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arXiv:2110.06857

Phase transitions in partial summation methods: Results from the 3D Hubbard model

Sergei Iskakov, Emanuel Gull

Published Oct 13, 2021• 0 citations
arXivHTMLPDF

Abstract

The accurate determination of magnetic phase transitions in electronic systems is an important task of solid state theory. While numerically exact results are readily available for model systems such as the half-filled 3D Hubbard model, the complexity of real materials requires additional approximations, such as the restriction to certain classes of diagrams in perturbation theory, that reduce the precision with which magnetic properties are described. In this work, we examine the description of magnetic properties in second order perturbation theory, GW, FLEX, and two TMatrix approximations to numerically exact CT-QMC reference data. We assess finite-size effects and compare periodic lattice simulations to cluster embedding. We find that embedding substantially improves finite size convergence. However, by analyzing different partial summation methods we find no systematic improvement in the description of magnetic properties, with most methods considered in this work predicting first-order instead of continuous transitions, leading us to the conclusion that non-perturbative methods are necessary for the accurate determination of magnetic properties and phase transitions.

DOI
10.48550/arxiv.2110.06857

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The accurate determination of magnetic phase transitions in electronic systems is an important task of solid state theory. While numerically exact results are readily available for model systems such as the half-filled 3D Hubbard model, the complexity of real materials requires additional approximations, such as the restriction to certain classes of diagrams in perturbation theory, that reduce the precision with which magnetic properties are described. In this work, we examine the description of magnetic properties in second order perturbation theory, GW, FLEX, and two TMatrix approximations to numerically exact CT-QMC reference data. We assess finite-size effects and compare periodic lattice simulations to cluster embedding. We find that embedding substantially improves finite size convergence. However, by analyzing different partial summation methods we find no systematic improvement in the description of magnetic properties, with most methods considered in this work predicting first-order instead of continuous transitions, leading us to the conclusion that non-perturbative methods are necessary for the accurate determination of magnetic properties and phase transitions.
A minimal model of a correlation-driven phase transition is the half-filled Hubbard model in three dimensions, for which numerically exact solutions are available. The authors compare exact results to results obtained from a range of diagrammatic perturbation theories, generalizable to real materials problems. They demonstrate that quantum embedding leads to an efficient elimination of finite-size effects. However, perturbative methods do not provide an accurate description of the phase transition even in the regime of weak-to-intermediate coupling.
We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as GWGWGW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GWGWGW in GW+DMFTGW + \mathrm{DMFT}GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
We present results of Monte Carlo simulations of the three-dimensional half-filled Hubbard model on simple-cubic 43{4}^{3}43 lattices and some 63{6}^{3}63 lattices with periodic boundary conditions, as well as of small clusters embedded in an effective medium. Magnetic properties were studied for values of the Hubbard repulsion U ranging from 0 to the bandwidth W=12t (t is the hopping matrix element). Our results show that antiferromagnetism is enhanced up to U\ensuremath{\sim}(5/6)W and then suppressed. The q=0 magnetic susceptibility is enhanced by U, although much less than predicted by the random-phase approximation. The transition temperature to an antiferromagnetic state is found to have a broad maximum around U\ensuremath{\sim}(5/6)W, with a maximum value of approximately W/18. The implications of our results for superconductivity in the attractive Hubbard model are also discussed.
We present important use cases and limitations when considering results obtained from cluster perturbation theory (CPT). CPT combines the solutions of small individual clusters of an infinite lattice system with the Bloch theory of conventional band theory to provide an approximation for the Green’s function in the thermodynamic limit. To this end, we are investigating single-band and multi-band Hubbard models in 1D and 2D systems. A special interest is taken in the supposed pseudogap regime of the 2D square lattice at half-filling and intermediate interaction strength (U≤3t)(U≤3t)(U≤3t) as well as the metal–insulator transition. We point out that the finite-size level spacing of the cluster limits the resolution of spectral features within CPT. This restricts the investigation of asymptotic properties of the metal–insulator transition, as it would require much larger cluster sizes that are beyond computational capabilities.
The antiferromagnetic Ne\'el temperature TN{T}_{N}TN​ is calculated as a function of the Coulomb interaction U for a half-filled 3D Hubbard model. Numerical results are given for the specific heat and staggered magnetic suspectibility on periodic lattices ranging from 43{4}^{3}43--103{10}^{3}103 sites, and used to determine TN{T}_{N}TN​. The size dependence of the results suggests that the transition is weakly first order at small to intermediate values of U.