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arXiv:1206.5103 · Physical Review B

Frustrated Bose-Einstein condensates with noncollinear orbital ordering

Zi Cai, Yu Wang, Congjun Wu

Published Aug 30, 2012• 11 citations
arXivHTMLPDF

Abstract

We investigate the unconventional Bose-Einstein condensations with the orbital degree of freedom in the three-dimensional cubic optical lattice, which gives rise to various exotic features absent in conventional scalar and spinor Bose-Einstein condensations. Orbital angular momentum moments are formed on lattice sites breaking time-reversal symmetry spontaneously. Furthermore, they exhibit orbital frustrations and develop a chiral ordering selected by the ``order-from-disorder'' mechanism.

DOI
10.1103/physrevb.86.060517

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Feynman's "no-node" theorem states that the conventional many-body ground state wavefunctions of bosons in the coordinate representation are positive definite. This implies that time-reversal symmetry cannot be spontaneously broken. In this article, we review our progress in studying a class of new states of unconventional Bose–Einstein condensations beyond this paradigm. These states can either be the long-lived metastable states of ultracold bosons in high orbital bands in optical lattices as a result of the "orbital Hund's rule" interaction, or the ground states of spinful bosons with spin-orbit coupling linearly dependent on momentum. In both cases, Feynman's argument does not apply. The resultant many-body wavefunctions are complex-valued and thus break time-reversal symmetry spontaneously. Exotic phenomena in these states include the Bose–Einstein condensation at nonzero momentum, the ordering of orbital angular momentum moments, the half-quantum vortex, and the spin texture of skyrmions.
The breaking of time-reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool, we use a matter-wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low-energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.
According to the “no-node" theorem, the many-body ground state wavefunctions of conventional Bose—Einstein condensations (BEC) are positive-definite, thus time-reversal symmetry cannot be spontaneously broken. We find that multi-component bosons with spin-orbit coupling provide an unconventional type of BECs beyond this paradigm. We focus on a subtle case of isotropic Rashba spin-orbit coupling and the spin-independent interaction. In the limit of the weak confining potential, the condensate wavefunctions are frustrated at the Hartree—Fock level due to the degeneracy of the Rashba ring. Quantum zero-point energy selects the spin-spiral type condensate through the “order-from-disorder" mechanism. In a strong harmonic confining trap, the condensate spontaneously generates a half-quantum vortex combined with the skyrmion type of spin texture. In both cases, time-reversal symmetry is spontaneously broken. These phenomena can be realized in both cold atom systems with artificial spin-orbit couplings generated from atom-laser interactions and exciton condensates in semi-conductor systems.
We predict the robust existence of a novel quantum orbital stripe order in the p-band Bose-Hubbard model of two-dimensional triangular optical lattices with cold bosonic atoms. An orbital angular momentum moment is formed on each site exhibiting a stripe order both in the superfluid and Mott-insulating phases. The stripe order spontaneously breaks time-reversal, lattice translation, and rotation symmetries. In addition, it induces staggered plaquette bond currents in the superfluid phase. Possible signatures of this stripe order in the time of flight experiment are discussed.