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Transport through a double barrier for interacting quasi one-dimensional electrons in a Quantum Wire in the presence of a transverse magnetic field

We discuss the Luttinger Liquid behaviour of a semiconducting Quantum Wire. We show that the measured value of the bulk critical exponent, $α_{bulk}$, for the tunneling density of states can be easily calculated. Then, the problem of the transport through a Quantum Dot formed by two Quantum Point Contacts along the Quantum Wire, weakly coupled to spinless Tomonaga-Luttinger liquids is studied, including the action of a strong transverse magnetic field $B$. The known magnetic dependent peaks of the conductance, $G(B)$, in the ballistic regime at a very low temperature, $T$, have to be reflected also in the transport at higher $T$ and in different regimes. The temperature dependence of the maximum $G_{max}$ of the conductance peak, according to the Correlated Sequential Tunneling theory, yields the power law $G_{max}\propto T^{2α_{end}-1}$, with the critical exponent, $α_{end}$, strongly reduced by $B$. This behaviour suggests the use of a similar device as a magnetic field modulated transistor.