Topology in many-body quantum systems in and out of equilibrium

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NON-TECHNICAL SUMMARY:This award supports theoretical and computational research and education to investigate the behavior of quantum systems with many particles, such as electrons in a crystalline solid. The large number of electrons and their inherent quantum mechanical nature can act in concert leading to new states of matter. Among these are superconductors, which can conduct electricity with zero resistance, and the quantum Hall states, in which a strong magnetic field causes electrons confined to two dimensions to execute tight circular orbits, thereby forcing net electrical current flow to thei edges of the system. The goal of the PI’s work is to combine tools from theoretical condensed matter and quantum information theory to gain an understanding of these states of quantum matter. In the last few years, quantum computing has motivated significant progress in condensed matter theory. Quantum computing deals with systems that contain many quantum bits, providing a new perspective on the physics of systems of many quantum mechanical particles that is complementary to the traditional one. The PI will combine new quantum computing inspired methods with standard condensed matter techniques to gain a better understanding of the landscape of condensed matter phases. One particular benefit of this program is that some of the exotic condensed matter phases studied by the PI may in turn have applications to the design of new quantum computing platforms.Under this award, the PI will support and mentor graduate students throughout their progress to a PhD. The PI will help educate the students in the relevant areas of condensed matter physics, and help them develop proper written and oral communication skills to facilitate the dissemination of their research.TECHNICAL SUMMARY: This award supports theoretical and computational research and education to classify topological quantum phases of matter, including those that are out of equilibrium and those protected by additional symmetries. The PI will combine ideas from condensed matter physics, quantum field theory, and quantum information theory to gain an understanding of these quantum many-body systems. Specifically, the PI will connect the topological features of quantum field theories, such as topological terms in continuum effective actions, to topological invariants that can in principle be directly extracted from a lattice Hamiltonian, such as braiding statistics of anyon or defect excitations. Because a lattice quantum many-body system is essentially a many-qubit system, it is natural that ideas from quantum information theory will naturally be involved in this work. One quantum information idea is that of a non-trivial quantum cellular automaton, which is a generalization of a shallow-depth circuit of local unitaries. The PI will explore the utility of using such quantum cellular automatons to disentangle exotic, “beyond-cohomology,” symmetry protected topological phases of matter.An additional tool that the PI plans to use is that of the conformal bootstrap. This is a numerical technique that constrains the possible conformal field theories that can exist, by putting bounds on their spectra of dimensions of local operators. Building on preliminary work, the PI's team will research how to constrain the conformal field theories that can exist at the boundaries of topological phases, by incorporating the corresponding 't Hooft anomalies into the conformal bootstrap. More generally, the aim of the PI will be to determine how to incorporate information about the non-local operators that appear in topological field-theories into the conformal bootstrap. Furthermore, the PI will also explore topological effects in non-equilibrium settings, such as the recently introduced measurement-induced entanglement transition. Preliminary work by the PI's team has uncovered, for a certain simple instance of this transition, a dual statistical-mechanics model which exhibits a topological phase transition. The PI plans to explore more general versions of this duality.Under this award, the PI will support and mentor graduate students throughout their progress to a PhD. The PI will help educate the students in the relevant areas of condensed matter physics and help them develop proper written and oral communication skills to facilitate the dissemination of their research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.