Probing Topological Phases in a Strongly Correlated Ladder Model via Entanglement

TL;DR

The paper addresses how symmetry-protected topology persists or deforms under strong inter-leg correlations in a two-leg ladder with p-wave-like hybridization. It combines DMRG with entanglement diagnostics (edge entanglement, entanglement spectrum) and spectral-gap/central-charge analyses to map the interacting phase diagram. A key finding is a dichotomy: the boundary between the trivial phase and the topological phase with winding number $ u=-2$ remains pinned to its non-interacting location, while boundaries separating $ u=-1$ or $ u=0$ from the topological sectors shift with interaction, accompanied by a reduction of the effective central charge from $c=2$ to $c=1$ along the robust line. These results illuminate how certain topological features can be preserved by interactions when symmetries enforce the transitions, with broader implications for correlated topological phases and potential extensions to topological Kondo-like systems.

Abstract

The interplay between non-trivial band topology and strong electronic correlations is a central challenge in modern condensed matter physics. We investigate this competition on a two-leg ladder model with a p-wave-like hybridisation between the legs. This model hosts a symmetry-protected topological phase in its non-interacting limit. Using the density-matrix renormalisation group algorithm, we compute the comprehensive quantum phase diagram in the presence of a repulsive inter-leg density-density interaction. Our analysis, based on entanglement entropy and the entanglement spectrum, reveals a fascinating dichotomy in the stability of the topological phase. We find a non-trivial change in the value of the edge entanglement entropy as we include interaction. Furthermore, we find that the phase boundary separating a trivial insulator phase and a topological one with winding number two remains robustly pinned at its non-interacting location, irrespective of the interaction strength. Variation of the effective conformal field theory's central charge near the critical line explains the robustness of the gap. In contrast, the transition to an insulating phase with winding number one is heavily renormalised, with the critical line shifting significantly as the interaction increases. By successfully mapping the phase diagram and identifying the distinct behaviours of the phase boundaries, our work clarifies how interactions can selectively preserve or destroy different aspects of a topological phase.

Figures (13)

• Figure 1: Schematic representation of the one-dimensional ladder model investigated in this work. Each unit cell (illustrated within a dotted square) consists of two sub-lattices, indicated by $a$ and $b$, the top (bottom) rung of each sub-lattice comprises the $c$ ($f$)-orbitals. The parameters: $t_{c_1}(t_{f_1})$ and $t_{c_2}(t_{f_2})$ represent the intra-lattice and inter-lattice hopping parameters for the $c\,(f)$ orbital, respectively; the orbitals hybridize via a $p$-wave-like hybridization energy given by $\pm v_{1}$ (between $c_b$ and $f_a$), and $\pm v_{2}$ (between $c_a$ and $f_b$).
• Figure 2: Solid colour regions represent the winding number $\nu$ calculated for $\mathcal{H}_k$ in the parameter space, $v_2/v_1=-1$. The colours blue, grey and red represent $\nu = 2, \, 1, \, 0$, respectively. We now calculate $\nu$ for two independent SSH chains, $H_L, \, H_R$ with hopping parameters defined in Equation \ref{['eq:effective_hopping']} irrespective of the condition $t_{c_1}+t_{f_1}+t_{c_2}+t_{f_2}=0$ but keeping $v_2/v_1=-1$. Hatched regions with right tilted lines (///) and left tilted lines (\\\\\\) represent the parameter space for which $H_L$ and $H_R$ are in the topological phase, respectively. The red dashed line represents the additional condition $t_{c_1}+t_{f_1}+t_{c_2}+t_{f_2}=0$ such that $\mathcal{H}_k=H_L\oplus H_R$. Hence, the winding number along the red dashed line can be written as ($\mathbb{Z}_L \oplus \mathbb{Z}_R$), and marked on the line. The left (right) plot is with $t_{c_1}=1$, $t_{f_1} = 0.9\,(-0.9)$.
• Figure 3: Phase diagram for the non-interacting ($V_{NN}=0$) case, in the $t_{c_2}/t_{c_1}$-$v_2/v_1$ plane, for different $t_{f_2}/t_{f_1}\approx 0.6, \, 1.0,\, 1.67$ in panels (a), (b), (c), respectively. The phase diagram consists of a trivial ($\nu=0$) and two different topological phases ($\nu=-2,\, -1$), characterized by different winding number ($\nu$) values as mentioned in the panels. Note that the slope of the phase separating line between phases with $\nu = -2, \, 0$ from the phase with $\nu = -1$ changes orientation depending on the value of $t_{f_2}/t_{f_1}$. This variation of the slope angle as a function of $t_{f_2}/t_{f_1}$ is shown in panel (d). The exact expression is shown in Equation \ref{['Eq: theta']}
• Figure 4: Cartoon of a finite-sized chain, with the subsystem $\mathcal{A}$ chosen such that it includes one of the edges.
• Figure 5: Phase diagram of the non-interacting system, in the $t_{c_2}/t_{c_1}$-$v_2/v_1$ plane, for $t_{f_2}/t_{f_1}=1$, reproduced in terms of the edge entanglement entropy, $S_{edge}/\ln(2)$. The phases are in excellent agreement with that derived using $\nu$ (Figure \ref{['fig:PhaseDiagram_nu']}(b)). The quantity, $S_{edge}/\ln(2)$, thus serves as an excellent order parameter for identifying topological phase transitions in the non-interacting system. Phases A, B, and C are labelled based on the $S_{edge}/\ln{2}$ values, which are equal to 0, 1, and 2, respectively.