Is there any phrase for the coldest days of the winter?

You can use the expression:

dead of winter

the middle of winter, when it is very cold:

• It was the dead of winter and the ground was covered in deep snow.

(Cambridge Dictionary)

The expression dead of is used to refer to:

The period of greatest intensity of something, such as darkness or cold. For example, I love looking at seed catalogs in the dead of winter, when it's below zero outside. The earliest recorded use of dead of night, for "darkest time of night," was in Edward Hall's Chronicle of 1548: "In the dead of the night ... he broke up his camp and fled." Dead of winter, for the coldest part of winter, dates from the early 1600s.

(The American Heritage Dictionary of Idioms by Christine Ammer)

Also, "depths of winter":

the middle of winter : the coldest part of winter.

(Merrian-Webster)

A common expression for the coldest days in British English is brass monkey weather.

Brass monkey weather: Extremely cold weather. _{[Cambridge English dictionary]}

Or it's cold enough to freeze the balls off a brass monkey.

Ice-cold, stone-cold,bone-chilling cold could also be used.

You could use

midwinter

This is probably most famously used in the Christmas hymn "In the bleak midwinter". It's not a phrase you meet very often, but it's something which anyone hearing/reading it would immediately understand.