Groundstate energy of the Heisenberg S=1/2 chain, ED “Hello World”

Computing the ground state energy of the S=1/2 chain

In this example we compute the ground state energy of the \(S=1/2\) Heisenberg chain on a periodic chain lattice in one dimension. The Hamiltonian is given by

\[H = J\sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j\]

where \(\mathbf{S}_i = (S_i^x, S_i^y, S_i^z)\) are the spin \(S=1/2\) operators and \(\langle i,j \rangle\) denotes summation over nearest-meighbor sites \(i\) and \(j\). We set \(J=1\). This is the “Hello World” application for ED.

#include <hydra/all.h>

int main() {
  using namespace hydra;

  int n_sites = 16;
  int nup = n_sites / 2;

  // Define the Hilbert space block
  auto block = Spinhalf(n_sites, nup);

  // Define the nearest-neighbor Heisenberg Hamiltonian
  BondList bonds;
  for (int i = 0; i < n_sites; ++i) {
    bonds << Bond("HB", "J", {i, (i + 1) % n_sites});
  }

  // Set the coupling constant "J" to one
  bonds["J"] = 1.0;

  // Compute and print the ground state energy
  double e0 = eig0(bonds, block);
  HydraPrint(e0);
  
  return EXIT_SUCCESS;
}