Hartree-Fock theory of spiral magnetic order in the 2-d Hubbard model

Volker Bach, Élliott H. Lieb, Jan Philip Solovej

Robin Scholle

The two-dimensional Hubbard model is one of the most fundamental models in condensed matter physics. It captures many properties of cuprate high-temperature superconductors. Despite its apparent simplicity, solving the Hubbard model is a challenging task. In this work, we employ the Hartree-Fock approximation, as well as functional renormalization group methods, to construct phase diagrams of the Hubbard model and to shed light on the zoo of magnetic, charge ordered, and superconducting states. The Hartree-Fock approximation can self-consistently determine the expectation values of local charge and magnetic order parameters. Mean-field theories like this can lead to a deeper understanding of the competition, coexistence and transition of different orders. We start by employing the Hartree-Fock approximation in a completely unrestricted form on finite-size lattices and complement the results with calculations in the thermodynamic limit. We construct mean-field phase diagrams of the two-dimensional Hubbard model in the thermodynamic limit for different values of the next-nearest neighbor hopping as a function of filling and temperature. We find that the dominant magnetic orders at moderate coupling strength are Néel order around half-filling, followed by spiral order, and stripe order as the hole doping is increased. We also clarify the nature of the transition from magnetic spiral to stripe order. We then extend the Hartree-Fock scheme by a functional renormalization group flow, which generates new effective interactions and allows us to also access the superconducting phase of the Hubbard model. We present an improved phase diagram, featuring Néel, spiral, and stripe order coexisting with d-wave superconductivity at low temperatures. We analyze the coexistence and competition between these orders. Finally, we perform unrestricted Hartree-Fock calculations of a Hubbard model on the graphene lattice and the Bernal bilayer graphene lattice. We also construct magnetic phase diagrams for these geometries, which feature Néel order around half-filling and various stripe orders in a secondary magnetic dome.

José Daniel Gouveia, R. G. Dias

Zheng-Yu Weng, C. S. Ting

Fakher F. Assaad

Two methods are used, Hartree-Fock and a variational wave function, to investigate spiral states in the three-band Hubbard model. Our variational wave function is a Gutzwiller-projected Hartree-Fock spiral state. We use quantum Monte Carlo methods to calculate the variational energy exactly. In order to avoid boundary defects in the spiral spin structure, we work in a frame where the spin-quantization axis is locally rotated. Both methods show that upon doping the antiferromagnetic structure present at half-band-filling is unstable against a (1,1) spiral state. For moderate (large) values of the Cu on-site Hubbard repulsion, there is a transition to a (1,0) spiral (ferromagnetic) state. The Hartree-Fock calculations show that most of the (1,1) spiral phase is unstable. On the other hand, the (1,0) spiral state is stable.

Robin Scholle, Walter Metzner, Demetrio Vilardi +1 more

We obtain an almost complete understanding of the mean-field phase diagram of the two-dimensional Hubbard model on a square lattice with a sizable next-nearest-neighbor hopping and a moderate interaction strength. In particular, we clarify the nature of the transition region between the spiral and the stripe phase. Complementing previous [] real-space Hartree-Fock calculations on large finite lattices, we solve the mean-field equations for coplanar unidirectional magnetic order directly in the thermodynamic limit, and we determine the nature of the magnetic states right below the mean-field critical temperature T * by a Landau free-energy analysis. While the magnetic order for filling factors n ≥ 1 is always of Néel type, for n ≤ 1 the following sequence of magnetic states is found as a function of increasing hole-doping: Néel, planar circular spiral, multispiral, and collinear spin-charge stripe states. Multispiral states are superpositions of several spirals with distinct wave vectors, and lead to concomitant charge order. We finally point out that nematic and charge orders inherited from the magnetic order can survive even in the presence of fluctuations, and we present a corresponding qualitative phase diagram. Published by the American Physical Society 2024