Efficient calculation of the antiferromagnetic phase diagram of the three-dimensional Hubbard model

Abstract

The Dynamical Cluster Approximation with Betts clusters is used to calculate the antiferromagnetic phase diagram of the 3D Hubbard model at half filling. Betts clusters are a set of periodic clusters which best reflect the properties of the lattice in the thermodynamic limit and provide an optimal finite-size scaling as a function of cluster size. Using a systematic finite-size scaling as a function of cluster space-time dimensions, we calculate the antiferromagnetic phase diagram. Our results are qualitatively consistent with the results of Staudt et al. [Eur. Phys. J. B 17 411 (2000)], but require the use of much smaller clusters: 48 compared to 1000.The accurate and efficient solution of lattice Hamiltonians such as the Hubbard model is a long standing challenge in the theoretical condensed matter community. These lattice models are routinely solved on a finite periodic lattice, for example with Monte Carlo, and the calculated properties extrapolated to the infinite limit. Due to the numerical expense in solving these models for large lattices, it is imperative to choose lattices that are efficient for the estimation and extrapolation of the physical properties of interest.In this paper we use the Dynamical Cluster Approximation (DCA) [1, 2, 3] (for a review see [4]) to explore the antiferromagnetic instability in the 3D Hubbard model at half filling, withwhere c ( †) iσ (creates) annihilates an electron with spin σ on site i, n iσ is the corresponding number operator, t the hopping amplitude between nearest neighbors i, j and U the on-site Coulomb repulsion. We solve this model on a series of finite clusters chosen according to the criteria proposed by Betts et al. [5,6]. We obtain converged results extrapolating from clusters of up to only 48 sites, which are in agreement with the calculations of Staudt et al. [7], who used conventional cubic lattices of up to 1000 sites and obtained the Néel temperature via the specific heat.To solve the Hamiltonian (1) we utilized the DCA [4]. For a 3D system the DCA maps the original lattice model onto a periodic cluster of size N c = L 3 c embedded in a selfconsistent host. Thus, correlations up to a range ξ < ∼ L c are treated directly, while the longer length scale physics is described at the mean-field level. With increasing cluster size, the DCA systematically interpolates between the single-site dynamical mean-field result and the exact result, while remaining in the thermodynamic limit. We solve the cluster problem using Quantum Monte Carlo (QMC) [8]. At half-filling there is no QMC sign problem; the only systematic error in the Monte Carlo is the time step error, which can be extrapolated away.In order to calculate the phase diagram of the system in the thermodynamic limit, we employ the scalingis the Néel temperature obtained from a DCA calculation with a cluster of linear cluster size L c . This form is justified if we envision the lattice as perfectly tiled by a periodic array of non-overlapping clusters. This system becomes ordered when the antiferromagnetic ...

Three-dimensional Hubbard model in the thermodynamic limit

“…6(a), we plot the Néel temperatures deduced in this way as a function of U as filled circles. We also plot T N for U ≤ 12 from the dynamical cluster approximation (DCA) [13,49] and DQMC, which match our results within the errorbars. The NLCE results are in very good agreement with the theoretical prediction for the large-U Heisenberg limit for U > 12 as well [35,51].…”

“…The 3D Hubbard model has long been a playground for numerical methods to study finite-temperature critical behavior in a system where the electronic correlations can be tuned. Its finite-temperature phase transition to the Néel ordered phase has been studied carefully by a variety of numerical techniques [11,13,16,26,32,33]. However, as these methods are often not well-equipped to handle the strong-coupling regime of the model (U 12t), the complete mapping of the ground state phase diagram in the temperature-interaction plane has been assisted by an asymptotic behavior based on the critical temperature of the low-energy theory in the limit of large interaction strengths [26,33].…”

Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory

“…It also improves over the twoparticle self-consistent theory (TPSC) (Vilk and Tremblay, 1997) which reaches a plateau for T N at large U . Moreover, in the most interesting regime of intermediate coupling 8t U 12t, both methods agree remarkably well with the DCA results by Kent et al, 2005 and diagrammatic determinant Monte Carlo (DDMC) by Kozik et al, 2013, in spite of the intrinsic differences between these methods. Minor quantitative deviations between DF and DΓA are observed only in the weak-and strongcoupling limits.…”

Breaking of SU(4) symmetry and interplay between strongly correlated phases in the Hubbard model

“…Given that DMFT overestimates the Néel temperature in the intermediate-and strong-coupling regimes for the singleband SU(2)-symmetric Hubbard model (see, e.g., Ref. 36 for comparison), we expect that the MIT second-order critical point in the SU(4)-symmetric system lies in the PM region and can thus be directly probed in experiment. In the ordered phase, the MIT is shifted towards smaller interaction strengths and coincides with the AFM transition line.…”