Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice

Tessa Cookmeyer, Johannes Motruk, and Joel E. Moore

Phys. Rev. Lett. 127, 087201 – Published 19 August 2021

Abstract

At strong repulsion, the triangular-lattice Hubbard model is described by $s = 1 / 2$ spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120° order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that, at physically relevant parameters, they stabilize a chiral spin liquid (CSL) of Kalmeyer-Laughlin (KL) type, clarifying observations in recent studies of the Hubbard model. We then present a self-consistent solution based on a mean-field rewriting of the interaction to obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL state, providing a physical understanding for the origin of the CSL.

Received 17 March 2021
Accepted 17 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.087201

Physics Subject Headings (PhySH)

Frustrated magnetism Quantum spin liquid Topological phases of matter

Strongly correlated systems

Density matrix renormalization group Hubbard model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tessa Cookmeyer ^1,2,*, Johannes Motruk ^1,2,3, and Joel E. Moore^1,2

¹Department of Physics, University of California, Berkeley, California 94720, USA
²Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
³Department of Theoretical Physics, University of Geneva, Quai Ernest-Ansermet 30, 1205 Geneva, Switzerland

^*tcookmeyer@berkeley.edu

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Vol. 127, Iss. 8 — 20 August 2021

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Images

Figure 1
(a) The different colored lines connect the spins involved in the different terms of Eq. (1). (b) The first Brillouin zone of the lattice showing several named points. (c) The proposed phase diagram from our ED results using the various orders in Fig. 2. For phase descriptions, see the Supplemental Material [73]. The phase boundaries were determined via the symmetry sector of the ground state and first excited state [46, 73]. The grayed out region is within the SFS parameter space found in [28], but also might have some dimer or plaquette ordering. (d) The phase diagram from the iDMRG results on the $L_{y} = 6$ cylinder on the $J_{2} = 0.00$ , 0.05 slices, which includes the CSL, perhaps suggested by ED. (e) From left to right, the 120°, collinear, zigzag (ZZ), and tetrahedral (whose spins, connected tail to tail, form a tetrahedron) classical spin orders are shown [73].
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Figure 2
(a)–(f) Various orders are shown in color vs $J_{2}$ and $J_{4}$ . The table in the upper right indicates the phase to which each order corresponds. (g) The overlap of the ground state with the manifold of KL states, which suggests that the CSL may appear for small $J_{2}$ and $J_{4}$ .
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Figure 3
We plot various order parameters that we extract from ground state wave function from iDMRG for the $L_{y} = 6$ cylinder, and $J_{2} = 0$ ( $J_{2} = 0.05$ ) for the left (right) column, respectively. (a),(b) [(g),(h)] We plot the spin-spin correlation at the $K$ ( $Y$ ) point, respectively. We see a jump in the value corresponds to a phase boundary. (c),(d) We plot $χ = ⟨ S_{i} \cdot (S_{j} \times S_{k}) ⟩$ averaged over all triangles of the lattice from the iDMRG results at varying bond dimension $χ_{BD}$ . (d) The jump in the nonzero value of $χ$ at $J_{4} = 0.19$ corresponds to whether the trivial ( $J_{4} \leq 0.19$ ) or semion ( $J_{4} \geq 0.20$ ) sector of the KL state is the ground state as evidenced by the entanglement spectra. We include an extrapolation [73] to $χ_{BD} \to \infty$ where it is nonzero. (e),(f) We plot the dimer-dimer correlation at the $M^{'}$ point for dimers in the $\hat{x}$ direction, which signals the VBS state. The phase boundaries estimated from these data are plotted in Fig. 1.
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Figure 4
(a) We plot the entanglement spectrum for the ground state at $(J_{2}, J_{4}) = (0.05, 0.18)$ on the $L_{y} = 8$ cylinder with $χ_{BD} = 1600$ . The $y$ axis is $- s \ln (s)$ , where $s$ are the Schmidt values. The color indicates the charge as specified in the legend, and different charges are offset slightly from each other to more clearly show the degeneracy. For each momentum, the counting of the lowest cluster of Schmidt values is shown for each of the $s_{z} \geq 0$ charges in color. They show the correct pattern for the Kalmeyer-Laughlin state. (b) We make the same plot as in (a) after adiabatically inserting one flux quantum through the cylinder. Although the Hamiltonian is the same, the entanglement spectrum has changed, indicating a topological degeneracy of the state. (c) During the flux insertion, we can monitor how much spin has flowed along the cylinder. We see that exactly a spin $1 / 2$ is pumped across the system, indicating a quantized fractional spin-Hall effect.
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