Numerical study of transport through a single impurity in a spinful Tomonaga–Luttinger liquid

The single impurity problem in a spinful Tomonaga–Luttinger liquid is studied numerically using path-integral Monte Carlo methods. The advantage of our approach is that the system allows for extensive analyses of charge and spin conductance in the nonperturbative regime. By closely examining the behavior of conductances at low temperatures, in the presence of a finite backward scattering barrier due to the impurity, we identified four distinct phases characterized by either perfect transmission or reflection of charge and spin channels. Our phase diagram for an intermediate scattering strength is consistent with the standard perturbative renormalization group (RG) analysis in the limit of weak and strong backward scatterings, in the sense that all our phase boundaries interpolate the two limiting cases. Further investigations show, however, that precise location and form of our phase boundaries are not trivially explained by the standard RG analysis, e.g., some part of the phase diagram looks much similar to the weak backscattering limit, whereas some other part is clearly derived from the opposite limit. In order to give a more intuitive interpretation of such behaviors, we also reconsidered our impurity problem from the viewpoint of a quantum Brownian motion picture.

Figure 1

Three-dimensional phase diagram in the $(K_{\rho },K_{\sigma },v)$ space. The phase boundaries are analytically obtained in the following two limits: for a weak impurity $(v\to 0)$, the phase boundaries are three straight lines $K_{\rho }+K_{\sigma }=2$, $K_{\rho }=1∕2$, and $K_{\sigma }=1∕2$; for a strong impurity $(v\to \infty )$, the phase boundaries are a hyperbola $K_{\rho }^{-1}+K_{\sigma }^{-1}=2$ and two straight lines $K_{\rho }=2$ and $K_{\sigma }=2$. The dotted lines connect the three points in the $v=0$ and $v=\infty$ planes, at which the phase boundaries coincide in the both planes.

Figure 2

Cluster updates of field variables ${\varphi }_{\rho j}$ and ${\varphi }_{\sigma j}$ in a ${\varphi }_{\rho }-{\varphi }_{\sigma }$ plane, where ${\phi }_{\nu j}\equiv {\varphi }_{\nu j}-{\varphi }_{\nu }^{\text{mirror}}$. The empty circles represent the potential minima, and the dashed lines represent the reference fields ${\varphi }_{\rho }^{\text{mirror}}$ and ${\varphi }_{\sigma }^{\text{mirror}}$. Only the field variables at the $j\text{th}$ time step in a cluster are shown. (a) During one period of a double-field cluster update, a point $({\phi }_{\rho j},{\phi }_{\sigma j})$ is subject to mirror reflection twice, i.e., once with respect to ${\varphi }_{\rho j}={\varphi }_{\rho }^{\text{mirror}}$, and subsequently to ${\varphi }_{\sigma j}={\varphi }_{\sigma }^{\text{mirror}}$. [The whole process is $({\phi }_{\rho j},{\phi }_{\sigma j})\to (-{\phi }_{\rho j},{\phi }_{\sigma j})\to (-{\phi }_{\rho j},-{\phi }_{\sigma j})$.] (b) As for a charge-field cluster update, ${\phi }_{\nu j}$ is subject to mirror reflection with respect to ${\varphi }_{\rho }={\varphi }_{\rho }^{\text{mirror}}$, i.e., $({\phi }_{\rho j},{\phi }_{\rho j})\to (-{\phi }_{\rho j},{\phi }_{\rho j})$. Similary, ${\phi }_{\sigma j}$ in a spin-field cluster is updated as $({\phi }_{\rho j},{\phi }_{\sigma j})\to ({\phi }_{\rho j},-{\phi }_{\sigma j})$.

Figure 3

(Color online) Charge conductance $G_{\rho }\left(i{\omega }_n\right)$ (upper) and spin conductance $G_{\sigma }\left(i{\omega }_n\right)$ (lower) for different values of $K_{\rho }$, plotted as a function of ${\omega }_n$. $K_{\sigma }$ is fixed at $K_{\sigma }=1.0$. Symmetric coupling case $(K_{\rho }\simeq K_{\sigma })$. $N=50$, 100, 200, and 400, and only the first ten points are shown for each Trotter number $N$. From top to bottom, the values of $K_{\rho }$ are 1.2, 1.1, 1.05, 1.025, 1.0, 0.975, 0.95, 0.9, and 0.8. Conductance curves (of both charge and spin) show an upward bend with decreasing ${\omega }_n$ when $K_{\rho }>1.025$, whereas they are bent downward when $K_{\rho }<0.975$.

Figure 4

(Color online) dc conductances of charge (upper) and spin (lower) channels for different values of $K_{\rho }$. $K_{\sigma }$ is fixed at $K_{\sigma }=1.0$. Symmetric coupling case $(K_{\rho }\simeq K_{\sigma })$. Conductances are plotted as a function of the inverse temperature $N$. As expected from Fig. 3, both charge and spin conductances increase with decreasing temperature (increasing $N$) for $K_{\rho }>1.025$, whereas they decrease for $K_{\rho }<0.975$.

Figure 5

(Color online) Charge conductance $G_{\rho }\left(i{\omega }_n\right)$ (upper) and spin conductance $G_{\sigma }\left(i{\omega }_n\right)$ (lower) for different values of $K_{\rho }$, plotted as a function of ${\omega }_n$. $K_{\sigma }$ is fixed at $K_{\sigma }=1.8$. Asymmetric coupling case $(K_{\rho }⪡K_{\sigma })$. $N=50$, 100, 200, and 400, and only the first ten points are shown for each Trotter number $N$. From top to bottom, the values of $K_{\rho }$ in the upper panel are 0.6, 0.55, 0.525, 0.5, 0.475, 0.45, and 0.4, while those in the lower panel are 0.5, 0.45, 0.425, 0.4, 0.375, 0.35, and 0.3. In this parameter regime, charge and spin channels behave quite differently.

Figure 6

(Color online) dc conductances of charge (upper) and spin (lower) channels for different values of $K_{\rho }$. $K_{\sigma }$ is fixed at $K_{\sigma }=1.8$. Asymmetric coupling case $(K_{\rho }⪡K_{\sigma })$. Conductances are plotted as a function of the inverse temperature $N$. Upper: the temperature dependence for the charge channel shows monotonic increase $(K_{\rho }>0.5)$ or decrease $(K_{\rho }<0.5)$ with decreasing temperature. Lower: the spin conductance behaves nonmonotonicly for $K_{\rho }=0.35$, 0.375, and 0.4, e.g., for $K_{\rho }=0.375$, the conductance increases from $N=50$ to $N=100$, whereas it decreases from $N=200$ toward $N=400$.

Figure 7

(Color online) Power-law decay of the dc conductance at lower temperatures. Charge and spin conductances at the point $(K_{\rho },K_{\sigma })=(1∕\sqrt{7},1)$ are plotted as a function of the inverse temperature $N$. Conductance for $K=1∕2$ in the spinless case is also shown. The dashed lines represent asymptotic power-law behaviors of the conductances in the $T=0$ limit: $\propto N^{1-\sqrt{7}}$ for the spinful case, while $\propto N^{-2}$ for the spinless case.

Figure 8

(Color online) Phase diagram obtained from our PIMC simulations for the impurity strength $v=4$. The solid lines stand for the boundaries between the I, III, and IV phases. In order to ease the comaprison with the RG results, we superposed the WBS results (dashed straight line) for the I-III and I-IV boundaries, and the SBS results (dotted hyperbolic curve) for the I-IV and III-IV boundaries, on top of our phase diagram.

Figure 9

Minima and maxima of the two-dimensional potential $v\cos {\varphi }_{\rho }\cos {\varphi }_{\sigma }$. The empty (filled) circles represent the potential minima (maxima). The solid (dotted) arrow stands for an example of tunneling of a massles particle between minima through a saddle point (over a maximum). For strongly anisotropic friction, e.g., $K_{\rho }^{-1}⪢K_{\sigma }^{-1}$, the diagonal process (through a saddle point) is suppressed so that the relevant process becomes the vertical one (over a maximum).