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arXiv:1006.4456 · Europhysics Letters (EPL)

Emergent spin liquids in the hubbard model on the anisotropic honeycomb lattice

Guangquan Wang, M. O. Goerbig, Christian Miniatura, Benoît Grémaud

Published Aug 1, 2011• 15 citations
arXivHTMLPDF

Abstract

We study the repulsive Hubbard model on an anisotropic honeycomb lattice within a mean-field and a slave-rotor treatment. In addition to the known semi-metallic and band-insulating phases, obtained for very weak interactions, and the anti-ferromagnetic phase at large couplings, various insulating spin-liquid phases develop at intermediate couplings. Whereas some of these spin liquids have gapless spinon excitations, a gapped one occupies a large region of the phase diagram and becomes the predominant phase for large hopping anisotropies. This phase can be understood in terms of weakly coupled strongly dimerized states.

DOI
10.1209/0295-5075/95/47013

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We study the repulsive Hubbard model on an anisotropic honeycomb lattice within a mean-field and a slave-rotor treatment. In addition to the known semi-metallic and band-insulating phases, obtained for very weak interactions, and the anti-ferromagnetic phase at large couplings, various insulating spin-liquid phases develop at intermediate couplings. Whereas some of these spin liquids have gapless spinon excitations, a gapped one occupies a large region of the phase diagram and becomes the predominant phase for large hopping anisotropies. This phase can be understood in terms of weakly coupled strongly dimerized states.
The Hubbard model encapsulates the physics of strongly correlated quantum systems in its most basic form. It has been studied intensively in the context of the high-temperature superconductivity. A number of novel phases were recently proposed for Hubbard-like models on the honeycomb lattice, the structure of graphene. We analyzed the Hubbard model of spin- 12\frac{1}{2}21​ fermions on the honeycomb lattice at half-filling using large-scale quantum Monte Carlo simulations. We find that the weak coupling semimetal and the antiferromagnetic Mott insulator at strong interaction are separated by an extended gapped phase in an intermediate coupling regime. Exploring excitation gaps, various correlation functions as well as probing for flux quantization, we conclude that a quantum spin liquid, lacking any conventional order, emerges with local charge and spin correlations, best described by a resonating valence bonds state.
A spin liquid is a novel quantum state of matter with no conventional order parameter where a finite charge gap exists even though the band theory would predict metallic behavior. Finding a stable spin liquid in two or higher spatial dimensions is one of the most challenging and debated issues in condensed matter physics. Very recently, it has been reported that a model of graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin liquid ground state in a wide region of the phase diagram, between a semi-metal (SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically exact quantum Monte Carlo simulations, we extend the previous study to much larger clusters (containing up to 2592 sites), and find, if any, a very weak evidence of this spin liquid region. Instead, our calculations strongly indicate a direct and continuous quantum phase transition between SM and AFMI.
Phase diagram of a Hubbard model for Majorana fermions on the honeycomb lattice is explored using a combination of field theory, renormalization group, and mean-field arguments, as well as exact numerical diagonalization. Unlike the previously studied versions of the model we find that even weak interactions break symmetries and lead to interesting topological phases. We establish two topologically nontrivial phases at weak coupling, one gapped with chiral edge modes and the other gapless with antichiral edge modes. At strong coupling a mapping onto a novel frustrated spin-12\frac{1}{2}21​ model suggests a highly entangled spin liquid ground state.
Motivated by recent experiments on the triangular lattice Mott-Hubbard system $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{C}\mathrm{N})}_{3}$, we develop a general formalism to investigate quantum spin liquid insulators adjacent to the Mott transition in Hubbard models. This formalism, dubbed the SU(2) slave-rotor formulation, is an extension of the SU(2) gauge theory of the Heisenberg model to the case of the Hubbard model. Furthermore, we propose the honeycomb lattice Hubbard model (at half filling) as a candidate for a spin liquid ground state near the Mott transition; this is an appealing possibility, as this model can be studied via quantum Monte Carlo simulation without a sign problem. The pseudospin symmetry of Hubbard models on bipartite lattices turns out to play a crucial role in our analysis, and we develop our formalism primarily for the case of a bipartite lattice. We also sketch its development for a general Hubbard model. We develop a mean-field theory to describe spin liquids and some competing states, and apply it to the honeycomb lattice. On the insulating side of the Mott transition, we find an SU(2) algebraic spin liquid (ASL), described by gapless S=1∕2S=1∕2S=1∕2 Dirac fermions (spinons) coupled to a fluctuating SU(2) gauge field. This result contrasts with that obtained via a U(1) slave-rotor approach, which instead found a U(1) ASL. That formulation does not respect the pseudospin symmetry, and is therefore not correct on the honeycomb lattice. We construct a low-energy effective theory describing the ASL phase, the conducting semimetal phase and the Mott transition between them. This physics can be detected in numerical simulations via the simultaneous presence of substantial antiferromagnetic and valence-bond solid correlations. In the SU(2) ASL, these observables have slowly decaying fluctuations in space and time, described by power laws with the same critical exponent. We recover the results of the U(1) slave-rotor formulation in the presence of a strong breaking of pseudospin symmetry. Our analysis suggests that both a third-neighbor electron hopping, and/or pseudospin-breaking terms such as a nearest-neighbor density interaction, may help to stabilize a spin liquid phase.
In a recent study [Phys. Rev. X 10, 021042 (2020)], we showed using large-scale density matrix renormalization group (DMRG) simulations on infinite cylinders that the triangular lattice Hubbard model has a chiral spin liquid phase. In this work, we introduce hopping anisotropy in the model, making one of the three distinct bonds on the lattice stronger or weaker compared with the other two. We implement the anisotropy in two inequivalent ways, one which respects the mirror symmetry of the cylinder and one which breaks this symmetry. In the full range of anisotropy, from the square lattice to weakly coupled one-dimensional chains, we find a variety of phases. Near the isotropic limit we find the three phases identified in our previous work: metal, chiral spin liquid, and ${120}^{\ensuremath{\circ}}$ spiral order; we note that a recent paper suggests the apparently metallic phase may actually be a Luther-Emery liquid, which would also be in agreement with our results. When one bond is weakened by a relatively small amount, the ground state quickly becomes the square lattice N\'eel order. When one bond is strengthened, the story is much less clear, with the phases that we find depending on the orientation of the anisotropy and on the cylinder circumference. While our work is to our knowledge the first DMRG study of the anisotropic triangular lattice Hubbard model, the overall phase diagram we find is broadly consistent with that found previously using other methods, such as variational Monte Carlo and dynamical mean field theory.
The phase diagram of the strongly correlated Hubbard model with intrinsic spin-orbit coupling on the honeycomb lattice is explored here. We obtain the low-energy effective model describing the spin degree of freedom. The resulting model is then studied by the Schwinger boson and Schwinger fermion approaches. The Schwinger boson method elucidates the boundary between the spin liquid phase and the magnetically ordered phases, Neel order, and incommensurate Neel order. Increasing the strength of the spin-orbit coupling is shown to narrow the width of the spin liquid region. The Schwinger fermion approach sheds further light on the nature of the spin liquid phase. We obtained three different candidates for the spin liquid phase within the mean-field approximation, namely, the gapless spin liquid, topological Mott insulator (fractionalized topological insulator), and chiral spin liquid phases. However, we argue that the gauge fluctuations and the instanton effect may suppress the first two spin liquids, while the chiral spin liquid is stable against gauge fluctuations due to its nontrivial topology.