Thermal Ising transition in two-dimensional SU(3) Fermi lattice gases with population imbalance

We focus on three-component SU(3) Fermi gases loaded into a square optical lattice, with population imbalance between one component and the others. At strong coupling the system is described by the SU(3) Heisenberg model with an external field that couples to the population imbalance. We discuss the ground state at the mean-field level and then analyze the thermal fluctuations with the semi-classical Monte Carlo method. The interplay of interactions, population imbalance and thermal fluctuations gives rise to a phase transition linked to the breaking of an emergent Ising symmetry, despite the absence of frustration. This represents a new scenario of discrete symmetry breaking in low-dimensional systems with continuous symmetries. Possible implementations with cold alkaline-earth(-like) atoms are discussed.

We focus on three-component SU (3) Fermi gases loaded into a square optical lattice, with population imbalance between one component and the others. At strong coupling the system is described by the SU(3) Heisenberg model with an external field that couples to the population imbalance. We discuss the ground state at the mean-field level and then analyze the thermal fluctuations with the semi-classical Monte Carlo method. The interplay of interactions, population imbalance and thermal fluctuations gives rise to a phase transition linked to the breaking of an emergent Ising symmetry, despite the absence of frustration. This represents a new scenario of discrete symmetry breaking in low-dimensional systems with continuous symmetries. Possible implementations with cold alkaline-earth(-like) atoms are discussed.
Introduction. Symmetry and its spontaneous breaking have been playing a central role for understanding the low-energy physics of many-body systems and classifying phase transition phenomena. According to the Mermin-Wagner-Hohenberg theorem [1,2], continuous symmetries cannot be spontaneously broken at nonzero temperature in one and two-dimensional systems with sufficiently short-range interactions. However, it has been found in certain systems with only continuously symmetric interactions that a spontaneous symmetry breaking with respect to a discrete order parameter can emerge through a nontrivial mechanism even in low dimensions at nonzero temperature. In the framework of Heisenberglike models, in a seminal work [3] Chandra, Coleman and Larkin introduced the idea that this scenario can be realized in the presence of frustration coming from competing exchange interaction; in fact, their proposal of an Ising transition in the frustrated J 1 -J 2 model on the square lattice has been confirmed in various subsequent studies [4][5][6]. An additional example of this kind is given by the Ising transition in the fully frustrated spin-1/2 Heisenberg ferromagnetic/antiferromagnetic square bilayer, which exhibits a finite temperature phase transition in the 2D Ising universality class [7,8], occurring at the endpoint of the discontinuous (first-order) phase transition between the singlet-dimer and fully polarized triplet phases. In the triangular lattice Heisenberg model it is the interplay of geometric frustration and magnetic field that stabilizes the up-up-down state, which breaks only a discrete translational Z 3 symmetry. In this Letter we aim to extend these concepts to SU(N )-symmetric Heisenberg models: we argue that, even without frustration coming from geometry or competing exchange interactions, the presence of a suitable "external field" [the SU(N ) symmetry admits N − 1 couplings that play the role of generalized magnetic fields] can indeed induce the breaking of an emergent discrete symmetry.
The optimal experimental platform to test the above idea is given by ultracold atoms in optical lattices, which represent invaluable quantum simulators of many-body physics. For instance, two-component Fermi atoms in an optical lattice can realize the Hubbard model [9], the simplest model for strongly correlated electrons. At half filling and strong coupling the system is well approximated by the SU(2) Heisenberg model [10], and recent experiments employing the quantum gas microscope technique have confirmed that in a square optical lattice short-range spin-spin correlations exhibit finite correlation length, as expected at low but finite temperatures (≳ 0.25 times the tunneling energy) [9]. Owing to the advances in the manipulation of cold alkaline-earth(-like) atoms, such as 173 Yb and 87 Sr, analogous experiments on systems with SU(N ) symmetry, N > 2, have been underway [11][12][13][14]. This has stimulated vast theoretical work, leading to numerous predictions of exotic ground states for various lattice geometries and and degrees N [15][16][17][18][19]. However, the effects of "external fields" that partially break the SU(N ) symmetry have been rarely investigated, as well as those of thermal fluctuations [20]. Note that in cold-atom experiments, such an external-field effect can be simulated by imposing a global imbalance of populations among the N components [21].
Model. Inspired by the previous considerations, in this Letter we study the strong coupling regime of threecomponent Fermi gases with SU(3)-symmetric interactions in a square optical lattice at 1/3 filling [15,16] with population imbalance between one component and the others, which breaks the original SU(3) symmetry down to SU(2)×U(1). The system is described by the antiferromagnetic SU(3) Heisenberg model with an external field:Ĥ whereλ i = (λ 1,i ,λ 2,i , · · · ,λ 8,i ) are the generators of the SU(3) Lie algebra in the defining representation [22], acting on the three local basis states, which we refer to as three "colors", |R i ⟩, |B i ⟩, and |G i ⟩ [15,16], at site i. In The SU(3) Heisenberg model corresponds to the spin-1 Hamiltonian with equal bilinear and biquadratic exchange couplings [23], which has been discussed in the context of spin liquid in NiGa 2 S 4 [24], under the iden- In the language of solid state physics, the imbalance field D corresponds to the intrinsic single-ion anisotropy of the magnetic material. In artificial quantum systems of alkaline-earth(-like) atoms in optical lattices, the Hamiltonian (1) can be realized more directly with no finetuning of coupling parameters. Those atoms possess SU(2I + 1)-symmetric repulsive interaction for nuclear spin I (I = 5/2 for 173 Yb and I = 9/2 for 87 Sr), and the technique of optical pumping allows for the preparation of any number N of components out of the 2I + 1 spin states. [25] The mean-field ground state. First, we consider the mean-field ground state of Eq. (1) at zero temperature, based on a variational wave function of the form |Ψ⟩ = vector parametrized as The amplitude and phases ( For D = 0, namely at the SU(3)-symmetric point, the minimization only imposes that the d i vectors on neighboring sites are orthogonal. This is not enough to uniquely determine the global configuration of the d i vectors, resulting in an accidental ground state degeneracy within the mean-field approximation. Previous studies have predicted that the thermal fluctuations favor the Néel configuration shown in Fig. 1(a) via entropic selection [15,16], while the threecolor stripe long-range order shown in Fig. 1(b) is chosen via a quantum order by disorder mechanism.
In the presence of an imbalance field D, this degeneracy is already lifted without taking fluctuations into account. In Figs. 1(c) and 1(d), we show the values of the energy per site and the global population imbalance P g , respectively, as functions of D/J, for several meanfield solutions, labeled (i)-(v). For D < 0, the imbalance field D tends to increase the global population of the |G⟩ state and competes with the antiferromagnetic coupling J, which tends to arrange different colors at neighboring sites. When D is sufficiently negative (D < −8J/ √ 3), the forced ferromagnetic state [(i) in Fig. 1] is formed (|ψ i ⟩ = |G⟩ for all i). For −8J/ √ 3 < D < 0, we found the "minority-united canted-Néel" (MUCN) phase [(ii) in Figs. 1 and 2], in which the variational parameters in , θ 0 , ϕ 0 , Q 2 · r i + α 0 ), as the ground state. Here, the phases θ 0 , ϕ 0 , and α 0 are arbitrary and Q 2 = (π, π) is the propagation vector. In the MUCN state, while the phase α i is alternating on the two sublattices (see Fig. 1(a)), interestingly the SU(2) sector of R and B exhibits ferromagnetic order with uniform θ 0 and ϕ 0 , despite the antiferromagnetic nature of the exchange interaction. This can be understood as a mechanism to minimize the interaction energy between the majority G and the minorities {R, B}. Notice that, the MUCN state can be mapped to the conventional canted-Néel state of quantum antiferromagnets under magnetic field through the mapping cos (θ/2)|R⟩ + e iϕ sin (θ/2)|B⟩ → | ↑⟩, |G⟩ → | ↓⟩. For D > 0, since the global population of {|R i ⟩, |B i ⟩} tends to increase, there is no competition with the antiferromagnetic interactions. Therefore, an infinitesimally small field stabilizes (iii), the standard Néel order (with R and B), which is of course the ground state of the SU(2) Heisenberg model.
Although the mean-field theory predicts the twosublattice ground states (MUCN and R-B Néel), besides the uniform state for D < −8J/ √ 3, we also found threesublattice solutions as metastable states [corresponding to (iv) and (v) in Figs. 1(c) and 1(d)]. While in state (iv) the three colors coexist with r(D < 0) = 1 , the component |G⟩ vanishes in state (v), that is, s i = 1. All these states, including the above-mentioned MUCN and R-B Néel, are massively degenerate, owing to the SU(2)×U(1) symmetry of the Hamiltonian. Upon choosing the gauge θ i = π/2, for which densities of B and R particles are identical and uniform on the lattice, we summarize the remaining two parameters ϕ and α of the mean-field solutions [(ii)-(v)] in Fig. 2.
Thermal phase diagram. Since the lowest temperatures reached in current experiments are in the range T ≃ 0.7J [SU(6) chain] [12] to T ≃ 0.9J [SU(2) square lattice] [9], addressing the finite temperature effects becomes important beyond a purely theoretical motivation. We hereby take into account thermal fluctuations from the meanfield ground state using semi-classical multi-color Monte Carlo (sMC) simulations [20,26,27].
We perform the standard Metropolis algorithm that allows the complex vectors d i on L × L sites under periodic boundary conditions to fluctuate thermally with the Boltzmann weight exp (−E MF /T ). In addition, we employ the relaxation acceleration (RA) technique [20] with local unitary transformations exp (icĤ loc i /∥Ĥ loc i ∥ F ) defined by the local mean-field HamiltonianĤ loc i = J 2 j∈NNi ⟨λ j ⟩ ·λ i − Dλ 8,i . Here the real number c is randomly generated from an uniform distribution on the interval [−1, 1], ∥ · · · ∥ F is the Frobenius norm, and the sum runs over all nearest-neighbor sites of site i. In the case of the SU(3) system, the eigenvalues ofĤ loc i can be calculated analytically, which contributes to the reduction of the calculation cost [28]. The single Monte Carlo step consists of the Metropolis updates for d σ,i over the entire lattice, followed by double RA sweeps. This procedure can enhance decorrelation and increase the convergence speed [20]. Figure 3 shows the sMC thermal phase diagram. The most surprising feature is a fluctuation-induced phase with true long-range order emerging for −0.8 ≲ D < 0. The R-B Néel and MUCN states, being continuoussymmetry-breaking states, only possess short-range correlations and a crossover connects them to the hightemperature disordered phase.
/L 2 to detect the correlations in the SU(2) and U(1) sectors, respectively. The R-B Néel and MUCN phases can be identified by S SU(2) (q) with ordering vectors q = Q 2 ≡ (π, π) and q = 0, respectively. We confirm that there is no crossing point of ξ SU(2) /L for different system sizes down to low temperatures [see a typical case in Fig. 4(a)], which indicates no long-range order at T > 0 as expected. In Fig. 3, we plot the crossover temperature T * c from disordered to short-range correlated state by choosing the condition ξ SU(2) = 10 sites [see Fig. 4(b)], which is comparable to the linear system size in typical cold-atom experiments in two dimensions [9]. The emergence of the nontrivial true long-range order between the regions with the R-B Néel and MUCN short-range correlations for T > 0 is of particular significance. In this phase, most of the snapshots in the sMC simulations show a checkerboard pattern where one sublattice is occupied mostly by |G⟩ and the other by random superpositions of |R⟩ and |B⟩: |ψ iA ⟩ ≃ |G⟩, |ψ iB ⟩ ≃ cos (θ iB /2)|R iB ⟩ + e iϕi B sin (θ iB /2)|B iB ⟩, where θ iB , ϕ iB are randomly chosen at each site in each sMC snapshot [i.e., disordered in the SU(2) sector]. This state breaks neither SU(2) nor U(1) symmetry, but only the discrete translational symmetry. Thus, the Mermin-Wagner-Hohenberg theorem [1,2] does not forbid a finitetemperature phase transition to this checkerboard partially disordered state' (CBPD) with true long-range order. In fact, as seen in Fig. 4(c), the scaled correlation lengths ξ U(1) /L, obtained from S U(1) (q = Q 2 ), for different linear sizes L cross each other at a certain critical point T c > 0. According to the finite size scaling theory the scaled magnetic susceptibility , should become sizeindependent at the critical temperature T c : Fig. 4(d) shows that this scaling law is reproduced with critical exponent η = 1/4, indicating the 2D Ising universality class, as expected from the Z 2 translational symmetry breaking of the checkerboard pattern.
Let us discuss the mechanism of the emergence of CBPD order. Within the mean-field approximation, this phase is included in the infinitely degenerate groundstate manifold at the SU(3)-symmetric point, since each pair of the d i vectors on neighboring sites is orthogo- nal. For T > 0 one sublattice is disordered in the SU(2) sector while the checkerboard pattern is kept. Since the partial disorder compensates the loss of entropy, thermal fluctuations select this phase from the zero-temperature degenerate manifold.
[30] Let us also recall that the threesublattice state selected by purely quantum fluctuations is unstable for any T > 0, according to linear flavor-wave theory [15], in favor of two-sublattice states, and therefore the CBPD phase is expected to be robust against the inclusion of quantum effects beyond single site. in the three-dimensional case, the entropic selection of twosublattice phases has also been reported [18], although the three-sublattice phase is not completely excluded near zero temperature due to the higher dimensionality.
Experimental detection. From an experimental perspective, it is convenient to describe the thermal phase diagram (Fig. 3) as a function of the global population imbalance P g defined by P g = √ 3 i ⟨λ 8,i ⟩ = NR+NB−2NG NR+NB+NG , where N µ represents the global population of color µ [21], because this quantity, rather than D, is controllable in cold atom experiments [21]. Figure 5 shows the T -P g phase diagram. Note that a phase separation region appears between CBPD and MUCN. The detection of the CBPD order may be realized as follows. First, the |G⟩ state is removed from the entire lattice via optical pumping [25] and horizontal pairs of neighboring lattice sites are merged into single sites so as to form a new square lattice. We then look for signatures of the Mott insulating state, in which ideally all sites are singly occupied by either R or B state, with the established techniques [31][32][33]. The experiment is repeated and the same protocol is operated with vertical pairs instead. If, in both instances, the Mott insulating state is detected, this would be proof of the realization of the CBPD order. In addition, the eventual extension of the quantum gas mi-croscope technique to SU(N ) systems [34] could detect each phase more directly. Note that the sMC method replaces the statistics of discrete quantum levels by approximate continuous classical statistics of the vectors d i . Since this treatment should overestimate the entropy of paramagnetic states, thus underestimating the transition/crossover temperatures, the lines of T c (T * c ) in Fig. 5 should be taken as lower bounds.
Conclusion. We have investigated the emergence of long-range order by thermal disorder in the square-lattice SU(3) Fermi gas at strong coupling with population imbalance and proposed experimental setups to realize and detect our predictions with alkaline-earth(-like) atoms. Applying mean-field theory and the sMC method to the SU(3) Heisenberg model, we found that the competition between antiferromagnetic interactions and the population imbalance gives rise to several interesting magnetic states, including a true long-range ordered state that breaks only a discrete symmetry. This state enjoys an entropic advantage coming from the abundance of color degrees of freedom and is therefore favored by thermal fluctuations. This scenario generalizes the idea of emergent discrete symmetry breaking in low-dimensional systems without the traditional forms of magnetic frustration. Furthermore, we recall that the quantum fluctuations at D = 0 (no population imbalance) select the three-color stripe order, which should then persist for sufficiently small D < 0. This creates the condition for a finite-temperature transition originating from the difference between quantum and thermal order-by disorder selection, a very rare phenomenon in magnetic systems [35,36].
In conclusion, our work offers new directions in SU(N > 2) magnetism, which is presently under intensive investigation in cold-atom experiments, showing further examples of the rich physics induced by population imbalance. It also provides a new insight into the phase transition phenomena of highly-symmetric systems, which can prove beneficial in other contexts, such as solid-state physics [24].
We would like to thank, Y. Takahashi