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arXiv:1105.3611 · Physical Review A

Gauge fields emerging from time-reversal symmetry breaking for spin-5/2 fermions in a honeycomb lattice

Gergely Szirmai, E. Szirmai, Alejandro Zamora, Maciej Lewenstein

Published Jul 29, 2011• 46 citations
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Abstract

We propose an experimentally feasible setup with ultracold alkaline-earth-metal atoms to simulate the dynamics of U(1) lattice gauge theories in 2 $+$ 1 dimensions with a Chern-Simons term. To this end we consider the ground-state properties of spin-5/2 alkaline-earth-metal fermions in a honeycomb lattice. We use the Gutzwiller projected variational approach in the strongly repulsive regime in the case of filling 1/6. The ground state of the system is a chiral spin-liquid state with $2\ensuremath{\pi}/3$ flux per plaquette, which violates time-reversal invariance. We demonstrate that due to the breaking of time-reversal symmetry the system exhibits quantum Hall effect and chiral edge states. We relate the experimentally accessible spin fluctuations to the emerging gauge-field dynamics. We discuss also properties of the lowest energy competing orders.

DOI
10.1103/physreva.84.011611

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We propose an experimentally feasible setup with ultracold alkaline-earth-metal atoms to simulate the dynamics of U(1) lattice gauge theories in 2 +++ 1 dimensions with a Chern-Simons term. To this end we consider the ground-state properties of spin-5/2 alkaline-earth-metal fermions in a honeycomb lattice. We use the Gutzwiller projected variational approach in the strongly repulsive regime in the case of filling 1/6. The ground state of the system is a chiral spin-liquid state with 2\ensuremathπ/32\ensuremath{\pi}/32\ensuremathπ/3 flux per plaquette, which violates time-reversal invariance. We demonstrate that due to the breaking of time-reversal symmetry the system exhibits quantum Hall effect and chiral edge states. We relate the experimentally accessible spin fluctuations to the emerging gauge-field dynamics. We discuss also properties of the lowest energy competing orders.
We study spin liquid phases of spin-5/2 alkaline-earth-metal atoms on a honeycomb lattice at finite temperatures. Our analysis is based on a Gutzwiller projection variational approach recast to a path-integral formalism. In the framework of a saddle-point approximation we determine spin liquid phases with lowest free energy and study their temperature dependence. We identify a critical temperature, where all the spin liquid phases melt and the system goes to the paramagnetic phase. We also study the stability of the saddle-point solutions and show that a time-reversal symmetry breaking state, a so-called chiral spin liquid phase, is realized even at finite temperatures. We also determine the spin structure factor, which, in principle, is an experimentally measurable quantity and is the basic tool to map the spectrum of elementary excitations of the system.
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 $N\ensuremath{\ge}5$ 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.
We analyze the physics of cold atoms in honeycomb optical lattices with on-site repulsion and spin-dependent hopping that breaks time-reversal symmetry. Such systems, at half filling and large on-site repulsion, have been proposed as a possible realization of the Kitaev model. The spin-dependent hopping breaks the spin degeneracy and, if strong enough, leads to four nonoverlapping bands in the noninteracting limit. These bands carry nonzero Chern number and therefore the noninteracting system has nonzero angular momentum and chiral edge states at 1/4 and 3/4 filling. We have investigated the effect of interactions on a quarter-filled system using the variational cluster perturbation theory and found that the critical spin-dependent hopping that separates the metal from the quantum Hall state is affected by interactions.
We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin-12\frac{1}{2}21​ system in a honeycomb lattice, to a chiral Z2{\mathbb{Z}}_{2}Z2​ spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral Z2{\mathbb{Z}}_{2}Z2​ state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.
We study a class of SU(N)\text{SU}(N)SU(N) Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-NNN limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter kkk, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.
Strongly interacting bosons that live in a lattice with degeneracy in its lowest-energy band experience frustration that can prevent the formation of a Bose-Einstein condensate. Such systems form an ideal playground to investigate spin-liquid behavior. We use the variational principle and the Chern-Simons technique of fermionization of hard-core bosons on the kagome lattice to find that below lattice filling fraction $\ensuremath{\nu}=\frac{1}{3}$ the system favors a topologically ordered chiral spin-liquid state that is gapped in bulk, spontaneously breaks time-reversal symmetry, and supports massless chiral bosonic edge mode. We construct the many-body variational wave function of the state and show that the corresponding energy coincides with the energy of the flat band. This result proves that the ground state of the system cannot stabilize a Bose condensate below $\ensuremath{\nu}=\frac{1}{3}$. The fermionization and variational scheme we outline apply to any non-Bravais lattice. We distinguish between the roles played by the Chern-Simons gauge field in lattices with a flat band and those exhibiting a moatlike dispersion (which is degenerate along a closed contour in the reciprocal space). We also suggest experimental probes to differentiate the proposed ground state from a condensate.