# Spin Transport in a Quantum Hall Insulator
## 1 Introduction  
The growing interest in spintronics [1, 2] has been incited by the prospect of building fast, energy-efficient devices utilizing the electron spin. In particular, there has been rapid development in magnonics [3], which uses spin waves to transport information [4, 5]. Magnonic systems are free of many disadvantages of electronic systems associated with energy dissipation. To accomplish dissipationless transport, it is possible to create the high-temperature Bose-Einstein condensation of magnons in yttrium-iron-garnet films [6, 7]. An even more innovative idea for the spin transfer is to include vortex-like spin structures (skyrmions) into the spin dynamics. There have been reported experimental studies involving the manipulation of skyrmions, measuring their mass and velocity [8]. Hence, it is expected that dissipative spin transport can be realized in a skyrmion system. This paper investigates other promising candidates for spin transport similar to magnons, the long-lived spin excitations in a Hall insulator.  
In Hall insulators, electron excitations are called magnetoexcitons, analogous to magnetoexcitons in semiconductors [9, 10, 11]. One of the simplest examples is a magnetoexciton formed by an excited electron at the first Landau level and a Fermi hole at a completely filled zero Landau level at the filling factor \(\
u=2\). In the excitation spectrum of a Hall insulator, two types of magnetoexcitons appear--a spin-singlet with the total spin S = 0 and a spin-triplet with the total spin S = 1 [9, 10, 11]. [PERSON]'s theorem states that the spin-singlet exciton's excitation energy is equal to the cyclotron energy [12]. At the same time, the excitation energy of the spin-triplet exciton is lower than the cyclotron energy by the value of Coulomb coupling due to electronic correlations in the electron system [13]. In this case, the electron-hole symmetry is achieved, when the excited electron has the same mass and charge as the effective Fermi hole. Since the local spatial density of electrons does not change, the transfer of spin excitons cannot be attributed to either charge or mass transfer. Thus, it has to be associated only with the energy and spin transfer.  
The spin-triplet exciton, unlike the spin-singlet exciton, is optically inactive (\"dark\" exciton). However, a non-equilibrium ensemble of spin-triplet excitons can be createdby optically exciting conduction-band electrons and valence-band holes to higher energy states [14]. The primary means of altering the spin of an electron system is to flip the spin of a photoexcited hole using the strong spin-orbit interaction in the valence band of GaAs. Consequently, when electrons from the zero Landau level recombine with a photoexcited hole in the valence band, the photoexcited electron system changes its spin. The relaxation of spin-triplet excitons to the ground state is accompanied by simultaneous changes in the orbital and spin quantum numbers of the electron system, along with the emission of a cyclotron photon. However, since such conditions are forbidden in the dipole approximation, it leads to an extremely long (\(\sim\)1 ms) lifetime of triplet excitons at low temperatures. On the other hand, the recombination times of a photoexcited hole are much shorter--approximately 100 ps in the heterostructures under study. Therefore, using photoexcitation, it is possible to achieve an enormous density of non-equilibrium spin excitations with an integer spin in a fermionic system to produce a magnetofermionic condensate. In the presented work, we explore the Bose-Einstein condensation of spin-triplet magnetoexcitons in a \(\
u\) = 2 quantum Hall insulator using photoluminescence (PL) and photo-induced resonant reflection (PRR) techniques.