correlation_matrix

Computes the two-point correlations of a pair of single-site operators between all pairs of sites of a State, returning the matrix $$ C_{ij} = \langle \psi | \mathcal{O}^{(1)}_i \, \mathcal{O}^{(2)}_j | \psi \rangle. $$ For example, choosing the operator types "Adag" and "A" yields the single-particle density matrix.

Sources: correlation_matrix.hpp · correlation_matrix.cpp

Definition

JuliaC++

correlation_matrix(state::State, type1::String, type2::String)

arma::mat correlation_matrix(State const &state, std::string type1, std::string type2);
arma::cx_mat correlation_matrixC(State const &state, std::string type1, std::string type2);

The function correlation_matrix returns a real result and raises an error if the result is complex; in that case use correlation_matrixC in C++. In Julia, correlation_matrix returns a real or complex matrix as appropriate.

Parameters

Name	Description
state	the State $
type1	the type of the first single-site operator \(\mathcal{O}^{(1)}\)
type2	the type of the second single-site operator \(\mathcal{O}^{(2)}\)

Returns

An \(N \times N\) matrix (with \(N\) the number of sites), whose entry \((i, j)\) is \(\langle \psi | \mathcal{O}^{(1)}_i \, \mathcal{O}^{(2)}_j | \psi \rangle\).

Info

The state is assumed to be normalized; the raw matrix elements are returned without dividing by \(\langle \psi | \psi \rangle\). The product of the two operators must keep the state within its block; a product that changes a conserved quantum number is reported as a block mismatch. On a symmetry-adapted block the operator is automatically symmetrized with the trivial irrep so that it acts within the block.

Usage Example

JuliaC++

let
    block = Spinhalf(8)
    ops = OpSum()
    for i in 1:8
    ops += "J" * Op("SdotS", [i, mod1(i + 1, 8)])
    end
    ops["J"] = 1.0
    e0, psi0 = eig0(ops, block)
    szsz = correlation_matrix(psi0, "Sz", "Sz")
end

auto block = Spinhalf(8);
auto ops = OpSum();
for (int i = 0; i < 8; ++i) {
  ops += "J" * Op("SdotS", {i, (i + 1) % 8});
}
ops["J"] = 1.0;
auto [e0, psi0] = eig0(ops, block);
arma::mat szsz = correlation_matrix(psi0, "Sz", "Sz");