A first order superfluid-Mott insulator transition for a Bose–Hubbard model in an emergent lattice

Yu Chen

Graduate School of China Academy of Engineering Physics,Beijing 100193,China

Abstract The combination of strong correlation and emergent lattice can be achieved when quantum gases are confined in a superradiant Fabry–Perot cavity.In addition to the discoveries of exotic phases,such as density wave ordered Mott insulator and superfluid,a surprising kink structure is found in the slope of the cavity strength as a function of the pumping strength.In this article,we show that the appearance of such a kink is a manifestation of a liquid–vapour-like transition between two superfluids with different densities.The slopes in the immediate neighborhood of the kink become divergent at the liquid–vapour critical points and display a critical scaling law with a critical exponent 1 in the quantum critical region.

Keywords: Bose–Hubbard model,emergent symmetry,cavity QED,liquid–gas transition

1.Introduction

The capability of confining quantum gases in a cavity enables us to realize strongly coherent coupling between atoms and light[1,2].With this technical advance,the Dicke model for superradiance has finally been achieved experimentally in a cold atom system [3] after decades of searching [4].What goes beyond the physics of the original Dicke model in these experiments is that an emergent lattice appears in concurrence with the superradiance and the atoms self-organize themselves into a density pattern.A roton mode softening across the superradiance transition is observed as a signature of this emergent property [5].

By imposing an optical lattice on an atomic gas,many models describing strongly correlated physics can be realized in a cold atomic setting,such as the Bose–Hubbard model and the Fermi Hubbard model[6].A novel aspect of the emergent optical lattice,compared to an externally imposed one,is that a long-range interaction between the atoms could be established.As a result,when strong interaction meets emergent lattices,competitions between local interactions and long range interactions appear.Recently,this combination has been achieved by Hamburg and ETH’s experimental groups by loading strongly interacting Bose gases into a strong coupling cavity [7,8].Exotic phases like density ordered superfluid and Mott insulator are observed as a manifestation of the competition between local onsite interactions and cavity mediated long range interactions,as is predicted by quite a few theoretical works [9–13].

Very recently,some pioneering works explored the phase boundaries and metastable states in the neighborhood of the phase transition between a homogenous Mott insulator and a density ordered Mott insulator [14,15].These metastable states are indications of the first order transitions predicted by recent theoretical studies of phase boundaries based on ETH’s set-up [16–18].Although present theoretical studies are satisfactory in many aspects,one striking feature in Hamburg’s experiment remains to be explained,that is the sharp kink structure (large slope change) of the superradiant cavity field against the pumping strength in the vicinity of SF-to-MI transition [7].In this paper,we construct an effective field theory close to the SF-to-MI transition point in the superradiant phase and give an explanation for the presence of these kinks,which is distinct from the other explanation of this kink as a result of density fluctuations [20].Our results are summarized as follows: (1) there is a liquid–vapour-like transition between two superfluids with density difference,which is similar to the liquid–gas transition in fermionic super-radiance due to the competition of two density order modes and p-band filling[19];(2)sharp kinks are present in a large region around the critical point which ends at the liquid–vapour-like transition;(3)the kink strength is divergent at the critical point with a critical exponent as 1.Our prediction of a liquid–vapour-like transition can be tested in current experimental set-ups and the appearance of divergent kinks in the superfluid phase serves as the smoking gun.

Figure 1.(a) Scheme of single mode cavity with interacting Bose gases.The pumping field is in the z direction,and the cavity field is in the x direction.(b)Emergent lattice configuration on xz plane in a superradiant cavity. t is along the diagonal direction andt′ is along the x direction.

2.Modeling

2.1.A model from experimental setup

The optical latticeVL(r)+VC(r) has a typical potential contour in thexzplane as shown in figure 1 (b).We project the motion of the atoms to the lowest band ofVL(r)+VC(r),and obtain a tight-binding model

where the field operatorbiannihilates a boson on site i[=(ix,iy,iz)(π/k0)].Here 〈ij〉and 〈〈ij〉〉denote the nearest neighbor(NN) and the next nearest neighbor (NNN) respectively;the additional subindicesx,yandrestrict the neighboring sites along these specific directions.The parameterstandtyare the NN hopping strength in thexzplane and along they-direction respectively,andt′ is the NNN hopping strength along thex-direction.The chemical potential μα=μ-η(α+α?)-V0|α|2includes the onsite energy shift due to the cavity field andUis the onsite interaction strength.1see supplementary material for details.The total lattice site number isNΛ.

2.2.Effective field theory derivation:a new method dealing with degenerate perturbation

To study phase transitions in the presence of the cavity field α,we proceed to derive from equation (1) an effective field theory involving α and the low energy degrees of freedom of the bosons.

To begin,we introduce a local superfluid order parameter φ ≡W〈bi〉whereW=4t+2t′+2ty.We note that due to the long range coherence,the superfluid phase φ is site-independent.Assuming we are in the vicinity of SF-to-MI transition where superfluid order is weak,we can obtain an effective mean field theory of φ perturbatively from equation (1) as follows [21].We first diagonalize the on-site Hamiltonianto obtain the Mott eigenstates |?〉,where ?is the atomic site occupation number.Then,by approximating the tunneling part in equation(1)at the mean field level bywe calculate the energy correction to the Mott eigenstates due to such a tunneling term.

Here we focus on the energetically degenerate point of two adjacent Mott insulator phases with occupation numbers?and ?+1.At such a point,??=??+1leading to μα=ν?=U?,whereTo obtain accurate results around μα≈ν?,we follow three steps.First,we carry out the nondegenerate perturbation in the subspace of {|?〉,|?-1〉,|?-2〉} and {|?+1〉,|?+2〉,|?+3〉} to find two ‘restricted ground states’|L〉and|H〉in each subspace.The ground state|L〉can be calculated as,

where ?j=-(μ+η(α+α?))j+Uj(j-1)/2.Similarly,the second dressed ground state|H〉in subspace{|?+1〉,|?+2〉,|?+3〉} can be obtained as

whereNLandNHare normalization factors.

In the second step,we write down the reduced hamiltonian in the Hilbert space spanned by |L〉and |H〉,which is

to reveal this hidden symmetry.Then by replacing α with θ,we get the energy density as a function of density order θ and superfluid order φ in the following form

3.A liquid–gas-like transition between superfluids

3.1.Phase transition prediction based on effective field theory

In the limit of large pumping field strengths,the optical lattice potential is deep and the hopping strengths are small such thatris positive and large.As a result the energy density? is minimized at φ=0 for any θ,namely the system is in the Mott insulator phase.In this case we have

As expected,? is further minimized at θ=1/2 (〈n〉=?+1)for positive detuning δ and at θ=-1/2(〈n〉=?)for negative δ.Right at δ=0,? is symmetric under the transformation θ →-θ.One could find the total symmetry of the low energy theory isat δ=0,which enlarges the exact symmetry of the original hamiltonian.A nonzero δ breaks the Z2symmetry explicitly and its sign change drives a first order transition between two Mott insulators.

To include a more general case for φ ≠0,we introduceThen the square root term in equation (11) becomesTo get an expansion of this term,there are two possible situations,one is Θ being finite,φ →0,another is φ being finite and Θ →0.The first case could be satisfied when Θ reflection symmetry is explicitly broken,that is,large δ case.In this limit we carry out a Taylor expansion in terms of φ in equation(11)and get

Figure 2.Phase diagram of Mott transition in superradiance phase.SF phase is in red while MI phases are in blue.Black lines represent second order transitions and the blue black dashed lines represent first order transitions.‘liquid–vapour’ critical point is marked in the red dot.White dashed lines are the paths for δ=0,0.005,0.011 and 0.0125.MI? labels Mott insulator phase with density ?,SF? labels superfluid phase with approximate density ?.

Figure 3.In(a),(b),(c),(d),the order parameters φ(dashed red)and〈n〉(solid blue) along paths δ=0,0.005,0.011 and 0.0125 (labeled as a,b,c,d in figure 2 as white dashed lines) are given.We use black dashed lines to label phase transition points.In (a),the SF-to-MI transition happens when φ becomes nonzero;in(b),a first order transition is labeled;in(c),there are two transitions;in(d),the right side dashed line is an SF-to-MI transition and the left side dashed line labels the kink.

3.2.Numerical mean field solution

Additional numerical simulation can be carried out based on our effective field theory up to|φ|4order.The phase diagram is shown in figure 2 where black and the black blue dashed lines represent the boundaries of the second order and the first order transitions,respectively;the red points represent the critical points.Four routesa,b,canddfor small δ around‘θ reflection symmetric’region are taken across these boundaries and the order parameters along them are shown in figure 3.Along route a,δ=0,a second order SF-to-MI transition is triggered by lowering the puming strength.For δ=0 and large pumping strength,the system is on the θ reflection symmetric line where two Mott insulators are degenerate.Along routeb,afirst order transition between the Mott insulator and superfluid is displayed by a jump of the order parameter in both φ andAlong routec,there are two transitions.From right to left,the first one is a second order SF-to-MI transition by a spontaneous breaking of the U(1)symmetry in φ;the second one is the liquid–vapour-like transition between two SFs.Finally,along pathd,there is only a second order MI-to-SF transition.But there is an obvious kink in both superfluid order and density order at pumping strengthVP=18.75Eras we labeled by vertical dashed line in figure 3(d).

Figure 4.(a)as a function of pumping field strength VP for different δ.δcr is the imbalance δ at critical point.(b) The minimal offor different δ as a function of VP.The correlation coefficient of linear scaling is 0.9997.(c)In this figure,we fix δ to sit on the liquid–vapour transition point at each pumping strength VP.On the two sides of the‘liquid–vapour transition’line,the density order is different.Here we label both density order parameters on each side of the transition point.If the density order is an integer,it is in MI phase and if the density order is not an integer,it is in superfluid phase.We observe three regions along the transition line,when VP/Er>23,the density difference between two phases is 1,this is MI-to-MI transition;when 20.2

Concerning experimental observation,the negative kink is more difficult to observe for technical reasons.As is shown by the white dashed line d in figure 2 for fixed δ,the route we take always bypasses the critical point from above,where only positive kinks are accessed.This is consistent with the experimental procedure where similar routes are taken.

4.Conclusion and outlook

To conclude,we construct an effective field theory from a microscopic model to study the Mott transition in emergent lattices and find a liquid–vapour-like transition between two superfluids.The liquid–vapour-like transition ends at a critical point within the superfluid phase,and a divergent density kink is predicted.This kink exists in a large region around the critical point and the maximal density slope scales asWe link the critical exponent extracted in the kink slope with the compressibility critical exponent γ,and this could be tested by leaking photon counting measurement in an experiment.Because the present liquid–vapour-like transition is driven by quantum fluctuations and the nature of this transition is non-equilibrium,the experimental discovery of critical exponent γ will not only include the mean field result but also show corrections from quantum fluctuations and the non-equilibrium effect.By quantum fluctuation effects,we mean those effects due to spatial quantum fluctuations in the quantum critical region,which we expect to have a difference in thermal fluctuations.By the non-equilibrium effect,we mean those effects from the atomic gas distribution deviation from the thermal equilibrium state.The situation is more severe when the cavity decay rate is comparable to the recoil energy,in which case the equilibrium is hard to establish.These studies will enrich studies for first order transitions in the quantum region.

Acknowledgments

We acknowledge Zhenhua Yu,Hui Zhai,and Andreas Hemmerich for inspiring discussions.We would like to thank Zhigang Wu,Ren Zhang and Juan Yao for careful reading of our manuscript and advice for presentation.YC is supported by Beijing Natural Science Foundation(Z180013),and NSFC under Grant No.12 174358 and No.11734010.

Appendix.Tight-binding model parameters

In this section,we present the construction of a tight-binding model from an emergent lattice.Assuming the condensed cavity field strength isα==Re(α)+iIm(α).Re(α)andIm(α) are real and imaginary part α.Then the potential for the atomic gas can be characterized as

Figure 5.Illustration of optical lattice configuration.The contour plot shows the equal potential energy lines.From red to blue,the potential becomes deeper.A sites are which are local minima of the lattice potential.B sites are saddle points of the lattice potential.t and t′ are nearest and next-to-nearest hoppings between two A sites.

In the tight-binding limit,we can only consider the nearest and next-to-nearest hoppings between two A sites.Here we denote the nearest hopping strength ast,and denote the nextto-nearest hopping strength ast′.tandt′are shown in figure 5.At the same time,we can definetyas the nearest hopping between site (m,l,n) and site (m,l±1,n).tycan be get by WKB approximation as

On the other hand,the onsite interaction energy can be given in terms of the Wannier wave function Φ(r) with

where ωy?s dependence has been absorbed in.In terms oft,t′,tyandU,we can then construct the tight-binding model equation (1) in the main text.