Abstract
One-dimensional topological superconductors are known to host Majorana zero modes at domain walls terminating the topological phase. Their non-Abelian nature allows for processing quantum information by braiding operations that are insensitive to local perturbations, making Majorana zero modes a promising platform for topological quantum computation. Motivated by the ultimate goal of executing quantum-information processing on a finite time scale, we study domain walls moving at a constant velocity. We exploit an effective Lorentz invariance of the Hamiltonian to obtain an exact solution of the associated quasiparticle spectrum and wave functions for arbitrary velocities. Essential features of the solution have a natural interpretation in terms of the familiar relativistic effects of Lorentz contraction and time dilation. We find that the Majorana zero modes remain stable as long as the domain wall moves at subluminal velocities with respect to the effective speed of light of the system. However, the Majorana bound state dissolves into a continuous quasiparticle spectrum after the domain wall propagates at luminal or even superluminal velocities. This relativistic catastrophe implies that there is an upper limit for possible braiding frequencies even in a perfectly clean system with an arbitrarily large topological gap. We also exploit our exact solution to consider domain walls moving past static impurities present in the system.
2 More- Received 20 May 2013
DOI:https://doi.org/10.1103/PhysRevX.3.041017
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Published by the American Physical Society
Popular Summary
The idea of Majorana fermions, particles that are their own antiparticles, first appeared within high-energy relativistic particle physics. For over 70 years, these exotic particles remained in the realm of theory, until it was understood that they were realizable in solid-state devices.
Majorana fermions in solid-state systems form as trapped states and lack the relativistic dynamics of free Majorana particles. But their exotic nature is on full display: Exchanging (or “braiding”) two Majorana fermions can change their quantum state in a robust way, independent of the details of the exchange. This feature makes them a promising candidate for qubits in fault-tolerant quantum computing. Fast braiding, however, may trigger competing processes in the solid-state systems that could destroy quantum coherence, particularly when impurities are present. How fast can we manipulate Majorana states without destroying their quantum coherence? In this paper, we show that there exists an effective speed of light that acts as an absolute speed limit for braiding Majorana fermions.
To show this, we must consider Majorana quasiparticles moving at arbitrary velocities. To our delight, we have found an elegant description of moving Majorana bound states in solid-state systems in the form of a variation of the Dirac equation of relativistic electrons. This description reconnects solid-state Majorana states to their relativistic cousins. With this insight, we have shown that the analogue of the speed of light in this solid-state system is the electronic spin-orbit coupling, which therefore sets a speed limit for braiding operations.
Looking beyond the context of topological quantum computing, our insights and analysis can also be viewed as a theoretical basis for creating realizations of extreme relativistic effects in solid-state labs.