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Chinese Physics Letters

Phase Diagram of the BCS–Hubbard Model in a Magnetic Field

Dong-Hong Xu, Yi-Cong Yu, Xing-Jie Han, Xi Chen, Kang Wang, Mingpu Qin, Haijun Liao, Tao Xiang

Published Jun 1, 2022• 1 citation
Publisher

Abstract

We propose an extended BCS–Hubbard model and investigate its ground state phase diagram in an external magnetic field. By mapping the model onto a model of spinless fermions coupled with conserving Z 2 variables which are mimicked by pseudospins, the model is shown to be exactly solvable along the symmetric lines for an arbitrary on-site Hubbard interaction on the bipartite lattice. In the zero field limit, the ground states exhibit an antiferromagnetic order of pseudospins. In the large field limit, on the other hand, the pseudospins are fully polarized ordered. With the increase of the applied field, a first-order phase transition occurs between these kinds of phases when the on-site Coulomb interaction is less than a critical value U c. Above this critical U c, a novel intermediate phase emerges between the fully polarized and antiferromagnetic phases. The ground states in this phase are macroscopically degenerate, like in a spin ice, and the corresponding entropy scales linearly with the lattice size at zero temperature.

DOI
10.1088/0256-307x/39/6/067403

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We propose an extended BCS–Hubbard model and investigate its ground state phase diagram in an external magnetic field. By mapping the model onto a model of spinless fermions coupled with conserving Z 2 variables which are mimicked by pseudospins, the model is shown to be exactly solvable along the symmetric lines for an arbitrary on-site Hubbard interaction on the bipartite lattice. In the zero field limit, the ground states exhibit an antiferromagnetic order of pseudospins. In the large field limit, on the other hand, the pseudospins are fully polarized ordered. With the increase of the applied field, a first-order phase transition occurs between these kinds of phases when the on-site Coulomb interaction is less than a critical value U c. Above this critical U c, a novel intermediate phase emerges between the fully polarized and antiferromagnetic phases. The ground states in this phase are macroscopically degenerate, like in a spin ice, and the corresponding entropy scales linearly with the lattice size at zero temperature.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
We analyze the possible types of ordering in a boson-fermion model. The Hamiltonian is inherently related to the Bose-Hubbard model for vector two-species bosons in optical lattices. We show that such a model can be reduced to the Kugel-Khomskii type spin-pseudospin model, but in contrast to the usual version of the latter model, we are dealing here with the case of spin S=1S=1S=1 and pseudospin 1/21/21/2. We show that the interplay of spin and pseudospin degrees of freedom leads to a rather nontrivial magnetic phase diagram including the spin-nematic configurations. Tuning the spin-channel interaction parameter Us{U}_{s}Us​ gives rise to quantum phase transitions. We find that the ground state of the system always has the pseudospin domain structure. On the other hand, the sign change of Us{U}_{s}Us​ switches the spin arrangement of the ground state within domains from a ferro- to antiferromagnetic one. Finally, we revisit the spin (pseudospin)-1/2 Kugel-Khomskii model and see the inverse picture of phase transitions.
The phase diagram of the q=3q=3q=3 state Potts antiferromagnet in a field is studied by exact mappings and Monte Carlo methods, and it is shown that systems with macroscopic ground-state degeneracy may exhibit conventional-type critical ordering. Our Monte Carlo data for the square-lattice case indicate a finite-temperature ordering in a range of field values where the entropy of the ground state is extensive. In a particular limit this phase transition can be shown to be of the Ising type by mapping the Potts model onto a colored, nearest-neighbor exclusion lattice gas. We also discuss the phase diagram of the body-centered-cubic Ising antiferromagnet in a field in order to show that renormalization-group arguments alone are not sufficient to restrict the diversity of possible low-temperature phases occurring in systems with a macroscopically degenerate ground state.
Motivated by the fascinating magnetic properties in the half-filled Hubbard model on a honeycomb lattice, we calculate the spin susceptibility and estimate the magnetic transition by applying the two-particle self-consistent (TPSC) method. The diagonalized spin-susceptibility matrix has its maximum at Q = 0 , which indicates that an uniform magnetic order develops between the unit cells. The sign of the spin susceptibility between same sublattices is opposite to that between different sublattices. Therefore, an antiferromagnetic state develops in this system. We calculate the temperature dependence of the Stoner factor and estimate the critical temperature extrapolation. By applying the same analysis to several values of on-site interactions, we present the phase diagram of the magnetic state in the half-filled Hubbard model on a honeycomb lattice. Based on the phase diagram, the critical on-site interaction U c /t above which magnetism occurs at the ground state can be estimated as being approximately 3.81.
We analyze the ground-state properties of mixtures consisting of scalar bosons and spin-12\frac{1}{2}21​ fermions using a mean-field treatment of the local boson-fermion interaction on a simple cubic lattice. In the deep superfluid limit of the boson sector and the BCS regime of the fermion sector, we derive BCS-type equations to determine the phase diagram of the system. We find a competition between a charge density wave and a superconducting phase. In the opposite limit, we study the Mott-insulator-to-superfluid transition of the boson sector in the presence of a staggered density-induced alternating potential provided by the fermions, and determine the mean-field transition line. In the two-superfluids phase of the mixture, we restrict to nearest-neighbor-induced interactions between the fermions and consider the extended Hubbard model. We perform a mean-field analysis of the critical temperature for the formation of boson-assisted sss-, extended ${s}^{\ensuremath{-}}$-, ddd-, and ppp-wave pairs at fermionic half-filling. We compare our results with a recent dynamical mean-field study [P. Anders et al., Phys. Rev. Lett. 109, 206401 (2012)].
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.