A minimal model is exactly solved for electron spin transport on a helix. Electron transport is assumed to be supported by well oriented p_z type orbitals on base molecules forming a staircase of definite chirality. In a tight binding interpretation, the spin-orbit coupling (SOC) opens up an effective π_z − π_z coupling via interbase p_x,y − p_z hopping, introducing spin coupled transport. The resulting continuum model spectrum shows two Kramers doublet transport channels with a gap proportional to the SOC. Each doubly degenerate channel satisfies time reversal symmetry; nevertheless, a bias chooses a transport direction and thus selects for spin orientation. The model predicts (i) which spin orientation is selected depending on chirality and bias, (ii) changes in spin preference as a function of input Fermi level and (iii) back-scattering suppression protected by the SO gap. We compute the spin current with a definite helicity and find it to be proportional to the torsion of the chiral structure and the non-adiabatic Aharonov-Anandan phase. To describe room temperature transport, we assume that the total transmission is the result of a product of coherent steps.

The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We can conclusively rule out that the model belongs to the 2D Ising model universality class. The dimensions of the relevant bulk and surface scaling fields are found to be $y_h=1.849\mathrm{}$, $y_t=0.977\mathrm{}$, and $y_{h_s}=0.54\mathrm{}$, in contrast to their Ising values $15/8$, $1$, and $1/2$, respectively.

Renormalization-group theory has stood, for over 40 years, as one of the pillars of modern physics. As such, there should be no remaining doubt regarding its validity. However, finite-size scaling, which derives from it, has long been poorly understood above the upper critical dimension $d_c$ in models with free boundary conditions. In addition to its fundamental significance for scaling theories, the issue is important at a practical level because finite-size, statistical-physics systems with free boundaries and above $d_c$ are experimentally relevant for long-range interactions. Here, we address the roles played by Fourier modes for such systems and show that the current phenomenological picture is not supported for all thermodynamic observables with either free or periodic boundaries. In particular, the expectation that dangerous irrelevant variables cause Gaussian-fixed-point scaling indices to be replaced by Landau mean-field exponents for all Fourier modes is incorrect. Instead, the Gaussian-fixed-point exponents have a direct physical manifestation for some modes above the upper critical dimension.