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arXiv:1702.00035 · Physical review. B./Physical review. B

Ising antiferromagnet in the two-dimensional Hubbard model with mismatched Fermi surfaces

Jan Gukelberger, Lei Wang, Lode Pollet

Published May 15, 2017• 4 citations
arXivHTMLPDF

Abstract

We study the phase diagram of the two-dimensional repulsive Hubbard model with spin-dependent anisotropic hopping at half filling. The system develops Ising antiferromagnetic long-range order already at infinitesimal repulsive interaction strength in the ground state. Outside the perturbative regime, unbiased predictions for the critical temperatures of the Ising antiferromagnet are made for representative interaction values by a variety of state-of-the-art quantum Monte Carlo methods, including the diagrammatic Monte Carlo, continuous-time determinantal Monte Carlo, and path-integral Monte Carlo methods. Our findings are relevant to ultracold atom experiments in the $p$ orbital or with spin-dependent optical lattices.

DOI
10.1103/physrevb.95.205121

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We study the phase diagram of the two-dimensional repulsive Hubbard model with spin-dependent anisotropic hopping at half filling. The system develops Ising antiferromagnetic long-range order already at infinitesimal repulsive interaction strength in the ground state. Outside the perturbative regime, unbiased predictions for the critical temperatures of the Ising antiferromagnet are made for representative interaction values by a variety of state-of-the-art quantum Monte Carlo methods, including the diagrammatic Monte Carlo, continuous-time determinantal Monte Carlo, and path-integral Monte Carlo methods. Our findings are relevant to ultracold atom experiments in the ppp orbital or with spin-dependent optical lattices.
We present a numerically exact study of the Hubbard model with spin-dependent anisotropic hopping on the square lattice using auxiliary-field quantum Monte Carlo method. At half filling, the system undergoes Ising phase transitions on cooling, leading to the formation of Ising-type antiferromagnetic order for repulsive interactions and charge-density wave order for attractive interactions at finite temperatures. By elegantly implementing the sign-problem-free condition and Hubbard-Stratonovich transformations, we achieve significant improvements in precision control of the numerical calculations and obtain highly accurate results of the transition temperatures from weak to strong interactions across representative anisotropies. We further characterize the system by examining the temperature dependence of various thermodynamic properties, including the energy, double occupancy, specific heat, and charge susceptibility. Specifically, we provide unbiased numerical results of the entropy map on temperature-interaction plane, the critical entropy, and the spin, singlon and doublon correlations, all of which are directly measurable in optical lattice experiments. Away from half filling, we explore the behavior of the sign problem and investigate the possible emergence of stripe spin-density wave order in the system with repulsive interaction.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
A two-dimensional model of interacting electrons on a lattice, the Hubbard model, has been studied by means of a numerical simulation technique. Results for the half-filled-band sector are presented for lattices of up to 6\ifmmode\times\else\texttimes\fi{}6 spatial sites, as a function of temperature and electron-electron repulsion UUU. In the ground state, the system is found to exhibit antiferromagnetic long-range order for all values of UUU, although greatly reduced from the mean-field-theory predictions.
Employing a combination of functional renormalization group calculations and projective determinantal quantum Monte Carlo simulations, we examine the Hubbard model on the square lattice bilayer at half filling. From this combined analysis, we obtain a comprehensive account of the ground-state phase diagram with respect to the extent of the system's metallic and (antiferromagnetically ordered) Mott-insulating as well as band-insulating regions. By means of an unbiased functional renormalization group approach, we exhibit the antiferromagnetic Mott-insulating state as the relevant instability of the free metallic state, induced by any weak finite on-site repulsion. Upon performing a careful analysis of the quantum Monte Carlo data, we resolve the difficulty of identifying this antiferromagnetic ground state for finite interlayer hopping in the weak-coupling regime, where nonmonotonic finite-size corrections are shown to relate to the two-sheeted Fermi surface structure of the metallic phase. On the other hand, quantum Monte Carlo simulations are well suited to identify the transition between the Mott-insulating phase and the band insulator in the intermediate-to-strong coupling regime. Here, we compare our numerical findings to indications for the transition region obtained from the functional renormalization group procedure.
Magnetic properties of the two-dimensional Hubbard-model are inferred from results of Monte Carlo simulations. Lattice sizes up to 8\ifmmode\times\else\texttimes\fi{}8 and temperatures down to T=t20T=\frac{t}{20}T=20t​ (t=hoppingt=\mathrm{hopping}t=hopping) were studied. The half-filled system is found to exhibit antiferromagnetic long-range order for all values of the Coulomb repulsion UUU. The low-temperature magnetic properties are found to be well described by spin-wave theory with renormalized local moment and spin-wave velocity. Numerical evidence presented suggests that when doped the system loses the long-range order immediately away from half filling.
Optical lattice experiments with ultracold fermion atoms and quantum gas microscopy have recently realized direct measurements of magnetic correlations at the site-resolved level. We calculate the short-range spin-correlation functions in the ground state of the two-dimensional repulsive Hubbard model with the auxiliary-field quantum Monte Carlo (AFQMC) method. The results are numerically exact at half filling where the fermion sign problem is absent. Away from half filling, we employ the constrained path AFQMC approach to eliminate the exponential computational scaling from the sign problem. The constraint employs unrestricted Hartree-Fock trial wave functions with an effective interaction strength UUU, which is optimized self-consistently within AFQMC. Large supercells are studied, with twist averaged boundary conditions as needed, to reach the thermodynamic limit. We find that the nearest-neighbor spin correlation always increases with the interaction strength UUU, contrary to the finite-temperature behavior where a maximum is reached at a finite UUU value. We also observe a change of sign in the next-nearest-neighbor spin correlation with increasing density, which is a consequence of the buildup of the long-range antiferromagnetic correlation. We expect the results presented in this paper to serve as a benchmark as lower temperatures are reached in ultracold atom experiments.