking to irreps, is dependent on the symmetry.

Classes
class	Sym::S1xS2< S1_, S2_ >
class	Sym::SU2< Kind, Scalar >
class	Sym::SUN< N, Scalar >
class	Sym::U0
class	Sym::U1< Kind, Scalar >
class	Sym::ZN< Kind, N, Scalar >