Can my landlord charge me for the rent after I moved out?

I followed my lease agreement for breaking the lease before the term end and provided a one month notice to my landlord - taking the rent responsibility in case the apartment is not leased till my lease expires. I took rent responsibility only because I was planning to not move-out if the apartment was not leased by a new tenant. Two weeks before vacating, my landlord informed me the apartment is leased to a new tenant and also told me that I cannot retract my move-out notice. 1 week after I