How to get_template_part using AJAX?

your callback function must be like this:

function cienna_show_slider_callback() {
    global $post, $product, $woocommerce; // just in case if your template file need this
    ob_start();
?>

<?php woocommerce_get_template( 'archive-product.php'); ?>

<?php
    $output = ob_get_contents();
    ob_end_clean();
    echo $output;
    die();
}