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Register a new userZero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization
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We introduce a second-quantized representation of the ring of symmetric functions to further develop a purely second-quantized -- or lattice -- approach to the study of zero modes of frustration free Haldane-pseudo-potential-type Hamiltonians, which in particular stabilize Laughlin ground states. We present three applications of this formalism. We start demonstrating how to systematically construct all zero-modes of Laughlin-type parent Hamiltonians in a framework that is free of first-quantized polynomial wave functions, and show that they are in one-to-one correspondence with dominance patterns. The starting point here is the pseudo-potential Hamiltonian in lattice form, stripped of all information about the analytic structure of Landau levels (dynamical momenta). Secondly, as a by-product, we make contact with the bosonization method, and obtain an alternative proof for the equivalence between bosonic and fermionic Fock spaces. Finally, we explicitly derive the second-quantized version of Reads non-local (string) order parameter for the Laughlin state, extending an earlier description by Stone. Commutation relations between the local quasi-hole operator and the local electron operator are generalized to various geometries.
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We provide a current perspective on the rapidly developing field of Majorana zero modes in solid state systems. We emphasize the theoretical prediction, experimental realization, and potential use of Majorana zero modes in future information processing devices through braiding-based topological quantum computation. Well-separated Majorana zero modes should manifest non-Abelian braiding statistics suitable for unitary gate operations for topological quantum computation. Recent experimental work, following earlier theoretical predictions, has shown specific signatures consistent with the existence of Majorana modes localized at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental findings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic Majorana zero modes in solid state systems. We also discuss fractional quantum Hall systems (the 5/2 state) in this context. We describe proposed schemes for carrying out braiding with Majorana zero modes as well as the necessary steps for implementing topological quantum computation.
Majorana fermions feature non-Abelian exchange statistics and promise fascinating applications in topological quantum computation. Recently, second-order topological superconductors (SOTSs) have been proposed to host Majorana fermions as localized quasiparticles with zero excitation energy, pointing out a new avenue to facilitate topological quantum computation. We provide a minimal model for SOTSs and systematically analyze the features of Majorana zero modes with analytical and numerical methods. We further construct the fundamental fusion principles of zero modes stemming from a single or multiple SOTS islands. Finally, we propose concrete schemes in different setups formed by SOTSs, enabling us to exchange and fuse the zero modes for non-Abelian braiding and holonomic quantum gate operations.
We study the surface states and chiral hinge states of a 3D second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighbouring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the $n = 0$ Landau level falls inside the surface Dirac gap, and not at the gap edge. The $n e 0$ levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels ($pm n$) which are symmetric around zero energy, the $n$ and $-n$ levels become asymmetric in our lattice model -- one of the pair may even be absent from the spectrum, or hybridized with the continuum. In order to resolve the issue, we extend the standard 2D massive Dirac surface theory, by including additional Hamiltonian terms at $mathcal{O} (k^2)$. While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar $mathcal{O}(k^2)$ correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.
We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This non-equilibrium dynamical phenomenon is a form of prethermalization, and occurs because of obstructions to the equilibriation of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a non-perturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes, and on trapped ion chains.
A real-space formulation is given for the recently discussed exciton condensate in a symmetrically biased graphene bilayer. We show that in the continuum limit an oddly-quantized vortex in this condensate binds exactly one zero mode per valley index of the bilayer. In the full lattice model the zero modes are split slightly due to intervalley mixing. We support these results by an exact numerical diagonalization of the lattice Hamiltonian. We also discuss the effect of the zero modes on the charge content of these vortices and deduce some of their interesting properties.
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