Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0606743 · Physical Review Letters

Quantum Stripe Ordering in Optical Lattices

Congjun Wu, W. Vincent Liu, Joel E. Moore, S. Das Sarma

Published Nov 10, 2006• 80 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.97.190406

Related papers

Exotic phases of interacting p -band bosons

F. Hébert, Zi Cai, V. G. Rousseau +3 more

We study a model of interacting bosons that occupy the first excited $p$-band states of a two-dimensional optical lattice. In contrast to the much studied single band Bose-Hubbard Hamiltonian, this more complex model allows for nontrivial superfluid phases associated with condensation at nonzero momentum and staggered order of the orbital angular momentum in addition to the superfluid-Mott insulator transition. More specifically, we observe staggered orbital angular momentum order in the Mott phase at commensurate filling and superfluidity at all densities. We also observe a transition between the staggered angular momentum superfluid phase and a striped superfluid, with an alternation of the phase of the superfluid along one direction. The transition between these two phases was observed in a recent experiment, which is then qualitatively well described by our model.

arXiv:1304.0554•Physical Review B•Jun 17, 2013•21 citations

Orbital order in a bosonic p -band triangular lattice

Hua Chen, Xiaoming Xie

We present a detailed study of the Bose-Hubbard model in a $p$-band triangular lattice by focusing on the evolution of orbital order across the superfluid-Mott insulator transition. Two distinct phases are found in the superfluid regime. One of these phases adiabatically connects the weak interacting limit. This phase is characterized by the intertwining of axial ${p}_{\ifmmode\pm\else\textpm\fi{}}={p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ and in-plane ${p}_{\ensuremath{\theta}}=cos\ensuremath{\theta}{p}_{x}+sin\ensuremath{\theta}{p}_{y}$ orbital orders, which break the time-reversal symmetry and lattice symmetries simultaneously. In addition, the calculated Bogoliubov excitation spectrum gaps the original Dirac points in the single-particle spectrum but exhibits emergent Dirac points. The other superfluid phase in close proximity to the Mott insulator with unit boson filling shows a detwined in-plane ferro-orbital order. Finally, an orbital exchange model is constructed for the Mott insulator phase. Its classical ground state has an emergent $\mathrm{S}\mathrm{O}(2)$ rotational symmetry in the in-plane orbital space and therefore enjoys an infinite degeneracy, which is ultimately lifted by the orbital fluctuation via the order by disorder mechanism. Our systematic analysis suggests that the in-plane ferro-orbital order in the Mott insulator phase agrees with and likely evolves from the latter superfluid phase.

arXiv:2102.05319•Physical review. B./Physical review. B•May 24, 2021•4 citations

Evidence for Quantum Stripe Ordering in a Triangular Optical Lattice

X. C. Wang, Guang-Quan Luo, Jin-Yu Liu +5 more

Understanding strongly correlated quantum materials, such as high-T_{c} superconductors, iron-based superconductors, and twisted bilayer graphene systems, remains as one of the outstanding challenges in condensed matter physics. Quantum simulation with ultracold atoms in particular optical lattices, which provide orbital degrees of freedom, is a powerful tool to contribute new insights to this endeavor. Here, we report the experimental realization of an unconventional Bose-Einstein condensate of ^{87}Rb atoms populating degenerate p orbitals in a triangular optical lattice, exhibiting remarkably long coherence times. Using time-of-flight spectroscopy, we observe that this state spontaneously breaks the rotational symmetry and its momentum spectrum agrees with the theoretically predicted coexistence of exotic stripe and loop-current orders. Like certain strongly correlated electronic systems with intertwined orders, such as high-T_{c} cuprate superconductors, twisted bilayer graphene, and the recently discovered chiral density-wave state in kagome superconductors AV_{3}Sb_{5} (A=K, Rb, Cs), the newly demonstrated quantum state, in spite of its markedly different energy scale and the bosonic quantum statistics, exhibits multiple symmetry breakings at ultralow temperatures. These findings hold the potential to enhance our comprehension of the fundamental physics governing these intricate quantum materials.

arXiv:2211.05578•Physical Review Letters•Nov 27, 2023•9 citations

Time-Reversal Symmetry Breaking of p -Orbital Bosons in a One-Dimensional Optical Lattice

Xiaopeng Li, Zixu Zhang, W. Vincent Liu

We study bosons loaded in a one-dimensional optical lattice of twofold p-orbital degeneracy at each site. Our numerical simulations find an anti-ferro-orbital p(x)+ip(y), a homogeneous p(x) Mott-insulator phase, and two kinds of superfluid phases distinguished by the orbital order (anti-ferro-orbital and paraorbital). The anti-ferro-orbital order breaks time-reversal symmetry. Experimentally observable evidence is predicted for the phase transition between the two different superfluid phases. We also discover that the quantum noise measurement is able to provide a concrete evidence of time-reversal symmetry breaking in the first Mott phase.

arXiv:1110.3364•Physical Review Letters•Apr 25, 2012•42 citations

Unconventional Superfluid Order in the F Band of a Bipartite Optical Square Lattice

Matthias Ölschläger, Georg Wirth, Andreas Hemmerich

We report on the first observation of bosons condensed into the energy minima of an F band of a bipartite square optical lattice. Momentum spectra indicate that a truly complex-valued staggered angular momentum superfluid order is established. The corresponding wave function is composed of alternating local F2x3-3x ± iF2y3-3y orbits and local S orbits residing in the deep and shallow wells of the lattice, which are arranged as the black and white areas of a checkerboard. A pattern of staggered vortical currents arises, which breaks time-reversal symmetry and the translational symmetry of the lattice potential. We have measured the populations of higher order Bragg peaks in the momentum spectra for varying relative depths of the shallow and deep lattice wells and find remarkable agreement with band calculations.

arXiv:1008.4752•Physical Review Letters•Jan 6, 2011•94 citations

Rotation-Symmetry-Enforced Coupling of Spin and Angular Momentum for p -Orbital Bosons

Yongqiang Li, Jian‐Min Yuan, Andreas Hemmerich +1 more

Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into p-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.

arXiv:1803.04429•Physical Review Letters•Aug 28, 2018•14 citations
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.
We study a model of interacting bosons that occupy the first excited ppp-band states of a two-dimensional optical lattice. In contrast to the much studied single band Bose-Hubbard Hamiltonian, this more complex model allows for nontrivial superfluid phases associated with condensation at nonzero momentum and staggered order of the orbital angular momentum in addition to the superfluid-Mott insulator transition. More specifically, we observe staggered orbital angular momentum order in the Mott phase at commensurate filling and superfluidity at all densities. We also observe a transition between the staggered angular momentum superfluid phase and a striped superfluid, with an alternation of the phase of the superfluid along one direction. The transition between these two phases was observed in a recent experiment, which is then qualitatively well described by our model.
We present a detailed study of the Bose-Hubbard model in a ppp-band triangular lattice by focusing on the evolution of orbital order across the superfluid-Mott insulator transition. Two distinct phases are found in the superfluid regime. One of these phases adiabatically connects the weak interacting limit. This phase is characterized by the intertwining of axial ${p}_{\ifmmode\pm\else\textpm\fi{}}={p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ and in-plane ${p}_{\ensuremath{\theta}}=cos\ensuremath{\theta}{p}_{x}+sin\ensuremath{\theta}{p}_{y}$ orbital orders, which break the time-reversal symmetry and lattice symmetries simultaneously. In addition, the calculated Bogoliubov excitation spectrum gaps the original Dirac points in the single-particle spectrum but exhibits emergent Dirac points. The other superfluid phase in close proximity to the Mott insulator with unit boson filling shows a detwined in-plane ferro-orbital order. Finally, an orbital exchange model is constructed for the Mott insulator phase. Its classical ground state has an emergent SO(2)\mathrm{S}\mathrm{O}(2)SO(2) rotational symmetry in the in-plane orbital space and therefore enjoys an infinite degeneracy, which is ultimately lifted by the orbital fluctuation via the order by disorder mechanism. Our systematic analysis suggests that the in-plane ferro-orbital order in the Mott insulator phase agrees with and likely evolves from the latter superfluid phase.
Understanding strongly correlated quantum materials, such as high-T_{c} superconductors, iron-based superconductors, and twisted bilayer graphene systems, remains as one of the outstanding challenges in condensed matter physics. Quantum simulation with ultracold atoms in particular optical lattices, which provide orbital degrees of freedom, is a powerful tool to contribute new insights to this endeavor. Here, we report the experimental realization of an unconventional Bose-Einstein condensate of ^{87}Rb atoms populating degenerate p orbitals in a triangular optical lattice, exhibiting remarkably long coherence times. Using time-of-flight spectroscopy, we observe that this state spontaneously breaks the rotational symmetry and its momentum spectrum agrees with the theoretically predicted coexistence of exotic stripe and loop-current orders. Like certain strongly correlated electronic systems with intertwined orders, such as high-T_{c} cuprate superconductors, twisted bilayer graphene, and the recently discovered chiral density-wave state in kagome superconductors AV_{3}Sb_{5} (A=K, Rb, Cs), the newly demonstrated quantum state, in spite of its markedly different energy scale and the bosonic quantum statistics, exhibits multiple symmetry breakings at ultralow temperatures. These findings hold the potential to enhance our comprehension of the fundamental physics governing these intricate quantum materials.
We study bosons loaded in a one-dimensional optical lattice of twofold p-orbital degeneracy at each site. Our numerical simulations find an anti-ferro-orbital p(x)+ip(y), a homogeneous p(x) Mott-insulator phase, and two kinds of superfluid phases distinguished by the orbital order (anti-ferro-orbital and paraorbital). The anti-ferro-orbital order breaks time-reversal symmetry. Experimentally observable evidence is predicted for the phase transition between the two different superfluid phases. We also discover that the quantum noise measurement is able to provide a concrete evidence of time-reversal symmetry breaking in the first Mott phase.
We report on the first observation of bosons condensed into the energy minima of an F band of a bipartite square optical lattice. Momentum spectra indicate that a truly complex-valued staggered angular momentum superfluid order is established. The corresponding wave function is composed of alternating local F2x3-3x ± iF2y3-3y orbits and local S orbits residing in the deep and shallow wells of the lattice, which are arranged as the black and white areas of a checkerboard. A pattern of staggered vortical currents arises, which breaks time-reversal symmetry and the translational symmetry of the lattice potential. We have measured the populations of higher order Bragg peaks in the momentum spectra for varying relative depths of the shallow and deep lattice wells and find remarkable agreement with band calculations.
Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into p-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.