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Mott Transition in the Π-Flux SU(4) Hubbard Model on a Square Lattice

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Mott Transition in the Π-Flux SU(4) Hubbard Model on a Square Lattice PHYSICAL REVIEW B 97, 195122 (2018) Mott transition in the π-flux SU(4) Hubbard model on a square lattice Zhichao Zhou,1,2 Congjun Wu,2 and Yu Wang1,* 1School of Physics and Technology, Wuhan University, Wuhan 430072, China 2Department of Physics, University of California, San Diego, California 92093, USA (Received 23 March 2018; published 14 May 2018) With increasing repulsive interaction, we show that a Mott transition occurs from the semimetal to the valence bond solid, accompanied by the Z4 discrete symmetry breaking. Our simulations demonstrate the existence of a second-order phase transition, which confirms the Ginzburg-Landau analysis. The phase transition point and the critical exponent η are also estimated. To account for the effect of a π flux on the ordering in the strong-coupling regime, we analytically derive by the perturbation theory the ring-exchange term, which is the leading-order term that can reflect the difference between the π-flux and zero-flux SU(4) Hubbard models. DOI: 10.1103/PhysRevB.97.195122 I. INTRODUCTION dimer orders exist in a projector QMC (PQMC) simulation [30]. With the rapid development of ultracold atom experiments, Considering the significance of density fluctuations in a the synthetic gauge field can be implemented in optical lattice realistic fermionic system, the SU(2N) Hubbard model is a systems [1–4]. Recently, the “Hofstadter butterfly” model prototype model for studying the interplay between density Hamiltonian has also been achieved with ultracold atoms of and spin degrees of freedom. Previous PQMC studies of the 87Rb [5,6]. When ultracold atoms are considered as carriers half-filled SU(2) Hubbard model with a π flux have demon- in optical lattices, they can carry large hyperfine spins. Due to strated a quantum phase transition from the massless Dirac the closed-shell electronic structure of alkaline-earth fermionic semimetal phase to a Mott-insulating phase, accompanied by atoms, their hyperfine spins are simply nuclear spins, and the appearance of long-range AF ordering [31–35]. As for thus the interatomic scatterings are spin-independent, leading the half-filled SU(4) Hubbard model without a flux, with to the SU(2N) symmetry [7–11]. A series of experimen- increasing Hubbard U, the AF order appears on a square lattice tal breakthroughs have been achieved with SU(2N) ultra- [36] while the valence bond solid (VBS) order emerges on a cold atoms [12–16]. Interestingly, an SU(6) Mott-insulating honeycomb lattice [37]. state has been observed with 173Yb atoms in an optical The PQMC method is basically unbiased, nonperturbative, lattice [14]. and asymptotically correct, and particularly sign-problem-free Intense curiosity has been piqued to explore the physics at half filling. In this paper, we shall conduct a PQMC study of when high symmetry meets the synthetic gauge field. Recent the ground-state properties of the half-filled SU(4) Hubbard theoretical studies reveal that multicomponent fermions sub- model with a π-flux gauge field on a square lattice, which ject to a gauge field can give rise to the spin liquid phases helps to unveil novel physics that is absent in both the SU(2) in the SU(N) Hubbard model at the mean-field level [17]as Hubbard model with a π flux and the SU(4) Hubbard model well as in the SU(N) Heisenberg model [18–20]. In solid-state without a flux. In the noninteracting limit, the ground state of physics, the SU(2N) Heisenberg model was first introduced to the system is the gapless Dirac semimetal. It is shown that the handle strong correlation physics by employing the systematic increase of the Hubbard U eventually drives the system into 1/N expansion [21–24]. The SU(2N) Heisenberg model is often considered as the low-energy effective model of the a Mott-insulating state accompanied by VBS ordering, which SU(2N) Hubbard model at strong coupling, where the density breaks the Z4 discrete symmetry. Since cubic terms are absent fluctuations are neglected. It is found that the filling number in the analytic part of Ginzburg-Landau (GL) free energy, the of particles per site can strongly affect relevant physics of semimetal-VBS phase transition on a square lattice should the SU(2N) Heisenberg model. At quarter filling, its ground be a continuous transition, in contrast to the semimetal-VBS state is the long-range antiferromagnetic (AF) order on a transitions on a honeycomb lattice [37–41]. Furthermore, the square lattice, which is confirmed by various quantum Monte critical exponent η is also extracted by finite-size scaling Carlo (QMC) studies [25–28]. At half-filling, different QMC analysis of the numerical data. methods, however, give rather conflicting results: The AF The rest of this paper is organized as follows. In Sec. II, order was found to be associated with the ground state by the model Hamiltonian and parameters of PQMC simulations a variational QMC simulation [29], whereas neither AF nor are introduced. The Mott gap opening mechanism is then studied in Sec. III. Subsequently in Sec. IV, the nature of quantum phase transitions is investigated. The ring-exchange processes are analyzed in Sec. V. The conclusions are drawn *[email protected] in Sec. VI. 2469-9950/2018/97(19)/195122(8) 195122-1 ©2018 American Physical Society ZHICHAO ZHOU, CONGJUN WU, AND YU WANG PHYSICAL REVIEW B 97, 195122 (2018) U=4 1.5 U=6 U=8 1.2 U=10 U=12 sp 0.9 Δ 0.6 0.3 0 FIG. 1. (a) The hopping integrals on a square lattice. Red and 0 0.04 0.08 0.12 0.16 0.2 blue lines correspond to −t and t, respectively, hence each plaquette 1/L is penetrated by a π flux. (b) The Brillouin zone of the π-flux model on a square lattice. The blue lines depict the Fermi surface at half- FIG. 2. The finite-size scalings of the single-particle gap sp for filling in the absence of flux. The red points are the Dirac points at the SU(4) π-flux Hubbard model as the Hubbard U varies. The = π ± π ± π π quadratic polynomial fitting is used. Error bars are smaller than k1,2 ( 2 , 2 ), and the points of ( 2 , 2 ) are equivalent to k1,2. symbols. II. MODEL AND METHOD A. The SU(4) π-flux Hubbard model the ground-state wave function of the noninteracting part of The SU(4) Hubbard model at half-filling is defined by the Eq. (1) with a small flux added for lifting the degeneracy at the lattice Hamiltonian as Dirac points. The simulation parameters are set to τ = 0.05 and β = 40. The measurements of physical observables are † U H =− t (c c + H.c.) + (n − 2)2, (1) performed around β/2 after projecting onto the ground state. ij iα jα 2 i ij ,α i where ij denotes the nearest neighbors, the sum runs over sites of a square lattice, α represents spin indices running from III. GAP OPENING MECHANISM 1to4,ni is the particle number operator on site i defined as In the weak-coupling regime, the system lies in the = 4 † = ni α=1 ciαciα and its average value ni 2intheSU(4) semimetal phase. With the increase of the coupling strength case, and U is the on-site repulsive interaction. U, the system undergoes a phase transition from semimetal For the nearest-neighbor hopping integral tij ,weusethe to Mott-insulating phase. In the SU(2) case, this transition x gauge in which tx = t and ty = (−1) t, such that the product is well-studied via the QMC method [31–35]. In the absence of phases of hopping integrals around a plaquette is eiπ =−1, of an intermediate spin-liquid phase, a second-order phase as illustrated in Fig. 1(a). At weak coupling, the low-energy transition occurs from the Dirac semimetal phase to the AF effective theory of the π-flux model on a square lattice can phase. Nevertheless, the ordering of the Mott-insulating phase be formulated in terms of Dirac fermions. In the weak- in the SU(4) case remains unclear. coupling limit of U/t → 0, the dispersion relations are ε(k) = 2 2 ±2t cos (kx ) + cos (ky ), hence there exist eight low-energy ± π ± π Dirac cones located at ( 2 , 2 ) when taking into account A. Single-particle gap opening the spin degeneracy, as shown in Fig. 1(b). In the atomic limit Since the Dirac cones are located at k = ( π , ± π ), we of U/t →∞, the system is in the Mott-insulating states at half 1,2 2 2 use the PQMC method to calculate the unequal-time Green’s filling. If a single particle is removed from one site and added to another site, the excitation energy is U, independent of the function from which the single-particle gap sp at k1,2 can fermion components. be extracted for various U. Then the parameter regime of the Mott-insulating phase can be determined accordingly. By finite-size scalings we find that the critical coupling strength Uc B. Parameters of PQMC simulations for a single-particle gap opening lies in the range from 8 to 10, We shall employ the zero-temperature PQMC method in the as shown in Fig. 2. This value of Uc lies in between the critical determinant formalism [42–44]. Recently, exciting progress couplings of the π-flux model with SU(2) symmetry and has been made in the PQMC algorithm for the sign-problem- the SU(4) honeycomb-lattice Hubbard model with zero flux.
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