Kagome chiral spin liquid in transition metal dichalcogenide moir\'e bilayers

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Kagome chiral spin liquid in transition metal dichalcogenide moiré bilayers

Johannes Motruk, Dario Rossi, Dmitry A. Abanin, and Louk Rademaker

Phys. Rev. Research 5, L022049 – Published 5 June 2023

Abstract

At $n = 3 / 4$ filling of the moiré flat band, transition metal dichalcogenide (TMD) moiré bilayers will develop kagome charge order. We derive an effective spin model for the resulting localized spins and find that its further neighbor spin interactions can be much less suppressed than the corresponding electron hopping strength. Using density matrix renormalization group simulations, we study its phase diagram and, for realistic model parameters relevant for $W {Se}_{2} / W S_{2}$ , we show that this material can realize the exotic chiral spin liquid phase and the highly debated kagome spin liquid. Our work thus demonstrates that the frustration and strong interactions present in TMD heterobilayers provide an exciting platform to study spin liquid physics.

Received 5 December 2022
Revised 27 April 2023
Accepted 2 May 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.L022049

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Frustrated magnetism Quantum spin liquid Topological phases of matter

Bilayer films Kagome lattice Strongly correlated systems Transition metal dichalcogenides

Density matrix renormalization group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Johannes Motruk ^1,*, Dario Rossi¹, Dmitry A. Abanin^1,2, and Louk Rademaker^1,3

¹Department of Theoretical Physics, University of Geneva, Quai Ernest-Ansermet 24, 1205 Geneva, Switzerland
²Google Research, Mountain View, California, USA
³Department of Quantum Matter Physics, University of Geneva, Quai Ernest-Ansermet 24, 1205 Geneva, Switzerland

^*johannes.motruk@unige.ch

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Vol. 5, Iss. 2 — June - August 2023

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Figure 1
(a) Ground-state phase diagram of the effective spin model (2) for $U / t_{1} = 75$ and $V / t_{1} = 10.5$ —corresponding to the interactions for $ɛ \approx 9.5$ in $W {Se}_{2} / W S_{2}$ at $θ = 0^{\circ}$ —as a function of $t_{2} / t_{1}$ and $t_{3} / t_{1}$ . The phases appearing are the chiral spin liquid (CSL), the kagome spin liquid (KSL) connected to the ground state of the nearest-neighbor Heisenberg Hamiltonian, a valence bond crystal (VBC), and $cuboc 1^{*}, q = {(0, 0)}^{*}$ , and $\sqrt{3} \times \sqrt{3}$ or ferromagnetically ordered phases. The $\sqrt{3} \times \sqrt{3}$ and ferromagnet are energetically degenerate. The red dot indicates the $t_{2} / t_{1}$ and $t_{3} / t_{1}$ values for $W {Se}_{2} / W S_{2}$ at $θ = 0^{\circ}$ . Arrows show how this point would shift qualitatively when tuning twist angle, pressure or changing the effective mass by a different material choice. (b) Phase diagram for $t_{2} / t_{1} \approx 0.145$ and $t_{3} / t_{1} \approx 0.08$ —corresponding to $W {Se}_{2} / W S_{2}$ at $θ = 0^{\circ}$ —as a function of $U / t_{1}$ and $V / U$ . Here, the $cuboc 2^{*}$ phase emerges as an additional magnetically ordered phase. The ratio of $V / U$ can be tuned by the gate distance while $U / t_{1}$ changes with different dielectric environment. Crucially, the system can be tuned into the CSL by merely changing the gate distance across almost the entire range of interaction strength. The hatched area denotes the region in which $J_{1}$ is negative (ferromagnetic) and the red dot indicates the interaction values of panel (a). The data underlying the phase diagrams have been obtained on an infinite YC8 cylinder. (c) Extended Hubbard model at $3 / 4$ filling with all charges localized on a kagome lattice. The unit cell of the charge ordering is indicated by the blue-lined box. When a spin- $↑$ particle hops from a site $j$ to a site $k$ along the direction of an arrow, it picks up a phase of $ϕ_{j k}$ . (d) Interactions of the resulting spin model on the kagome lattice.
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Figure 2
Contributions of the different orders of the effective Hamiltonian from our DFT estimates for $W {Se}_{2} / W S_{2}$ and dielectric screening with $ɛ = 9.5$ as a function of twist angle. Dashed lines denote negative values.
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Figure 3
CSL region as a function of $t_{2} / t_{1}$ and $t_{3} / t_{1}$ for various combinations of $U$ and $V$ from DMRG on an infinite YC8 cylinder as detected by the absolute value of the chiral order parameter $〈 S_{i} \cdot (S_{j} \times S_{k}) 〉$ averaged over all small nearest-neighbor triangles on the kagome lattice. (a) $U / t_{1} = 75$ and $V / U = 0.14$ corresponding to the phase diagram in Fig. 1. (b) For decreased $U / t_{1} = 70$ with $V / U = 0.14$ unchanged, the CSL region moves slightly to the lower right, but narrows. (c) Opposite effect when increasing to $U / t_{1} = 70$ , still at $V / U = 0.14$ . (d) Changing $V / U = 0.13$ has a similar effect as decreasing $U$ , (e) larger $V / U = 0.15$ behaves comparable to increased $U$ .
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Figure 4
(a) Momentum-resolved entanglement spectrum in the CSL phase on a YC12 cylinder. The $S^{z}$ sectors of the levels show the counting pattern expected from the CFT describing the edge states $(1, 1, 2, 3, 5, 7, ...)$ . (b) Expectation value $〈 S_{L}^{z} 〉$ value of the left half of the cylinder under spin flux insertion. $Δ 〈 S^{z} 〉 = 1 / 2$ is pumped from the right to the left half of the cylinder under $2 π$ flux insertion indicating a spin Hall conductivity of $σ_{x y}^{spin} = 1 / 2$ .
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Figure 5
Absolute values of $J$ ratios from our DFT estimates for $W {Se}_{2} / W S_{2}$ as a function of twist angle for different values of $ɛ$ . For $ɛ = 7$ and 11, the ratio $J_{2} / J_{1}$ changes sign due to $J_{1}$ becoming negative indicated by the blue (positive) and red (negative) shading. At $ɛ = 15, J_{1}$ is negative over the entire twist angle range. $J_{2}$ and $J_{3}$ are positive everywhere while $J_{3}^{'}$ is always negative. We chose the absolute values here for better presentation clarity on a log scale. We also include the ratio of $V / t_{1}$ (red line) and $V / t_{1} = 5$ (red dashed line). Above this value, almost the entire particle density is localized on the kagome lattice, ensuring the validity of our spin model description.
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Physical Review Research

Kagome chiral spin liquid in transition metal dichalcogenide moiré bilayers

Johannes Motruk, Dario Rossi, Dmitry A. Abanin, and Louk Rademaker

Phys. Rev. Research 5, L022049 – Published 5 June 2023

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