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arXiv:1510.00380 · Physical Review A

Bose-Einstein condensation in a frustrated triangular optical lattice

Peter Janzen, Wen-Min Huang, Ludwig Mathey

Published Dec 15, 2016• 2 citations
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Abstract

The recent experimental condensation of ultracold atoms in a triangular optical lattice with a negative effective tunneling parameter paves the way for the study of frustrated systems in a controlled environment. Here, we explore the critical behavior of the chiral phase transition in such a frustrated lattice in three dimensions. We represent the low-energy action of the lattice system as a two-component Bose gas corresponding to the two minima of the dispersion. The contact repulsion between the bosons separates into intra- and intercomponent interactions, referred to as ${V}_{0}$ and ${V}_{12}$, respectively. We first employ a Huang-Yang-Luttinger approximation of the free energy. For ${V}_{12}/{V}_{0}=2$, which corresponds to the bare interaction, this approach suggests a first-order phase transition, at which both the $\mathrm{U}(1)$ symmetry of condensation and the ${\mathbb{Z}}_{2}$ symmetry of the emergent chiral order are broken simultaneously. Furthermore, we perform a renormalization-group calculation at one-loop order. We demonstrate that the coupling regime $0<{V}_{12}/{V}_{0}\ensuremath{\le}1$ shares the critical behavior of the Heisenberg fixed point at ${V}_{12}/{V}_{0}=1$. For ${V}_{12}/{V}_{0}>1$ we show that ${V}_{0}$ flows to a negative value, while ${V}_{12}$ increases and remains positive. This results in a breakdown of the effective quartic-field theory due to a cubic anisotropy and, again, suggests a discontinuous phase transition.

DOI
10.1103/physreva.94.063614

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The recent experimental condensation of ultracold atoms in a triangular optical lattice with a negative effective tunneling parameter paves the way for the study of frustrated systems in a controlled environment. Here, we explore the critical behavior of the chiral phase transition in such a frustrated lattice in three dimensions. We represent the low-energy action of the lattice system as a two-component Bose gas corresponding to the two minima of the dispersion. The contact repulsion between the bosons separates into intra- and intercomponent interactions, referred to as V0{V}_{0}V0​ and V12{V}_{12}V12​, respectively. We first employ a Huang-Yang-Luttinger approximation of the free energy. For V12/V0=2{V}_{12}/{V}_{0}=2V12​/V0​=2, which corresponds to the bare interaction, this approach suggests a first-order phase transition, at which both the U(1)\mathrm{U}(1)U(1) symmetry of condensation and the Z2{\mathbb{Z}}_{2}Z2​ symmetry of the emergent chiral order are broken simultaneously. Furthermore, we perform a renormalization-group calculation at one-loop order. We demonstrate that the coupling regime 0<V12/V0\ensuremath≤10<{V}_{12}/{V}_{0}\ensuremath{\le}10<V12​/V0​\ensuremath≤1 shares the critical behavior of the Heisenberg fixed point at V12/V0=1{V}_{12}/{V}_{0}=1V12​/V0​=1. For V12/V0>1{V}_{12}/{V}_{0}>1V12​/V0​>1 we show that V0{V}_{0}V0​ flows to a negative value, while V12{V}_{12}V12​ increases and remains positive. This results in a breakdown of the effective quartic-field theory due to a cubic anisotropy and, again, suggests a discontinuous phase transition.
We study the critical behavior of Bose–Einstein condensation in the second band of a bipartite optical square lattice in a renormalization group framework at one-loop order. Within our field theoretical representation of the system, we approximate the system as a two-component Bose gas in three dimensions. We demonstrate that the system is in a different universality class than the previously studied condensation in a frustrated triangular lattice due to an additional Umklapp scattering term, which stabilizes the chiral superfluid order at low temperatures. We derive the renormalization group flow of the system and show that this order persists in the low energy limit. Furthermore, the renormalization flow suggests that the phase transition from the thermal phase to the chiral superfluid state is first order.
The study of geometrically frustrated many-body quantum systems is of central importance to uncover novel quantum mechanical effects. We design a scheme where ultracold bosons trapped in a one-dimensional state-dependent optical lattice are modeled by a frustrated Bose-Hubbard Hamiltonian. A derivation of the Hamiltonian parameters based on Cesium atoms, further show large tunability of contact and nearest-neighbor interactions. For pure contact repulsion, we discover the presence of two phases peculiar to frustrated quantum magnets: the bond-order-wave insulator with broken inversion symmetry and a chiral superfluid. When the nearest-neighbor repulsion becomes sizable, a further density-wave insulator with broken translational symmetry can appear. We show that the phase transition between the two spontaneously symmetry-broken phases is continuous, thus representing a one-dimensional deconfined quantum critical point not captured by the Landau-Ginzburg-Wilson symmetry-breaking paradigm. Our results provide a solid ground to unveil the novel quantum physics induced by the interplay of nonlocal interactions, geometrical frustration, and quantum fluctuations.
We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2{\mathbb{Z}}_{2}Z2​ symmetry, coupled by effective density-density interactions, the global symmetry is ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$. At the BEC transition, it may break into ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when both components condense simultaneously, or to $\mathrm{U}(1)\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly decaying scaling corrections arising from the onsite interspecies interaction.
We study a spinor (two-component) Bose gas confined in a one-dimensional double-valley optical lattice which has a double-well structure in momentum space. Based on field theory analysis, it is found that spinor bosons in the double-valley band may form a spin-charge mixed chiral spin quasicondensate under certain conditions. Our numerical calculations in a concrete $\ensuremath{\pi}$-flux triangular ladder system confirm the robustness of the chiral spin order against interactions and quantum fluctuations. This exotic atomic Bose-Einstein condensate exhibits spatially staggered spin loop currents without any charge dynamics despite the complete absence of spin-orbit coupling in the system, creating an interesting approach to atom spintronics. The entanglement entropy scaling allows us to extract conformal-field-theory central charge and establish the low-energy effective field theory for the chiral spin condensate as a two-component Luttinger liquid. Our predictions should be detectable in atomic experiments through spin-resolved time-of-flight techniques.
The physics of quantum degenerate cold gases in optical lattices has rapidly grown to an extremely dynamic field over the past few years. In the experimental realizations, however, mainly cubic symmetries (or their two- and one-dimensional projections) are considered. We have implemented an optical lattice with an underlying triangular symmetry in order to investigate strongly correlated ultra-cold atoms in a novel experimental geometry. Exhibiting an explicit polarization dependence (compare figure 1) the optical lattice realized here should allow for the creation and analysis of thus far unexplored magnetic phases. Experiments attending to the quantum phase transition from a superfluid to a Mott-insulating state in a three- as well as in a two-dimensional system with triangular symmetry have been performed. Similarities as well as differences to the findings obtained in cubic lattices will be discussed and can be attributed to the inherent differences in the crucial lattice parameters such as tunneling energy J and on-site interaction U.
We study the superfluid and insulating phases of interacting bosons on the triangular lattice with an inverted dispersion, corresponding to frustrated hopping between sites. The resulting single-particle dispersion has multiple minima at nonzero wave vectors in momentum space, in contrast to the unique zero-wave-vector minimum of the unfrustrated problem. As a consequence, the superfluid phase is unstable against developing additional chiral order that breaks time-reversal (T\mathcal{T}T) and parity (P\mathcal{P}P) symmetries by forming a condensate at nonzero wave vector. We demonstrate that the loss of superfluidity can lead to an even more exotic phase, the chiral Mott insulator, with nontrivial current order that breaks T,P\mathcal{T},\mathcal{P}T,P. These results are obtained via variational estimates, as well as a combination of bosonization and density-matrix renormalization group of triangular ladders, which, taken together, permit a fairly complete characterization of the phase diagram. We discuss the relevance of these phases to optical lattice experiments, as well as signatures of chiral symmetry breaking in time-of-flight images.