Hi, you may be interested in this recent related post

• Why the Japanese really surrendered? - note also the link to the FAQ at the bottom. Please click!

Since you have a linked answer that hits on most the pertinent points, I will basically follow its structure but add some details that may be of interest. Thanks to /u/PantsTime for writing up his good answer and /u/Searocksandtrees for linking to it. I should also say that I tend to fall into the more traditional school of thought, thinking that the atomic bombs were a major factor in the Japanese surrender. Naturally I will try and keep the facts of this post as objective as possible and I will mark statements that are my own opinion.

It is certainly true that Japan was being defeated on many fronts. I am not an expert on the history of the Pacific Theatre, but my understanding is that many historians hold the Battle of Midway to be a major turning point. This battle was quite far removed from Japan, but it is the beginning of a bunch of naval successes. By 1945, the Americans were occupying the Mariana Islands (the Little Boy bomb was assembled here). Other acquisitions south of Japan allowed America to bomb Ryukyu and other population centres in Japan (perhaps most notably Tokyo). Furthermore, these acquisitions allowed America to continue its blockade and, as Hasegawa notes, to divide Japanese forces on the mainland from the Home Islands and vice versa. One final thing to note is that America was not the sole actor in battling Japan during this period. The surrender of Germany relieved European troops and allowed them to relocate in Asia. Commonwealth troops played a particularly important role in taking back British colonies.

As the commenter also notes, Japan's leadership was basically divided into two camps. One included prominent politicians, such as Shigenitsu Togo, the Minister for Foreign Affairs (the other commenter is wrong to say that the Russian invasion was decisive for Togo). It also included Admiral Yonai (there was a long history of animosity between the Japanese Navy and Army) and eventually Emperor Hirohito himself. This was the peace party. Unfortunately, they were working essentially covertly to pursue peace. One factor behind this is the dogmatism of the military faction, notably Korechiko Anami. Nonetheless, the peace party did attempt to get the Soviets to mediate peace talks as they realised how dire the situation was becoming. There are interesting episodes preceding Potsdam in which the Japanese ambassador was rebuffed by the Soviets. It often strikes me as surprising that the Japanese seem not to have caught onto the fact that the Soviets were turning even though their ambassadors were received so coldly and that the Soviets refused to refresh the Soviet-Japan Neutrality Pact.

The reasons for the Soviets turning are several. For one, they had agreed and Yalta - and again at Potsdam - that the ending of World War II should be expeditious. Secondly, the Chinese were making overtures to the Soviets. Both countries were united in their anger at the Japanese for their historic hubris and militarism. Thirdly, the Soviets probably wanted to invade before Japan was defeated so that they could occupy territories like Manchuria. This would give them a gambit in claiming the territories during post-World War II peace negotiations. Because the United States were somewhat unwilling to concede certain contested locations, like Darien, it was important for the Soviets that they make the land-grab instead.

What was the effect of the bomb? The other commenter is correct to note that the two main responses from the hawkish faction of Japanese politics either refused to believe it was an atomic bomb or did not think the US would have enough to continue their bombing campaign. On the point of the Japanese military not believing that the Americans had used an atomic bomb, I recommend this fascinating article which describes how the Japanese sent out a scientific team to Hiroshima to discover what it had been hit with and how logistical problems prevented them from giving the news to Japan's Supreme War Council until the morning that Nagasaki was hit. The author attempts to link this idea into the fact that the Japanese had not got very far in its nuclear research - which is true. Not only had the Japanese abandoned its nuclear programmes, it also destroyed most of its equipment, like the all-important cyclotrons, and so could not produce nuclear weapons even if it had learnt that the manufacture of atomic bombs was a possibility.

As the commenter also notes, the Japanese did not believe the Americans had enough bombs to sustain an atomic bombing campaign. There is truth to the fact that the Americans did not have many bombs on hand. However, the production of bombs was occuring at an impressive pace. If the war had not ended, the Americans would have had a third by the 20th of August. I think this is a point that is often missed: that even if one denies that the atomic bombs used at Hiroshima and Nagasaki were not decisive in the Japanese surrender, the atomic bombs did have the potential to be decisive. I think the Japanese probably would have surrendered if they had realised sooner what they were being hit with or if they had suffered a more sustained series of atomic attacks.

What was the effect of Soviet invasion? It is very important to note two things. Firstly, the Russian invasion put an end to any hope for a mediated peace. As the other commenter notes, this pushed the peace faction towards peace all the more. For the Army it was devastating for another reason: it had moved most of its troops to Ryukyu in expectation of Operation OLYMPIC, the American invasion timetabled for November. To facilitate this, they had vacated Hokkaido. In short, their armies were in the wrong place to defend against a Soviet attack on the Home Islands. Furthermore, as I've noted, the Americans could essentially prevent troops being transported from the mainland to the Home Islands or vice versa, so there was no option to buff the Soviet front. The Soviet invasion was damning for the Japanese and pushed both factions towards peace.

So what do historians argue about?

One is that the atomic bombs did not kill so many people. The firebombing of Tokyo was absolutely devastating; Hiroshima less so. The bombing of Nagasaki was even less destructive than Hiroshima because the bomb missed its target. Personally, I do not find this argument entirely convincing, but the author of your piece, Ward Wilson, has been a big proponent of it. Firstly, I have never seen good evidence from Japanese military personalities that the loss of life was totally disregarded. If it is, it tends to be a function of the belief that their civilians were not being killed by atomic bombs, which I've touched on above. Furthermore, I think the atomic bomb changed the ways that war was conducted in significant ways besides simply killing lots of people - the main one being it exposed US forces almost not at all. In short, the Americans could inflict Hiroshima-scale casualties with a single aeroplane. This has to be borne in mind if one's thinking about what the atomic weapons achieved qua weapons.

A second argument draws on strategic reports. One major source is the US Strategic Bombing Survey Reports of 1946. /u/restricteddata can probably say more on this than I can because he is much smarter. In short, there were multiple reports produced by the United States which said that the bombs had been strategically unnecessary. Historians often use these reports to say that America was impetuous and acted in an ethically unjustifiable way by deploying weapons when they were not necessary. Perhaps less controversially, it might be said that the American acknowledgement that it could have won the war with conventional means is a step in the direction of admitting that the Russian intervention could have been a decisive factor.

Another set of reports that have been interrogated for historians tackling this question are the casualty reports that were produced. This is once again a debate that I can't throw much weight behind. In short, it follows the same pattern as the one above: if Truman received inflated figures for the casualties attendant on Operation OLYMPIC, he used the bomb believing it to be necessary when it was not in fact necessary, and this is unjustifiable. This is a very muddy debate and often risks falling into counterfactual territory. I think most historians acknowledge that the figures weren't perfect, but that Truman had honest concerns about American loss of life if the war was not ended sooner rather than later. As I say, I think using these reports to say that Japan did not surrender because of the bomb is a logically tenuous line of argument.

Thirdly, I mentioned the strategic effects of the Soviet attack above. Revisionist historians have put a lot of weight on both factions of Japanese government being caught like sitting ducks by the Soviet intervention. The author of your piece pithily encapsulates how revisionists read this point: "After Hiroshima was bombed on Aug. 6, both options [pursuing war or mediated peace] were still alive". The Soviet intervention, they argue, denied this, and so was a major factor in the Japanese decision to surrender. I tend to think this is one of the stronger arguments in the revisionist's arsenal.

I feel like this answer has kind of tapered off. I'll edit it later because I feel as if I'm missing something. For further reading, there are three heavyweights in the current debate: Robert Maddox Ford on the traditionalist side, Gar Alperovitz on the revisionist side, and Hasegawa as centre-revisionist.