## Abstract

Quantum criticality emerges when a many-body system is in the proximity of a continuous phase transition that is driven by quantum fluctuations. In the quantum critical regime, exotic, yet universal properties are anticipated; ultracold atoms provide a clean system to test these predictions. We report the observation of quantum criticality with two-dimensional Bose gases in optical lattices. On the basis of in situ density measurements, we observe scaling behavior of the equation of state at low temperatures, locate the quantum critical point, and constrain the critical exponents. We observe a finite critical entropy per particle that carries a weak dependence on the atomic interaction strength. Our experiment provides a prototypical method to study quantum criticality with ultracold atoms.

In the vicinity of a continuous quantum phase transition, a many-body system enters the quantum critical regime, where quantum fluctuations lead to nonclassical universal behavior (*1*, *2*). Quantum criticality not only provides novel routes to new material design and discovery (*1*, *3*–*6*), but also provides a common framework for problems in condensed matter, nuclear physics (*7*, *8*), and cosmology (*1*, *9*). Quantum criticality plays a central role in strongly correlated systems such as heavy-fermion materials (*5*), spin dimer systems (*10*), Ising ferromagnets (*11*), and chromium at high pressure (*12*).

Ultracold atoms offer a clean setting for quantitative and precise investigation of quantum phase transitions (*13*–*16*) and critical phenomena (*17*). For example, the superfluid-to-Mott insulator quantum phase transition can be realized by loading atomic Bose-Einstein condensates into optical lattices (*13*). In recent experiments, scaling behavior of physical observables was reported in interacting Bose gases in three (*17*) and two dimensions (*18*), and in Rydberg gases (*19*), where collective behavior is insensitive to microscopic details. In addition, suppression of the superfluid critical temperature near the Mott transition was observed in three-dimensional (3D) optical lattices (*20*). Studying quantum criticality in cold atoms on the basis of finite-temperature thermodynamic measurements, however, remains challenging and has attracted increasing theoretical interest in recent years (*21*–*24*).

We report the observation of quantum critical behavior of ultracold cesium atoms in a two-dimensional (2D) optical lattice across the vacuum-to-superfluid transition. This phase transition can be viewed as a transition between a Mott insulator with zero occupation number and a superfluid, and can be described by the Bose-Hubbard model (*25*). Our measurements are performed on atomic samples near the normal-to-superfluid transition, connecting to the vacuum-to-superfluid quantum phase transition in the zero-temperature limit.

The quantum phase transition and quantum critical regime in this study are illustrated in Fig. 1. The zero-temperature vacuum-to-superfluid transition occurs when the chemical potential μ approaches its critical value μ_{0}. Sufficiently close to the quantum critical point, the critical temperature *T*_{c} for the normal-to-superfluid transition is expected to decrease according to the following scaling (*25*)

_{kBTct=c(μ−μ0t)zν}(1)

where *k*_{B} is the Boltzmann constant, *t* is the tunneling energy, *z* is the dynamical critical exponent, ν is the correlation length exponent, and *c* is a constant. In the quantum critical regime (shaded area in Fig. 1), the temperature *T* provides the sole energy scale, and all thermodynamic observables are expected to scale with *T* (*25*). Thus, the equation of state is predicted to obey the following scaling (*21*)

_{N˜=F(μ˜)}(2)

in which *F*(*x*) is a generic function, and

_{N˜=N−Nr(kBTt)Dz+1−1zν and μ˜=μ−μ0t(kBTt)1zν}(3)

are the scaled occupation number and scaled chemical potential, respectively. Here, *N* is the occupation number, *D* is the dimensionality, and *N*_{r} is the nonuniversal part of the occupation number. For the vacuum-to-superfluid transition in the 2D Bose-Hubbard model, we have *N*_{r} = 0 and *D* = 2, and the predicted critical exponents are *z* = 2 and ν = 1/2, characteristics of the dilute Bose gas universality class (*2*, *22*, *25*). We note that in a 2D system, there can be logarithmic corrections to scaling functions, including those in this study, near the quantum critical point (*2*). Within the temperature range of our experiment, however, the measurement is consistent with the above scaling laws in the absence of logarithmic corrections. Scaling behavior of *T*_{c} in the quantum critical regime was also observed in 2D condensates of spin triplets (*10*).

Our experiment is based on 2D atomic gases of cesium-133 in 2D square optical lattices (*26*, *27*). The 2D trap geometry is provided by the weak horizontal (*r*-) confinement and strong vertical (*z*-) confinement (*27*), with envelope trap frequencies *f*_{r} = 9.6 Hz and *f*_{z} = 1940 Hz, respectively. Typically, 4000 to 20,000 atoms are loaded into the lattice. The lattice constant is *d* = λ/2 = 0.532 μm and the depth is *V*_{L} = 6.8 *E*_{R}, where *E*_{R} = *k*_{B} × 63.6 nK is the recoil energy, λ = 1064 nm is the lattice laser wavelength, and *h* is the Planck constant. In the lattice, the tunneling energy is *t* = *k*_{B} × 2.7 nK, the on-site interaction is *U* = *k*_{B} × 17 nK, and the scattering length is *a* = 15.9 nm. The sample temperature is controlled in the range of 5.8 to 31 nK.

We determine the equation of state *n*(μ,*T*) of the sample from the measured in situ density distribution *n*(*x*,*y*) (*18*, *26*). The chemical potential μ(*x*,*y*) and the temperature *T* are obtained by fitting the low-density tail of the sample where the atoms are normal. The fit is based on a mean-field model that accounts for interaction (*28*–*30*). Equation of state measured near the quantum critical point can reveal essential information on quantum criticality, as proposed in (*28*).

We locate the quantum critical point by noting that at the critical chemical potential μ = μ_{0}, the scaled occupation number _{0} = −4.5(6)*t* (Fig. 2A, inset). We identify this point as the critical point for the vacuum-to-superfluid transition, and our result agrees with the prediction −4*t* (*28*).

To test the critical scaling law, we compare the equation of state at different temperatures. On the basis of the expected exponents *z* = 2 and ν = 1/2, we plot the scaled occupation number *t*/*k*_{B} = 2.7 nK. Deviations become obvious at higher temperatures.

We examine the range of critical exponents *z* and ν that allow the scaled equation of state at low temperatures to overlap within experimental uncertainties. Taking μ_{0} = −4.5*t* and various values of *z* and ν in the range of 0 < *z* < 4 and 0 < ν < 1, we compute the corresponding scaled occupation numbers *T* = 5.8 to 15 nK can collapse to a single curve by computing the reduced chi-squared (*30*). The best-fit exponents (Fig. 2B) are determined as _{0} = −4*t*, we find the exponents to be *z* = 2, ν = 1/2, and μ_{0} = −4.5*t*.

Our measurements at different temperatures allow us to investigate the breakdown of quantum criticality at high temperatures. To quantify the deviations, we focus on the temperature dependence of the scaled occupation number _{0} (Fig. 3). Deviations from the low-temperature value are clear when the temperature exceeds *T** = 22 nK ≈ 8*t*/*k*_{B}. From this, we conclude that at μ = μ_{0}, the upper bound of thermal energy for the quantum critical behavior in our system is *k*_{B}*T** ≈ 8*t*. Our result is in fair agreement with the prediction of 6*t* based on quantum Monte Carlo calculations (*23*).

From the equation of state, one can derive other thermodynamic quantities in the critical regime. We derive the entropy per particle *S*/*N* based on measurements in the temperature range of 5.8 to 15 nK, using a procedure similar to (*31*). The measured entropy per particle depends only on the scaled chemical potential *a* = 1.8(1), *b *= 1.1(1). From this linear dependence, we derive an empirical equation of state analogous to the ideal gas law (*30*)

_{P=Cnx(kBT)y}(4)

where *P* is the pressure of the 2D gas, *x* = 2/(1 + *b*) = 0.95(5), *y* = 2*b*/(1 + *b*) = 1.05(5), *C* = 0.8(2)(*td*^{2})* ^{w}* is a constant, and

*w*= (1 −

*b*)/(1 +

*b*) = −0.05(5).

Finally, we observe a weak dependence of the critical entropy per particle on the atomic interaction. Noting that a weakly interacting 2D Bose gas follows similar scaling laws near μ = 0 (*18*) because it belongs to the same underlying dilute Bose gas universality class (*2*, *32*), we apply similar analysis and extract the critical entropy per particle *S*_{c}/*N* at four interaction strengths *g* ≈ 0.05, 0.13, 0.19, 0.26, shown together with the lattice data (*g* ≈ 2.4) in Fig. 4B. We observe a slow growing of *S*_{c}/*N* with *g*, and compare the measurements with mean-field calculations. The measured *S*_{c}/*N* is systematically lower than the mean-field predictions, potentially as a consequence of quantum critical physics. The weak dependence on the interaction strength can be captured by a power-law fit to the data as *S*_{c}/*Nk*_{B} = 1.6(1)*g*^{0.18(2)}.

In summary, on the basis of in situ density measurements of Bose gases in 2D optical lattices, we confirm the quantum criticality near the vacuum-to-superfluid quantum phase transition. Our experimental methods hold promise for identifying general quantum phase transitions, and prepare the tools for investigating quantum critical dynamics.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1217990/DC1

Materials and Methods

Fig. S1

## References and Notes

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**Acknowledgments:**We thank N. Prokof'ev and D.-W. Wang for discussions and numerical data; Q. Zhou, K. Hazzard, and N. Trivedi for discussions; and N. Gemelke and C. Parker for discussions and reading of the manuscript. The work was supported by NSF (grants PHY-0747907 and NSF-MRSEC DMR-0213745), the Packard foundation, and a grant from the Army Research Office with funding from the Defense Advanced Research Projects Agency Optical Lattice Emulator program. The data presented in this paper are available upon request sent to cchin@uchicago.edu.