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 definition, their required number of sites, and the blocks for which they are available.

The precise definition of an operator can depend on the block it is applied to (for example, a Hop term sums over both spin species on a tJ block, but acts on a single species on a Fermion block). For the block-specific formulas, see the "Operators" section on the respective block page: Spinhalf, tJ, Electron, Boson, Fermion.

Throughout, \(c^\dagger_{i\sigma}, c_{i\sigma}\) denote fermionic creation/annihilation operators, \(a^\dagger_i, a_i\) bosonic ones, \(n_i\) a number operator, and \(\mathbf{S}_i = (S^x_i, S^y_i, S^z_i)\) a spin operator with \(S^\pm_i = S^x_i \pm i S^y_i\).

Spin operators

Type	Description	No. of sites	Blocks
Sz	local magnetic moment \(S^z_i\)	1	Spinhalf, tJ, Electron, Boson
Sx	local magnetic moment \(S^x_i\)	1	Spinhalf, tJ, Electron, Boson
Sy	local magnetic moment \(S^y_i\)	1	Spinhalf, tJ, Electron, Boson
S+	spin raising operator \(S^+_i\)	1	Spinhalf, tJ, Electron, Boson
S-	spin lowering operator \(S^-_i\)	1	Spinhalf, tJ, Electron, Boson
SdotS	Heisenberg interaction \(\mathbf{S}_i \cdot \mathbf{S}_j\)	2	Spinhalf, tJ, Electron, Boson
SzSz	Ising interaction \(S^z_i S^z_j\)	2	Spinhalf, tJ, Electron, Boson
Exchange	spin exchange \(\frac{1}{2}(S^+_i S^-_j + S^-_i S^+_j)\)	2	Spinhalf, tJ, Electron, Boson
ExchangeAsym	antisymmetric exchange \(\frac{1}{2}(S^+_i S^-_j - S^-_i S^+_j)\)	2	Spinhalf, tJ, Electron, Boson
ScalarChirality	scalar chirality \(\mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)\)	3	Spinhalf, Boson
TotalSz	total magnetization \(\sum_i S^z_i\)	0	Spinhalf, tJ, Electron

On a Boson block the spin operators refer to the spin \(S = (d-1)/2\) associated with the local dimension \(d\).

Bosonic operators

Type	Description	No. of sites	Blocks
Adag	bosonic creation operator \(a^\dagger_i\)	1	Boson
A	bosonic annihilation operator \(a_i\)	1	Boson

Fermionic operators

Type	Description	No. of sites	Blocks
Cdag	creation operator \(c^\dagger_i\) (spinless)	1	Fermion
C	annihilation operator \(c_i\) (spinless)	1	Fermion
Cdagup	creation operator \(c^\dagger_{i\uparrow}\)	1	tJ, Electron
Cup	annihilation operator \(c_{i\uparrow}\)	1	tJ, Electron
Cdagdn	creation operator \(c^\dagger_{i\downarrow}\)	1	tJ, Electron
Cdn	annihilation operator \(c_{i\downarrow}\)	1	tJ, Electron

Hopping operators

The hopping terms are the hermitian combination \(-(c^\dagger_i c_j + c^\dagger_{j\sigma}c_{i\sigma})\) carrying an overall minus sign. Note that the coupling multiplies the whole term, so a complex coupling makes it non-hermitian (see Complex couplings). The *Asym variants provide the antisymmetric combination \(-(c^\dagger_i c_j - c^\dagger_{j\sigma}c_{i\sigma})\).

Type	Description	No. of sites	Blocks
Hop	hopping over all species, \(-\sum_\sigma (c^\dagger_{i\sigma}c_{j\sigma} + c^\dagger_{j\sigma}c_{i\sigma})\)	2	tJ, Electron, Boson, Fermion
Hopup	hopping of \(\uparrow\) electrons, \(-(c^\dagger_{i\uparrow}c_{j\uparrow} + c^\dagger_{j\uparrow}c_{i\uparrow})\)	2	tJ, Electron
Hopdn	hopping of \(\downarrow\) electrons, \(-(c^\dagger_{i\downarrow}c_{j\downarrow} + c^\dagger_{j\downarrow}c_{i\downarrow})\)	2	tJ, Electron
HopAsym	antisymmetric hopping over all species, \(-\sum_\sigma (c^\dagger_{i\sigma}c_{j\sigma} - c^\dagger_{j\sigma}c_{i\sigma})\)	2	tJ, Electron, Boson, Fermion
HopupAsym	antisymmetric \(\uparrow\) hopping, \(-(c^\dagger_{i\uparrow}c_{j\uparrow} - c^\dagger_{j\uparrow}c_{i\uparrow})\)	2	tJ, Electron
HopdnAsym	antisymmetric \(\downarrow\) hopping, \(-(c^\dagger_{i\downarrow}c_{j\downarrow} - c^\dagger_{j\downarrow}c_{i\downarrow})\)	2	tJ, Electron

Density and interaction operators

Type	Description	No. of sites	Blocks
N	number operator \(n_i\) (spinless fermion / boson)	1	Boson, Fermion
NN	density-density interaction \(n_i n_j\) (spinless)	2	Fermion
Nup	number of \(\uparrow\) electrons \(n_{i\uparrow}\)	1	tJ, Electron
Ndn	number of \(\downarrow\) electrons \(n_{i\downarrow}\)	1	tJ, Electron
Ntot	total number \(n_i = n_{i\uparrow} + n_{i\downarrow}\)	1	tJ, Electron
Nupdn	double occupancy \(n_{i\uparrow} n_{i\downarrow}\)	1	Electron
NtotNtot	density-density interaction \(n_i n_j\)	2	tJ, Electron
NupNup	\(n_{i\uparrow} n_{j\uparrow}\)	2	tJ, Electron
NupNdn	\(n_{i\uparrow} n_{j\downarrow}\)	2	tJ, Electron
NdnNup	\(n_{i\downarrow} n_{j\uparrow}\)	2	tJ, Electron
NdnNdn	\(n_{i\downarrow} n_{j\downarrow}\)	2	tJ, Electron
NupdnNupdn	double-occupancy correlation \(n_{i\uparrow}n_{i\downarrow} n_{j\uparrow}n_{j\downarrow}\)	2	Electron
HubbardU	on-site interaction: \(\sum_i n_{i\uparrow}n_{i\downarrow}\) (Electron), \(\frac{1}{2}\sum_i n_i(n_i-1)\) (Boson)	0	Electron, Boson

\(t\)-\(J\) operators

Type	Description	No. of sites	Blocks
tJSzSz	\(t\)-\(J\) Ising interaction \(S^z_i S^z_j - \frac{n_i n_j}{4}\)	2	tJ
tJSdotS	\(t\)-\(J\) Heisenberg interaction \(\mathbf{S}_i \cdot \mathbf{S}_j - \frac{n_i n_j}{4}\)	2	tJ

Total quantum numbers

Type	Description	No. of sites	Blocks
TotalN	total particle number \(\sum_i n_i\)	0	tJ, Electron, Boson, Fermion
TotalNup	total \(\uparrow\) number \(\sum_i n_{i\uparrow}\)	0	tJ, Electron
TotalNdn	total \(\downarrow\) number \(\sum_i n_{i\downarrow}\)	0	tJ, Electron

Generic operators

Type	Description	No. of sites	Blocks
Id	the identity operator \(\mathbb{1}\)	0	Spinhalf, tJ, Electron, Boson, Fermion
Matrix	a generic interaction defined by an explicit matrix (see below)	any	Spinhalf, Boson

Distributed blocks

The distributed blocks (SpinhalfDistributed, tJDistributed, ElectronDistributed) support only the subset of operators that have a dedicated distributed kernel. The supported types are listed on each distributed block's page.

Matrix type

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

C++

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

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

C++

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.