Open Wilson chains for quantum impurity models: Keeping track of all bath modes

When constructing a Wilson chain to represent a quantum impurity model, the effects of truncated bath modes are neglected. We show that their influence can be kept track of systematically by constructing an “open Wilson chain” in which each site is coupled to a separate effective bath of its own. As a first application, we use the method to cure the so-called mass-flow problem that can arise when using standard Wilson chains to treat impurity models with asymmetric bath spectral functions at finite temperature. We demonstrate this for the strongly sub-Ohmic spin-boson model at quantum criticality where we directly observe the flow towards a Gaussian critical fixed point.

Figure 1

(a) Impurity model. (b) Initialization. (c) Open Wilson chain (OWC). (d) Renormalized Wilson chain (RWC).

Figure 2

DHO susceptibility $\chi \left(T\right)$ as function of temperature, computed by NRG on C1-RWCs (circles) and by VMPS on C2-RWCs (squares), for $\alpha =0.1$, 0.19, 0.199, and 0.1999 (from bottom to top). Solid lines show exact results.

Figure 3

(a) Critical exponent $x$ for the sub-Ohmic SBM, as a function of $s$, computed by VMPS using RWCs of type C1 (circles) and C2 (squares). Examples of $\chi \left(T\right)$ curves used to extract these exponents are shown in (b) for $s=0.3$ and (c) for $s=0.6$. Error bars in (a) are derived by varying the fitting ranges, e.g., as indicated by dark and light shading in (b) and (c).

Figure 4

Energy-level flow diagrams for the sub-Ohmic SBM with $s=0.6$ (left column) and $s=0.4$ (right column), computed by VMPS techniques [5, 19] on C1-RWCs (top row) and C2-RWCs (bottom row). Dashed lines depict flow to delocalized ($\alpha <{\alpha }_{\mathrm{c}}$) or localized fixed points ($\alpha >{\alpha }_{\mathrm{c}}$), and solid lines depict critical flow ($\alpha ={\alpha }_{\mathrm{c}}$). For the latter, the C2 flow in (d) is characteristic of a Gaussian fixed point.