The Question / Problem
This is a common problem for people who have to leave accommodation and apply as homeless due to domestic abuse or some other reason which means that their home is no longer reasonable for them to continue to occupy. They may be provided with alternative accommodation by the local authority under Part 7 of the Housing Act 1996 but until they can terminate their previous tenancy they have two lots of rent to pay. This will be a problem because the DWP and/or Housing Benefit will only pay the rent for one tenancy.
Most assured shorthold tenancies in the private sector are for fixed periods or usually a year or more with no break clause allowing for earlier termination. This means that the tenant who has to leave faces being stuck with two tenancies until the fixed term of their first tenancy expires. If the tenant asks the council who is rehousing them for help in getting out the first tenancy or paying the rent for it they will usually not get it on the basis that the council see their role as limited to providing safe accommodation but not sorting out the termination of the first tenancy.
There may be a further problem where the applicant has a partner who may have been the perpetrator of the domestic abuse who is still living at the first property and does not want to leave. The partner may be a joint tenant with the applicant.
The Answer /Solution
Unfortunately the local authority are entitled to say that helping the applicant sort out their previous tenancy and avoid rent arrears is not their responsibility. Their duties are concerned with securing that suitable new accomodation is provided.
There are are two primary ways for a tenancy to be terminated by the tenant before the end of the fixed term. These are (1) a surrender of the tenancy and (2) An occupier's notice to quit.
A surrender of a tenancy is the best solution to this problem and should be available where the landlord for the first property is sympathetic. They way it works is that the tenant invites the landlord to accept a surrender of the tenancy. This should be set out in writing and if accepted will have the effect of terminating the tenancy with immediate effect.
If the landlord is not sympathetic and refuses to accept a surrender of the tenancy then the tenant will need to serve an occupier's notice to quit as soon as possible. My view (please leave a comment correcting me if you disagree) is that the once the fixed term has ended the notice can be served and will take effect after a month or so depending on the tenancy. If the fixed term has not ended the tenant should serve a notice to take effect as soon as the fixed term ends.
As regards the partner / joint tenant left behind at the property the law provides that termination of a joint tenancy by one of the joint tenants with or without the agreement of the other tenant ends the tenancy. This is because the joint tenancy is treated as one combined tenancy which can be ended by either tenant as was confirmed by the Court of Appeal in the case or Hammersmith LBC v Monk 
In practice a landlord faced with a tenant having to move out in an emergency such as this would be well advised to accept a surrender of the tenancy. This is because they would be better of reletting the property as soon as possible. Until they do they face not receiving any rent. They may be able to take legal action to recover the rent but in they may have difficulty find out where the former tenant has moved to and may face difficulty in enforcing a judgment for rent arrears against a former tenant on low income. Most landlords will end up deciding to write off the rent arrears. Accepting a tenant's surrender of the tenancy also has the advantage of allowing them to re-let the property without the risk of the former tenant reappearing and making a claim for unlawful eviction.
A tenant under a fixed term tenancy which has not expired has no legal right to have the tenancy terminated before the end of the fixed term. If the landlord refuses to accept the surrender they remain liable to pay the rent up to the end of the fixed term. They won't have to pay it if the landlord does manage to re-let the property but will be liable to pay if the property can't be re-let. In practice I have found that it is very rare for landlords to pursue tenants who move out early for rent arrears.