Quantum impurity in a Luttinger liquid: Universal conductance with entanglement renormalization

We study numerically the universal conductance of Luttinger-liquid wire with a single impurity via the multiscale entanglement renormalization ansatz (MERA). The scale-invariant MERA provides an efficient way to extract scaling operators and scaling dimensions for both the bulk and the boundary conformal field theories. By utilizing the key relationship between the conductance tensor and ground-state correlation function, the universal conductance can be evaluated within the framework of the boundary MERA. We construct the boundary MERA to compute the correlation functions and scaling dimensions for the Kane-Fisher fixed points by modeling the single impurity as a junction (weak link) of two interacting wires. We show that the universal behavior of the junction can be easily identified within the MERA and argue that the boundary MERA framework has tremendous potential to classify the fixed points in general multiwire junctions.

Figure 1

Ternary MERA with periodic boundary condition for lattice length $N=18$ lie in the $x$ axis. Blue triangles are isometries $w_{\tau }$ and orange squares are disentanglers $u_{\tau }$. ${\mathcal{L}}_{\tau }$ indicates various lattice layers with $\tau =0,1,T$. The gray vertical lines separate three lattice sites with green circles as one foundation block in each lattice layer. Each lattice spacing is renormalized by the corresponding RG transformation, and the length scales of effective lattices are changed along the $y$ axis.

Figure 2

Diagrammatic representation of the constraints for the isometries $w$ and disentanglers $u$ of the ternary MERA.

Figure 3

(a) Four pairs of $(w_{\tau },u_{\tau })$ in buffer layers $(\tau =1,2,3,4)$ and the same copies of the scale-invariant pair $(w_s,u_s)$ in the scale-invariant layers are employed. Along the RG transformation axis, the pseudoscaling dimension ${\Delta }_{\sigma }\left(\tau \right)$ of buffer layers flows into the real scaling dimension ${\Delta }_{\sigma }$ of the scale-invariant RG fixed point. (b) Both ${\Delta }_{\sigma }\left(\tau \right)$ and the scaling dimension of the spin primary field in the CFT are calculated, and the pseudoscaling dimension approaches the exact value as the layer $\tau$ is close to the scale-invariant layers. The exact value of the scaling dimension of the spin primary field from the CFT is $\frac{1}{8}$ for the transverse Ising model [9]. The bound dimension used here is $\chi =8$.

Figure 4

Sketch of the boundary scale-invariant MERA structure for three layers. The central tensors $w_{\tau }^B$ are used to represent the boundary state inside the causal cone (light green shaded area) of the green impurity. ${\tilde{h}}_{\tau ,\tau +1}^{\mu }$ is an effective two-site boundary Hamiltonian which is obtained by inhomogeneous coarse graining, related to the causal cone of $K_{\tau ,\tau +1}^{\mu }$ such as the red shaded area. Here, $w_{\tau }^{\mu },u_{\tau }^{\mu }$, and $V_{\tau }^{\mu }$ are bulk isometries, bulk disentanglers, and boundary truncation tensors, respectively. The effective impurity Hamiltonian ${\tilde{h}}_B$ is constructed from the impurity Hamiltonian $h_B$ and two-site Hamiltonian $h_{0,1}^I$ and $h_{0,1}^{II}$ in the bulk wires. Two quantum wires are connected in the junction, and we also label the site indices for each wire.

Figure 5

(a) Graphic representation of ${\tilde{h}}_B$ including the impurity Hamiltonian $h_B$ and the inhomogeneous coarse graining of the first two-site Hamiltonian $h_{0,1}^{\mu }$ in wire $\mu \in I,II$. (b), (c) Using inhomogeneous coarse graining to obtain the boundary Hamiltonian ${\tilde{h}}_{0,1}^{\mu }$ with a scaling factor $\frac{1}{3}$ for wires $I$ and $II$, respectively.

Figure 6

The current-current correlation function as a function of the distance from the boundary. (a) Universal behavior with the same $A=0.032,\alpha \approx 2$ for $g=1.5$ with various $t=0.5,0.8,0.9$ is observed. (b) Nonuniversal behavior with distinct $A$ and $\alpha >2$ for $g\approx 0.8$ and $t=0.9,0.5,0.3,0.1$. The calculations are carried out using $\chi =12$ and ${\chi }^B=24$.

Figure 7

(a) The spin-spin correlation function in Eq. (24) as a function of the distance between two spin operators for a bulk wire with $g\approx 0.79,0.9,1,1.24,1.5$. (b) The exponent $\beta$ of the spin-spin correlation function as a function of $g$. The calculations are carried out using $\chi =16$.

Figure 8

The spin-spin correlation function in Eq. (24) as a function of the distance between two spin operators crossing the junction for various $g$, and we chose two spin operators that have the same distance far away from the boundary. (a) The universal behavior at the perfect transmission fixed point for $g=1.5$ with $t=0.3,0.5,0.9$, the exact solution [47, 48] describes the character of the bulk. (b) The nonuniversal behavior at the total reflection fixed point for $g\approx 0.8$ with $t=0.1,0.5,0.9$, and the lines are the fitting results of $C_{S^{+}{S}^{-}}\left(r\right)\approx a{r}^{-2\beta }$.

Figure 9

The lowest few nonvanishing scaling dimensions of primary fields of the bulk and boundary as a function of Luttinger parameters $g$. The calculations are carried out using $\chi =12$ and ${\chi }^B=24$.

Figure 10

Ternary MERA for three layers with an impurity at the junction described by $h_B$. The shaded green is the causal cone for the junction. The two-site Hamiltonian $h_{i,i+1}^{\mu }$ for layer $\tau =0$ (pink bars) is described in Eq. (A3). We also label site indices in each layer running from zero to infinity for both wires. The light blue triangles and the light yellow squares represent bulk isometries and bulk disentanglers, respectively.

Figure 11

(a) Graphic representation of Eq. (A1). An isometry is decomposed into two tensors $V_{\tau }^{\mu }$ and $B_{\tau }^{\mu }$ linked in the truncated bond dimension ${\chi }^B$. (b) The isometric conditions of $V_{\tau }^{\mu }$ for $\mu \in I,II$. The lines at the right-hand side of equal signs are identity matrices. (c) Construct a central tensor $w_{\tau }^B$ from three tensors $B_{\tau }^I,B_{\tau }^{II}$, and $u_{\tau }^B$.

Figure 12

Graphic representation of ${\tilde{h}}_{\tau ,\tau +1}^{\mu }$ (a) for wire $I$ and (b) for wire $II$.

Figure 13

The red bars represent the effective boundary Hamiltonian consisting of ${\tilde{h}}_{\tau ,\tau +1}$ and ${\tilde{h}}_B$. $w_{\tau }^B$ are the central tensors indicated by blue triangles. ${\rho }_{\tau }$ and ${\tilde{h}}_{\tau }^{\mathrm{c}}$ are central density matrices and central Hamiltonian, respectively.

Figure 14

(a) The average central ascending process for ${\tilde{h}}_{\tau +1}^{\mathrm{c}}$. (b) ${\tilde{h}}_1^{\mathrm{c}}$ is carried out using the ascending operation of the effective Hamiltonian ${\tilde{h}}_B$, described in Fig. 5. (c) The descending superoperator $D^{\mathrm{c}}$ composes of $w_{\tau }^B$ and $w_{\tau }^{B†}$, and it acts on ${\rho }_{\tau +1}$ to obtain the central density matrix ${\rho }_{\tau }$.

Figure 15

Graphic representation of Eqs. (B1) and (B2).

Figure 16

The missing triangle corresponds to $w_{\tau }^B$, and its corresponding environment $Y_{\tau }$ is defined by the sum of five tensor networks with certain weights.

Figure 17

The missing triangle corresponds to $w_0^B$, and its environment $Y_0$ includes three tensor networks, where the effective Hamiltonian ${\tilde{h}}_B$ is defined in Fig. 5.

Figure 18

Both the missing and the blue triangles represent the scale-invariant central tensor $w_s^B$. The environment $Y_s$ is a sum of tensor networks composed of the scale-invariant density matrix ${\rho }_s$ and the effective scale-invariant Hamiltonian ${\tilde{h}}_s^{\nu }$ with $\nu \in I,II,c$ defined in Eqs. (18) and (B5).