Chiral Spin-Density Wave, Spin-Charge-Chern Liquid, and $d+id$ Superconductivity in $1/4$-Doped Correlated Electronic Systems on the Honeycomb Lattice

Recently, two interesting candidate quantum phases—the chiral spin-density wave state featuring anomalous quantum Hall effect and the $d+id$ superconductor—were proposed for the Hubbard model on the honeycomb lattice at $1/4$ doping. Using a combination of exact diagonalization, density matrix renormalization group, the variational Monte Carlo method, and quantum field theories, we study the quantum phase diagrams of both the Hubbard model and the $t\text{−}J$ model on the honeycomb lattice at $1/4$ doping. The main advantage of our approach is the use of symmetry quantum numbers of ground-state wave functions on finite-size systems (up to 32 sites) to sharply distinguish different quantum phases. Our results show that for $1\lesssim U/t<40$ in the Hubbard model and for $0.1<J/t<0.80(2)$ in the $t\text{−}J$ model, the quantum ground state is either a chiral spin-density wave state or a spin-charge-Chern liquid, but not a $d+id$ superconductor. However, in the $t\text{−}J$ model, upon increasing $J$, the system goes through a first-order phase transition at $J/t=0.80(2)$ into the $d+id$ superconductor. Here, the spin-charge-Chern liquid state is a new type of topologically ordered quantum phase with Abelian anyons and fractionalized excitations. Experimental signatures of these quantum phases, such as tunneling conductance, are calculated. These results are discussed in the context of $1/4$-doped graphene systems and other correlated electronic materials on the honeycomb lattice.

Popular Summary

A reliable determination of quantum phase diagrams of correlated electronic systems has long been a central issue in quantum condensed matter physics. Doping yields stronger quantum entanglement, so it is more difficult to determine quantum phases in doped materials. By attributing different lattice quantum numbers to different quantum phases, we can show that our technique is powerful even for small lattices. Here, we investigate 1/4-doped correlated systems arranged in a honeycomb lattice. This level of doping has not yet been experimentally realized, but new thin-film technologies make it likely that such doping will be possible in the near future.

The spin-charge-Chern liquid state is identified as a candidate of a new type of quantum phase in certain correlation regimes. This state represents a novel topologically ordered quantum phase, which respects spin-rotation and lattice translation and rotation symmetries, but it is gapped in the bulk and has specific fractionalized quasiparticle excitations (anyons) and several unexpected properties. We show that experimental signatures, such as tunneling conductance, can clearly distinguish the proposed quantum phases, including the new quantum liquid. We analytically investigate symmetric quantum wavefunctions of different candidate quantum phases on finite-size systems—we conduct 8-, 24-, and 32-site sample calculations—and study their characteristic symmetry quantum numbers. We compare these values with results from unbiased numerical simulations that have been greatly advanced in the past decade, such as exact diagonalization and density matrix renormalization group theory. This comparison allows, to a large degree, a sharp distinction between quantum phases, using the finite system sizes as an advantage rather than a limitation. Our approach shows that, in general, a combination of different quantum many-body techniques allows, to a certain level, a sharp determination of quantum phases on lattices with some commensurate filling of electrons.

We propose that some materials, such as transition metal heterostructures and graphene, could be used to realize the doped honeycomb correlated models analyzed in this work.

Figure 1

(a) Symmetries of the honeycomb lattice. (b) Nested Fermi surface at $1/4$ doping.

Figure 2

(a) The chiral spin-density wave and (b) $d+id$ pairing order parameters in real space.

Figure 3

The phase diagrams of the correlated electronic systems on the honeycomb lattice at $1/4$ doping. (a) In the Hubbard model, the ground state is found to be either in a CSDW phase or in a SCCL phase over the majority of the parameter range $1\lesssim U/t<40$. (b) In the $t\text{−}J$ model, the CSDW phase or the SCCL phase is identified in the regime $0.1<J/t<0.80(2)$, separated from the $d+id$ superconductor phase at larger $J/t$ by a first-order phase transition.

Figure 4

The real space pattern of the slave-fermion amplitudes describing the CSDW or SCCL phases. The dashed line encircles the doubled unit cell. (a) The nearest-neighbor (NN) and next-nearest-neighbor (NNN) boson pairing amplitudes $A_{ij}$ are directional (labeled by arrows) since $A_{ij}=-A_{ji}$. $A_{ij}$ on the NN (NNN) bonds have the same magnitude, respectively. Their different phases are represented by different colors. Black, 1; violet, $e^{i\pi /2}$; green, $e^{i5\pi /6}$; orange, $e^{i\pi /6}$; red, $e^{i\pi /3}$; blue, $e^{i2\pi /3}$. (b) The NN (NNN) boson/fermion hopping amplitudes $B_{ij}/{\chi }_{ij}$ also have uniform magnitudes, respectively. When they are complex, the amplitudes are directional $B_{ij}=B_{ji}^{*}$, ${\chi }_{ij}={\chi }_{ji}^{*}$ (labeled by arrows). The phases are illustrated by colors. Black, $\pm 1$; blue, $e^{i\varphi }$; red, $-e^{i\varphi }$. Here, the real number $\varphi ={\varphi }_b$ for bosons and $\varphi ={\varphi }_f$ for fermions. ${\varphi }_b$ and ${\varphi }_f$ can be viewed as two variational parameters. The above pattern is for one of the two degenerate ground states while the other one is its time-reversal image, which can be obtained by sending these amplitudes to their complex conjugates: $A_{ij}/B_{ij}/{\chi }_{ij}\to A_{ij}^{*}/B_{ij}^{*}/{\chi }_{ij}^{*}$. Sites numbered 1 to 8 label the quadrupled unit cell used in Appendix pp3.

Figure 5

Three samples marked by dashed lines are used in numerical calculations: (a) 8-site and 32-site samples; (b) 24-site sample. Periodic boundary conditions are applied for each of them.

Figure 6

Energy per site of $t\text{−}J$ model on 8-site sample obtained by exact diagonalization. Bottom inset: Crossover between two different ground states occurs at $J/t=0.089(1)$. Numbers show the state degeneracy. Top inset: Lowest states with spin zero at fixed $J/t=0.7$.

Figure 7

Ground-state energy per site of the Hubbard model. Red line, 32-site sample, obtained by DMRG. Blue line, 8-site sample, obtained by ED. For 8-site sample, ED shows a twofold degenerate ground state.

Figure 8

The optimal value of pairing amplitude $\mathrm{\Delta }$, which is the only variational parameter in the projected $d+id$ wave function, for 24-site (blue circles) and 32-site (red squares) samples.

Figure 9

Expectation value of 60-deg rotation operator ($C_6$) in DMRG ground state: (a) $t\text{−}J$ model on 32-site sample (triangles) with projection of wave function to center-of-mass momentum $\mathrm{\Gamma }$, and on 24-site sample (disks) without projection. (b) Hubbard model on the 32-site sample, with (black triangles) and without (smaller gray triangles) projection to $\mathrm{\Gamma }$ momentum.

Figure 10

The lowest ground-state energy of projected $d+id$ variational wave function, obtained by VMC calculations, is shown as a fraction of the DMRG ground-state energy on the same sample. Blue line is for the 24-site sample, red line is for 32-site sample. Inset: DMRG energy per site of 24-site (blue line) and 32-site (red line) samples.

Figure 11

Spin-spin correlation function $⟨S_z(i)S_z(j)⟩$ measurement. Blue is positive, red negative, and disk radius is proportional to amplitude. Site $i$ is fixed at green circle, while every bond $i$, $j$ is averaged over translations and rotations to increase the number of sampled observable values in MC runs and thereby reduce statistical error. “max” labels absolute amplitude of largest shown disk. The measurement on 32-site sample uses the DMRG ground state projected into sector with center-of-mass momentum $\mathrm{\Gamma }$ and $C_6$ rotation eigenvalue $\exp (-i\pi /3)$: (a) $t\text{−}J$ model, $J/t=0.5$, (b) Hubbard model, $U/t=8$. The spin-spin correlation is longer ranged and consistent with tetrahedral pattern throughout CSDW or SCCL phase. On 24-site sample, the $t\text{−}J$ model DMRG ground state is projected into sector with $C_6$ eigenvalue $\exp (-i2\pi /3)$. (c) Same correlation behavior is found deep in small-$J$ regime of 24-site sample, while (d) spin pattern is lost in large-$J$ regime, even as short-range correlations grow.

Figure 12

Value of spin-spin correlation function $⟨S_z(i)S_z(j)⟩$ for farthest pair of sites $i$, $j$ (see insets), with DMRG ground state projected into sector having center-of-mass momentum $\mathrm{\Gamma }$, and $C_6$ eigenvalue $\exp (-i\pi /3)$ on 32-site sample: (a) $t\text{−}J$ model, (b) Hubbard model. (c) On 24-site sample, the $t\text{−}J$ model DMRG ground state is projected to $\exp (-i2\pi /3)$ eigenvalue sector of $C_6$: the spin correlation vanishes with crossover to ($d+id$)-like regime. Averaging over translationally and rotationally related pairs is included in all measurements to reduce the statistical error. The correlation consistently grows throughout CSDW or SCCL phase with larger $U$ (smaller $J$).

Figure 13

Nature of spin-spin correlation pattern $⟨S_z(i)S_z(j)⟩$ in the CSDW or SCCL phase, with DMRG ground state of 32-site sample projected into sector having center-of-mass momentum $\mathrm{\Gamma }$, and $C_6$ eigenvalue $\exp (-i\pi /3)$: (a) $t\text{−}J$ model, (b) Hubbard model. Three values of correlation with sites chosen as $i=0$, $j=3$, 5, 13 (see inset) are averaged, and that average is divided by correlation between $i=0$, $j=4$. Tetrahedral pattern predicts this ratio to be $-1/3$ (blue line). Correlation for each site pair $i$, $j$ is averaged over all translationally related pairs to reduce statistical error.

Figure 14

Spin chirality of a characteristic triangle (inset) in 32-site sample, from $t\text{−}J$ model DMRG ground state projected into sector having center-of-mass momentum $\mathrm{\Gamma }$, and $C_6$ eigenvalue $\exp (-i\pi /3)$. The chirality is averaged over all translations and rotations of the triangle to reduce statistical error.

Figure 15

Symmetry quantum numbers for $C_6$ rotation, calculated analytically (Appendix pp3) for the fourfold topologically degenerate ground-state sector of the SCCL phase on different lattice sizes. $C_6$ eigenvalues are plotted on the unit circle in complex plane. The sets of eigenvalues differ by overall phase between different system sizes ($N$ is integer and $X\times Y$ labels number of unit cells along $a_1$, $a_2$), although all systems are topologically a torus.

Figure 16

Top: Point contact measurement with weak tunneling ($G\ll e^2/h$) into edge states can distinguish between the CSDW and SCCL phase. Bottom: Temperature dependence of tunneling conductance through point junction between metal or SC lead and CSDW or SCCL, exhibiting different values of power-law exponent $\alpha$.

Figure 17

Line junction modeled as the large-$N$ limit of array of point contacts. In weak tunneling regime having $T\ll V$ or $V\ll T$, with $T$ temperature and $V$ voltage, the conductance can distinguish between the CSDW and SCCL phases.

Figure 18

DMRG energy of Hubbard model with $U/t=6$ on 32-site sample, as function of limiting MPS matrix size $m$.

Figure 19

Spin-spin correlation function $⟨S_z(i)S_z(j)⟩$ in the DMRG ground state projected into sector with center-of-mass momentum $\mathrm{\Gamma }$ and $C_6$ rotation eigenvalue $\exp (-i\pi /3)$. Blue is positive, red negative, and disk radius is proportional to amplitude. Site $i$ is fixed at green circle, while every bond $i$, $j$ is averaged over translations and rotations to reduce statistical error by increasing the number of sampled observable values in MC runs. “max” labels absolute amplitude of largest shown disk. All four parameters are chosen within the CSDW or SCCL phase: (a),(b) Hubbard model, (c),(d) $t\text{−}J$ model.