Quantum many-body physics and non-Hermiticity

Physics is the part of science whose goal is to reveal how Nature works. There are two pillars needed to gain such understanding: (i) find out what are the elementary constituent parts (e.g. particles) and their mutual interactions, and (ii) understand how those particles and interactions lead to the world we observe. Both pillars are paramount; high-energy physics to a large extend deals with (i) and provides us with elementary particles and 4 fundamental interactions. Physics though does not end with (i). How e.g. electromagnetic force leads to the distribution of photons in the sunlight, or a ball rolling down an incline, is a job for pillar (ii). The proposed project belongs to (ii). More specifically, we are going to study different new emergent phenomena that appear when the number of constituent parts is large -- the many-body physics -- and, in particular, on ***what effects can non-Hermitian mathematics, that often emerges in the effective description, have on the relevant physics***. Let us explain briefly what we mean by an effective description, how the underlying non-Hermiticity comes about, and what can be its surprising consequences. To describe a many-body system, even when the number of particles is macroscopically large, the fundamental description is quantum mechanical. For a toy spin-1/2 chain the dimension of the Hilbert space grows exponentially as 2^n with the number of spins n, and therefore quickly gets prohibitively large even for moderate n. As one says, Hilbert space is a big place, and this can simultaneously be a curse (e.g., for numerical studies of many-body systems) and a blessing (e.g., for quantum computation because it allows for qualitatively new effects compared to classical physics). However, often we are not interested in all 2^n degrees of freedom (DOF), but only few observable ones. An effective or a coarse grained description is therefore sought in which one traces out most DOFs. It turns out that such an effective description is often not Hermitian (unitary) anymore, even if one starts with a Hermitian description of the whole system. Some examples that we plan to address are (i) Nonequilibrium physics as described by the Lindblad master equation, (ii) Markovian processes, (iii) Quantum computation. One of the questions is that of relaxation, and closely related transport, in the thermodynamic limit. Standard reasoning, based on intuition from Hermitian systems is, that the relaxation rate will be given by the spectral gap of the corresponding propagator (e.g. of a Lindbladian, or a Markovian matrix). Recent discovery [1], that we are very excited about, though shows that the non-Hermitian matrices can have a so-called phantom eigenvalue, where the standard ``folklore'' of the gap determining the relaxation rate is not correct. Because non-Hermitian matrices appear in different fields of physics, as well as more broadly (in mathematics of Markov chains, or computational science and the speed of stochastic Monte Carlo type algorithms), the proposed project has potentially also a multidisciplinary impact. Several very recent findings in single-particle systems and their non-Hermitian topology also have certain similarities to phantom relaxation. We therefore judge to be at a very particular point of time when there is likely that in the next couple of years a significant and important new discoveries will be made in this emerging field. With our past experience we are uniquely fit to deliver on the project. While we currently have one PhD student working on some aspects of the discovery, to really stay afront and explore all the interesting questions listed in the proposal, a successful project application would allow us to employ additional research forces. [1] J.Bensa and M.Znidaric, Phys.Rev.X 11, 031019 (2021).