Kicking semimetals to reveal quantum ghosts
Like the name suggests, a phase transition occurs when a system changes from one phase to another as some parameter is modified. We probably don't think about it much, but we are literally surrounded by phase transitions in our daily life. When we're boiling pasta, the water undergoes a transition from liquid to gas. When we put ice in our drink, it melts undergoing a solid to liquid phase transition. But phase transitions are even more widespread than we think. They occur in biological systems at all scales, from protein and DNA folding, to bird flocking patterns. They exist in the stock market, and even in social media.
The so-called Landau theory was thought to describe all continuous phase transitions until the 1980s, when topological phase transitions were first discovered. In topological matter, the quantum mechanical ground state is degenerate, meaning that more than one possible ground state with the same energy exists. Because of this, topological matter does not have a local order parameter, nor spontaneously break symmetries. Topological phases are instead described by a different set of numbers – called topological invariants – that rely on the topology of the ground state. At first sight, then, topological phase transitions don't fit well in the paradigm developed by Landau. Or do they?
In fact, there exist a method - called the Curvature Renormalization Group (CRG) method - which allows to extract critical exponents and scaling laws from the behavior of the curvature function, which is the integrand of topological invariants mentioned above. The CRG method works brilliantly for point-like gap closures in any dimension, for interacting systems, and periodically driven systems. But what about more exotic topology? In one of our earlier studies, we found preliminary evidence that the scaling above is modified when the gap closures are extended, for instance from 1 to 1/2 in nodal loop semimetals. These are topological materials (CuTeO3 is a possible candidate) in which the energy gap closes along a closed loop (hence the name!) in momentum space. Nodal loop semimetals are expected to give rise to very interesting properties, such as anisotropic electron transport or unusual optical response. Knowing how they behave close to a phase transition could thus be very useful to understand and possibly harness these properties. However, while the CRG is a very useful theoretical tool, it relies on the measurement of the curvature function, which is typically hard to do in real settings. Are there other approaches that we can take to determine the scaling laws of topological phase transitions?
In our study, we explored this question by employing quenches, i.e. sudden or continuous changes in system parameters. The connection to scaling laws and critical exponents relies on the Kibble-Zurek mechanism, which connects defect generation to scaling laws. The Kibble-Zurek mechanism was first developed for Landau-type phase transitions, but people have applied it also to topological phase transitions because it ultimately relies of the fundamental phenomenon of fluctuations. Fluctuations are minute random deviations of a system from its average state and occur everywhere. In classical systems they are mainly due to thermal noise, i.e. the random motion of particles. In quantum systems – including topological matter – they are typically due to quantum noise stemming from Heisenberg's uncertainty principle. In a nutshell, the quantum vacuum is not a static entity, but buzzes with a continuous creation and annihilation of particle/antiparticle pairs which leads to temporary random changes in the amount of energy in space. If the Universe was a TV, even when turned off it wouldn't show a black screen, but a snow screen of white noise.
We took a prototypical model for a 2D nodal loop semimetal which exhibits both Dirac-type and nodal loop gap closures and we quenched the parameters across both transitions. By calculating the number of topological defects as a function of the quench rate, we were able to determine the critical exponents from the Kibble-Zurek scaling and compare them with the previous CRG predictions. As expected, we found that while the Dirac gap closure (at a point) gives a critical exponent of 1, having an extended gap closure like in a nodal loop reduces the critical exponent to 1/2. That means that the correlation length diverges much more slowly for nodal loop semimetals, which should impact electronic and optical properties of such materials.
But there is more! To gain more insight in the difference between the Dirac and nodal loop gap closures, we decided to investigate how the defects behave as a function of time. We then employed another kind of quench, a sudden quench where the system starts from a given phase but it is immediately kicked into a different phase across a topological phase transition. This is a bit like dropping an ice cube into boiling water instead of letting it slowly melt in the glass. Sudden quenches have been prominently employed to investigate dynamical quantum phase transitions (DQPTs), which are extensions of the notion of criticality to time-evolving systems and are characterized by sudden changes in their macroscopic properties as a function of time. Because of the sudden change in parameters, a lot of energy is injected into the system, which is thrown out of equilibrium and can again generate defects. In the case of our nodal loop semimetal, the defects appear as phase vortices, where the phase of the quantum state suddenly "slips" by 2π. Some vortices are static, meaning that they are fixed at certain high-symmetry points in momentum space and do not evolve in time. However, some defects also appear and disappear dynamically.
By tracking the defect generation as a function of time, we revealed several surprising features. Even when we quench across the Dirac gap closure, the dynamical vortices are always created along nodal loop lines. This is quite remarkable, because the nodal loop gap closures occur elsewhere in the parameter space. It is like invisible "ghosts" of the nodal loop gap closure permeate the entire system and are what carry the dynamical defects. The trajectories themselves change as a function of time, with the nodal loops acquiring different shapes at time goes on (see figure). Another unexpected result is that the location of such defects does not coincide with the points where DQPTs take place, which goes against the predictions of current literature. Somehow, the defect generation along the nodal loop ghosts decouples from the nonanaliticities in the wave function that signal the DQPTs.
We hope that our work will shed some light on new ways of probing exotic topological matter like nodal loop semimetals and the complexities of their topological phase transitions. The low-energy description of a model might not be enough to describe the full dynamics of defect generation and we might need to come up with more general methods beyond the Kibble-Zurek mechanism to describe DQPTs correctly.
Quench dynamics and scaling laws in topological nodal loop semimetals, Karin Sim, R. Chitra, and Paolo Molignini, Phys. Rev. B 106, 224302 (2022)