Research project Locality vs. topology in quantum matter

This project brings together several frontiers of basic science - from materials science and quantum optics to the theory of topological phases and modern mathematics - with a potential to provide a basis of future technology.

Spectacular ongoing developments in fabricating atomic precision multilayer transition metal
oxide structures as well as newly-devised optical lattices loaded with cold atoms provide
intriguing new platforms for the study and design of strongly correlated phases of matter. At
the same time, the recent discovery of topological insulators is transforming the way we think
of — and search for — collective quantum phenomena. In this theory proposal, I specifically
target two particularly novel setups, namely (i) thin films of geometrically frustrated
materials featuring strong Coulomb interactions and spin-orbit coupling, notably including
the pyrochlore iridates, A2Ir2O7, and (ii) open and driven cold atom systems featuring a subtle
interplay between Hamiltonian and Liouvillian dynamics. The synergy justifying the parallel
search for topological phases in these seemingly disparate settings stems from the fact that
they pose the same underlying — and largely unresolved — challenge, namely to understand
the interplay between band structure topology and local properties. These include the
combined effects of interactions, disorder, dislocations, and the underlying lattice leading to
local constraints and Berry curvature fluctuations, as well as limits imposed by global
topology on the locality in terms of Wannier functions and tensor networks. An improved
understanding of these matters will greatly facilitate the design and control of exotic
topological phases and their concomitant quasiparticles. A salient goal of this proposal is to
identify candidate systems harbouring non-Abelian anyons at elevated temperatures making
use of the unprecedented control and tunability offered by (i) and (ii). This brings together
several frontiers of basic science, while at the same time having the potential to provide the
basis of future technological advances.

While the majority of research is tending towards an ever higher degree of specialization, a desirable goal of science is to unify the description of the world around us at vastly different time, length and energy scales — and at levels of complexity ranging from elementary particles to biological systems. Contemporary physics provides a particularly beautiful example of such a unification between seemingly opposite frontiers, namely high energy particle physics on the one end, and condensed matter physics concerned with the emergent low energy collective behavior of ~10^24 constituents at the other end. Most saliently, it has been realized that many elusive particles hypothesized in the context of high-energy physics, such as Weyl and Majorana fermions, can emerge in the form of quasi-particles in the condensed matter setting. In fact, it turns out that condensed matter quasi-particles can be even more enigmatic as in the case of the fractional quantum Hall states where they are “anyons” possessing charge and statistics fundamentally different from what is expected given the constituent particles. Of special interest are non-Abelian anyons, whose hallmark is that a quantum state involving them is degenerate and can turn into an orthogonal state by adiabatically braiding quasi-particles. This has opened the prospect of storing quantum information in the global properties of these states, and has inspired ideas of using them as building blocks for quantum computers immune to any local disturbance, and thereby to decoherence which are the major obstacles to building large quantum computers. Still, no true “table-top” realizations of the needed states exist, and practically useful “topological quantum computation” remains a dream. Scrutinizing the fine-print it becomes clear that the extreme conditions — temperature of less than 1 Kelvin and magnetic fields of more than 10 Tesla, etc. — needed to realize fractional quantum Hall states in conventional semiconductor heterostructures will almost certainly exclude them from any practical use in computational devices.
However, the past years have witnessed a true paradigm shift in condensed matter physics whereby theory has taken the driver’s seat, feeding experimentalists with novel ideas for realizing topological states in a rich variety of systems. At the same time, the experimental technology has developed immensely in fields ranging from oxide interfaces to shaken optical lattices leading to an unparalleled experimental development. Particularly exciting are new ideas on how to engineer lattice analogues of fractional quantum Hall states, so-called fractional Chern insulators. These phases do not require an external magnetic field and may potentially persist — and harbor non-Abelian anyons — even at room temperature. They however pose qualitatively new theoretical (and practical) challenges including understanding the effects of Berry curvature fluctuations and various lattice specific competing instabilities, but also facilitate exciting new developments due to the possibility of having higher Chern numbers. During the past two years, the first integer (non-interacting) Chern insulators were experimentally realized without the need of magnetic fields in systems as diverse as magnetic topological insulators and optical lattices. The pursuit to find the more exotic fractional Chern insulators is ongoing, and is of high relevance to the present project.

Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems - Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis) appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.

Exceptional Spin Liquids from Couplings to the Environment - We establish the appearance of a qualitatively new type of spin liquid with emergent exceptional points when coupling to the environment. We consider an open system of the Kitaev honeycomb model generically coupled to an external environment. In extended parameter regimes, the Dirac points of the emergent Majorana fermions from the original model are split into exceptional points with Fermi arcs connecting them. In glaring contrast to the original gapless phase of the honeycomb model that requires time-reversal symmetry, this new phase is stable against all perturbations. The system also displays a large sensitivity to boundary conditions resulting from the non-Hermitian skin effect with telltale experimental consequences. Our results point to the emergence of new classes of spin liquids in open systems that might be generically realized due to unavoidable couplings with the environment.

Non-Hermitian Topological Sensors - We introduce and study a novel class of sensors whose sensitivity grows exponentially with the size of the device. Remarkably, this drastic enhancement does not rely on any fine-tuning, but is found to be a stable phenomenon immune to local perturbations. Specifically, the physical mechanism behind this striking phenomenon is intimately connected to the anomalous sensitivity to boundary conditions observed in non-Hermitian topological systems. We outline concrete platforms for the practical implementation of these non-Hermitian topological sensors ranging from classical metamaterials to synthetic quantum materials.

Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States - We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

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