Abstract
Recent experiments on porbital atomic bosons have suggested the emergence of a spectacular ultracold superfluid with staggered orbital currents in optical lattices. This raises fundamental questions concerning the effects of thermal fluctuations as well as possible ways of directly observing such chiral order. Here we show via Monte Carlo simulations that thermal fluctuations destroy this superfluid in an unexpected twostep process, unveiling an intermediate normal phase with spontaneously broken timereversal symmetry, dubbed a ‘chiral Bose liquid’. For integer fillings (n≥2) in the chiral Mott regime, thermal fluctuations are captured by an effective orbital Ising model, and Onsager’s powerful exact solution is adopted to determine the transition from this intermediate liquid to the paraorbital normal phase at high temperature. A lattice quench is designed to convert the staggered angular momentum, previously thought by experts difficult to directly probe, into coherent orbital oscillations, providing a timeresolved dynamical signature of chiral order.
Introduction
Orbital degrees of freedom and interactions play a crucial role in the emergence of many complex phases in solid state materials. High temperature superconductivity in the cuprates^{1} and pnictides^{2}, colossal magnetoresistance observed in Mn oxides^{3}, and chiral pwave superconductivity proposed in Sr_{2}RuO_{4} (ref. 4), are all nucleated by strong correlation effects in a multiorbital setting^{5}. For ultracold atomic gases, interaction effects combined with the band topology of p orbitals^{6} have been argued to lead to exotic topological or superfluid (SF) phases for fermions^{7,8,9,10,11,12,13} as well as bosons^{14,15,16,17,18,19,20,21,22,23,24,25}. Interactions are predicted to drive a semimetal to topological insulator quantum phase transition in twodimensions (2D) for fermions in p_{x}, p_{y} and orbitals^{12}, while interacting porbital atomic fermions in threedimensions (3D) could lead to axial orbital order^{26}. For weakly interacting 2D lattice bosons in p_{x} and p_{y} orbitals the ground state is proposed to be a SF with staggered p_{x}±ip_{y} order^{15}; such order is also found for onedimensional (1D) strongly interacting porbital bosons^{24}. For bosons, these exotic phases can result from a particularly simple effect: repulsive contact interactions favour a maximization of the local angular momentum _{z}, a bosonic variant of the atomic Hund’s rule for electrons^{15,22,25}.
Here we study porbital phases beyond their ground state properties, focusing on open fundamental issues such as thermal and dynamical effects, to be elaborated below. Our work is motivated by recent experiments that have successfully prepared longlived metastable phases of weakly interacting ^{87}Rb atoms in p orbitals^{21,27}. In the deep lattice regime, this experimental system is well approximated by a tight binding model on a checkerboard optical lattice with bosons in the p_{x}, p_{y} and s orbital degrees of freedom (see Fig. 1). The Hamiltonian of the model is obtained by extending the early theoretical studies^{14,15,22,28} to the checkerboard lattice configuration used in the recent experiments of Wirth et al.^{21} and Ölschläger et al.^{27} Restricting ourselves to nearestneighbor tunnelling, the Hamiltonian is H=H_{tun}+H_{loc}, with tunnelling and local terms^{27},
Here, b_{x}(x), b_{y}(x) and b_{s}(x) are bosonic annihilation operators of p_{x}, p_{y} and s orbitals at site x. The position vectors r=r_{x}a_{x}+r_{y}a_{y}, with integers r_{x} and r_{y}. The vector a_{x} (a_{y}) is the primitive vector of the square lattice in the x (y) direction (Fig. 1). The positions of s orbitals are , , and . The density operators are defined as and . The angular momentum operator is
Note that a square lattice C_{4} rotational symmetry has been assumed.
With an analysis of the timeofflight momentum distribution, Wirth et al.^{21} and Ölschläger et al.^{27} found evidence suggesting a staggered p_{x}±ip_{y} SF. However, a direct measurement of its key property—the angular momentum order—remains a challenge. This is especially crucial in the absence of SF coherence, since quantum or thermal fluctuations may kill superfluidity while preserving chiral angular momentum order. This issue is also relevant to chiral electronic fluids proposed to emerge in Sr_{2}RuO_{4} (refs 29, 30, 31), doped graphene^{32} and the pseudogap state of hightemperature superconductors^{33,34,35}.
While previous work has focused on the ground state properties of such unconventional Bose SFs, there remain important outstanding issues to be addressed. First, how do thermal or quantum fluctuations, which are important in any experimental setting, melt these unconventional SF states? Second, which experiment can directly detect the spatially modulated angular momentum underlying these unusual quantum states? And finally, what can one learn from the timeresolved dynamics of such complex orbital states?
This brings us to two central results responding to the above open questions. First, using classical Monte Carlo simulations of an effective model of interacting p_{x} and p_{y} bosons, we show that thermal fluctuations lead to a twostep melting of the staggered p_{x}±ip_{y} SF ground state. Sandwiched between a lower temperature Berezinskii–Kosterlitz–Thouless (BKT) transition at which superfluidity is lost, and a higher temperature Ising transition at which timereversal symmetry is restored, lies a ‘chiral Bose liquid’ with spontaneously broken timereversal symmetry. In other words, it is a remarkable state of matter that is chiral but not SF. For large Hubbard repulsion at integer fillings, n≥2, a strong coupling expansion yields Mott insulating states with staggered p_{x}±ip_{y} order. As shown schematically in Fig. 1, this opens up a wide window in the phase diagram where staggered angular momentum order persists even in the absence of superfluidity. Our work represents the first study of the finite temperature properties of staggered p_{x}±ip_{y} chiral Bose fluids. Second, mapping the p_{x}, p_{y} orbitals onto an effective pseudospin1/2 degree of freedom, we show that one can simulate the spin dynamics in magnetic solids by orbital dynamics of pband bosons. Specifically, we numerically study a particular lattice quench, using timedependent matrix product and Gutzwiller states, which is shown to convert the angular momentum order of such chiral fluids into timedependent oscillations of the orbital population imbalance, analogous to Larmor spin precession. These oscillations directly reveal the experimentally hardtodetect ‘hidden order’ associated with spontaneous timereversal symmetry breaking. This quench is analogous to nuclear magnetic resonance schemes in liquids or solids, which tip the nuclear moment vector and study its subsequent precession using radiofrequency probes. This noninterferometric, and timeresolved, route to measuring the angular momentum order works in SF as well as nonSF regimes, and it could be implemented using recent experimental innovations^{36,37,38}.
Results
Chiral Bose liquid and Mott insulator at strong coupling
We begin with the strong coupling regime, where atoms can localize to form a Mott insulator (MI) ground state. When the s orbitals in one of the sublattices (Fig. 1) are largely mismatched in energy with the p orbitals, with a gap Δ_{sp}, the nearestneighbor tunnelling Hamiltonian for bosons dominantly residing in the p orbitals is given by
with hopping amplitudes being mediated by the s orbitals. At integer filling, with n≥2, a strong porbital Hubbard repulsion in equation (2) favors a local state with a fixed particle number with nonzero angular momentum in order to minimize the interaction energy^{22,24,25,28}, leading to a MI with a twofold degeneracy of orbital states p_{x}±ip_{y} at each site. This extensive degeneracy is lifted by virtual boson fluctuations within second order perturbation theory in the boson hopping amplitudes. This effect is captured, by setting _{z}(r)=σ_{z}(r)_{z}(r), and deriving an effective exchange Hamiltonian between the Ising degrees of freedom σ_{z}(r),
where . The chiral MI ground state thus supports a staggered (antiferromagnetic) angular momentum pattern, with a nonzero order parameter out to arbitrarily strong coupling. Such staggered timereversal symmetry broken MIs, albeit for far more delicate plaquette currents, are known to emerge in frustrated Bose Hubbard models without orbital degrees of freedom, but only in an extremely small parameter window of interactions^{39,40,41}. As schematically shown in Fig. 1, heating this MI leads to a ‘chiral Bose liquid’ with spontaneously broken timereversal symmetry. It only reverts to a conventional normal fluid above a symmetry restoring thermal phase transition of , which occurs at k_{B}T_{I}≈2.27 adopted from the exact result^{42}. Taking =0.02 and U=1 in units of the recoil energy of the optical lattice, we estimate the Ising transition temperature in the experimental system^{21,27} to be ~1nK, well within the reach of cooling techniques in optical lattices.
Monte Carlo simulations at weak coupling
At low temperature, and with decreasing interaction strength, we expect both the chiral Bose liquid and the chiral MI to become unstable to superfluidity^{25}. Equivalently, decreasing temperature at weak coupling is expected to lead to a sequence of transitions from normal to chiral Bose liquid and subsequently to chiral SF as shown in Fig. 1b. To address this physics in the weak coupling limit, we begin with the Hamiltonian , supplemented with local porbital interactions
For small , the band structure of pband bosons has minima at (π, 0) and (0, π). Interactions scatter boson pairs from one minimum into the other, leading the bosons to condense into a superposition state of the two modes, phaselocked with a relative phase ±π/2. This gives rise to a p_{x}±ip_{y} SF ground state with a spontaneously broken timereversal symmetry and nonzero staggered angular momentum order.
To study the impact of thermal fluctuations on this weakly correlated SF, we make the reasonable assumption that classical phase fluctuations dominate the universal physics in the vicinity of the thermal phase transitions of this SF. This allows us to ignore the subdominant density fluctuations, and to replace , with ρ being the boson density, arriving at an effective classical phaseonly Hamiltonian
where Δ_{j}θ_{α}(r)=θ_{α}(r+a_{j})–θ_{α}(r) with j=x, y, and U≈ρ^{2}U_{p}/6. A more powerful symmetrybased effective field theory (EFT) approach (see Methods) confirms exactly the same latticeregularized model. Using Monte Carlo simulations (see Methods), we have studied the thermal phase diagram of this model for fixed U/, with =1.
As shown in Fig. 2a, the Binder cumulant^{43} B_{L}(_{z}) for the staggered angular momentum order, computed for U/=1 on L × L systems with various L, exhibits a unique crossing point at T/=2.088(3), signalling a critical point with a diverging correlation length. The critical value of this Binder cumulant is B*≈0.61, very close to the universal 2D Ising value ≈0.61069 for the aspect ratio of unity and periodic boundary conditions used in our simulations. This suggests that the staggered angular momentum order disappears at T_{I}/=2.088(3) via a thermal transition in the 2D Ising universality class. We confirm this from the scaling collapse of the order parameter curves shown in Fig. 2b using 2D Ising exponents ν=1 and β=1/8.
To track the destruction of superfluidity, we have computed the atom momentum distribution function , which is accessible via timeofflight measurements, as well as the SF stiffness ρ_{s} (see Methods). On general grounds, a 2D quasicondensed SF is expected to lose its superfluidity via a BKT transition driven by vortex unbinding, and it has been observed in trapped ultracold atom SFs. The BKT transition is signalled by a jump in ρ_{s} satisfying the universal relation ρ_{s}(T_{BKT})=2T_{BKT}/π.
As seen from Fig. 3a, ρ_{s} indeed appears to have a sharp drop in the temperature range 2.05–2.10, rounded by finite size effects. To confirm the BKT nature of the SF transition, and to precisely locate T_{BKT}, we resort to three distinct approaches. The first uses the finitesize scaling of ρ_{s}. Weber and Minnhagen^{44} have derived the finitesize scaling of the SF density at T_{BKT}. An unbiased fit to this logscaling form (see Methods) shows that the fit error (Fig. 3c) exhibits the expected deep minimum at the BKT transition, enabling us to estimate T_{BKT}/=2.072(3), and ρ_{s}(T_{BKT})/T_{BKT}≈0.64. The latter is close to the universal value 2/π=0.6366…, providing further concrete evidence of a BKT transition. The second uses finitesize scaling of the transition temperature. We identify the finitesize BKT transition temperature T_{c}(L) via ρ_{s}(T_{c}(L),L)=2T_{c}(L)/π. Above a BKT transition, the correlation length , yielding a finitesize scaling form (σ and L_{0} are nonuniversal numbers). An extrapolation using this formula, as shown in Fig. 3b,d, yields T_{BKT}/=2.065(4). The third uses the momentum distribution n(k). As shown in Fig. 3e, n(k) exhibits equal height peaks at k_{1}=(π, 0) and k_{2}=(0, π). This is consistent with the weak coupling analysis, which shows degenerate pband dispersion minima at these momenta. At a BKT transition, n(k_{i})∝L^{7/4} (in contrast to scaling as ρ_{c}L^{2} for a true Bose condensate with a condensate density ρ_{c}). This implies that the scaled momentum distribution n(k_{i})L^{−7/4} should cross at T_{BKT} for varying system sizes L. The plot of this scaled distribution (Fig. 3f) indeed reveals a unique crossing point, which yields T_{BKT}/≈2.067(3).
All three methods to locate T_{BKT} yield consistent answers. Combining these results, our best estimate for the SF transition temperature is T_{BKT}/≈2.069(4). For =U=0.3, in units of recoil energy of the experimental optical lattice^{21,27}, we estimate T_{BKT}~50 nK. Our numerical study unambiguously shows that the chiral p_{x}±ip_{y} SF undergoes a twostep destruction: a lower temperature BKT transition at which superfluidity is lost, followed by a higher temperature Ising transition at which timereversal symmetry is restored, leading to an unconventional ‘chiral Bose liquid’ at intermediate temperatures 2.069(4)≲T/≲2.088(3). Such multiple transitions are also proposed to be relevant for 2D spinor condensates^{45}. With increasing correlations, the BKT transition temperature is expected to get suppressed, eventually vanishing at the Mott transition (for integer fillings n≥2), while the Ising transition remains nonzero for arbitrarily large repulsion as seen from our strong coupling results. Correlation effects thus enhance the window where one realizes a ‘chiral Bose liquid’ as shown in the schematic temperatureinteraction phase diagram in Fig. 1.
Quantum quench and singlesite orbital dynamics
One can draw a fruitful analogy between the two orbital states at each site p_{x}, p_{y} and a pseudospin1/2 degree of freedom ↑,↓. This suggests that one can simulate spin dynamics in solid state materials by studying orbital dynamics of pband bosons. As we will see, this also suggests a route to directly detecting the angular momentum order in the p_{x}±ip_{y} SF and ‘chiral Bose liquid’ of the type we have obtained. In our analogy, the p_{x}±ip_{y} state corresponds to a pseudospin pointing along the y direction in spin space. Applying a ‘magnetic field’ along the x direction to this pseudospin should then induce Larmor precession, leading to periodic oscillations of the zmagnetization, corresponding to oscillations in the orbital population imbalance N(p_{x})–N(p_{y}). Let us imagine we prepare the system in a certain initial state, and then suddenly quench to a state where we set U_{p}=U_{s}=0, turn off all hoppings so t=0, and turn on a ‘magnetic field’ term
at time τ=0; we later discuss how to realize such a term in optical lattice experiments. The staggered sign leads to a staggered coupling between the p_{x} and p_{y} orbitals. If initially a staggered superposition p_{x}±e^{iθ}p_{y} is prepared, this results in a rectification of all local Larmor precessions such that they add up to produce a macroscopic oscillation of the populations of the p_{x} and p_{y} orbitals. The porbital imbalance, , evolves, within a Heisenberg picture, as
where is the staggered angular momentum operator whose evolution is in turn given by
This leads to periodic oscillations of ΔN(r,τ)=‹ΔN(r,τ)› as
where ΔN(r, 0) and (r, 0) denote the initial orbital magnetization and staggered angular momentum, respectively. Neglecting possible spatial inhomogeneity in λ(r) and φ(r), by focusing at the trap centre, we can set λ(r)=λ and φ(r)=φ, and extract the initial angular momentum order from the amplitude A(r) and the phase shift φ in the dynamics of the averaged number difference with N_{s} being the number of lattice sites at the trap centre, and denoting the spatial average of (…). The coefficient λ can be directly readoff from the oscillation period τ_{Q} ≡ π/λ. We emphasize here that suitably averaged over the entire trap can be measured in timeofflight experiments^{21}.
For a state with nonzero staggered angular momentum order, but no initial orbital population imbalance, that is, ΔN(r,0)=0, such as our chiral fluids, we expect (τ) to oscillate with a nonzero amplitude, and a phase shift φ=±π/2 whose sign will fluctuate from realization to realization, reflecting the spontaneous nature of timereversal symmetry breaking. The amplitude of the signal will then be a direct measure of the staggered angular momentum order parameter, vanishing in a singular manner at the Ising phase transition, which restores timereversal symmetry. In contrast, a completely thermally disordered conventional normal fluid would have (τ)=0. A state with an initial orbital population imbalance but no angular momentum order, obtained by explicitly breaking the square lattice C_{4} symmetry in the initial Hamiltonian as achieved in recent experiments, would exhibit oscillations with a nonsingular amplitude and a phase shift φ=0. Finally, if spontaneous timereversal symmetry breaking exists in a system without C_{4} symmetry, the amplitude of (τ) will be nonsingular while its phase shift will change in a singular manner, going from φ=±π/2 in a completely ordered state to φ=0 at the timereversal symmetry restoring phase transition. Since this quench induced orbital magnetization dynamics is inherently a noninterferometric probe of the angular momentum order, it suggests a simple and powerful method for measuring timereversal symmetry breaking in SF as well as nonSF chiral states. Our proposal thus significantly extends the earlier proposed quench dynamics approach for probing generic current orders^{46,47}. In the presence of SF order, our real space quench is analogous to the recent proposal of Cai et al.^{48}, which proposes to extract the relative phase between the p_{x} and p_{y} orbitals by studying momentum spectra after applying a Raman pulse to the Bose condensate. However, our proposal differs in showing that the angular momentum order can be probed irrespective of long range phase coherence or sharp momentum peaks.
Numerical simulations of quench dynamics
Our above analysis assumed that the quantum quench was complete, that is, all tunnelings (t) and interactions (U) were entirely switched off when H_{mag} was switched on. We now show, using numerical simulations, that the coherent orbital oscillations are robust even with small nonzero tunnelings and interactions present after the quench, that is, for an incomplete quench.
Since the scheme we are proposing here directly measures the local angular momentum order, it does not rely on the system dimensionality (beyond the assumption of longrange order). We therefore numerically simulate the zero temperature quench dynamics of a 1D model of porbital bosons, using both timedependent Gutzwiller mean field theory^{49} and timedependent matrix product states (tMPS), finding good agreement at both weak and strong couplings and qualitatively similar conclusions at intermediate interaction strength. We then use the Gutzwiller mean field theory to also simulate the dynamics for the 2D case relevant to current experiments.
The Hamiltonian of the 1D system is^{24}
where j index lattice sites. The ground state phase diagram of this 1D system includes two types of Mott states at strong coupling: a chiral Mott with staggered angular momentum order, and a nonchiral MI. For weak correlations, it supports two types of SF ground states: a chiral SF with staggered angular momentum order and a nonchiral SF^{24}. The chiral states have an order parameter , with , which is analogous to the 2D case. We start with different ground states of H_{1D} and study their time evolution under a quantum quench that suddenly changes the Hamiltonian to H_{1D}+ΔH_{1D}, where
The oscillatory dynamics of and is confirmed even for these entangled manybody states (Fig. 4). Since the 1D geometry does not possess C_{4} symmetry, we expect the different states to be distinguished by the phase shift φ, not the amplitude, of the oscillatory dynamics. The chiral Mott and SF states develop a periodic motion with nonzero phase shift. The dynamics of nonchiral states indicate zero phase shift. In this way the chiral states can be distinguished from nonchiral states by measuring the phase shift, which is directly related to the angular momentum order parameter. Deep in the chiral SF state, the phase shift is φ=±π/2, and it decreases in magnitude on approaching the chiral–nonchiral critical point. The phase shift vanishes in a singular fashion at this quantum critical point, signalling that this phase transition associated with timereversal symmetry can be probed by measuring the order parameter via the phase shift φ.
Comparing timedependent Gutzwiller^{50} and tMPS methods (Fig. 4), we find that the Gutzwiller approach captures orbital dynamics fairly well. We thus apply this approach to study orbital dynamics of the 2D system, as in experiments^{21}. We have verified that a partial quench leads to longlived ΔN oscillations as long as t and U are weak compared with the quench strength, that is, and . These results are shown in Fig. 5.
Experimental proposal for quench
To engineer the Hamiltonian H_{mag} of Equation (8), we implement a quench potential V_{mag}(x) modulated in the (1,1) direction in addition to the lattice potential giving rise to the quench Hamiltonian
with
which is valid in the tight binding regime when the quench potential is weak as compared with the original optical lattice (see Supplementary Note 1). Here, ω_{0} is the harmonic oscillator frequency of the lattice wells hosting the p orbitals, and a≡a_{x}=a_{y} is the lattice constant. Since the local density operator of the p orbitals n_{p}(r) commutes with ΔN(r) and , it does not contribute to the dynamics of ΔN(r), and hence may be neglected. This is verified in our Gutzwiller simulations. We choose the quench potential as
with some integer ν≥0, a positive amplitude Γ, and K_{x}, K_{y} denoting the primitive vectors of the reciprocal lattice (a_{i}·K_{j}=2πδ_{ij} with i,jε{x,y}). The quench potential provides a lattice along the (1,1) direction, and it breaks both C_{4} symmetry and mirror symmetries in the x and y directions. The potential is minimal at every second site of the Bravais lattice at positions r(=r_{x}a_{x}+r_{y}a_{y}) with even r_{x}+r_{y} and it is maximal for all adjacent sites specified by odd r_{x}+r_{y}. Hence, the second derivative in Equation (15) produces the alternating sign required for realization of the quench Hamiltonian in Equation (8). Combining Equations (16) and (15) yields
with E_{rec}≡ħ^{2}k^{2}/2m denoting the single photon recoil energy for photons with wave number .
The lattice potential together with the corresponding quench potential can be realized by superlattice techniques demonstrated in several experiments^{20,21,36,51}. For example, following Wirth et al.^{21}, the lattice potential arises via two optical standing waves oriented along the (1,1) and (1,−1) axes with a wave number =2π/1,064 nm. The corresponding quench lattice requires an additional standing wave along the (1,1) axis with wave number . Hence, the case ν=1 requires ≈2π/709 nm, which is experimentally readily provided by diode laser sources. Both standing waves along the (1, 1)direction may be derived by retroreflecting two parallelly propagating laser beams with wave numbers k and k′ by the same mirror. To prepare the required spatial relative phase of the two lattices, k′ may be slightly detuned from the precise ratio k′/k=
Discussion
In most cold atom experiments, the trap potential can induce a slowly varying inhomogeneity in the ‘magnetic field’ λ as δλ=max{λ(r)}–min{λ(r)}. We expect the oscillations of the trap averaged number difference to decay over a timescale . Finally we emphasize that besides the angular momentum order parameter, the quantum quench proposal and a subsequent study of ΔN(r, τ) using insitu microscopy can also yield correlation functions of (r), from which the diverging correlation length near the transition from chiral Bose liquid to normal can be extracted.
Methods
Monte Carlo simulations
We carry out the Monte Carlo study of the Hamiltonian in Equation (7) using a Metropolis sampling of the phase configurations {θ_{x}(r), θ_{y}(r)}, with 10^{7} sweeps to equilibrate the system at each temperature, and averaging all observables over 10^{8} configurations.
To study the chiral order, we construct the angular momentum order parameter and compute its Binder cumulant^{43} . The universal order parameter distribution at renormalization group fixed points leads to universal values of _{L→∞}; on finite size systems, this yields Binder cumulant curves, which cross at the critical point associated with angular momentum ordering. The critical value B* of the Binder cumulant is wellknown to be universal, independent of lattice structure and details of the Hamiltonian, and depending only on the aspect ratio and boundary conditions used in the simulations. For periodic boundary conditions on L × L lattices, B*≈0.61069 for the 2D Ising universality class.
The SF stiffness ρ_{s} is defined as the change in the free energy density in response to a boundary condition twist; for , it is explicitly given by
where ‹⋯› refers to the thermal average. At a BKT transition, ρ_{s}(T) jumps to zero, with ρ_{s}(T_{BKT})/T_{BKT}=2/π, a universal value. On finite size systems, the universal SF stiffness jump gets severely rounded, and a careful finitesize scaling is required to extract T_{BKT}. On the basis of the KT renormalization group equations, Weber and Minnhagen have shown^{44} that ρ_{s}(T_{BKT}, L) scales as
where c is a nonuniversal number. It is wellknown that fitting to this logscaling form at different temperatures leads to an error, which exhibits a steep minimum at T_{BKT}, enabling us to extract T_{BKT} from our simulations. An unbiased fit to ρ_{s}(T)/T, using a twoparameter scaling form,
further confirms the universal jump at the T_{BKT} identified by the error minimum. Using this, we find k≈0.64 from our simulations, in very good agreement with the KT value 2/π=0.6366....
Effective field theory description
We now show that the Monte Carlo study in the Results section is a lattice model simulation of the EFT for weakly interacting bosons in the experiments^{21}, which does not rely on tight binding approximation. In the band structure calculation for the lattice rotation symmetric case^{21}, dispersion of the relevant pband E_{p}(k) has two degenerate minima at Q_{x}=(π,0) and Q_{y}=(0,π), around which low energy modes can be excited due to quantum and thermal fluctuations. This leads to a twocomponent EFT, where the fields are introduced as
with Λ a momentum cutoff and annihilation operators for the Bloch modes near the band minima. The form of EFT is determined by considering lattice rotation and reflection symmetries, under which the fields φ_{α} transform as
and
respectively. The Hamiltonian density of the EFT consistent with these symmetries is
with effective couplings , and g’s. At tree level, they are approximated as , and
where a_{s} is the 3D scattering length, a is the lattice constant, {u}_{{\text{Q}}_{\alpha}} is the periodic Bloch wavefunction, and the integral ∫d^{3}x is over one unit cell. Details of calculation for the coupling constants are provided in the Supplementary Note 2. Although the connection of the EFT to microscopic origins is made clear in the tree level estimate, the form of EFT resulting from symmetry analysis does not rely on this estimate. In the vicinity of thermal phase transitions of the SF phases, classical phase fluctuations are expected to dominate the universal physics, which allows us to ignore the subdominant density fluctuations and to replace φ_{α} by with ρ the total density. In terms of phases θ_{α}, the Hamiltonian density is rewritten as
Bearing in mind the periodic nature of the phases θ_{α}, a proper lattice regularization of this EFT leads precisely to Equation (7).
Additional information
How to cite this article: Li, X. et al. Proposed formation and dynamical signature of a chiral Bose liquid in an optical lattice. Nat. Commun. 5:3205 doi: 10.1038/ncomms4205 (2014).
References
Bednorz, J. & Müller, K. Possible high Tc superconductivity in the BaLaCuO system. Zeitschrift für Physik B Condensed Matter 64, 189–193 (1986).
Kamihara, Y. et al. Ironbased layered superconductor: LaOFeP. J. Am. Chem. Soc. 128, 10012–10013 (2006).
von Helmolt, R., Wecker, J., Holzapfel, B., Schultz, L. & Samwer, K. Giant negative magnetoresistance in perovskitelike La2/3Ba1/3MnOx ferromagnetic films. Phys. Rev. Lett. 71, 2331–2333 (1993).
Luke, G. M. et al. Timereversal symmetrybreaking superconductivity in Sr2RuO4 . Nature 394, 558–561 (1998).
Tokura, Y. & Nagaosa, N. Orbital physics in transitionmetal oxides. Science 288, 462–468 (2000).
Lewenstein, M. A. & Liu, W. V. Optical lattices: orbital dance. Nat. Phys. 7, 101–103 (2011).
Zhao, E. & Liu, W. V. Orbital order in mott insulators of spinless pband fermions. Phys. Rev. Lett. 100, 160403 (2008).
Zhang, Z., Hung, H.H., Ho, C. M., Zhao, E. & Liu, W. V. Modulated pair condensate of porbital ultracold fermions. Phys. Rev. A 82, 033610 (2010).
Cai, Z., Wang, Y. & Wu, C. Stable FuldeFerrellLarkinOvchinnikov pairing states in twodimensional and threedimensional optical lattices. Phys. Rev. A 83, 063621 (2011).
Hung, H.H., Lee, W.C. & Wu, C. Frustrated Cooper pairing and fwave supersolidity in coldatom optical lattices. Phys. Rev. B 83, 144506 (2011).
Zhang, Z., Li, X. & Liu, W. V. Stripe, checkerboard, and liquidcrystal ordering from anisotropic porbital fermi surfaces in optical lattices. Phys. Rev. A 85, 053606 (2012).
Sun, K., Liu, W. V., Hemmerich, A. & Das Sarma, S. Topological semimetal in a fermionic optical lattice. Nat. Phys. 8, 67–70 (2012).
Li, X., Zhao, E. & Liu, W. V. Topological states in a ladderlike optical lattice containing ultracold atoms in higher orbital bands. Nat. Commun. 4, 1523 (2013).
Isacsson, A. & Girvin, S. M. Multiflavor bosonic hubbard models in the first excited bloch band of an optical lattice. Phys. Rev. A 72, 053604 (2005).
Liu, W. V. & Wu, C. Atomic matter of nonzeromomentum BoseEinstein condensation and orbital current order. Phys. Rev. A 74, 013607 (2006).
Kuklov, A. B. Unconventional strongly interacting BoseEinstein condensates in optical lattices. Phys. Rev. Lett. 97, 110405 (2006).
Lim, L.K., Smith, C. M. & Hemmerich, A. Staggeredvortex superfluid of ultracold bosons in an optical lattice. Phys. Rev. Lett. 100, 130402 (2008).
Stojanović, V. M., Wu, C., Liu, W. V. & Das Sarma, S. Incommensurate superfluidity of bosons in a doublewell optical lattice. Phys. Rev. Lett. 101, 125301 (2008).
Zhou, Q., Porto, J. V. & Das Sarma, S. Condensates induced by interband coupling in a doublewell lattice. Phys. Rev. B 83, 195106 (2011).
SoltanPanahi, P., Lühmann, D.S., Struck, J., Windpassinger, P. & Sengstock, K. Quantum phase transition to unconventional multiorbital superfluidity in optical lattices. Nat. Phys. 8, 71–75 (2012).
Wirth, G., Ölschläger, M. & Hemmerich, A. Evidence for orbital superfluidity in the pband of bipartite optical square lattice. Nat. Phys. 7, 147–153 (2011).
Li, X., Zhao, E. & Liu, W. V. Effective action approach to the pband mott insulator and superfluid transition. Phys. Rev. A 83, 063626 (2011).
Cai, Z. & Wu, C. Complex and real unconventional BoseEinstein condensations in high orbital bands. Phys. Rev. A 84, 033635 (2011).
Li, X., Zhang, Z. & Liu, W. V. Timereversal symmetry breaking of porbital bosons in a onedimensional optical lattice. Phys. Rev. Lett. 108, 175302 (2012).
Hébert, F. et al. Exotic phases of interacting pband bosons. Phys. Rev. B 87, 224505 (2013).
Hauke, P. et al. Orbital order of spinless fermions near an optical feshbach resonance. Phys. Rev. A 84, 051603 (2011).
Ölschläger, M. et al. Interactioninduced chiral px±ipy superfluid order of bosons in an optical lattice. New J. Phys. 15, 083041 (2013).
Wu, C. Unconventional BoseEinstein condensations beyond the ‘no node’ theorem. Mod. Phys. Lett. B 23, 1–24 (2009).
Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr2RuO4 and the physics of spintriplet pairing. Rev. Mod. Phys. 75, 657–712 (2003).
Kallin, C. Chiral pwave order in Sr2RuO4 . Rep. Prog. Phys. 75, 042501 (2012).
Nandkishore, R. Prediction and description of a chiral pseudogap phase. Phys. Rev. B 86, 045101 (2012).
Nandkishore, R., Levitov, L. S. & Chubukov, A. V. Chiral superconductivity from repulsive interactions in doped graphene. Nat. Phys. 8, 158–163 (2012).
Varma, C. M. Proposal for an experiment to test a theory of hightemperature superconductors. Phys. Rev. B 61, R3804–R3807 (2000).
Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001).
Fauqué, B. et al. Magnetic order in the pseudogap phase of highTc superconductors. Phys. Rev. Lett. 96, 197001 (2006).
Folling, S. et al. Direct observation of secondorder atom tunnelling. Nature 448, 1029–1032 (2007).
Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).
Ölschläger, M., Wirth, G., Kock, T. & Hemmerich, A. Topologically induced avoided band crossing in an optical checkerboard lattice. Phys. Rev. Lett. 108, 075302 (2012).
Dhar, A. et al. Bosehubbard model in a strong effective magnetic field: Emergence of a chiral mott insulator ground state. Phys. Rev. A 85, 041602 (2012).
Dhar, A. et al. Chiral mott insulator with staggered loop currents in the fully frustrated BoseHubbard model. Phys. Rev. B 87, 174501 (2013).
Zaletel, M. P., Parameswaran, S. A., Rüegg, A. & Altman, E. Chiral Bosonic Mott Insulator on the Frustrated Triangular Lattice. Preprint at http://arxiv.org/abs/1308.3237 (2013).
Onsager, L. Crystal statistics. i. a twodimensional model with an orderdisorder transition. Phys. Rev. 65, 117–149 (1944).
Binder, K. Finite size scaling analysis of Ising model block distribution functions. Zeitschrift für Physik B Condensed Matter 43, 119–140 (1981).
Weber, H. & Minnhagen, P. Monte carlo determination of the critical temperature for the twodimensional XY model. Phys. Rev. B 37, 5986–5989 (1988).
Mukerjee, S., Xu, C. & Moore, J. E. Topological defects and the superfluid transition of the s=1 spinor condensate in two dimensions. Phys. Rev. Lett. 97, 120406 (2006).
Killi, M. & Paramekanti, A. Use of quantum quenches to probe the equilibrium current patterns of ultracold atoms in an optical lattice. Phys. Rev. A 85, 061606 (2012).
Killi, M., Trotzky, S. & Paramekanti, A. Anisotropic quantum quench in the presence of frustration or background gauge fields: a probe of bulk currents and topological chiral edge modes. Phys. Rev. A 86, 063632 (2012).
Cai, Z., Duan, L.M. & Wu, C. Phasesensitive detection for unconventional BoseEinstein condensation. Phys. Rev. A 86, 051601 (2012).
Seibold, G. & Lorenzana, J. Timedependent Gutzwiller approximation for the Hubbard model. Phys. Rev. Lett. 86, 2605–2608 (2001).
Snoek, M. & Hofstetter, W. Twodimensional dynamics of ultracold atoms in optical lattices. Phys. Rev. A 76, 051603 (2007).
Anderlini, M. et al. Controlled exchange interaction between pairs of neutral atoms in an optical lattice. Nature 448, 452–456 (2007).
Acknowledgements
We acknowledge helpful discussions with Youjin Deng, Subroto Mukerjee and Sankar Das Sarma. This work is supported by the NSERC of Canada (AP), NSF PHY1125915 (X.L.), AFOSR (FA95501210079), ARO (W911NF1110230), DARPA OLE Program through ARO and the Charles E. Kaufman Foundation of The Pittsburgh Foundation (X.L. and W.V.L.), the National Basic Research Program of China (Grant No 2012CB922101) and Overseas Collaboration Program of NSF of China (11128407) (W.V.L.), and the German Research Foundation DFGSFB 925 and the Hamburg Centre for Ultrafast Imaging (A.H.). X.L. would like to thank KITP at UCSB for hospitality. W.V.L. and A.H. acknowledge partial support by NSFPHYS1066293 and the hospitality of the Aspen Center for Physics.
Author information
Authors and Affiliations
Contributions
X.L. and A.P. conceived and evolved the theoretical ideas in discussion with W.V.L. A.H. examined and improved the experimental protocol. X.L. and A.P. performed numerical simulations. All authors worked on theoretical analysis and contributed in completing the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Notes 12 (PDF 195 kb)
Rights and permissions
About this article
Cite this article
Li, X., Paramekanti, A., Hemmerich, A. et al. Proposed formation and dynamical signature of a chiral Bose liquid in an optical lattice. Nat Commun 5, 3205 (2014). https://doi.org/10.1038/ncomms4205
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms4205
This article is cited by

Chiral magnetism and spontaneous spin Hall effect of interacting Bose superfluids
Nature Communications (2014)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.