Phases of Quasi-One-Dimensional Fractional Quantum (Anomalous) Hall - Superconductor Heterostructures

TL;DR

This work analyzes quasi-one-dimensional heterostructures formed from fractional quantum Hall states and superconductors under strong order-parameter fluctuations. By combining a phenomenological field theory, a mapped $\,\mathbb{Z}_3$ parafermion topological Josephson chain (rotor model), and large-scale DMRG, the authors map a phase diagram featuring Mott insulators and two gapless Luttinger liquids carrying charges $2e$ and $2e/3$. They identify Berezinskii–Kosterlitz–Thouless transitions and a continuous $\,\mathbb{Z}_3\times U(1)$ transition with central charge $c=9/5$, supported by CFT arguments and numerical central-charge extraction. The results illuminate parafermion edge physics and ground-state degeneracy in fluctuating SC environments and have implications for moiré-material FCI-SC realizations where strong order-parameter fluctuations are observed.

Abstract

Motivated by recent observations of fractional Chern insulators (FCIs) in the vicinity of superconducting (SC) phases, we study fractional quantum (anomalous) Hall-superconductor heterostructures in the presence of $U(1)$ order-parameter fluctuations and particularly focus on the case of $ν= 2/3$ quantum Hall states leading to $\mathbb Z_3$ parafermions. We first employ a phenomenological field theory to qualitatively determine the phase diagram. Furthermore, we generalize a previously established alternating pattern of superconductor and tunneling regions, coupled to fractional quantum Hall edge states, to map the problem onto a topological Josephson junction chain involving lattice parafermions. Using density matrix renormalization group simulations, we establish a phase diagram composed of Mott insulating phases and two different Luttinger liquids whose fundamental excitations carry charges 2e and $2e/3$, respectively. In agreement with analytical considerations using conformal field theory, we numerically find transitions of Berezinskii-Kosterlitz-Thouless (BKT) type as well as a continuous $\mathbb Z_3 \times U(1)$ second-order phase transition characterized by central charge c = 9/5. We finally extract information about a possible ground state degeneracy and comment on the stability of parafermionic edge states in the presence of fluctuations. These theoretical foundations can be expected to be of practical importance for gate-defined FCI-SC heterostructures in moiré materials, in which broad superconducting transitions indicative of strong order parameter fluctuations were observed.

Figures (10)

• Figure 1: a): Graphical representation of the system. A FQ(A)H-superconductor heterostructure with strong fluctuations is modeled by an array of floating superconductors placed into an FQ(A)H system. The edge states (indicated by the black arrows) encircle the superconductors. b): Schematic close-up of the system in panel a. $\mathbb{Z}_3$ parafermions $\alpha_{x_j}$ emerge at the interfaces at the tunneling (t) and superconducting ($\Delta$) strips. The parafermions hybridize via the tunnelling and pairing regions via $J_t$ and $J_\Delta$, respectively. Whole Cooper pairs can tunnel between superconductors via the Josephson coupling $J_J$. c): Phase diagram obtained via DMRG. The phase boundaries are a guide to the eye based on numerical data such as the central charge, the Luttinger parameter, or the energy spread of the ground-state wavefunction. The dashed white lines denote the phase boundaries predicted by mean field theory. The top left sector is the 2$e$ Luttinger liquid (LL), the bottom left sector is the Mott insulator (MI), and the right sector is the 2$e$/3 LL.
• Figure 2: a) Phase diagram obtained by RG arguments from the phenomenological field theory. b) Close up of the transition between the 2$e$ LL and the 2$e$/3 LL. The thin colored lines indicate where operators become relevant/irrelevant. The $\mathbb{Z}_3 \times U(1)$ phase transition (red line) is defined as the line where the scaling dimension of $H_\Theta$ and $H_{\text{FQH+SC}}$ are equal. The BKT transition from the Mott insulator is defined by the line where the operator $H_\varphi$ becomes irrelevant while the operator $H_{\Theta}$ is still relevant and the operator with the lowest scaling dimension.
• Figure 3: Phase diagram for varying gate charge $Q_g$ for $J_\Delta/E_c=0$ (panel a) and $J_\Delta/E_c=0.1$ (panel b). All lobes centered around an even integer are Mott insulators with a multiple of 2$e$ charges fixed on each lattice site (2$e$ MI). Lobes that are not centered around an even integer have a multiple of the fractional charge $2e/3$ at each lattice site ($2e/3$ MI). The height and position of the lobes for the 2/3 MI are denoted by the bold and dashed lines, respectively.
• Figure 4: Log-log plot of correlation functions $\langle B_i B_j^\dagger\rangle$ (red triangles) and $\langle b_i b^\dagger_j\rangle$ (blue dots) as a function of spatial distance for the 2e LL ($J_t=0.0032$, $J_J=3.8065$) and the $2e/3$ ($J_t=0.021$, $J_J=3.8065$) LL phase. The black lines correspond to a fit with the function $f(|i-j|)= \frac{A}{|i-j|^\alpha}$, where $A$ and $\alpha$ are free fit parameters.
• Figure 5: Luttinger parameter across the two BKT transitions as a function of the coupling constants on the axis of the phase diagram \ref{['fig:summaryPlot']}c, where $J_t=0$ and $J_J=0$ correspond to panels a) and b), respectively. The critical $K^*_\varphi = 4$ and $K^*_\rho = 4/3$ estimate the critical coupling constants for different system sizes $L$. A linear extrapolation vs. $1/L$ for the five largest systems leads to $J^*_J/E_c = 3.416(2)$ and $J^*_t/E_c = 0.3794(2)$, marked by dashed lines.