Assuming your new TV supports Miracast, Settings > xperia connectivity > Miracast. YMMV as to the quality of this, I have a few TVs and handsets but universally for some reason I can't make it anywhere near as smooth as a chromecast - I so wish I could. This is involving eth/802n minimum networks. Many people get it working fine for the record.
You might be able to use DLNA too which isn't casting but more file sending / playback. This will always have 100% quality as it's not realtime.