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arXiv:1007.3266 · Physical Review B

Z2 spin liquid and chiral antiferromagnetic phase in the Hubbard model on a honeycomb lattice

Yuan-Ming Lu, Ying Ran

Published Jul 15, 2011• 79 citations
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Abstract

In a Schwinger-fermion representation we classify all 128 possible spin liquids that preserves $\mathit{SU}(2)$ spin-rotational symmetry, honeycomb lattice group symmetry, and time-reversal symmetry. Among them we identify a ${Z}_{2}$ spin liquid called the sublattice-pairing state (SPS) as the spin liquid phase discovered in recent numerical study on a honeycomb lattice [Meng et al., Nature (London) 464, 847 (2010)]. Our method provides a systematic way to identify spin liquids close to Mott transition. We also show that the SPS is identical to the zero-flux ${Z}_{2}$ spin liquid in Schwinger-boson representation [Wang, Phys. Rev. B 82, 024419 (2010)]. through an explicit duality transformation. SPS is connected to an unusual antiferromagnetic ordered phase, which we term the chiral-antiferromagnetic (CAF) phase, by an $O(4)$ critical point. The CAF phase breaks the $\mathit{SU}(2)$ spin rotational symmetry completely and has three Goldstone modes. Our results indicate that there is likely a hidden phase transition between the CAF phase and the simple antiferromagnetic phase at large $U/t$. We also propose numerical measurements to reveal the CAF phase and the hidden phase transition.

DOI
10.1103/physrevb.84.024420

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In a Schwinger-fermion representation we classify all 128 possible spin liquids that preserves SU(2)\mathit{SU}(2)SU(2) spin-rotational symmetry, honeycomb lattice group symmetry, and time-reversal symmetry. Among them we identify a Z2{Z}_{2}Z2​ spin liquid called the sublattice-pairing state (SPS) as the spin liquid phase discovered in recent numerical study on a honeycomb lattice [Meng et al., Nature (London) 464, 847 (2010)]. Our method provides a systematic way to identify spin liquids close to Mott transition. We also show that the SPS is identical to the zero-flux Z2{Z}_{2}Z2​ spin liquid in Schwinger-boson representation [Wang, Phys. Rev. B 82, 024419 (2010)]. through an explicit duality transformation. SPS is connected to an unusual antiferromagnetic ordered phase, which we term the chiral-antiferromagnetic (CAF) phase, by an O(4)O(4)O(4) critical point. The CAF phase breaks the SU(2)\mathit{SU}(2)SU(2) spin rotational symmetry completely and has three Goldstone modes. Our results indicate that there is likely a hidden phase transition between the CAF phase and the simple antiferromagnetic phase at large U/tU/tU/t. We also propose numerical measurements to reveal the CAF phase and the hidden phase transition.
Spin liquids are novel states of matter with fractionalized excitations. A recent numerical study of Hubbard model on a honeycomb lattice\cite{Meng2010} indicates that a gapped spin liquid phase exists close to the Mott transition. Using Projective Symmetry Group, we classify all the possible spin liquid states by Schwinger fermion mean-field approach. We find there is only one fully gapped spin liquid candidate state: "Sublattice Pairing State" that can be realized up to the 3rd neighbor mean-field amplitudes, and is in the neighborhood of the Mott transition. We propose this state as the spin liquid phase discovered in the numerical work. To understand whether SPS can be realized in the Hubbard model, we study the mean-field phase diagram in the J1−J2J_1-J_2J1​−J2​ spin-1/2 model and find an s-wave pairing state. We argue that s-wave pairing state is not a stable phase and the true ground state may be SPS. A scenario of a continuous phase transition from SPS to the semimetal phase is proposed. This work also provides guideline for future variational studies of Gutzwiller projected wavefunctions.
Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum J(1)-J(2) spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases-antiferromagnetic, an unusual Z(2) SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the Z(2) SL state as the likely candidate for the SL phase of the Hubbard model.
Motivated by the recent numerical evidence [Z. Meng, T. Lang, S. Wessel, F. Assaad, and A. Muramatsu, Nature (London) 464, 847 (2010)] of a short-range resonating valence bond state in the honeycomb lattice Hubbard model, we consider Schwinger boson mean field theories of possible spin liquid states on honeycomb lattice. From general stability considerations the possible spin liquids will have gapped spinons coupled to Z2{Z}_{2}Z2​ gauge field. We apply the projective symmetry group method to classify possible Z2{Z}_{2}Z2​ spin liquid states within this formalism on honeycomb lattice. It is found that there are only two relevant Z2{Z}_{2}Z2​ states, differed by the value of gauge flux, zero or $\ensuremath{\pi}$, in the elementary hexagon. The zero-flux state is a promising candidate for the observed spin liquid and continuous phase transition into commensurate N\'eel order. We also derive the critical field theory for this transition, which is the well-studied O(4) invariant theory [A. V. Chubukov, T. Senthil, and S. Sachdev, Phys. Rev. Lett. 72, 2089 (1994); A. V. Chubukov, S. Sachdev, and T. Senthil, Nucl. Phys. B 426, 601 (1994); S. V. Isakov, T. Senthil, and Y. B. Kim, Phys. Rev. B 72, 174417 (2005)], and has an irrelevant coupling between Higgs and boson fields with cubic power of spatial derivatives as required by lattice symmetry. This is in sharp contrast to the conventional theory [S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)], where such transition generically leads to incommensurate magnetic order. In this scenario the Z2{Z}_{2}Z2​ spin liquid could be close to a tricritical point. Soft boson modes will exist at seven different wave vectors. This will show up as low-frequency dynamical spin susceptibility peaks not only at the $\ensuremath{\Gamma}$ point (the N\'eel order wave vector) but also at Brillouin-zone-edge center MMM points and twelve other points. Some simple properties of the $\ensuremath{\pi}$-flux state are studied as well. Symmetry allowed further neighbor mean field ansatz are derived in appendices which can be used in future theoretical works along this direction.
We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin-12\frac{1}{2}21​ system in a honeycomb lattice, to a chiral Z2{\mathbb{Z}}_{2}Z2​ spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral Z2{\mathbb{Z}}_{2}Z2​ state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.
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 this paper we generalized the slave-particle technique to study the phase diagram of the Hubbard model on honeycomb lattice which may contain charge fluctuations. For large UUU, we have antiferromagnetic order phase. As we decrease UUU below Uc2≃3tU_{c2}\simeq 3tUc2​≃3t, the system undergoes a first order phase transition into a gapped spin rotation invariant phase. Under a semiclassical approximation of the slave-particle approach, we find that such phase breaks the translation symmetry, the parity and the lattice rotation symmetry. However, beyond the semiclassical approximation, a Z2Z_2Z2​ spin liquid that does not break any lattice symmetry is also possible.