Time-reversal symmetry breaking Abelian chiral spin liquid in Mott phases of three-component fermions on the triangular lattice

We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude $-t\left(t>0\right)$ and on-site interaction $U$. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform $\pi$ flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers of $t/U$ for general $\mathrm{SU}\left(N\right)$ and arbitrary uniform charge flux per plaquette, which are subsequently studied using exact diagonalizations and variational Monte Carlo simulations for $N=3$, as well as exact diagonalizations of the $\mathrm{SU}\left(3\right)$ Hubbard model on small clusters. Up to third order in $t/U$, and for the time-reversal symmetric cases (flux 0 or $\pi )$, the low-energy description is given by the $J\text{−}K$ model with Heisenberg coupling $J$ and real ring exchange $K$. The phase diagram in the full $J\text{−}K$ parameter range contains, apart from three already known, magnetically long-range ordered phases, two previously unreported phases: (i) a lattice nematic phase breaking the lattice rotation symmetry and (ii) a spontaneous time-reversal and parity symmetry breaking Abelian chiral spin liquid. For the Hubbard model, an investigation that includes higher-order itinerancy effects supports the presence of a phase transition inside the insulating region, occurring at ${(t/U)}_c\approx 0.07[{(U/t)}_c\approx 13]$ between the three-sublattice magnetically ordered phase at small $t/U$ and this Abelian chiral spin liquid.

Figure 1

Comparison of the predicted phase diagram for the triangular lattice $J\text{−}K$ model on the 12-, 21- and 27-site clusters from ED (top three rows) and the 144-site cluster from VMC (bottom row). Around the Heisenberg point $\left(\alpha =0\right)$ the three-sublattice ordered phase (3-SL LRO,green) is present. For increasing values of $\alpha$ up to $\alpha =\pi$ a $\pi /3$-flux chiral spin liquid (CSL,orange), a lattice nematic phase (LN,blue), a ${120}^{\circ }$ LRO phase (white) and a ferromagnetically ordered phase (FM,gray) occur. The stripe state from VMC is expected to be closely related to the LN phase from ED as discussed at the end of Sec. 4.

Figure 2

Phase diagram of the SU(3) Hubbard model with $\mathrm{\Phi }=\pi$ on the triangular lattice. In the Mott-insulating phase for small $t/U$ the 3-SL LRO and $\pi /3$-CSL are found by ED and VMC. The metal-insulator transition is estimated in Sec. 5b as ${(t/U)}_c^{\text{mi}}\approx 1/8.5\approx 0.12$ (ED on 12 sites for the Hubbard model). The large uncertainty on this value and the nature of the Mott phase in that area is indicated by the grey area.

Figure 3

Illustration of the considered hopping configurations in the VMC calculations. (a) For the all-up plaquette order the magnitude of the hopping amplitudes on the thick purple bonds was changed between 0.5 and 2. The phases ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ were either chosen to be uniform with values $m\pi /6\left(m\in \mathbb{Z}\right)$, or set to a combination of 0 or $\pi$. For the 3-SL LRO (shown by the color of the vertices) the on-site chemical potential was changed from 0 to 2. For the (b) stripe and (c) zigzag stripe orders, the strengths of the hopping on the purple bonds are set to 1, and the strengths on the thin black bonds between the stripes are tuned between 0 and 2. In both stripe cases we considered either uniform flux configurations or opposite fluxes on up and down triangles with $\varphi =\{0,\pi /3,\pi /2,2\pi /3,\pi \}$.

Figure 4

Ground-state energies from ED on clusters with $N_s=\{12,21,27\}$ sites for the $J\text{−}K$ model with the coupling constants $J=cos\alpha$ and $K=sin\alpha$. Different point shapes indicate different symmetry sectors. The singlet sector $[N_s/3,N_s/3,N_s/3]$ yields the ground states of the 3-SL LRO, CSL, and LN phase below $\alpha /\pi \approx 0.7$. For $0.7\lesssim \alpha /\pi \lesssim 0.85$, the ${120}^{\circ }$ color order state, which contains only two colors from the sector $[N_s/2,N_s/2,0]$ or $[(N_s+1)/2,(N_s-1)/2,0]$, is present. In the FM phase for $\alpha /\pi \gtrsim 0.85$, where the energy per site is identical for all clusters, only a single color is present and therefore the ground states lie in the symmetry sectors $[N_s,0,0]$. Wherever the phase boundaries are not the same for different lattice sites $N_s$, the color of the boundary indicates the corresponding cluster size.

Figure 5

Spectrum of the $J\text{−}K$ model on the 21-site cluster from ED. The marker size corresponds to the overall chirality signal and is plotted for the lowest three states (the degeneracy of these states is 1, 4, and 1). The three (six when we count the degeneracies) low-lying singlets with strong chiral signal indicate the presence of a CSL phase in an extended parameter space at intermediate values of $K/J$.

Figure 6

Structure factor at the $K$ point (top) and chirality signal per lattice site (bottom) for the $J\text{−}K$ model from ED on the 12-, 21-, and 27-site clusters. Coming from the 3-SL LRO phase at small ratios $K/J$, the structure factors at the ordering momentum $K$ decrease as the chirality signals increase, indicating the CSL. In the regime, where the chirality signal decreases, a LN phase occurs.

Figure 7

Real space connected chirality correlator for the $J\text{−}K$ model from ED on the 21-site cluster at $K/J=0.455$. The degeneracy of the first three singlet states is 1-4-1. While the lowest three singlet levels show long-range chiral ordering, the fourth one does not.

Figure 8

Spin structure factor (top) and dimer-dimer correlations (bottom) of the $J\text{−}K$ model in the LN phase from ED on the 27-site cluster. The structure factor peak is close to the $X$ point. The dimer-dimer correlations clearly show that the rotational symmetry of the lattice is broken in this phase.

Figure 9

Spectrum of the SU(3) $J\text{−}K$ model at $\alpha =3\pi /4$ from ED on the 21-site cluster as a function of the quadratic Casimir $C_2$ of each irrep. The ground state lies in the [11,10] irrep of SU(3), where only two of the three possible colors are present. The black line is a guide to the eye illustrating that the ground state is indeed not a singlet. The red line is a linear fit through the lowest eigenvalue in each two-color irrep corresponding to the effective $\mathrm{SU}(2)$ tower of states in the spectrum. Further information on the quadratic Casimir operator is given in Appendix pp3.

Figure 10

Variational energies for the $J\text{−}K$ model with real $K$ of the most competitive states for the 144-site system compared to the results by Bieri et al. [33] (based on Fig. 3 of their paper).

Figure 11

Effective couplings in units of $U$ of the Hubbard model as a function of $t/U$ for $\mathrm{\Phi }=\pi$ using bare fifth-order series. Plotted are the largest contributions in every order with a pictogram sketching the associated permutations. The dark yellow background indicates the area where the CSL is observed within ED and VMC, whereas the light yellow corresponds to its stability according to VMC only (compare Sec. 5c). The inset shows the nearest-neighbor exchange in different orders and Padé extrapolations.

Figure 12

Ratio of effective coupling constants $K/J$ depending on $t/U$ for $\mathrm{\Phi }=\pi$ using bare series up to order 5 (green) as well as Padé extrapolations (cyan, blue). The inset shows a similar plot for the three-site ring exchange $K$ in units of $U$ by itself. The background colors are defined as in Fig. 11.

Figure 13

Comparison of the excitation spectra between the $J\text{−}K$ model (left), the Hubbard model (middle) and the $\mathcal{O}\left(4\right)$ (order 4) effective model (right) on the 12-site cluster from ED. Since the couplings in the effective descriptions are polynomials in $t/U$, the energies from the spin models are multiplied by $U$ to be comparable to the energies of the Hubbard model. For $U\approx 30t$, the spectra agree very well, but differences become noticeable as $U$ decreases. The grey regions of the spectra are not in the Mott-insulating phase of the Hubbard model and the effective models are not valid.

Figure 14

Scaling of the differences in the ground-state energies between various spin models and the Hubbard model on the 12-site cluster from ED. The effective second order Heisenberg model is abbreviated by HB model. A line is fitted for each spin model to estimate the scaling behavior. The slopes are in good agreement with the expectations.

Figure 15

Particle-hole gap for the SU(3) Hubbard model with $\mathrm{\Phi }=\pi$ from ED on the 12-site cluster indicating the estimation of the metal-insulator transition.

Figure 16

Spectrum of the fourth- and fifth-order effective model for $\mathrm{\Phi }=\pi$ from ED on the 21-site cluster. The marker size corresponds to the overall chirality signal and is plotted for the lowest three states. Three low-lying singlets with strong chirality signal indicate the presence of a CSL phase around $U/t\approx 12$. The grey regions of the spectra are not in the Mott-insulating phase of the Hubbard model and the effective models are not valid. The grey (black) solid line indicates the lowest energy eigenstate in the adjoint irrep [8,7,6] of the fourth-(fifth-)order effective model. The inset is a zoom into the CSL region.

Figure 17

Structure factor at the $K$ point (top) and chirality signal per lattice site (bottom) for the fourth- and fifth-order effective model for $\mathrm{\Phi }=\pi$ from ED on the 21-site cluster. Coming from the 3-SL LRO phase at large ratios $U/t$, the structure factors at the ordering momentum K decrease as the chirality signals increase around $U/t\approx 12$. The grey regions of the structure factors and the chirality signals are not in the Mott-insulating phase of the Hubbard model and the effective models are not valid.

Figure 18

All possible SU(3) Young tableaus for $N_s=3$ and the corresponding labels.

Figure 19

Triangular lattice clusters with 12, 21, and 27 sites used to derive the ED results presented in this work.

Figure 20

Scaling of the prefactors ${\mu }^{\left(S\right)}$ and ${\mu }^{\left(T\right)}$ as a function of the squared system size $L^2$. We show data for $L=\{6,9\}$. The extrapolated value of $\text{Im}{\mu }^{\left(T\right)}$ gives an estimate on the chiral central charge of $c\approx -2.34$. The value $\text{Im}{\mu }^T(L=0)=\pi /6$ would correspond to central charge $c=-2$.

Figure 21

Triangular lattice cluster with 12 sites (yellow) and PBCs (white). The chosen gauge is illustrated by the arrows on the bonds: a hopping in the direction of the arrow (respectively, in the opposite direction) has a positive phase factor $e^{i\mathrm{\Phi }}$ (respectively, the complex conjugate $e^{-i\mathrm{\Phi }}$.

Figure 22

Effective couplings in units of $U$ of the Hubbard model as a function of $t/U$ for $\mathrm{\Phi }=\pi /2$ in a double logarithmic plot. Shown are the nonvanishing real (left) and imaginary (right) contributions in bare orders. The diamonds encircled in black (gray) give the Padé extrapolants with the exponents [3,2] ([2,1]). Note that all uneven (even) terms in the real (imaginary) part vanish. The background colors are defined as in Fig. 12 in the main text.

Figure 23

Imaginary part of the ratio of effective coupling constants $-K/J$ depending on $t/U$ for $\mathrm{\Phi }=\pi /2$ using bare series up to order 5. The ratios of Padé extrapolants with the exponents [3,2] as well as the direct Padé extrapolation of the ratio $-\text{Im}K/J$ are indicated. The insets show similar plots for the imaginary part of the negative three-site ring exchange $-\text{Im}K$ and the nearest-neighbor exchange $J$. Extrapolations of $J$ with different pairs of exponents [2,2] and [2,3] are identical at $\mathrm{\Phi }=\pi /2$. The background colors are defined as in Fig. 12 in the main part of the manuscript.