Renormalization group flows for Wilson-Hubbard matter and the topological Hamiltonian

Understanding the robustness of topological phases of matter in the presence of interactions poses a difficult challenge in modern condensed matter, showing interesting connections to high-energy physics. In this work, we leverage these connections to present a complete analysis of the continuum long-wavelength description of a generic class of correlated topological insulators: Wilson-Hubbard topological matter. We show that a Wilsonian renormalization group (RG) approach, combined with the so-called topological Hamiltonian, provide a quantitative route to understand interaction-induced topological phase transitions that occur in Wilson-Hubbard matter. We benchmark two-loop RG predictions for a quasi-1D Wilson-Hubbard model by means of exhaustive numerical simulations based on matrix product states (MPS). The agreement of the RG predictions with MPS simulations motivates the extension of the RG calculations to higher-dimensional topological insulators.

Figure 1

Phase diagram of the imbalanced Creutz-Hubbard ladder. Phase diagram displaying a topological insulator (TI) phase, and two other nontopological phases, namely, an orbital phase with a long-range ferromagnetic Ising order (oFM) and an orbital paramagnetic phase (oPM). The blue circles label numerical results, and the colored phase boundaries are a guide to the eye.

Figure 2

One-loop tadpole diagrams around $k_1$. (a) (Left) Forward scattering of $({\eta }_1,{\eta }_2)=\left(\mathrm{L},\mathrm{R}\right)$ fermions onto $({\eta }_3,{\eta }_4)=\left(\mathrm{L},\mathrm{R}\right)$. (Right) Tadpole diagrams by joining an incoming and outgoing line of the same $\eta \in \{\mathrm{L},\mathrm{R}\}$ flavor into a loop. The momentum inside the loop is constrained to a small region around the cutoff ${\mathrm{\Lambda }}_{\mathrm{c}}/s\le \tilde{k}\le {\mathrm{\Lambda }}_{\mathrm{c}}$. The loop integrals over the fast modes are labeled by ${\eta }_{\mathrm{f}}(\tilde{k},\tilde{\omega })$ and depicted by dashed closed lines. (b) (Left) Backward scattering of $({\eta }_1,{\eta }_2)=\left(\mathrm{R},\mathrm{L}\right)$ fermions onto $({\eta }_3,{\eta }_4)=\left(\mathrm{L},\mathrm{R}\right)$. (Right) Corresponding tadpole diagrams.

Figure 3

One-loop tadpole diagrams involving $k_1$ and $k_0$. (a) (Left) Scattering of $({\eta }_1,{\eta }_2)=\left(\mathrm{L},\mathrm{u}/\mathrm{d}\right)$ fermions onto $({\eta }_3,{\eta }_4)=\left(\mathrm{u}/\mathrm{d},\mathrm{R}\right)$. (Right) Tadpole corrections to the mass of the light fermions by joining an incoming and outgoing line of the heavy fermions (we use the same conventions as in Fig. 2). (b) Same as for (a) but for the scattering of $({\eta }_1,{\eta }_2)=\left(\mathrm{R},\mathrm{u}/\mathrm{d}\right)$ fermions onto $({\eta }_3,{\eta }_4)=\left(\mathrm{u}/\mathrm{d},\mathrm{L}\right)$.

Figure 4

Benchmark of the RG prediction for the critical line. Comparison of the one-loop (49) (yellow line) and two-loop (52) (red line) predictions of the critical line. The blue circles label numerical results of Fig. 1 in the region of weak to moderate interactions.

Figure 5

Two-loop diagrams for the Wilson mass term. [(a) and (b)] Possible two-loop diagrams for the correction to the mass of the light fermions ${\eta }_j\in \{\mathrm{L},\mathrm{R}\}$ by virtual creation-annihilation of light/heavy fermions in fast modes $\eta ,\tilde{\eta },\widehat{\eta }\in \{\mathrm{R},\mathrm{L},\mathrm{u},\mathrm{d}\}$.

Figure 6

Entanglement spectrum of the Creutz-Hubbard ladder. Lowest $N_{\mathrm{ES}}=22$ eigenvalues of the entanglement spectrum ${\varepsilon }_n\in \mathrm{ES}\left({\rho }_{\ell }\right)$ for the Creutz-Hubbard ladder (36) for $\mathrm{\Delta }\varepsilon =2.2t$, $V_{\mathrm{v}}=0.2t$, which lies inside the TI region of Fig. 4. We compare the results obtained via the MPS numerical simulations (squares), and via the RG-corrected topological Hamiltonian (circles) (53).

Figure 7

Entanglement entropies of the critical Creutz-Hubbard ladder: Von Neumann and Rényi block entanglement entropies are represented as a function of the block chord length. The Hubbard interaction strengths are set to (a) $V_{\mathrm{v}}=0$, (b) $0.2t$, and (c) $0.4t$, while the corresponding energy imbalance $\mathrm{\Delta }\varepsilon$ is set to the critical value of Eq. (52) predicted by the RG calculations.

Figure 8

Luttinger parameter at criticality. The Luttinger parameter $K$ is extracted by fitting the bipartite fluctuations $\mathcal{F}(\ell ,L)$ obtained by the MPS numerical simulations to the linear logarithmic scaling with the chord length of Eq. (62). For each critical point $(\mathrm{\Delta }\varepsilon ,V_{\mathrm{v}})$, we extract the thermodynamic value of the Luttinger parameter by extracting the corresponding $K_L$ values for various lengths $L/a\in [64,192]$, and fitting them to $K_L=K+c_1/L+c_2/L^2$.