A systematic low-energy effective field theory for hole-doped antiferromagnets on the honeycomb lattice is constructed. The formalism is then used to investigate spiral phases in the staggered magnetization as well as the formation of two-hole bound states.
Based on a symmetry analysis of the microscopic Hubbard and t-J models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the SU(2)s symmetry, just as nucleons in Quantum Chromodynamics (QCD) pick up their mass from spontaneous chiral symmetry breaking. In the broken phase, the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory, we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits f-wave symmetry.
Identifying the correct low-energy effective theory for magnons and holes in an antiferromagnet has remained an open problem for a long time. In analogy to the effective theory for pions and nucleons in QCD, based on a symmetry analysis of Hubbard and $t\text{\ensuremath{-}}J$-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$. The effective theory is based on a nonlinear realization of the spontaneously broken spin symmetry and makes model-independent universal predictions for the entire class of lightly doped antiferromagnetic precursors of high-temperature superconductors. The predictions of the effective theory are exact, order by order in a systematic low-energy expansion. We derive the one-magnon exchange potentials between two holes in an otherwise undoped system. Remarkably, in some cases the corresponding two-hole Schr\"odinger equations can even be solved analytically. The resulting bound states have d-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry, while for two holes in the same pocket the symmetry resembles dxy.
We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the single- and two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends strong support to the quantitative success of the systematic low-energy effective field theory method not only in the ferromagnetic but also in the physically most interesting antiferromagnetic case.
In view of quite recent experimental activities on magnetic and superconducting properties of honeycomb and hexagonal lattice based materials, in this thesis we have used field-theoretic and many-body methods to investigate magnetic and superconducting properties of the large-U Hubbard model on the honeycomb lattice at half-filling and in the hole-doped regime. Within the framework of a functional-integral approach, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. The Hamiltonian related to the charge degrees of freedom is exactly diagonalized. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the (quantum) magnetic and superconducting phases at half-filling and in the hole-doped regime. At half-filling, we deal with the underlying spin degrees of freedom of the quantum antiferromagnetic (AF) Heisenberg model by employing a second-order spin-wave analysis, in which case we have calculated the ground-state energy and the staggered magnetization; the results are in very good agreement with previous studies. Further, in the continuum, we derive a nonlinear σ-model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. In the challenging hole-doped regime, our approach allows the derivation of a t-J Hamiltonian, and the analysis of the role played by charge and spin quantum fluctuations on the ground-state energy and, particularly, on the breakdown of the AF order at a critical hole doping; the results are benchmarked against recent Grassmann tensor product state simulations. In addition, we have performed an extensive study of the electronic structure of the doped system for each competing phase: AF, ferromagnetic (FM), and (spin-singlet pairing) s-, dx₂₋ʏ₂ – and idxʏ -wave superconducting (SC) state induced by purely electronic effects. In this context, an energetic analysis of the ground state of these phases reveal that the AF order prevails for low hole doping, while a dominantly chiral dx₂₋ʏ₂ + idxʏ superconducting state was found in the vicinity of the Van Hove singularity (high hole doping). We also stress that a thermodynamic analysis of the superconducting phase shows that the critical temperature is directly related to the exchange constant J = 4t²/U, in which t denotes the hopping amplitude and U the on-site Coulomb repulsion of the Hubbard model (purely electronic origin). Remarkably, the competition between the AF and dx₂₋ʏ₂ + idxʏ SC phases takes place by the occurrence of a first-order transition accompanied by a spatial phase separation of the referred phases.
Magnetic phases in the J1-J2 antiferromagnetic XY model on the honeycomb lattice
We study ground-state properties and phase diagram of the J1-J2 antiferromagnetic XY model on the honeycomb lattice by means of the developed corner transfer matrix renormalization group algorithm with the two-site unit cell and the infinite spiral projected entangled pair states ansatz. We identify the main phases: Néel, Ising, collinear, and incommensurate spiral phases, as well as the transitions between them, as functions of the ratio J2/J1. In the regime of competing types of ordering, we show that the energies of the dimerized states are systematically higher than the energies in the collinear phase. This collinear phase transforms to the incommensurate spiral phase through the second-order phase transition upon a further increase of J2/J1.
Variational Monte Carlo Study of the Doped t-J Model on Honeycomb Lattice
The ground state of the bipartite t-J model must satisfy a specific sign structure, based on which the single-hole and two-hole ground state Ansa¨tze on honeycomb lattice are constructed and studied by a variational Monte Carlo (VMC) method. The VMC results are in good agreement with the exact diagonalization (ED) calculation. For the single-hole case, the degenerate ground states are characterized by quantum numbers of a spin-1/2 and an orbital angular momentum Lz=±2. The latter is associated with the emergent chiral spin/hole currents mutually surrounding the hole/spin-1/2 as a composite object or ``twisted hole''. A vanishing quasiparticle spectral weight is shown in the large-sample limit. In the two-hole ground state, the holes form a spin-singlet pairing with d+id symmetry in the Cooper channel, but are of s-wave symmetry as a tightly bound pair of the ``twisted holes''. Such a pairing mechanism of dichotomy can be attributed to eliminating the local spin currents which has nothing to do with the long-range antiferromagnetic correlation. Superconducting ground state at finite doping is briefly discussed in terms of the tightly bound hole pairs as the building blocks.