Domain Wall Dynamics in a Spin-$1/2$ Chain

Author Paul Ebert

This example demonstrates the time-evolution of quantum states in XDiag for the dynamics of a domain wall in the ferromagnetic spin-$1/2$ XXZ chain

\[ \mathcal{H} = \sum_{i=1}^{N-1} \Big(- J \bm{S}^z_i \cdot \bm{S}^z_{i+1} + \frac{\Delta}{2}\big( \bm{S}^+_i \cdot \bm{S}^-_{i+1} + \bm{S}^-_i \cdot \bm{S}^+_{i+1} \big) \Big), \]

where $0 < J$. Note that here we use open instead of periodic boundary conditions, allowing to have a single domain wall in the system.

We choose as initial state the following eigenstate of the associated Ising chain (i.e. the model with only $S^zS^z$ interactions) $$ \ket{\psi_0} = \ket{\uparrow\ldots\uparrow\downarrow\ldots\downarrow}, $$ having a domain wall in the middle of the chain. We expect the exchange terms in the XXZ Hamiltonian above to dissolve the domain wall over time as $\ket{\psi_0}$ is not an eigenstate of the XXZ chain. This process may also be viewed as suddenly turning on exchange interactions in an Ising chain.

The code below demonstrates how the the initial domain wall slowly dissolves. The following figure was created using the Julia version of the code, which includes a plotting command:

Dynamics of a domain wall

C++Julia

#include <xdiag/all.hpp>

using namespace xdiag;


int main() 
try {

    // define open ferromagnetic XXZ chain
    int N = 16;
    double J = 0.1;
    double Delta = 0.5;

    auto H = OpSum();
    for (int i=0; i<(N-1); i++){
        H += "J" * Op("SzSz", {i, i+1});
        H += "Delta" * Op("Exchange", {i, i+1});
    }

    H["J"] = J;
    H["Delta"] = Delta;

    // define initial state with domain wall
    auto block = Spinhalf(N);
    std::vector<std::string> psi0_vec (N);
    for (int i=0; i<N/2; i++) {psi0_vec[i] = "Up";}
    for (int i=N/2; i<N; i++) {psi0_vec[i] = "Dn";}
    auto psi = product_state(block, psi0_vec);

    // time evolve psi0 and measure Sz expectation value
    double dt = 0.5;
    int Nt = 30;
    std::vector<std::vector<double> > Sz_expectation(Nt, std::vector<double>(N));
    for (int t_step=0; t_step<Nt; t_step++){
        time_evolve_inplace(H, psi, dt);
        for (int i=0; i<N; i++){
            Sz_expectation[t_step][i] = innerC(Op("Sz", {i}), psi).real(); // result must be real
        }
    }

    // do something here with Sz_expectation (see Julia version for plotting routine)

    return 0;
} catch(Error e){
    error_trace(e);
}

using XDiag
using Plots 

function main()

    # define open ferromagnetic XXZ chain
    N = 16
    J = 0.1
    Delta = 0.5

    H = OpSum()
    for i in 1:(N-1)
        H += "J" * Op("SzSz", [i, i+1])
        H += "Delta" * Op("Exchange", [i, i+1])
    end

    H["J"] = J
    H["Delta"] = Delta

    # define initial state with domain wall
    block = Spinhalf(N)
    psi0_vec = vcat(repeat(["Up"], N÷2), repeat(["Dn"], N÷2))
    psi = product_state(block, psi0_vec)

    # time evolve and measure Sz expectation value
    dt = 0.5
    Nt = 30
    Sz_expectation = Matrix{Float64}(undef, Nt, N)
    for t_step in 1:Nt
        time_evolve_inplace(H, psi, dt)
        for i in 1:N
            Sz_expectation[t_step, i] = real(inner(Op("Sz", [i]), psi))
        end
    end

    # plot Sz expectation value
    @show heatmap(
        Sz_expectation,
        c = :thermal,
        xlabel="chain coordinate",
        ylabel="time step",
        title = "Sz expectation value over time")

end


main()

Domain Wall Dynamics in a Spin-\(1/2\) Chain