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Zig-zagSpinChain

The problem

In this project we study frustrated spin chains, that is spinchains with nearest-neighbour and next-nearest neighbour exchange couplings J1 and J2.
The Hamiltonian for this problem is

{%h, H = J_1 \sum_{j=1}^{L-1} \vec{S}_j, \vec{S}_{j+1} + J_2 \sum_{j=1}^{L-2} \vec{S}_j, \vec{S}_{j+2} - h \sum_{j=1}^{L} S_j^z %}

We want to use mptoolkit to study the chirality correlator for this problem.

Preliminaries

To create the lattice run spinchain-zigzag-u1 (U(1) Symmetry) or spinchain-zigzag-su2 (SU(2)) Symmetry). The syntax is the same for both programs:

spinchain-zigzag-u1 <L> <EdgeSpin> <BulkSpin> <J1> <J2> <outfile>

We proceed by calculating the ground state of the system. First, create a random wave-function with mp-random:

mp-random <lattice> <quantum number> <count> <outfile>

The Parameter <quantum number> is the total spin in z-direction of the random wave function.
Now we look at the chirality operator {%chir, <\vec{\kappa}_i> := <\vec{S}_i \times \vec{S}_{i+1} > %}

Chiral correlation

To calculate the chirality correlator {%kap, <\kappa_i \kappa_j> %} use the mp-local-fourpoint program when your lattice was created with spinchain-zigzag-u1. If you used spinchain-zigzag-su2 you want to use mp-correlator.

Using mp-local-fourpoint

The mp-local-fourpoint program calculates a generic 4-point correlator {$ \langle A(i-1) B(i) C(j) D(j+1) \rangle $}

Thus, we need to expand the components of {$\kappa$}. These are:

{%kz,\kappa^z_i = \frac{i}{2}(S^+_i S^-_{i+1} - S^-_i S^+_{i+1})%}

{%kp,\kappa^+_i = i(S^z_i S^+_{i+1} - S^+_i S^z_{i+1})%}

{%km,\kappa^-_i = i(S^-_i S^z_{i+1} - S^z_i S^-_{i+1})%}

We are mostly interested in the {$\langle \kappa^z_i \kappa^z_j \rangle$} correlator, which expands to

{%kexp,-\frac{1}{4}(S^+_i S^-_{i+1} S^+_j S^-_{j+1} \; - \; S^+_i S^-_{i+1} S^-_j S^+_{j+1} \; - \; S^-_i S^+_{i+1} S^+_j S^-_{j+1} \; + \; S^-_i S^+_{i+1} S^-_j S^+_{j+1} )%}

In this case our Problem is symmetric under U(1) transformations, therefore

 <Sp(i)Sm(i) Sm(j)Sp(j)> = <Sm(i)Sp(i) Sm(j)Sp(j)> and
 <Sm(i)Sp(i) Sm(j)Sp(j)> = <Sp(i)Sm(j) Sp(j)Sm(j)> holds and we can save time by only calculating these terms once.

So your command will look like

The operators S^+ and S^- translate into Sp, or respectively, Sm in mptoolkit syntax.

The kappa[p|m|z] operators are also added to the lattice file, so the four-point correlator can also be obtained using mp-correlation.

Using mp-correlation

When using mp-correlation you have to be aware that it uses bond indices. When your spinchain is of length L you would use:

This will calculate the correlator <kappa_i kappa_j> for j = 1..L-1 and write the results in the file lattice.correl.