Heat Transport via Low-Dimensional Systems with Broken Time-Reversal Symmetry

We consider heat transport via systems with broken time-reversal symmetry. We apply magnetic fields to the one-dimensional charged particle systems with transverse motions. The standard momentum conservation is not satisfied. To focus on this effect clearly, we introduce a solvable model. We exactly demonstrate that the anomalous transport with a new exponent can appear. We numerically show the violation of the standard relation between the power-law decay in the equilibrium correlation and the diverging exponent of the thermal conductivity in the open system.

Figure 1

Schematic picture of the periodic boundary condition. The $x$ components of variables are shown.

Figure 2

Numerical calculation of the space-time correlation $C_{\epsilon \epsilon }(i,t)=⟨\delta {\epsilon }_{i+1}(t){\epsilon }_1{⟩}_{\mathrm{eq}}$ for the dynamics Eqs. (8)–(10). The fifth-order Runge-Kutta algorithm with $dt=0.001$ is used for the deterministic dynamics, and $10^9$ initial states were taken from the canonical distribution. Parameters: $B=1$, $N=512$, $T=1$, and $\gamma =0.1$. One can clearly see the absence of sound waves in case (I), while case (II) has finite ballistic peaks.

Figure 3

Numerical check of the exponent $\beta$. Parameters: $N=2048$, $T=1$, $\gamma =0.1$, and $B=1$ for cases (I) and (II). Numerical procedure is the same as in Fig. 1.

Figure 4

Numerical calculation of the system-size dependence of the thermal conductivity. The fifth-order Runge-Kutta algorithm with $dt=0.001$ is used and the steady state was checked from the uniformity of local energy current. Parameters: $(T_1,T_N)=(2,1)$, $\gamma =0.1$, ${\gamma }^{\prime }=1$, and $B=1$ for cases (I) and (II). The exponent $\alpha$ in case (I) is different from known exponents. The error bar of the exponent for case (I) is estimated using Gnuplot for the range $N=[512,8192]$.