Tower Of States $\alpha \text{XX}$ model

Author Rafael Soares

We perform a tower of states (TOS) analizes [1] of the $\alpha \text{XX}$ model [2]. This model consists of an array of $N$ spin-$\frac{1}{2}$ sites with long-range $\text{XY}$ couplings that decay as a power law with exponent $\alpha$, $$ \mathcal{H} = -\dfrac{J}{2} \sum_{i<j} \dfrac{S^+_iS^-_j+S^-_iS^+_j}{|i - j|^\alpha}. $$

The TOS analysis provides strong evidence for spontaneous symmetry breaking (SSB) in the thermodynamic limit, as the ground state of a finite system is completely symmetric. The spectrum of the $\alpha \text{XX}$ model can be labeled by total magnetization since $\left[\mathcal{H}, S^z \right]=0$.

To perform the TOS analysis, we gather the eigenvalues using either full exact diagonalization (ED) or the Lanczos algorithm in each symmetry sector.

Image title

In the figure above for $\alpha=1$, the lowest energy in each $S^z$ sector clear forms a tower of states. These energies scale as $\propto 1 / N$, so in the thermodynamic limit there the collapse of the tower of states forming a highly degenerate ground state manifold [3]. As such, in the thermodynamic limit SSB occurs. In contrast, for a finite system the ground state always lies in $S^z=0$, and so it is symemtric (it is also an eigenstate of the Hamiltonian's symmetry).

SSB does not occur for all values of $\alpha$. For example, when $\alpha \rightarrow +\infty$, the system becomes a one-dimensional $\text{XX}$ spin chain with only short-range interactions. In this case, the ground state is always in the $S^z=0$ sector.

C++Julia

#include <xdiag/all.hpp>
using namespace xdiag;

int main() {
  int Nsites = 14;    // total number of sites
  double alpha = 1.0; // Exponent power-law decay

  auto ops = OpSum(); // create OpSum

  for (int i = 0; i < Nsites; i++) {
    for (int j = i + 1; j < Nsites; j++) {
      double J = -1.0 / (sqrt(Nsites) * pow(abs((double)(i - j)), alpha));
      ops += J * Op("Exchange", {i, j});
    }
  }

  // Diagonalize in each magnetization sector using FullED
  int total_eigen = pow(2, Nsites);
  arma::mat energies = arma::mat(
      total_eigen, 2,
      arma::fill::zeros); // Collect the energies with the number of spins up.
  arma::vec eig;

  int count = 0;
  for (int nup = 0; nup <= Nsites; nup++) {
    auto block = Spinhalf(Nsites, nup);
    arma::mat H = matrix(ops, block);
    eig_sym(eig, H);
    for (int j = 0; j < eig.n_elem; j++) {
      energies(count, 0) = nup;    // Magnetization sector
      energies(count, 1) = eig(j); // Energy eigenvalue
      count++;
    }
  }
  // Construct the filename
  std::string flstring = "energies_tos_XXmodel.Nsites." +
                         std::to_string(Nsites) + ".alpha" +
                         std::to_string((double)alpha) + ".outfile.h5";

  auto save_fl = FileH5(flstring, "w!");
  save_fl["energies"] = energies;
  return 0;
}

using XDiag
using LinearAlgebra
using Printf
using HDF5

function main()
    Nsites = 24 ## Number of Sites 
    alpha = 1.0 ## Exponent power-law decay
    ops = OpSum() ## create OpSum

    for i in 1:Nsites
        for j in (i+1):Nsites
        J = -1.0 / (sqrt(Nsites) * (abs(i - j))^(alpha))
        ops += J * Op("Exchange", [i, j])
        end
    end

    energies = Vector{Float64}[] ## Collect the energies

    Diagonalize in each magnetization sector using FullED
    for nup in 0:Nsites
        block = Spinhalf(Nsites, nup)
        H = matrix(ops, block)
        eig = eigvals(Hermitian(H))
        for e0 in eig
            push!(energies, [nup, e0])
        end
    end

    #Diagonalize in each magnetization sector using Lanczos to get the first few eigenvalues
    # for nup in 0:Nsites
    #     block = Spinhalf(Nsites, nup)
    #     r = eigvals_lanczos(ops, block, neigvals = 1);
    #     for e0 in r.eigenvalues
    #         push!(energies, [nup, e0])
    #     end
    # end

    filename = @sprintf("energies_tos_XXmodel.Nsites.%d.alpha.%d.outfile.h5", Nsites, alpha)
    h5open(filename, "w") do file
        write(file, "energies", hcat(energies...))
    end
end

main()

references

[1] P. W. Anderson, An Approximate Quantum Theory of the Antiferromagnetic Ground State, Phys. Rev. 86, 694 (1952)

[2] Tommaso Comparin, Fabio Mezzacapo and Tommaso Roscilde, Robust spin squeezing from the tower of states of U(1)-symmetric spin Hamiltonians, Phys. Rev. A 105, 02262 (2022)

[3] Alexander Wietek, Michael Schuler and Andreas M. Läuchli, Studying Continuous Symmetry Breaking using Energy Level Spectroscopy, arXiv:1704.08622 (2017).

Tower Of States \(\alpha \text{XX}\) model

references