Ground-state phase diagram of the square lattice Hubbard model from density matrix embedding theory

Abstract

We compute the ground-state phase diagram of the Hubbard and frustrated Hubbard models on the square lattice with density matrix embedding theory using clusters of up to 16 sites. We provide an error model to estimate the reliability of the computations and complexity of the physics at different points in the diagram. We find superconductivity in the ground state as well as competition between inhomogeneous charge, spin, and pairing states at low doping. The estimated errors in the study are below T c in the cuprates and on the scale of contributions in real materials that are neglected in the Hubbard model. DOI: 10.1103/PhysRevB.93.035126 The Hubbard model [1][2][3] is one of the simplest quantum lattice models of correlated electron materials. Its one-band realization on the square lattice plays a central role in understanding the essential physics of high-temperature superconductivity [4,5]. Rigorous, near-exact results are available in certain limits [6]: at high temperatures from series expansions [7][8][9][10], in infinite dimensions from converged dynamical mean-field theory [11][12][13][14], and at weak coupling from perturbation theory [15] and renormalization group analysis [16,17]. Further, at half filling, the model has no fermion sign problem, and unbiased determinantal quantum Monte Carlo simulations can be converged [18]. Away from these limits, however, approximations are necessary. Many numerical methods have been applied to the model at both finite and zero temperatures, including fixed-node, constrained path, determinantal, and variational quantum Monte Carlo (QMC) [19][20][21][22][23][24][25][26][27][28][29], density matrix renormalization group (DMRG) [30][31][32], dynamical cluster (DCA) [33,34], (cluster) dynamical mean-field theories (CDMFT) [35,36], and variational cluster approximations (VCA) [37,38]. (We refer to DCA/CDMFT/VCA collectively as Green's function cluster theories.) These pioneering works have suggested rich phenomenology in the phase diagram including metallic, antiferromagnetic, and d-wave (and other kinds of) superconducting phases, a pseudogap regime, inhomogeneous orders such as stripes, and charge, spin, and pairdensity waves, as well as phase separation [6,19,20,24,25,[27][28][29]32,35,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. However, as different numerical methods have yielded different pictures of the ground-state phase diagram, a precise quantitative picture of the ground-state phase diagram has yet to emerge.It is the goal of this paper to produce such a quantitative picture as best as possible across the full Hubbard model phase diagram below U = 8. Our method of choice is density matrix embedding theory (DMET), which is very accurate in this regime [59][60][61][62][63][64][65][66], employed together with clusters of up to 16 sites and thermodynamic extrapolation. We carefully calibrate errors in our calculations, giving error bars to quantify the remaining uncertainty in our phase diagram. These error bars also serve,...

Dynamical mean-field theory study of stripe order and d -wave superconductivity in the two-dimensional Hubbard model

“…Zheng et al [44] have also performed DMET calculations to study the U -doping phase diagram. At U = 6, they find antiferromagnetic order at half-filling, and superconducting order partially coexisting with inhomogeneous magnetic order roughly from doping 0.1 to 0.3.…”

“…Our study thus extends existing single-site real-space DMFT studies of the stripe state and plaquette DMFT studies of the d-wave superconductivity by simultaneously allowing both types of order. The advantage of this real-space treatment over the full cluster and finite-size size methods employed, e.g., by Zheng et al [43,44] is that the computational time for a single iteration of the self-consistency loop scales only linearly with the unit cell size, allowing us to study stripe wavelengths up to λ = 16 for multiple values of the doping. The approximation neglects correlation effects beyond neighboring sites, and thus it can be regarded as a type of mean-field solution for the d-wave superconductivity.…”

“…We find a wide region of coexistence between striped order and spatially modulated d-wave superconductivity, and analyze the behavior of the superconducting order parameter in the striped states. Compared to other recent ground-state studies, we can reach longer stripe wavelengths [44] and study a wider doping region [43]. This allows us to more carefully study the stripe energetics and to determine the ground-state stripe wavelengths.…”

“…At the fixed doping 1 8 , superconducting states were found within DMET, iPEPS, and DMRG, but the states seemed to be metastable [43]. The topic has also been touched in other DMET [44] and functional renormalization group calculations [68], as well as in variational calculations of finite width chains [44,69], but has not been taken into consideration in the DMFT-based Green's function methods. In addition, an analysis of the energetics for the different stripe wavelengths for a wide region of doping values has been missing.…”

“…A study comparing these four methods consistently found striped states at 1 8 doping [43]. The DMET has also been used to map the density-interaction phase diagram of the Hubbard 2469-9950/2018/97(7)/075112 (10) 075112-1 ©2018 American Physical Society model showing (co)existence of stripe states and d-wave superconductivity [44], and AFQMC has been used to study long wavelength stripes at low doping [45]. The existence of the stripe state is also being debated for the closely related t-J -model, where stripes have been studied using iPEPS [46], variational monte carlo [47], and fixed-node Monte Carlo (FN) methods [48].…”