Final Report Summary - FTIBOUNDARY (Boundary theories of fractional topological insulators)
*Summary and Objectives
Topological phases represent a novel form of matter with the characteristic property that low energy modes are forbidden in their bulk but are present in a remarkably robust form at their boundaries. This project aimed at exploring this boundary physics in the case of fractional topological insulators (FTIs, variants of topological phases emerging from strong interactions), including cases involving boundary superconductivity when new exotic particles called Majorana fermions can emerge.
Specific project objectives included: (I) studying the boundary physics of two-dimensional (2D) FTIs through assessing their robustness to certain physically relevant perturbations and obtaining observable predictions in linear and point contact geometries; (II) formulating the surface theory of 3D FTIs using gauge theories analogous to those in high energy physics and using this formulation to derive observable signatures of the FTI phase; (III) studying strongly interacting forms of Majorana fermions that can arise on the boundaries of FTIs.
*Description of Work, Results, and Impact
Objective (I) was addressed in the interim period between submitting the proposal and the start of the project. One of the central questions was how the boundary modes respond if one breaks time-reversal invariance, the defining symmetry of FTIs. We found that the edge modes are much more robust than anticipated: unlike 'ordinary' topological insulators, where such perturbations always eliminate the edge modes (by making low energy propagation forbidden), in FTIs this can happen only if the intermode interaction or the perturbation itself is stronger than a critical value. In addition, we have calculated the temperature/voltage scaling of the conductance both in linear and point contact geometries, thereby delivering key observable predictions that can be used in future experiments for identifying the FTI phase.
Just before the start of the project, significant progress was made that fundamentally reshaped Objective (III): we discovered the topological Kondo effect, a new form of strong interaction physics that can emerge in Majorana fermion devices very close to those in ongoing experiments, without requiring FTIs. The exploration of this physics formed the central goal of the first year of the project.
The topological Kondo effect arises from coupling leads of conduction electrons to Majorana fermions hosted on a mesoscopic superconducting island. The adjective topological indicates that unlike the usual Kondo effect, which arises from coupling conduction electrons to a spin qubit, the coupling here is to a topological qubit formed of the Majorana fermions. The Kondo effect is one of the central paradigms in strongly interacting physics. Similarly, the topological Kondo effect is the first paradigm for Majorana fermion induced strong interaction phenomena. As the presence of a topological qubit in the system indicates, the potential impact of these studies however go beyond mere strong interactions: a topological Kondo device could be used to test the health of these qubits in prospective quantum computer elements using simple conduction measurements.
The first work after the start of the project explored how the topological Kondo effect depends on the number of Majorana fermions coupled to leads and on conduction electron interactions. The main finding was that even with noninteracting leads, the topological Kondo device realises a series of instances of exotic strongly interacting phases as the number of Majorana-lead couplings increases. These phases are robust against moderate conduction electron interactions. Tuning these interactions or the Majorana lead couplings across a certain point, a transition between the Kondo phases and more familiar Luttinger liquid phases can be achieved.
The second work, a collaboration with researchers at the University of Oxford, aimed at exploring the topological Kondo effect using the numerical renormalisation group - a numerically exact method tailor made for the Kondo problem. The main objectives this work achieved included (i) going beyond regimes of the temperature and voltage which were accessible to the previous two analytical studies and (ii) studying the effects of the small, nonlocal, Majorana-Majorana couplings neglected in our initial works. In particular, the simulations provided the full crossover curve of the differential conductance of the experimentally most relevant three lead topological Kondo device. Having access to this crossover curve may be crucial once quantitative comparisons to experiments can be made.
Progress has also been made in a direction related to Objective (II) by exploring a connection between topological insulators and gauge theories. The subject of this study was a 2+1 dimensional SU(2) Yang-Mills-Higgs theory in the presence of a fixed background SU(2) magnetic field. The gauge dynamics in this case gives rise to a spatial lattice structure. It was shown that coupling massive fermions to this lattice background realises a phase with topological insulator physics, providing the first example of how topological insulating phases currently studied in condensed matter can emerge in dynamical gauge theory models familiar in high energy physics. Studying further instances (e.g. 3D topological insulators, topological superconductors) and analogues of strongly interacting phases such as FTIs form natural further directions of research. This was a collaborative work with researchers at DAMTP, Cambridge.
Topological phases represent a novel form of matter with the characteristic property that low energy modes are forbidden in their bulk but are present in a remarkably robust form at their boundaries. This project aimed at exploring this boundary physics in the case of fractional topological insulators (FTIs, variants of topological phases emerging from strong interactions), including cases involving boundary superconductivity when new exotic particles called Majorana fermions can emerge.
Specific project objectives included: (I) studying the boundary physics of two-dimensional (2D) FTIs through assessing their robustness to certain physically relevant perturbations and obtaining observable predictions in linear and point contact geometries; (II) formulating the surface theory of 3D FTIs using gauge theories analogous to those in high energy physics and using this formulation to derive observable signatures of the FTI phase; (III) studying strongly interacting forms of Majorana fermions that can arise on the boundaries of FTIs.
*Description of Work, Results, and Impact
Objective (I) was addressed in the interim period between submitting the proposal and the start of the project. One of the central questions was how the boundary modes respond if one breaks time-reversal invariance, the defining symmetry of FTIs. We found that the edge modes are much more robust than anticipated: unlike 'ordinary' topological insulators, where such perturbations always eliminate the edge modes (by making low energy propagation forbidden), in FTIs this can happen only if the intermode interaction or the perturbation itself is stronger than a critical value. In addition, we have calculated the temperature/voltage scaling of the conductance both in linear and point contact geometries, thereby delivering key observable predictions that can be used in future experiments for identifying the FTI phase.
Just before the start of the project, significant progress was made that fundamentally reshaped Objective (III): we discovered the topological Kondo effect, a new form of strong interaction physics that can emerge in Majorana fermion devices very close to those in ongoing experiments, without requiring FTIs. The exploration of this physics formed the central goal of the first year of the project.
The topological Kondo effect arises from coupling leads of conduction electrons to Majorana fermions hosted on a mesoscopic superconducting island. The adjective topological indicates that unlike the usual Kondo effect, which arises from coupling conduction electrons to a spin qubit, the coupling here is to a topological qubit formed of the Majorana fermions. The Kondo effect is one of the central paradigms in strongly interacting physics. Similarly, the topological Kondo effect is the first paradigm for Majorana fermion induced strong interaction phenomena. As the presence of a topological qubit in the system indicates, the potential impact of these studies however go beyond mere strong interactions: a topological Kondo device could be used to test the health of these qubits in prospective quantum computer elements using simple conduction measurements.
The first work after the start of the project explored how the topological Kondo effect depends on the number of Majorana fermions coupled to leads and on conduction electron interactions. The main finding was that even with noninteracting leads, the topological Kondo device realises a series of instances of exotic strongly interacting phases as the number of Majorana-lead couplings increases. These phases are robust against moderate conduction electron interactions. Tuning these interactions or the Majorana lead couplings across a certain point, a transition between the Kondo phases and more familiar Luttinger liquid phases can be achieved.
The second work, a collaboration with researchers at the University of Oxford, aimed at exploring the topological Kondo effect using the numerical renormalisation group - a numerically exact method tailor made for the Kondo problem. The main objectives this work achieved included (i) going beyond regimes of the temperature and voltage which were accessible to the previous two analytical studies and (ii) studying the effects of the small, nonlocal, Majorana-Majorana couplings neglected in our initial works. In particular, the simulations provided the full crossover curve of the differential conductance of the experimentally most relevant three lead topological Kondo device. Having access to this crossover curve may be crucial once quantitative comparisons to experiments can be made.
Progress has also been made in a direction related to Objective (II) by exploring a connection between topological insulators and gauge theories. The subject of this study was a 2+1 dimensional SU(2) Yang-Mills-Higgs theory in the presence of a fixed background SU(2) magnetic field. The gauge dynamics in this case gives rise to a spatial lattice structure. It was shown that coupling massive fermions to this lattice background realises a phase with topological insulator physics, providing the first example of how topological insulating phases currently studied in condensed matter can emerge in dynamical gauge theory models familiar in high energy physics. Studying further instances (e.g. 3D topological insulators, topological superconductors) and analogues of strongly interacting phases such as FTIs form natural further directions of research. This was a collaborative work with researchers at DAMTP, Cambridge.