Chinese demographic history is uniquely intriguing, and is full of fluctuations, but one that sticks out is the more than doubling of the population over a relatively short time during the Song Dynasty.
That kind of sustained growth for a century seems out of character for the pre-modern era, where population growth was usually very low, with birth rates not much higher than death rates. Was there a conspicuous increase in fertility rate during the Song, a decline in infant mortality, both?
The population then continuously declined back to 60 million, barely higher than the population before the spike, and roughly equal to the Han Dynasty centuries earlier. Even the devastation in the 19th century, caused in part by the Taiping and Dungan rebellions, didn't result in that kind of demographic collapse, and the population was back to its 1850 level in about half a century. Following the peak during the Song, the population maintained a significant downward trajectory for nearly two centuries. What caused that level of decline so soon after a rapid increase?
The population rose so high in the Song period because of a combination of intensive and extensive growth, i.e. increased density in established areas of settlement as well as expansion into new areas. The Han and Tang populations both peaked at about 60 million in the entire empire, with high density mostly restricted to a narrow wedge around the lower Yellow River in North China. By the 12th century the Chinese population had roughly doubled, with the majority now living south of the Huai River. For a number of reasons, that population shrunk significantly over the course of the Yuan period (although the reported 60 million is almost certainly a serious undercount), but by the establishment of the Ming in the 14th century the Chinese population was still at about 80 million, and still mostly in the South. In other words, the pattern of extensive growth- expansion into the South- held, but the pattern of intensive growth- unprecedented levels of density in core areas- receded.
Why exactly did Song China experience so much intensive population growth? It was a reflection of the period's considerable intensive economic growth, part of what has been called an "efflorescence" or an "commercial revolution." A number of favorable trends coincided to produce this economic boom. Politically, the Song regime maintained the efficient administration that had characterized Han and Tang governance, yet was more relaxed towards economic activity and the merchant class. Socially, the Confucian literati finally eclipsed the aristocratic elements which had been so influential in Han and Tang, leading to the creation of a social elite which was both larger and more open to upward mobility. An open intellectual climate also allowed for the invention and dissemination of new technologies and commercial practices, and an international environment of expanding seaborne trade created new opportunities for Chinese producers and merchants.
Taken together, these factors enabled an economic boom that generated both rising incomes and rising population densities. The peasants, who made up the overwhelming majority of the population, were increasingly able to turn away from simple subsistence and begin specializing for the market; food production shifted decisively towards southern rice production (which produced the most calories per unit of land), opening up more farmland for profitable crops like tea, sugar and cotton and freeing up more labor for manufacturing in the cities. This new urban labor force moved into fast-expanding sectors like textiles, porcelain, shipbuilding and iron work- which were fueled in large part by foreign demand- and the growing class of small businessmen were often able to join the ranks of the Confucian gentry, thereby legitimizing business and staving off any potential conservative reaction. Each element reinforced the other, so that average incomes rose significantly. This in turn led to extreme population density, as it was now economical to feed larger urban populations with crops shipped from further afield.
The Jin conquest of North China caused serious short-term disruption in this system but did not derail it in the long term, allowing for continued population growth into the period of Jin-Song division. Decline set in instead over the course of Mongol rule. While the Chinese market economy did not collapse, the serious damage inflicted during the Mongol conquest, the burden of ill-fated major invasions of Japan and Vietnam, ruinous inflation in the paper currency and a system of official discrimination against southern Chinese all contributed to the system's deterioration. Outbreaks of plague in the early 14th century dealt yet another demographic blow. Still, scholars doubt the reports of about 60 million in the Yuan census of 1290 and the early Ming census of 1393, and the actual population was most likely in the ballpark of 85 million instead (the Mongol case was likely a simple under count, and the 1393 census actually recorded the number of "households" rather than people, allowing for a wide range of extrapolations).
The population exploded to even greater heights under the Ming (maybe as high as 300 million by 1650) because of a second, even larger commercial revolution, made possible by the Columbian Exchange. The changes brought on by American food crops, American silver, and global economic integration where what set the pattern for Chinese population from the late Ming through Qing periods.
Sources:
Fairbank & Goldman, China: A New History (particularly useful for its Han, Tang and Song population density maps)
Hansen, The Open Empire: A History of China to 1600 (for census numbers and analysis thereof)