A recent research activity in theoretical condensed matter physics concerns the topological states of matter. These are many body states of electrons giving rise to fascinating quantum macroscopic phenomena. A new kind of order appears that is called topological order and measures the influence of the topology of space on the collective behavior of electrons. The topological states are gapped in the bulk and possess massless boundary excitations. A classification of topological phases in ten universality classes has been achieved for free electron (band) systems in any space dimensions. This follows from the analysis of general free-fermion quadratic Hamiltonians and their symmetry under time-reversal and charge-conjugation. Each class is characterized by a topological invariant integer number. The main topic of this thesis is to extend this characterization to topological states made by interacting electrons. To describe also interacting system, we used the methods of effective field theory. This approach involves topological gauge theories that do not have a local dynamics but reproduce the global properties of the ground state. Furthermore, when defined on manifolds with boundaries, these theories involve massless bosons living at the edge whose dynamics is described by a conformal field theory (CFT). By means of these methods, in this thesis we discussed the effective action of both quantum Hall states and three-dimensional topological insulators. The aim of this thesis is to characterize these topological phases by looking at the properties of their boundary conformal invariant theories. In many cases, this approach allows to discuss universal features of the bulk that are robust and independent of microscopic details. We applied these methods to investigate the universality of bulk responses in the quantum Hall states. In particular, our analysis of the edge theory clarified the universality of the response involving the electron ‘intrinsic orbital spin’ s that parameterizes the Hall viscosity. We first built the theory of edge excitations for general integer Hall effect, by taking a straightforward limit of the microscopic states near the edge. We discussed the edge spectrum and showed that the orbital spin causes a shift in the dispersion relation of excitations. For particular boundary conditions, we found that this shift entails non-vanishing ground-state values of the edge charge and conformal spin. We explained that these Casimir-like effects at the edge are indeed universal and robust. We also showed how to generalize these results to fractional Hall states by using the symmetry of incompressible Hall fluids. Finally, we briefly discussed the possibility of measuring these edge effects by a tunneling experiment in the Coulomb blockade regime and by quadrupole deformations of the droplet of Hall fluid. Recently, interacting three-dimensional topological insulators were also introduced and theoretically analyzed, showing that they possess fractional charge and vortex excitations. In this thesis, we introduced the effective field theory description of interacting topological insulators: namely the BF topological gauge theory. We thus studied the corresponding bosonic surface theory and the dynamics it can support. We proposed a non-trivial conformal dynamics for the surface degrees of freedom. This amounts to a non-local, scale invariant version of the Abelian gauge theory in (2+1) dimensions, also called loop model. This theory has appeared in a number of recent research topics such as the duality transformations in (2 + 1) dimensions. Indeed, we showed that the loop model provides a neat example of a calculable self-dual massless theory. By means of the qualitative determination of the phase diagram using energy-entropy Peierls estimates, we found that the loop model possesses a critical line. Its solitonic excitations cor- respond to order-disorder fields with fermionic or anyonic statistical phases. These features remind of the compactified boson theory in (1 + 1) dimensions and makes the loop model a good candidate for bosonization of free and interacting fermions in (2+1) dimensions. We thus quantized the loop model and obtained the spectrum of the excitations at the surface of three-dimensional topological insulators. In order to solve the issue of non-locality, we reformulated the model as a higher dimensional local theory with excitations in (2 + 1) and in (3 + 1) dimensions. We considered two different boundary spatial geometries, the torus T 2 and the sphere S 2 , and we evaluated the partition function of the theory by considering both the solitonic and oscillating spectra. In the sphere case, the geometry is conformally flat and the Hamiltonian maps into the dilatation operator. Therefore, the solitonic energies determine the spectrum of conformal dimensions of the fields in the theory. Our expression of the partition function in this geometry explicitly confirmed the conformal invariance and self-duality of the loop model. Finally, we also verified that the spectrum of conformal fields reproduces the expected fractional statistics of bulk excitations.

Conformal symmetry in topological states of matter / Lorenzo Maffi. - (2021).

Conformal symmetry in topological states of matter

Lorenzo Maffi
2021

Abstract

A recent research activity in theoretical condensed matter physics concerns the topological states of matter. These are many body states of electrons giving rise to fascinating quantum macroscopic phenomena. A new kind of order appears that is called topological order and measures the influence of the topology of space on the collective behavior of electrons. The topological states are gapped in the bulk and possess massless boundary excitations. A classification of topological phases in ten universality classes has been achieved for free electron (band) systems in any space dimensions. This follows from the analysis of general free-fermion quadratic Hamiltonians and their symmetry under time-reversal and charge-conjugation. Each class is characterized by a topological invariant integer number. The main topic of this thesis is to extend this characterization to topological states made by interacting electrons. To describe also interacting system, we used the methods of effective field theory. This approach involves topological gauge theories that do not have a local dynamics but reproduce the global properties of the ground state. Furthermore, when defined on manifolds with boundaries, these theories involve massless bosons living at the edge whose dynamics is described by a conformal field theory (CFT). By means of these methods, in this thesis we discussed the effective action of both quantum Hall states and three-dimensional topological insulators. The aim of this thesis is to characterize these topological phases by looking at the properties of their boundary conformal invariant theories. In many cases, this approach allows to discuss universal features of the bulk that are robust and independent of microscopic details. We applied these methods to investigate the universality of bulk responses in the quantum Hall states. In particular, our analysis of the edge theory clarified the universality of the response involving the electron ‘intrinsic orbital spin’ s that parameterizes the Hall viscosity. We first built the theory of edge excitations for general integer Hall effect, by taking a straightforward limit of the microscopic states near the edge. We discussed the edge spectrum and showed that the orbital spin causes a shift in the dispersion relation of excitations. For particular boundary conditions, we found that this shift entails non-vanishing ground-state values of the edge charge and conformal spin. We explained that these Casimir-like effects at the edge are indeed universal and robust. We also showed how to generalize these results to fractional Hall states by using the symmetry of incompressible Hall fluids. Finally, we briefly discussed the possibility of measuring these edge effects by a tunneling experiment in the Coulomb blockade regime and by quadrupole deformations of the droplet of Hall fluid. Recently, interacting three-dimensional topological insulators were also introduced and theoretically analyzed, showing that they possess fractional charge and vortex excitations. In this thesis, we introduced the effective field theory description of interacting topological insulators: namely the BF topological gauge theory. We thus studied the corresponding bosonic surface theory and the dynamics it can support. We proposed a non-trivial conformal dynamics for the surface degrees of freedom. This amounts to a non-local, scale invariant version of the Abelian gauge theory in (2+1) dimensions, also called loop model. This theory has appeared in a number of recent research topics such as the duality transformations in (2 + 1) dimensions. Indeed, we showed that the loop model provides a neat example of a calculable self-dual massless theory. By means of the qualitative determination of the phase diagram using energy-entropy Peierls estimates, we found that the loop model possesses a critical line. Its solitonic excitations cor- respond to order-disorder fields with fermionic or anyonic statistical phases. These features remind of the compactified boson theory in (1 + 1) dimensions and makes the loop model a good candidate for bosonization of free and interacting fermions in (2+1) dimensions. We thus quantized the loop model and obtained the spectrum of the excitations at the surface of three-dimensional topological insulators. In order to solve the issue of non-locality, we reformulated the model as a higher dimensional local theory with excitations in (2 + 1) and in (3 + 1) dimensions. We considered two different boundary spatial geometries, the torus T 2 and the sphere S 2 , and we evaluated the partition function of the theory by considering both the solitonic and oscillating spectra. In the sphere case, the geometry is conformally flat and the Hamiltonian maps into the dilatation operator. Therefore, the solitonic energies determine the spectrum of conformal dimensions of the fields in the theory. Our expression of the partition function in this geometry explicitly confirmed the conformal invariance and self-duality of the loop model. Finally, we also verified that the spectrum of conformal fields reproduces the expected fractional statistics of bulk excitations.
2021
Andrea Cappelli
ITALIA
Lorenzo Maffi
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1245203
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