Turf shoes are better and have more material around the entire shoe that helps when inadvertently catching a ball with a foot. There are field hockey specific shoes (see Longstreth's website) but they are often just slightly modified lacrosse turf shoes.
Nike, Asics, and UA all have lacrosse turf shoes. If you are looking to find them in a physical store, try PJ Sports in Bethesda or sometimes a Dicks will have a selection of turf shoes. (Do not get ankle "boot" lacrosse shoes, get the ones that are low since you need to get the shinguards down low on the ankle).
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