After reciting Al Fatihah you can recite another sura in rak'ats 1 and 2. Can that sura be sura Fatihah?
Basically there seem to be a consensus among scholars that repeating the recitation of al-Fatihah in the prayer without a valid reason is at least makrooh (there are opinions saying it invalidates the prayer).
For example read in:
In the following I will translate from Arabic, these translations are of my own -unless I quoted a source- take them carefully
- in Radd al-Mohtar رد المحتار of ibn 'Abdiyn (hanafi) he considers it as wajib:
والرابع عشر ترك تكرير الفاتحة قبل سورة الأوليين
And the fourteenth is omissing the repetition of al-Fatihah before the surah of the two first (raka's).
- in al-Khulassa al-Fiqhya 'ala madhhab as-sadah alk-Malikiya الخلاصة الفقهية على مذهب السادة المالكية when quoting what invalidates the prayer excluding some verbal acts- (*):
كتكرير الْفَاتِحَة فَلَا يُبْطِلهَا وَإِنَّمَا يحرم إِن كَانَ عمدا وَيسْجد سُجُود السَّهْو إِن كَانَ سَهوا
Like reciting al-Fatihah, it doesn't invalidate the prayer, but it is considerd haram if made intentionally, and if it was unintentionally one should do sujud sahw.
(*) In a commentary of ar-Rissala of ibn Abi Zayd al-Qayrawani the opinion about a person intentionally repeating the recitation was close to that quoted by imam an-Nawawi in his al-Majmo' and if a person nevertheless didn't perform sujud as-Sahw the prayer still would be considered as valid.
- in imam an-Nawawi's al-Majmo' المجموع شرح المهذب:
فإن قرأ الفاتحة مرتين سهوا لم يضر ، وإن تعمد فوجهان الصحيح المنصوص لا تبطل ; لأنه لا يخل بصورة الصلاة ( والثاني ) : تبطل كتكرار الركوع
So if one reads al-Fatihah twice unintentionally (or out of forgetfulness) it wouldn't harm, but if he did so willingly there are two opinions, the soundest and established is that it doesn't invalidate the prayer as it doesn't change (badly) or deviate from the -usual- practice of the prayer and the second is it does invalidate as if one would repeat (re-do)the ruku'.
- in al-Insaaf of the hanbali scholar 'ala' ad-Dyn al-Mardaawi علاء الدين المرداوي:
It says in al-Insaaf (2/99): It is makrooh to repeat al-Faatihah. This is our opinion as it is the view of the majority of our companions, and it was stated definitively by many of them. And it was said that it invalidates (the prayer). End quote. (Source islamqa #116917)
Some valid reasons are for example you recited it in a manner that you made big mistakes and repeated it in a more correct manner or you recited it silently in a prayer or prayer part where you should have recited it loudly and corrected your mistake.
Reciting al-fatihah (once and) alone is enough
On the other hand it is not necessary to recite a surah after al-Fatihah so your prayer would still be valid if you stick and stopped after the fatihah and moved to ruku':
If I add nothing to the (recitation) of the Umm al Qur'an (Surat al-Fatiha), would it make the prayer incomplete? He (AbuHuraira) said: If you add to that (if you recite some of verses of the Qur'an along with Surat at-Fatiha) that is better for you. But if you are contented with it (Surat al-Fatiha) only, it is sufficient for you. (Sahih Muslim 1 & 2)
For details on the fact that reciting al-Fatiha is enough refer to fatwa islamqa #6422. Basically the majority considers recitation after fatihah as sunnah in the two first raka'a only imam a-Shafi'i in his new madhhab considers it as mustahabb, while other scholars considered it makrooh which imam an-Nawawi considered as more correct. When it comes to optional prayers the matter is considered as more open by scholars (see for example on how the prophet used to pray the fajr -sunnah- in Sahih Muslim which leads to another evidence that praying without reciting a surah after fatihah -in this case for an optional prayer- is valid)