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Physical review. B./Physical review. B

Variational Monte Carlo approach to the SU(4) spin-orbital model on the triangular lattice

Chun Zhang, Hui-Ke Jin, Yi Zhou

Published Mar 4, 2024• 4 citations
Publisher

Abstract

Previous investigations have suggested that the simplest spin-orbital model on the simplest frustrated lattice can host a nematic quantum spin-orbital liquid state. Namely, the orbital degeneracy of the SU(4) Kugel-Khomskii model tends to enhance quantum fluctuations and stabilize a quantum spin-orbital liquid exhibiting stripy features on the triangular lattice, as revealed by the state-of-the-art method of the density matrix renormalization group boosted by Gutzwiller projected wave functions. In this work, using the variational quantum Monte Carlo method, we study several spin-orbital liquid states, including a uniform $\ensuremath{\pi}$-flux state, three stripy states, and a plaquette state, on the $L\ifmmode\times\else\texttimes\fi{}L$ torus up to $L=24$. It turns out that one of these stripy states, called the ``stripe-II'' state, is energetically favored. This ground state breaks the ${C}_{6}$ symmetry of the lattice, resulting in a reduced ${C}_{2}$ symmetry and doubled unit cells while preserving the SU(4) spin-orbital rotation symmetry. Such a nematic quantum spin-orbital liquid state can be characterized by a parton Fermi surface (FS) consisting of open orbits in the Brillouin zone, in contrast to the circular FS of the uniform $\ensuremath{\pi}$-flux state.

DOI
10.1103/physrevb.109.125103

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Previous investigations have suggested that the simplest spin-orbital model on the simplest frustrated lattice can host a nematic quantum spin-orbital liquid state. Namely, the orbital degeneracy of the SU(4) Kugel-Khomskii model tends to enhance quantum fluctuations and stabilize a quantum spin-orbital liquid exhibiting stripy features on the triangular lattice, as revealed by the state-of-the-art method of the density matrix renormalization group boosted by Gutzwiller projected wave functions. In this work, using the variational quantum Monte Carlo method, we study several spin-orbital liquid states, including a uniform \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux state, three stripy states, and a plaquette state, on the L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL torus up to L=24L=24L=24. It turns out that one of these stripy states, called the ``stripe-II'' state, is energetically favored. This ground state breaks the C6{C}_{6}C6​ symmetry of the lattice, resulting in a reduced C2{C}_{2}C2​ symmetry and doubled unit cells while preserving the SU(4) spin-orbital rotation symmetry. Such a nematic quantum spin-orbital liquid state can be characterized by a parton Fermi surface (FS) consisting of open orbits in the Brillouin zone, in contrast to the circular FS of the uniform \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux state.
Previous investigations have suggested that the simplest spin-orbital model on the simplest frustrated lattice can host a nematic quantum spin-orbital liquid state. Namely, the orbital degeneracy of the SU(4) Kugel-Khomskii (KK) model tends to enhance quantum fluctuations and stabilize a quantum spin-orbital liquid exhibiting stripy features on the triangular lattice, as revealed by the state-of-the-art method of the density matrix renormalization group boosted by Gutzwiller projected wave functions. In this work, using the variational quantum Monte Carlo method, we have studied several spin-orbital liquid states, including a uniform πππ flux state, three stripy states, and a plaquette state, on the L×LL\times{}LL×L torus up to L=24L=24L=24. It turns out that one of these stripy states, called the "stripe-II" state, is energetically favored. This ground state breaks the C6C_6C6​ symmetry of the lattice, resulting in a reduced C2C_2C2​ symmetry and doubled unit cells, while preserving the SU(4) spin-orbital rotation symmetry. Such a nematic quantum spin-orbital liquid state can be characterized by a parton Fermi surface (FS) consisting of open orbits in the Brillouin zone, in contrast to the circular FS of the uniform πππ-flux state.
By combining the density matrix renormalization group (DMRG) method with Gutzwiller projected wave functions, we provide clear evidence that the ground state of the SU(4) Kugel-Khomskii spin-orbital model on the triangular lattice can be well described by a ``single Gutzwiller projected wave function with an emergent parton Fermi surface, despite it exhibits strong finite size effect and even-odd discrepancy in quasi-one-dimensional cylinders. This ground state preserves SU(4) symmetry, but spontaneously breaks translational symmetry by doubling the unit cell along one of the lattice vector directions. The finite size effect and even-odd discrepancy can be resolved by the fact that the parton Fermi surface consists of open orbits in the reciprocal space. Thereby, a nematic spin-orbital liquid state is expected in the two-dimensional limit. Furthermore, our DMRG results indicate that the fluctuating stripes are critical and the central charge of each stripe is c=3c=3c=3, in agreement with the SU(4)1_11​ Wess-Zumino-Witten conformal field theory. All these results are consistent with the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem.
We argue for the existence of a liquid ground state in a class of square lattice models of orbitally degenerate insulators. Starting with the SU(4)-symmetric Kugel-Khomskii model, we utilize a Majorana Fermion representation of spin-orbital operators to access unusual phases. Variational wave functions of candidate liquid phases are thus obtained, whose properties are evaluated using variational Monte Carlo. These states are disordered and are found to have excellent energetics and ground state overlap (>40(>40%)(>40 when compared with exact diagonalization on 16-site clusters. We conclude that these are spin-orbital liquid ground states with emergent nodal fermions and Z2{\text{Z}}_{2}Z2​ gauge fields. Connections to spin-3/2 cold-atom systems and properties in the absence of SU(4) symmetry are briefly discussed.
We study the energetics of Gutzwiller projected BCS states of various symmetries for the triangular lattice antiferromagnet with a four-particle ring exchange using variational Monte Carlo methods. In a range of parameters the energetically favored state is found to be a projected ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ paired state which breaks lattice rotational symmetry. We show that the properties of this nematic or orientationally ordered paired spin-liquid state as a function of temperature and pressure can account for many of the experiments on organic materials. We also study the ring-exchange model with ferromagnetic Heisenberg exchange and find that among the studied ans\"atze, a projected fff-wave state is the most favorable.
By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-1/21/21/2 Heisenberg model with the first-neighbor (J1{J}_{1}J1​), second-neighbor (J2{J}_{2}J2​), and additional scalar chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ on the triangular lattice. In the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model with $0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B 92, 041105(R) (2015) and Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403(R) (2015)] find a possible gapped spin liquid with the signal of a competition between a chiral and a Z2{Z}_{2}Z2​ spin liquid. Motivated by the DMRG results, we consider the chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ as a perturbation for this nonmagnetic phase. We find that with growing ${J}_{\ensuremath{\chi}}$, the gapless U(1) Dirac spin liquid, which has the best variational energy for ${J}_{\ensuremath{\chi}}=0$, exhibits the energy instability towards a gapped spin liquid with nontrivial magnetic fluxes and nonzero chiral order. We calculate topological Chern number and ground-state degeneracy, both of which identify this flux state as the chiral spin liquid with fractionalized Chern number C=1/2C=1/2C=1/2 and twofold topological degeneracy. Our results indicate a positive direction to stabilize a chiral spin liquid near the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model.
We study the effective spin-orbital model that describes the magnetism of 4d14{d}^{1}4d1 or 5d15{d}^{1}5d1 Mott insulators in ideal tricoordinated lattices. In the limit of vanishing Hund's coupling, the model has an emergent SU(4) symmetry, which is made explicit by means of a Klein transformation on pseudospin degrees of freedom. Taking the hyperhoneycomb lattice as an example, we employ parton constructions with fermionic representations of the pseudospin operators to investigate possible quantum spin-orbital liquid states. We then use variational Monte Carlo (VMC) methods to compute the energies of the projected wave functions. Our numerical results show that the lowest-energy quantum liquid corresponds to a zero-flux state with a Fermi surface of four-color fermionic partons. In spite of the Fermi surface, we demonstrate that this state is stable against tetramerization. A combination of linear flavor wave theory and VMC applied to the complete microscopic model also shows that this liquid state is stable against the formation of collinear long-range order.