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arXiv:1501.04086 · Physical Review A

Synthetic-gauge-field stabilization of the chiral-spin-liquid phase

Gang Chen, Kaden R. A. Hazzard, Ana María Rey, Michael Hermele

Published Jun 9, 2016• 37 citations
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Abstract

We explore the phase diagram of the $\mathrm{SU}(N)$ Hubbard models describing fermionic alkaline-earth-metal atoms in a square optical lattice with, on average, one atom per site, using a slave rotor mean-field approach. We find that the chiral spin liquid (CSL) predicted for $N\ensuremath{\ge}5$ and large interactions passes through a fractionalized state with a spinon Fermi surface as interactions are decreased before transitioning to a weakly interacting metal. We show that by adding a uniform artificial gauge field with $2\ensuremath{\pi}/N$ flux per plaquette, the CSL becomes the ground state for all $N\ensuremath{\ge}3$ at intermediate interactions, persists to weaker interactions, and exhibits a larger spin gap. For $N\ensuremath{\ge}5$ we find the CSL is the ground state everywhere the system is a Mott insulator. The gauge field stabilization of the CSL at lower interactions, and thus at weaker lattice depths, together with the increased spin gap, can relax the temperature constraints required for its experimental realization in ultracold atom systems.

DOI
10.1103/physreva.93.061601

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•Jan 1, 2020•0 citations
We explore the phase diagram of the SU(N)\mathrm{SU}(N)SU(N) Hubbard models describing fermionic alkaline-earth-metal atoms in a square optical lattice with, on average, one atom per site, using a slave rotor mean-field approach. We find that the chiral spin liquid (CSL) predicted for N\ensuremath≥5N\ensuremath{\ge}5N\ensuremath≥5 and large interactions passes through a fractionalized state with a spinon Fermi surface as interactions are decreased before transitioning to a weakly interacting metal. We show that by adding a uniform artificial gauge field with 2\ensuremathπ/N2\ensuremath{\pi}/N2\ensuremathπ/N flux per plaquette, the CSL becomes the ground state for all N\ensuremath≥3N\ensuremath{\ge}3N\ensuremath≥3 at intermediate interactions, persists to weaker interactions, and exhibits a larger spin gap. For N\ensuremath≥5N\ensuremath{\ge}5N\ensuremath≥5 we find the CSL is the ground state everywhere the system is a Mott insulator. The gauge field stabilization of the CSL at lower interactions, and thus at weaker lattice depths, together with the increased spin gap, can relax the temperature constraints required for its experimental realization in ultracold atom systems.
We study an optical Raman square lattice with U(1)\mathrm{U}(1)U(1) synthetic gauge flux to show a chiral spin liquid (CSL) phase for cold atoms based on slave-rotor and spinon mean-field theories, respectively. An effective U(1) gauge flux generated by Raman potentials plays a major role in realizing the CSL phase. By using slave-rotor techniques we find the CSL phase at a strong on-site Fermi Hubbard interacting regime. For large value of Hubbard interaction, we derive an effective spin model including up to the four spin interactions. By spinon mean-field analysis it is shown that the CSL phase is stabilized in the case of strong magnetic frustration. Two mean-field approximation methods give consistent phase diagrams and provide qualitative numerical evidence of the CSL phase.
We study Mott insulators of fermionic alkaline earth atoms, described by Heisenberg spin models with enhanced SU(N) symmetry. In dramatic contrast to SU(2) magnetism, more than two spins are required to form a singlet. On the square lattice, the classical ground state is highly degenerate and magnetic order is thus unlikely. In a large-N limit, we find a chiral spin liquid ground state with topological order and Abelian fractional statistics. We discuss its experimental detection. Chiral spin liquids with non-Abelian anyons may also be realizable with alkaline earth atoms.
Motivated by recent progress in the realization of artificial gauge fields and SU(NNN) Mott insulators using alkaline-earth-like atoms in optical lattices, we develop an unbiased SU(NNN) real-space dynamical mean-field theory approach to study the effect of spin-orbit coupling and on-site Hubbard interaction UUU on SU(3) fermionic systems. We investigate the behavior of the local magnetization, double occupancies, and the triple occupancy versus the Hubbard interaction across the metal to Mott insulator transition. We map out the magnetic phase diagram in the large-UUU limit and show that the spin-orbit coupling can stabilize long-range orders such as ferromagnets, spirals, and stripes with different orientations in SU(3) Mott insulators.
We show that, in the presence of a π/2 artificial gauge field per plaquette, Mott insulating phases of ultracold fermions with SU(N) symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by an approximate ground space of N low-lying singlets for periodic boundary conditions, and by chiral edge states described by the SU(N)_{1} Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for N between 3 and 9, and by a parton construction based on a set of N Gutzwiller projected fermionic wave functions with flux π/N per triangular plaquette. Experimental implications are briefly discussed.
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.
In this thesis, the Hubbard model with hopping amplitude, t, and on-site interaction, U, in the Mott-insulating regime is explored for unconventional magnetic phases at T = 0 K. These phases are captured within effective low-energy descriptions, and are triggered by quantum fluctuations due to finite hopping. We aim for unconventional phases from two mechanisms, an increased symmetry SU(N) with N > 2 compared to the most common SU(2), and highly frustrated interactions. For SU(N)-symmetric fermions in artificial gauge fields, chiral spin liquid phases (CSLs) are investigated. They are determined within effective spin models, which we derive in order six in t/U for the honeycomb lattice for general values of N, and study in detail for the triangular lattice up to order five in t/U for SU(3) and SU(7) by exact diagonalisations and variational Monte Carlo simulations. On the triangular lattice, the third-order effective model is the J -K model with nearest-neighbour coupling, J ∈ R, and ring exchange, K ∈ C. For SU(3) and Im K = 0, the three-sublattice long-range ordered phase is ruled out by a spontaneously time-reversal symmetry breaking π/3-flux CSL through � nite hopping. Further, for Im K = 0, a lattice nematic, a 120◦ long-ranged ordered, and the ferromagnetically ordered phase occur. The CSL is found to be extended to Re K = 0. It is also present in the fourth- and fifth-order model in the most relevant π-flux case, where time-reversal symmetry is spontaneously broken. Together with the estimated metal-insulator transition point at larger hoppings, this CSL is therefore expected to be realised in the SU(3) Hubbard model. The analogous CSL is also predicted for SU(7)-symmetric fermions. Experimental realisations with ultra-cold atoms in optical lattices are described. SrCu2(BO3)2 is a highly frustrated quantum magnet. At ambient pressure it is captured by the spin-1/2 Shastry-Sutherland model, which yields the effective strong-coupling description of the associated SU(2) Hubbard model. To describe SrCu2(BO3)2 under higher pressure, we extend the Shastry-Sutherland model by asymmetric coupling strengths, and find that asymmetries of a few percent are sufficient to replace the empty plaquette singlet phase (EPP) of the Shastry-Sutherland model with the filled plaquette singlet phase (FPP) that is found to be identical to a quasi one-dimensional Haldane phase. These results are achieved by series expansions and infinite projected entangled-pair states. The dispersions and dynamic structure factors of both plaquette phases are derived. The EPP naturally hosts one low-lying triplon, whereas the FPP has two. Therefore, the FPP appears to be the likely candidate for SrCu2(BO3)2 under pressure. The excitations of both phases show some agreement with experimental data if the exchange amplitudes change more than expected. Further measurements are suggested. At last, an orthogonal-plaquette model of spins-1/2 is introduced and proven to host two exact valence bond crystals based on plaquette singlets with an extensive degeneracy at the phase transition. This model exhibits an extensive number of conserved quantities, which is used for further investigations of the phase diagram.