It has long been believed that central banks cannot push their policy rates much below zero. This is because of a no-arbitrage relation with physical cash. By holding physical cash, you can guarantee yourself a 0% nominal return: if you have 100 dollars in cash in your wallet today, you can ensure that you'll have 100 nominal dollars in your wallet next year, therefore guaranteeing a 0% nominal return. Thus, if the central bank tried to set its policy rate to cause the nominal bank deposit rate to be (say) -20%, then most everyone would pull their money out of the banking system and hold cash instead, to obtain the higher interest rate. (This is essentially a form of Gresham's Law: central banks peg physical cash and bank reserves at a 1:1 rate; but the rate of return on the two need not be the same.)
This zero lower bound (ZLB) on nominal interest rates however is not a hard limit: because of the costs and inconvenience of storing cash and preventing theft, individuals and firms are willing to accept a somewhat negative nominal interest rate. Indeed, in Europe, the policy rate has been slightly negative since 2014, and this has spilled over to interest rates facing households, such as mortgage rates in Denmark.
In the United States, unlike the ECB in the Eurozone, the Fed has been unwilling to target a policy rate even modestly below zero. Various Fed officials have argued (among other things) that they lack the legal authority to do so; or that benefits would not be large enough to outweigh some perceived risks.
One policy that has been proposed to overcome the ZLB is abolition of cash. Removing cash would remove easy access to a financial instrument which provides a guaranteed, safe 0% nominal return – and would therefore break the no-arbitrage argument which enforces the zero lower bound on nominal interest rates. There are other arguments for abolishing cash: most prominently, to reduce tax evasion. On the flip side, some argue that abolishing cash would force individuals to conduct all of their transactions via the banking system, reducing privacy. (Plausibly, the development of privacy-preserving cryptocurrency technology reduces this concern.) Under this argument, then, the efficiency case for abolishing cash depends on the tradeoff of costs versus benefits.