Collective motion is ubiquitous in nature on all scales: from flocking starlings to schooling fish to swarming bacteria. These animal groups seem like a single entity governed by a collective mind. Taking this analogy to the realm of physics, the interplay between the microscopic and macroscopic worlds is intensively investigated in statistical mechanics: simple interactions between the microscopic constituents can explain the measurable properties of macroscopic systems. Once at the nanoscale, the bizarre world of quantum mechanics takes a big hold. Take for example fractionalisation: the collective excitations of a system cannot be constructed as combinations of its elementary constituents. While several techniques have been developed to understand similar phenomena in equilibrium systems, their study is much harder for out-of-equilibrium setups. The EU-funded ThermOutOfEq project devised engineering setups to increase theoretical understanding of collective phenomena at the nanoscale. It also developed theoretical approaches for studying how chaotic systems behave.
The arrow of time points forward
Researchers described findings that demonstrate the reversible and irreversible nature of quantum dynamics under weak drives. Use of matrix product states helped in the analysis of the quantum (Heisenberg spin chain model). This popular model describes the interactions of magnetic spins in a 1D array and its properties can be exactly computed in equilibrium. Although the microscopic components are individual spins, collective excitations correspond to spin waves. The latter behave essentially as particles: when two spin waves travelling in opposite directions meet each other, they can join in a ‘bound state’, a bigger ‘particle’. Pairs of bound states can also interact, generating a complex spectrum of ‘particles’ of different sizes. “To better understand the dynamics of spin waves and their bound states out of equilibrium, we proposed a setup where a magnetic field accelerates the particles. We observed that some bound states are not stable when they move too fast. There is a critical velocity at which they break in smaller particles,” explains project coordinator Andrea De Luca. Furthermore, the researchers noticed that there is only one way to form a bound state once two or more spin waves merge: e.g. the energy of the bound state is completely fixed because energy is conserved. On the contrary, once a bound state is broken, there are many ways of distributing its energy between the outgoing particles. “This kind of ‘lack of information’ is at the origin of the generation of entropy and irreversibility. We noticed that before spin waves reach a critical velocity, the system is completely reversible: once we reverse the magnetic field, the spin waves go back to their initial state. When they exceed the critical velocity, there is no memory of the broken bound states,” adds De Luca.
Random matrices and quantum chaos
Understanding how chaos manifests in the smooth and wavelike nature of phenomena on the quantum scale is challenging. “In the classical world, a small perturbation to the motion of the particle (ball) in billiards could cause a big change on its trajectory. But, in the quantum realm, defining any trajectories for quantum systems is impossible. The only thing that we know is that chaotic quantum systems statistically behave as random matrices,” explains De Luca. Using a Floquet random circuit, researchers confirmed this long-standing conjecture. They succeeded in providing an exactly solvable example of a system that is quantum and chaotic and is composed by a large number of degrees of freedom and showed how chaos and random matrix behaviour appear together.
ThermOutOfEq, spin waves, bound state, irreversible, collective excitations, out of equilibrium, quantum chaos, Heisenberg spin chain model, Floquet