Switches the theme, only for the homepage. May not help if widgets/menus also need to be switched.

switch-theme.php

<?php
add_action( 'setup_theme', 'this_setup_theme' );
/**
 * Sets a different theme on the homepage from the rest of the site.
 *
 * Just checks the REQUEST_URI. Works, but hasn't been tested thoroughly.
 * Currently it will only change if the query string is also empty, but if there
 * needs to be query strings (e.g. utm_campaign, etc.) that can be dropped.
 *
 * Code from https://github.com/Rarst/toolbar-theme-switcher
 *
 * @return void
 */
function this_setup_theme() {
	// Hack to tell if we're on the homepage
	if ( '/' === $_SERVER['REQUEST_URI'] && empty( $_SERVER['QUERY_STRING'] ) ) {
		$theme = wp_get_theme( 'twentythirteen' ); // @todo: use theme slug for homepage here
		add_filter( 'pre_option_template', array( $theme, 'get_template' ) );
		add_filter( 'pre_option_stylesheet', array( $theme, 'get_stylesheet' ) );
		add_filter( 'pre_option_stylesheet_root', array( $theme, 'get_theme_root' ) );

		$parent = $theme->parent();

		if ( empty( $parent ) ) {
			add_filter( 'pre_option_template_root', array( $theme, 'get_theme_root' ) );
		} else {
			add_filter( 'pre_option_template_root', array( $parent, 'get_theme_root' ) );
		}

		add_filter( 'pre_option_current_theme', '__return_false' );
	}
}