Projects
"More is different"
-P.W. Anderson
Quantum Phase Transitions
It has been more that 80 years since the first categorical study of classical phases of matter was proposed by Ehrenfest. Classical phases of thermodynamically large systems are characterized by an 'order parameter' which changes value sharply about a critical point at which the system is said to go through a phase transition. For example, pure water goes through a phase transition from being a liquid to a solid (ice) at the critical temperature of zero degree Celsius under atmospheric pressure; the order parameter being the specific density. Just as thermal phase transitions can be shown to be driven by thermal statistical fluctuations near the critical point, a new class of phase transitions have been proposed which may be probed when the system is in equilibrium at zero absolute temperature. The driving force behind these phase transitions are nothing but the inherent quantum fluctuations arising out of non-commutativity in a quantum many-body system. Analogous to standard phase transitions, these quantum phase transitions are also characterized by an order parameter and have been seen to exhibit universal critical scaling near the critical point. A very popular example of a quantum phase transition is the external magnetic field driven transition between a quantum ferromagnet to a paramagnet characterized by the local magnetization density (serves as an order parameter) and routinely used in the designing of magnetic memory storage devices.
Often in nature and also in the quantum realm, one comes across systems evolving in time rather than being in equilibrium. A few of the interesting questions to ask and the focus of my studies is whether these quantum phase transitions can be probed in systems thrown out of equilibrium. Can quantum criticality and the different phases be studied in time-evolving observables characteristic to non-equilibrium quantum systems?
Topological Quantum Phase transitions
Topology of a geometrical object is its property which allows one to deform the object and modify its shape continuously without cutting or gluing any of its points. In theoretical physics, one may define the topology of mathematical functions describing the behavior of physical systems. In this context, certain quantum many-body systems can indeed be classified by its topology which can only change when the system goes through a topological quantum phase transition. Therefore the quantities characterizing the topology of the system in a phase is the appropriate order parameter in this scenario. The importance of the study of topological phases of matter is intricately intertwined with two very significant properties of topology itself:
Topological classifications are NOT local protocols. That is, the topological phase a quantum system is dependent on the complete/whole system collectively rather than its spatially local structures.
A major consequence of the above fact is that, the topological properties of a system in a topological phase are often robust against local perturbations which are not sufficient to change the topology of the complete system as a whole.
It is this promise of topologically protected phases that makes it worth the attention, as topological protection can also make the phases robust against local environmental perturbations. Such applications will also have a huge impact on the designs of quantum computers and simulators which must be robust against continuous environmental interactions and decoherence.
My primary interest in this field is to understand the response of topological quantum phases against perturbations throwing the system out-of-equilibrium. Most of the time I try to address the question of whether it is possible to dynamically tune the topological phases of a system and drive it between the different topological phases through viable procedures. The dynamical preparation of a topologically protected quantum state starting from a trivial one is one of the challenges I look forward to overcome. How such systems respond to external dissipative processes is also a question which I try to address in my studies.
Dynamical Quantum Phase Transitions
Analogous to equilibrium quantum phase transitions as discussed previously, a completely new class of quantum phase transitions have been recently discovered in thermodynamically large systems. Just as the critical behavior of stationary quantum systems are associated to a critical value of a parameter describing it, these new phase transitions are a manifestly non-equilibrium phenomena. To elaborate, following a quench in a quantum many-body system (a sudden change in any of its parameters throwing the system out-of-equilibrium), one looks at the probability of the system to return to its initial state, through a so called measure of "echo". Precisely when the time-evolving state becomes orthogonal to the initial state manifold, the system exhibits critical phenomena at critical instants of time. The complete early-time dynamics of the system can then be classified in a series of "dynamical phases" separated by a series of "critical times" rather than a system parameter or temperature. Similar to equilibrium quantum phase transitions, a time evolving quantum system shows universal critical scaling in the vicinity of these critical instants.
Although there have been many somewhat thorough studies on these emergent criticalities, one of the open challenges in this area is to discover observables that might serve as an order parameter characterizing these dynamical phases.
My focus in this area is precisely to construct many-body observable quantities that capture these non-equilibrium phase transitions and which at the same time can also be probed through direct experiments. I also work to understand the robustness of the dynamical quantum phases against various interactions within the system, interactions with an external environment (open quantum systems) and also in finite temperatures.
Entanglement and Thermalization
Entanglement is one of the mind-twisting concepts which takes a quantum system far away from a classical one and is also intricately connected to the dynamics of quantum many-body systems. To be specific, entangled quantum systems cannot be thought of to exist independently, regardless of how far they are separated from each other in space-time. This forces one to altogether forfeit the concept of "locality" in the process of understanding a quantum theory. However counter-intuitive it may seem, countless experiments have already proven quantum entanglement to be an inherent part of our Universe and is at the heart of numerous modern day technologies and the quantum computer.
Theoretically, many measures have been proposed to quantify and study entanglement as a mathematical theory which will be helpful to truly understand the phenomena in connection to more established laws of nature. One such measure is the entanglement entropy of a subpart of a larger entangled system. Specifically this amounts to the missing information about a subsystem (hence, the name "entropy") one encounters when it is only the subsystem which is focused upon while completely disregarding the other part to which it is entangled. It is now almost unanimously accepted that the entanglement entropy is a good measure of entanglement in isolated quantum systems and is directly proportional to the degree two quantum systems are entangled. Using this measure it is also understood now how a quantum many-body system initially in a scarcely entangled state gradually becomes more and more entangled over time following a sudden quench in one of its parameters.
This dynamical growth of entanglement is deeply connected to a very fundamental question in physics, i.e., "Can/How does equilibrium statistical mechanics emerge in isolated quantum systems?". This question is intrinsically connected to the theoretical foundation of statistical mechanics and the emergence of irreversibility in many-body quantum systems (i. e., the arrow of time), which ultimately leads to all of thermodynamics, a theory explaining the working of engines, refrigerators and countless other natural processes. The process of emergence of a statistical ensemble picture in quantum many-body systems is popularly known as "Thermalisation". Over time an integrable quantum system can be described as a generalised statistical ensemble characterized by an extensive number of conserved quantities through which the system preserves the intial state information completely. Due to the dynamical evolution of the system, every part of a sub-system becomes entangled with the rest of the system and therefore shows an extensive saturation of entanglement entropy with respect to the size of the sub-system.
On the other hand, non-integrable quantum systems following a quench tends to undergo chaotic dynamics in which the initial state information is quickly scrambled throughout the thermodynamically large system over time. This leads to the emergence of irreversible dynamics in chaotic systems. Remarkably, according to Eigenstate Thermalisation Hypothesis, every eigenstate of the system having a finite energy density, behaves like a statistical ensemble with respect to expectations of few-body observables. The study of entanglement growth in such systems is an exciting area of recent research which is expected to elucidate the process of thermalisation in microscopic detail. More precisely, due to thermalization after sufficiently long times, the entanglement entropy in such systems coincide with that of a microcanonical ensemble (as energy is the only conserved quantity).
My area of research in this context is to understand how non-integrable and open quantum systems thermalize at late-times by probing the growth of entanglement and local measures in a quantum many-body system. I try to construct good measures of quantum entanglement in mixed state ensembles and open quantum systems. I also study the emergence of chaotic dynamics in much simpler single-particle quantum systems when they are perturbed/driven externally and start evolving non trivially in time thereafter.