There was once an extremely rational donkey. Once, when it was very hungry, two bales of hay were placed before it. The bales were identical in every respect, both at the same distance. The donkey starved to death. - a retelling of Buridan’s ass

Two roads diverged in a yellow wood

And sorry I could not travel both

And be one traveler, long I stood – The Road Not Taken by Robert Frost

காவலரைத் தன்சேடி காட்டக்கண் டீரிருவர்

தேவர் நளனுறுவாச் சென்றிருந்தார் - பூவரைந்த

மாசிலாப் பூங்குழலாள் மற்றவரைக் காணநின்று

ஊசலா டுற்றா ளுளம் - naLaveNbA of pugazhendi pulavar

We rarely encounter a fork in the road where both paths are identical. The choices that we make are often between two options that differ, if only in minute details. We may then choose one over the other due to some special feature that marks it out as superior. However, if faced with two identical choices, what do we do? Imagine a contestant in a game show who has to choose between two doors that are identical in appearance - one hiding a car and the other an empty room. How does she make a choice? In all likelihood, she will choose one at random. Although we may not recognize it, this offers a window into our own human nature – we are irrational beings who are happy to make random choices. However, suppose an ideal being exists who is perfectly rational. Faced with such a dilemma, what course of action is available to it? We will see that certain magnets that obey the rules of quantum mechanics throw interesting light on this question.

In various contexts within physics, we encounter a choice between two equivalent outcomes. We may imagine a ball perched atop a ridge that may roll down on either side. However, the outcomes here are not quite equally likely. We invariably have small effects -- a mismatch in the slopes, a weak gust of wind, a tilt in the initial placement of the ball, etc. Thanks to these effects, one outcome is favoured over the other and the choice is made. Such decision-making-by-small-nudges is interesting in its own right. However, we cast this aside as there is actually no decision to be made. It is as if our game show contestant is perceptive enough to detect faint differences between the doors. She will then choose the door that bears telltale marks of a car being driven through.

We next consider a situation where two outcomes are perfectly equivalent. Typically, this occurs in small systems (at the atomic scale) which do not leave room for an imbalanced force such as roughness. Invariably, with small size comes quantum mechanics. Indeed, there are many examples of decision making in the quantum realm. Perhaps the best known is the celebrated Stern-Gerlach experiment, first performed in 1922. A silver atom is fired towards a detector that measures its ‘spin’ (crudely, the sense in which the atom rotates about a given axis). The silver atom is initially prepared with its spin along the x direction, but the detector is configured to measure spin along the z direction. The rules of quantum mechanics dictate that there are two possible outcomes, crudely clockwise or anticlockwise. If the experiment is done carefully, both outcomes are precisely equal in likelihood. How does the silver atom grapple with the horns of this dilemma? It chooses one of the two outcomes at random! If the experiment is repeated several times, each outcome is realized in half the runs. Several sophisticated versions of this phenomenon have been devised and tested. It is even used in cutting edge technology as a ‘random number generator’. Crudely, each silver atom yields one of two spin values – this can be interpreted as a random ‘bit’ that takes two values, 0 or 1. By repeating the experiment, we can generate a steady stream of random bits. This is useful in several applications – for instance, to encrypt messages.

We have discussed a paradigm of decision making wherein a quantum system randomly chooses one of two outcomes. This raises several interesting questions, e.g., is the choice truly random? Can we rule out the presence of small nudges that imbalance the outcome? The current state of knowledge suggests that the choice is indeed truly random. Regardless, we set this aside as well. Going back to our game show analogy, this sort of decision making is akin to having a whimsical contestant – a random choice is made without rhyme or reason. Is there a different way to handle decision making?

Before proceeding further, we recollect some basic rules of quantum mechanics. Consider a particle that is constrained to move on a circular track. Quantum mechanics tells us that the particle can be viewed as a wave. Crudely, it is useful to imagine a ring-shaped tank of water. The motion of the particle is akin to a wave on the water’s surface. The wave can move clockwise or anti-clockwise. If we neglect friction-like dampening forces, the wave continues to circle around the ring forever. Likewise, the particle can move around the circle in a clockwise or anti-clockwise fashion. If the particle is in a state of motion, it will keep going around the circle indefinitely.

Let us now bring in the element of decision making. To do this, we place the particle on a figure-of-eight track. This track is essentially similar to the circle except that it intersects with itself at a node. Imagine the following situation. The particle moves along one leg of the track and approaches the node. When it reaches the intersection, it faces a four-fold decision: it can go straight, turn left, turn right or even turn back. The four choices are on the same footing as there is nothing that favours one over the others. How does the particle respond? Remarkably, if the energy of the particle is low, it is unable to decide! It sticks to the node as if it were pondering the choices. It moves short distances along each leg to explore the paths, but does not go deep in either direction. In the language of quantum mechanics, it forms a ‘bound state’. This is a consequence of the wave-like nature of particles in quantum mechanics. It has no equivalent in the classical world of our day-to-day life. We have a quantum contestant in the game show who is afflicted with decision paralysis!

In recent work in my research group, we have demonstrated this phenomenon in certain magnets. We take small magnets that are composed of a few spins – say four in number. For our purposes, we can imagine each spin to be an arrow that can point in an arbitrary direction. The orientations of the four spins encode the state of magnet. There are a large number of possible states; in fact, the number of possibilities is infinite. These states have different energies depending on the angles that the spins make with one another. We focus on the behaviour of the magnet at very low temperatures. In this regime, the magnet only accesses states that have minimum energy. In turn, the states that have minimum energy form a smaller set – they can be imagined as points on a curve. For example, in a certain magnet, the minimum energy states correspond to points on a circle. In a different magnet, they form points on a figure-of-eight-like space. This picture leads to a remarkable ‘emergent’ phenomenon – the magnet behaves as if it were a particle that moves on a curve! Moreover, this particle follows the rules of quantum mechanics. We can now immediately study decision-making. When the magnet resembles a particle on a figure-of-eight curve, it encounters nodes where it has to choose a direction. And lo and behold, the particle freezes at the node as it cannot make a decision. This has elegant consequences – the magnet freezes in a certain state and stays put, unless it receives a strong kick.

‘Quantum indecision’ can be a fairly general phenomenon. We have brought it into sharp relief in the physics of magnets. In fact, we could dig deeper to add further complexity to this problem. In the discussion so far, we have considered a four-fold choice – go straight, turn left, turn right or turn back. By systematically tweaking the magnet, we can increase the number of choices. The particle can choose to proceed along any one of a large number of paths. Remarkably, larger the number of choices, the more indecisive the particle becomes. The particle attaches itself even more tightly to the intersection!

Our papers on this topic:

• Effective theories for quantum spin clusters: Geometric phases and state selection by singularity

Physical Review B 100, 134411 (2019); arXiv version here.

• Order by singularity in Kitaev clusters

Physical Review Research 2, 023212 (2020); arXiv version here. arxiv:1912.04341