Is it permissible for a woman to seek divorce from her husband who treats her with coldness and cruelty, while she feels attraction towards other men, for fear of falling into sin?

If a husband treats his wife with harshness and coldness, that is not a righteous companionship (mu'ashara bi'l-ma'roof). However, it is not a justification for the wife to incline towards other non-mahram (definition) men, because the wife is entrusted with her husband’s honor, as Allah the Almighty says: ﴿So righteous women are devoutly obedient, guarding in [the husband’s] absence what Allah would have them guard.﴾ The wife must fear Allah and rely on Him, and openly discuss with her husband her need for affection and love. If he responds, then that is good and excellent. If he does not respond, then she may seek a khul‘ (divorce initiated by the wife). She must also know that modesty brings nothing but good, and that adherence to religion and abiding by its limits are the cause of success and happiness in this world and the Hereafter.