What happens if a magic item gets broken? Do they lose their magic power?
The easiest example is a wand. For example, a wand is broken into two. Can the top half be used to cast the spell? Or do the characters need to bring it to a "magic shop" to get it repaired?
I'm asking about magic items in general. For this question I'm particularly interested in wands and weapons, but will appreciate a more general guidance on treating broken magic items.
Many magic items (namely staves) specify what happens when they are broken. Most don't. If we look at the DMG (pg 141):
Most magic items are objects of extraordinary artisanship. Thanks to a combination of careful crafting and magical reinforcement, a magic item is at least as durable as a nonmagical item of its kind. Most magic items, other than potions and scrolls, have resistance to all damage. Artifacts are practically indestructible, requiring extraordinary measures to destroy.
So, besides artifacts all magic items are broken like anything else, just that they have resistance to all damage. This is basically so they cannot easily be broken, and most likely since it's intended that they cannot be easily repaired. In the case of potions and scrolls there isn't even that, and due to their small size (usually) they are extremely easy to break.
So if by broken you mean dealing enough damage that its hit points are reduced to 0 (the base assumption), then a magic item's magic is lost. Your +1 sword would become a simple, well made normal sword after breaking and repairing by any non-magic-granting means (I guess a wish would easily repair a +1 sword, but not Mending or similar spells).
This is supported by the quote @V2Blast found from Chris Perkins:
What happens if an Eversmoking Bottle is broken? Does the smoke cloud expand on and on forever?
If a magic item breaks, it ceases to function. The eversmoking bottle would stop producing smoke.
So reducing to 0hp = broken, and "If a magic item breaks, it ceases to function". If it's merely damaged, most likely a simpler fix could restore it to a "perfect" state, though many DMs could rule that the magic becomes unstable. No hard rule on "bruised but not broken" magical items, AFAIK.
If you want to realistically "repair" it, just use the crafting magic items rules in DMG 128. If your DM is generous, you might get a discount. Maybe you can homerule something to make magic items easier to repair, but nothing official as far as I know.
Some could argue that "ceases to function" is not the same as "loses its magic". Barring a definitive Sage Advice or an unfound quote, I think RAW is hard to specify. I do think RAI points to broken -> needs to be reenchanted.
There are also no general rules that says or even implies that magic items lose their magic when broken. There are specific rules for artifacts and the like, but nothing for magic items in general. As such, this is a DM decision. If a magic sword is cracked somehow, the DM decides if the sword loses its magic entirely or if it is completely mundane now.
Chris Perkins (story designer for D&D) has suggested that magic items stop functioning when broken:
What happens if an Eversmoking Bottle is broken? Does the smoke cloud expand on and on forever?
If a magic item breaks, it ceases to function. The eversmoking bottle would stop producing smoke.
The fact that magic items stop functioning when broken makes sense, but that is not the same thing as magic items losing their magical nature necessarily. Also note that Chris is not an official source for RAW or even RAI, so at best this is just advice from a very experienced DM.
In the end there are no rules that cover this. A DM must decide for themselves what way they wish to handle broken magic items. In my experience, this does not come up almost ever but that may vary from table to table.
A relevant response from Chris Perkins (story designer for D&D):
What happens if an Eversmoking Bottle is broken? Does the smoke cloud expand on and on forever?
If a magic item breaks, it ceases to function. The eversmoking bottle would stop producing smoke.
That said, by RAW, I don't think there are any rules about whether any break in a magic item causes it to lose its magic, other than the descriptions of specific magic items that state this property of those particular items (e.g. the Staff of Power or Staff of the Magi's Retributive Strike ability). Ostensibly, if you totally broke a magic item, it would not longer be able to function as normal.
In the official adventure Dungeon of the Mad Mage (2018), there is a description of a broken magic item
on the first level, in location 6a, "Hall of Three Lords"; it is a "a white wooden staff broken in half."
Although clearly indicated as broken, the item is still described as registering with a "detect magic spell reveal[ing] the faintest, lingering trace of magic within both fragments..."
We are also told that the item "can be repaired with a mending cantrip."
Note that mending "can physically repair a magic item or construct, but the spell can't restore magic to such an object." RAW, repairing this item should not restore functionality. In the adventure, though, after it is repaired...
The first time the intact staff is held, it wails, “Help! Thief! Criminal!” before its magic fades forever.
So in this example, we have a broken magic item that still detects as magical. Regardless of its actual function, breaking it has not removed its magic completely, although it has apparently diminished it.
As far as whether the broken item is still fully functional, unfortunately, the adventure does not tell us what its original function was. If its only function was
to detect its own theft, once - perhaps from a single-use magic mouth
then it had no function while broken but mending completely restored its full functioning. If, however, it had other powers, then it does not have them either while broken or after being fixed.
As a caveat, the item is written more for dramatic effect in a module that pays homage to the old school 'random dungeon' vibe. It could easily be taken as an exception rather than as establishing a general rule about the nature of broken magic items.
