Hall effect in strongly correlated low-dimensional systems

We investigate the Hall effect in a quasi-one-dimensional system made of weakly coupled Luttinger liquids at half filling. Using a memory function approach, we compute the Hall coefficient as a function of temperature and frequency in the presence of umklapp scattering. We find a power-law correction to the free-fermion value (band value), with an exponent depending on the Luttinger parameter $K_{\rho }$. At sufficiently high temperature or frequency, the Hall coefficient approaches the band value.

Figure 1

Schematics of the model. The chains and the current $I$ go along the $x$ axis, the magnetic field $H$ is applied along the $z$ axis, and the Hall voltage is measured along the $y$ axis.

Figure 2

Correction of the high-temperature/high-frequency Hall coefficient $R_H$ by the umklapp scattering in weakly coupled Luttinger liquids. $R_H^0$ is the value of the Hall coefficient in the absence of umklapp scattering, Eq. (13), and $W$ is the electron bandwidth. Our approach breaks down below some crossover scale (dashed line, see text). In this figure, we have assumed that $A$ in Eq. (19) is negative.

Figure 3

Example of a diagram appearing in Eq. (17) at first order in $t_{\perp }$ and for $g_2=0$. The full (dashed) lines correspond to free right- (left-) moving fermions, $j$ is the chain index, and the arrows represent up and down spins. The magnetic field increases the momentum of the electron by $\delta k=eH{a}_y$.