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Operator types

List of operator types

Generic operators in XDiag are represented as OpSum objects made up of a coupling, which can be a real/complex number or a string, and Op objects. Every Op is defined by a type. Here we list all the available types implemented in XDiag, their required number of sites, and the blocks for which they are available.

Type Description No. of sites Blocks
Hop A hopping term for \(\uparrow\) and \(\downarrow\) spins of the form $$ \textcolor{red}{-}\sum_{\sigma=\uparrow\downarrow} (tc^\dagger_{i\sigma}c_{j\sigma} + \textrm{h.c.})$$ 2 tJ, Electron, tJDistributed
Hopup A hopping term for \(\uparrow\) spins of the form $$ \textcolor{red}{-}(tc^\dagger_{i\uparrow}c_{j\uparrow} + \textrm{h.c.})$$ 2 tJ, Electron, tJDistributed
Hopdn A hopping term for \(\downarrow\) spins of the form $$ \textcolor{red}{-}(tc^\dagger_{i\downarrow}c_{j\downarrow} + \textrm{h.c.})$$ 2 tJ, Electron, tJDistributed
HubbardU A uniform Hubbard interaction across the full lattice of the form $$ \sum_i n_{i\uparrow}n_{i\downarrow}$$ 0 Electron
Cdagup A fermionic creation operator for an \(\uparrow\) spin \(c^\dagger_{i\uparrow}\) 1 tJ, Electron, tJDistributed
Cdagdn A fermionic creation operator for an \(\downarrow\) spin \(c^\dagger_{i\downarrow}\) 1 tJ, Electron, tJDistributed
Cup A fermionic annihilation operator for an \(\uparrow\) spin \(c_{i\uparrow}\) 1 tJ, Electron, tJDistributed
Cdn A fermionic annihilation operator for an \(\downarrow\) spin \(c_{i\downarrow}\) 1 tJ, Electron, tJDistributed
Nup A number operator for an \(\uparrow\) spin \(n_{i\uparrow}\) 1 tJ, Electron, tJDistributed
Ndn A number operator for an \(\downarrow\) spin \(n_{i\downarrow}\) 1 tJ, Electron, tJDistributed
Ntot A number operator \(n_i = n_{i\uparrow} + n_{i\downarrow}\) 1 tJ, Electron, tJDistributed
NtotNtot A density-density interaction \(n_i n_j\) 2 tJ, Electron, tJDistributed
SdotS A Heisenberg interaction of the form $$ \mathbf{S}_i \cdot \mathbf{S}_j = S^x_iS^x_j + S^y_iS^y_j + S^z_iS^z_j$$ 2 Spinhalf, tJ, Electron, SpinhalfDistributed, tJDistributed
SzSz An Ising interaction of the form $ S^z_i S^z_j $ 2 Spinhalf, tJ, Electron, SpinhalfDistributed, tJDistributed
Exchange A spin exchange interaction of the form $$ \frac{1}{2}(JS^+_i S^-_j + J^*S^-_iS^+_j)$$ 2 Spinhalf, tJ, Electron, SpinhalfDistributed, tJDistributed
Sz A local magnetic moment in the \(z\)-direction $ S^z_i$ 1 Spinhalf, tJ, Electron, SpinhalfDistributed, tJDistributed
S+ A local spin raising operator \(S^+_i\) 1 Spinhalf, SpinhalfDistributed
S- A local spin lowering operator \(S^-_i\) 1 Spinhalf, SpinhalfDistributed
ScalarChirality A scalar chirality interaction of the form $$ \mathbf{S}_i \cdot ( \mathbf{S}_j \times \mathbf{S}_k)$$ 3 Spinhalf
tJSzSz An Ising interaction as encountered in the \(t-J\) model of the form $$ S^z_i S^z_j - \frac{n_i n_j}{4}$$ 2 tJ, tJDistributed
tJSdotS An Heisenberg interaction as encountered in the \(t-J\) model of the form $$ \mathbf{S}_i \cdot \mathbf{S}_j - \frac{n_i n_j}{4}$$ 2 tJ, tJDistributed
Matrix A generic spin interaction no an arbitrary number of sites defined via a coupling matrix arbitrary Spinhalf

Matrix type

The Matrix interaction type is a special type with whom one can define generic interactions for the Spinhalf block. In addition to the type and sites argument, also a numerical matrix is provided when constructing the Op object. The matrix describes the operator acting on the \(2^n\) dimensional space spanned by the \(n\) sites of the operator. For example, we can represent a \(S^x\) spin operator as,

auto sx = arma::mat({{0, 1},{1, 0}});
auto op = Op("Matrix", 0, sx);

More generically, we can use this mechanism to construct arbitary spin interactions, e.g. 

auto sx = arma::mat({{0, 1},{1, 0}});
auto sz = arma::mat({{0.5, 1},{0, -0.5}});

arma::mat sxsz = arma::kron(sx, sz);
arma::mat sxszsxsz = arma::kron(sxsz, sxsz);

auto op_sxsz = Op("Matrix", {0, 1}, sxsz);
auto op_sxszsxsz = Op("Matrix", {0, 1, 2, 3}, sxsz);

Here we have been using the Kronecker product function kron.