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## Zig-zagSpinChain## The problemIn this project we study frustrated spin chains, that is spinchains with nearest-neighbour
and next-nearest neighbour exchange couplings J1 and J2. {%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. ## PreliminariesTo create the lattice run 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 <lattice> <quantum number> <count> <outfile> The Parameter ## Chiral correlationTo calculate the chirality correlator {%kap, <\kappa_i \kappa_j> %} use the ## Using mp-local-fourpointThe 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 `mp-local-fourpoint lattice-zigzag lattice.zigzag.gs.psi "Sp" "Sm" "Sp" "Sm" 1 L >> lattice.zigzag.gs.correl.pmpm`
The operators S^+ and S^- translate into The ## Using mp-correlationWhen using `mp-correlation lattice groundstate i "Kappa" L-1 "Kappa" >> lattice.correl`
This will calculate the correlator <kappa_i kappa_j> for j = 1..L-1 and write the results in the file lattice.correl. |

Page last modified on August 25, 2013, at 01:04 PM