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Absence of a Spin Liquid Phase in the Hubbard Model on the Honeycomb Lattice

Scientific Reports volume 2, Article number: 992 (2012) | Download Citation

Abstract

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.

Introduction

A spin liquid can be considered as a Mott insulator that is not adiabatically connected to any band insulator and does not break any symmetry even at zero temperature. Recently, much attention has been focused on a possible spin liquid in two or three spatial dimensions1,2,3,4,5. On one hand, it has been suggested experimentally that several organic materials represent good candidates for spin liquids6,7,8. On the other hand, the existence of a spin liquid has so far been demonstrated only in very few and particularly simplified models9,10.

In order to understand this issue, let us consider a model Hamiltonian on a lattice describing the insulating state of electrons at half-filling, i.e., one electron per lattice site. Since the charge gap is assumed, only spin degrees of freedom remain and can be described by the spin-1/2 Heisenberg model. Since any spin-1/2 model corresponds to a well defined interacting hard core boson model, the crucial question is to understand how – at zero temperature – these bosonic degrees of freedom can avoid Bose-Einstein condensation and/or crystallization, necessary conditions for a stable spin liquid with no long-range order of any kind.

In this report, we study the ground state of the half-filled Hubbard model on the honeycomb lattice (see Fig. 1) defined by the following Hamiltonian, where is a creation (annihilation) operator of spin up/down (σ = ↑,↓) electrons on lattice site i, , and t (U) denotes the nearest-neighbor hopping (on-site repulsion). This is known to be a model Hamiltonian for graphene with U/t ≈ 311. More importantly, it is not geometrically frustrated, namely, as seen in Fig. 1, the neighboring sites of any site on A sublattice belong to B sublattice (and vice versa). Indeed, it is well known that the ground state becomes an antiferromagnetic insulator (AFMI), i.e., classically Néel ordered, from a semi-metal (SM) with increasing U/t12.

Figure 1: The honeycomb lattice.
Figure 1

Primitive lattice vectors and are denoted by red arrows. As an example, the honeycomb lattice with L = 3 is indicated by dashed blue lines. Solid and open circles indicate sites on A and B sublattices, respectively.

Very recently, Meng et al.2 has reexamined the ground state phase diagram of this model and found a possible spin liquid phase in the range between SM and AFMI. Their finding is rather surprising because it is widely believed that a stable spin liquid occurs most likely in frustrated quantum systems where strong quantum fluctuations destroy the long-range magnetic order even at zero temperature1. Their study was particularly successful because, with the auxiliary field technique13, there is no sign problem in the corresponding quantum Monte Carlo simulations, and an accurate finite size scaling was possible by using numerically exact results for clusters containing up to 648 sites. So far, their results represent the most important numerical evidence for a possible spin liquid ground state in a “realistic” electronic model in two dimensions (2D), because, to our knowledge, only a particularly simplified quantum dimer model on the triangular lattice9 and the Kitaev model10, built ad hoc to have an exact solution, allow a spin liquid ground state in 2D. Furthermore, their results were considered to be a clear violation of the “Murphy's Law”: in a too simple model, not vexed by the “fermion sign problem”, nothing interesting can occur5.

Here, by performing simulations for much larger clusters containing up to 2592 sites, we show that antiferromagnetic order concomitantly occurs once the insulating behavior sets in, supporting the more conventional Hartree-Fock (HF) transition from SM to AFMI12. Although our results agree with the previous study for the same clusters up to 648 sites, we have reached a quite different conclusion, as the possible spin liquid region reduces substantially to a small interval , if it ever exists. This reminds us similar claims on spin liquid behaviors in different systems in 2D14,15, which have been corrected later on by much larger cluster simulations, showing instead antiferromagnetic long-range order16,17.

Results

We use finite size clusters of N = 2L2 sites (thus containing L × L unit cells) with periodic boundary conditions (see Fig. 1), which satisfy all symmetries of the infinite lattice18 (also see Supplementary information). Here L is the linear dimension of the cluster and we take L up to 36. We use the well established auxiliary field Monte Carlo technique13, which allows the statistical evaluation of the following quantity, where Ô is a physical operator, |ψR〉 (|ψL〉) is the right (left) trial wave function (not orthogonal to the exact ground state), and τ is the projection time. The exact ground state expectation value 〈Ô〉 of the operator Ô is then obtained by adopting the limit of τ → ∞ and Δτ → 0 for O(τ), where Δτ is the short time discretization of τ. This approximation – the so called Trotter approximation – is necessary to introduce the auxiliary fields13 and implies a systematic error, negligible for small Δτ (see Supplementary information).

First, we study both the spin structure factor and the spin-spin correlations Cs(R) = 〈Sr,A·Sr+R,A〉 at the maximum distance |R| = Lmax of each cluster for U/t = 4, where the strongest evidence of a spin liquid behavior was found in Ref. 2. Here Sr,A (Sr,B) is the spin operator at unit cell r on A (B) sublattice. As shown in Fig. 2b, our results show consistently a finite value of the antiferromagnetic order parameter for L → ∞, in sharp contrast to the existence of a spin liquid, i.e., spin disordered, ground state reported in Ref. 2.

Figure 2: Finite size scaling of spin-spin correlation functions for the Hubbard model on the honeycomb lattice at half-filling.
Figure 2

Spin structure factor, SAF, and spin-spin correlations at the maximum distance, Cs(Lmax), are denoted by triangles and circles, respectively. Here, L is the linear size of clusters containing N = 2L2 sites. Antiferromagnetic order parameter squared, , is estimated by finite size extrapolating SAF and Cs(Lmax) to L → ∞, namely, . Solid curves are fit of the data by cubic polynomials in 1/L. It is clearly seen that a consistent extrapolated value is obtained for both quantities SAF and Cs(Lmax), indicated respectively by triangles and circles at 1/L = 0. Error bars of the extrapolated values are computed with a resampling technique described in Methods. Insets show the expanded plots for large L. The fits are stable upon removal of the data for the largest (i.e., L = 36) or the smallest (i.e., L = 6) size, and the extrapolated value of Cs(Lmax) is always consistent with SAF/N within two standard deviations. Including the largest size calculations in the fits increases the extrapolated values slightly and at the same time gives more consistent values of Cs(Lmax) and SAF/N in L → ∞, thus clearly indicating that our present estimate provides an accurate lower bound for the AF order parameter ms. All data presented in this figure refers to Δτt = 0.1, because the Trotter Δτ error is essentially negligible (see Fig. 3). More details are found in Supplementary information.

By doing similar calculations for several U/t values (see Fig. 2 and Supplementary information), we find in Fig. 3 that ms approximately scales linearly with respect to U/t, i.e., ms |UUc|β, with a critical exponent , which is similar to the critical behavior (β = 1) predicted by the HF theory12. Although corrections to this almost linear critical behavior are obviously expected, they do not change much the critical value Uc at which the antiferromagnetic order melts, as clearly shown in Fig. 3. Our best estimate of the critical value is Uc/t = 3.869± 0.013, which is much smaller than the one (≈ 4.3) reported in Ref. 2. Note, however, that the critical exponent β may be different from the present estimate if the critical region is very close to Uc. In such case the accurate determination of β obviously requires much larger clusters which are not feasible at present.

Figure 3: The ground state phase diagram for the half-filled Hubbard model on the honeycomb lattice.
Figure 3

Antiferromagnetic order parameter ms (open squares) as a function of U/t. The error, due to the finite Δτ in the evaluation of SAF, is removed by quadratic extrapolations for Δτt = 0.1, Δτt = 0.2, and Δτt = 0.4 (see Supplementary information for details). The antiferromagnetic order parameter ms is obtained by finite-size extrapolating the square root of SAF/N, , as shown in Fig. 2. For comparison, ms estimated by finite-size extrapolating SAF for Δτt = 0.1 without the Δτ correction is also plotted (solid circles). SM and AFMI stand for semi-metal and antiferromagnetic insulator, respectively. Solid lines are fit of ms with the critical behavior ms = (UcU)β, for selected critical exponents β. β = 1 for the HF theory12, β = 0.3362 for the classical critical theory of quantum magnets19, and β = 0.80 ± 0.04 is the best fit of our data. In any case, the critical Uc ranges from Uc/t = 3.8 (β = 1) to Uc/t = 3.9 (β = 0.3362). Our best estimate is Uc/t = 3.869 ± 0.013.

Let us now evaluate the spin gap Δs. In order to avoid possible errors in extrapolating the imaginary time displaced spin-spin correlation functions, here we calculate directly the total energies in the singlet and the triplet sectors, with improved estimators, which dramatically reduce their statistical errors20 (also see Supplementary information). We can see clearly in Fig. 4a that the extrapolated spin gaps for different U/t values are always zero within statistical errors (e.g., the statistical error as small as 0.004t for U/t = 4).

Figure 4: Finite size scaling of spin gap and charge-charge correlation functions for the Hubbard model on the honeycomb lattice at half-filling.
Figure 4

(a) Spin gap Δs = E(S = 1) − E(S = 0) for various U/t, where E(S) is the lowest energy for a given spin S. Solid curves are fits of data by quadratic polynomials in 1/L. The extrapolated values are also indicated at 1/L = 0. Error bars of the extrapolated values are computed with the resampling technique. In the semi-metallic region, the spin gap scales to zero with increasing the resolution in momentum space, namely as 1/L. In the antiferromagnetic region, the spin gap should instead vanish as 1/L2. This explains why for U/t = 4.3 the gap extrapolates to negative values, as we are well inside the antiferromagnetic phase (see Fig. 3). In any case, a sizable spin gap is not found for any value of U/t. (b) Charge-charge correlation function at the maximum distance |R| = Lmax for several values of U/t. In the semi-metallic phase, ρ(R) ~ 1/R4 and L4ρ(Lmax) should converge to a finite value for L → ∞. Instead, when a charge gap opens, the charge-charge correlations should decay exponentially and L4ρ(Lmax) converges to zero in this limit. Indeed, a quadratic extrapolation to L → ∞ of this quantity, which is clearly appropriate in the semi-metallic phase, appears to vanish in the interval between U/t = 3.8 and U/t = 3.9, in remarkable agreement with the critical value Uc = 3.869 ± 0.013 estimated for the antiferromagnetic transition (see Fig. 3). Obviously, a polynomial fit is not consistent in the insulating region and this explains why the extrapolated value to 1/L = 0 seems slightly positive in this case. For the spin gap and the charge-charge correlation functions, the Trotter Δτ error is negligible, and all data shown here refers to Δτt = 0.14 and 0.1, respectively.

Next, we investigate whether the system is metallic or insulating, namely, whether there exists a zero or a finite charge gap. For this purpose, it is enough to simply study the long distance behavior of charge-charge correlations, . Here nr,A is the density operator at unit cell r on A sublattice (see Fig 1). They should change from power law to exponential behavior at a critical U where the charge gap opens up. This change of behavior is evident in Fig. 4b and appears consistently around the onset of the antiferromagnetic transition (Uc), within a remarkably small uncertainty < 0.1t on the value of U. Our results therefore strongly support the more conventional scenario of a direct and continuous quantum phase transition between SM and AFMI12.

Discussion

Let us now discuss here why we have not found any evidence of a spin liquid phase. As shown in Ref. 21, by applying one of the theorems by Lieb22, it is easily proved that the exact ground state of this model for U ≠ 0 satisfies the Marshall sign rule23 in the sector of no doubly occupied sites, accounting for low energy spin excitations. Indeed, the phases coincide with those of the simple antiferromagnetic Néel state ordered along the x-spin quantization axis, , where |↑〉R and |↓〉R are spin configurations (along the z-spin quantization axis) at site R. The expansion of this state in terms of |↑〉R and |↓〉R yields the simple Marshall sign, namely, it is negative if the number of spin down in the A sublattice is odd. Thus, the phases of the ground state are trivial in the bosonic spin 1/2 sector. Therefore, Bose-Einstein condensation can hardly be avoided and a magnetic long-range order occurs once the charge gap becomes finite.

At this point, one could be tempted to assume the general validity of the above observation for generic S = 1/2 model Hamiltonians with SU(2) invariance and use this criteria based on the phases of the ground state wave function as a powerful guideline in the search of spin liquids for model systems as well as for real materials. Indeed, in all unfrustrated spin-1/2 Heisenberg and Hubbard models in the sector of no doubly occupied sites, the phases of the ground state wave function are not at all entangled in real space as they factorize into independent contributions relative to each site. Instead, the phases of the ground state wave functions are highly non trivial in well established spin liquid models such as, for instance, the Kitaev's model10, and the celebrated quantum dimer model on the triangular lattice9, because they are described by paired wave functions, which couple in a non trivial way the phases of nearest neighbor spins24.

Therefore, we conclude that in a true spin liquid in 2D, the phases of the ground state wave function should be highly non trivial and entangled, otherwise any seed of spin liquid behavior would be most likely destabilized. To our best knowledge, the above observation is valid so far for all spin-1/2 models with SU(2) invariance. Notice that the restriction to SU(2) invariant models appears to be important because the spin-1/2 easy-axis Heisenberg model on the Kagome lattice most likely display spin liquid behavior25,26,27. Here, however, the calculations have not been confirmed on fairly low temperatures yet. Therefore, further numerical study is required for understanding what are the key ingredients that stabilize a spin liquid phase in “realistic” electronic models.

Methods

Here we employ the standard auxiliary field Monte Carlo algorithm13 with a more efficient implementation20 by using different left and right trial functions |ψL〉 and |ψR〉 in equation (2). We include also in the trial wave function a Gutzwiller type projection, exp(−gΣi nini), where g is the Gutzwiller variational parameter, to optimize the efficiency. As reported in Ref. 20, the statistical error in evaluating the energy E(S) for a given spin S is dramatically reduced for appropriate values of g. Thus we can evaluate the spin gap ΔS = E(S = 1) − E(S = 0) with high accuracy, without facing the negative sign problem, by directly simulating the singlet S = 0 and the triplet S = 1 sectors separately (see Supplementary information).

In order to accelerate the convergence to the ground state, we use for |ψR〉 a Slater determinant with a definite spin S, by breaking only spatial symmetries to remove the degeneracy at momenta K and K′ for the clusters chosen (a similar strategy was adopted in Ref. 2). Conversely, we use for |ψL〉 a rotational and translationally invariant Slater determinant obtained by diagonalizing a mean field Hamiltonian, containing an explicit antiferromagnetic order parameter directed along the x-spin quantization axis. In this way, the left and the right trial wave functions break different symmetries (spin and spatial ones, respectively), and for any symmetric operator Ô the convergence to the ground state is expected to be much faster because it is dominated by the singlet gap in the symmetric sector Δgap, that is clearly much larger than, e.g., the lowest triplet excitation in the magnetic phase. Since Δgap is expected to scale to zero (if indeed zero) at most as , we use a projection time τ = (L + 4)/t, which we have tested carefully to be large enough for well converged results (see Supplementary information). We have also checked that the systematic error due to discretizing τ is basically negligible with Δτt = 0.14 for the spin gap calculations and with Δτt = 0.1 for the correlation functions (see Fig. 3 and Supplementary information).

In order to evaluate the statistical errors of the finite size extrapolations, we use a straightforward resampling technique. This resampling technique is used, for example, when values of SAF calculated for finite sizes are extrapolated to L → ∞ to estimate ms in Fig. 3. Let us denote in general the calculated Monte Carlo data f(L) and the corresponding statistical error δf(L) obtained for a cluster of size L. The main goal of this resampling technique is to estimate the finite-size extrapolated value c0 and its statistical error δc0 when the Monte Carlo data are fitted by, e.g., cubic polynomials, In this resampling technique, we first generate for each L a “fictitious sample” which is normally distributed around f(L) with its standard deviation δf(L), which is also an output of the quantum Monte Carlo simulation. Then, we fit these “fictitious” data to equation (3), by using the weighted (with 1/(δf(L))2) least square fit, and estimate c0. We repeat this Mrs times so that we have now Mrs samples for c0, i.e., distributed according to a probability distribution consistent with the Monte Carlo simulations. Finally, we simply average for , and the standard deviation of gives an estimate of the statistical error δc0 of the extrapolated value 〈c0〉. We take Mrs = 200. We have checked that the resultant 〈c0〉 and δc0 are not dependent on Mrs as long as Mrs is large enough.

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Acknowledgements

We acknowledge E. Tosatti, F. Becca, and T. Li for useful discussions. We are also grateful to A. Muramatsu and F. F. Assaad for valuable comments and providing us some of their numerical data reported in Ref. 2. This work is supported by a PRACE grant 2010PA0447 and by MIUR-COFIN2012. Part of the results is obtained by the K computer at RIKEN Advanced Institute for Computational Science.

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Affiliations

  1. SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy

    • Sandro Sorella
  2. Democritos Simulation Center, CNR – IOM Instituto Officina dei Materiali, 34151 Trieste, Italy

    • Sandro Sorella
  3. Computational Materials Science Research Team, RIKEN AICS, Kobe, Hyogo 650-0047, Japan

    • Sandro Sorella
    • , Yuichi Otsuka
    •  & Seiji Yunoki
  4. Computational Condensed Matter Physics Laboratory, RIKEN ASI, Saitama 351-0198, Japan

    • Seiji Yunoki
  5. CREST, Japan Science and Technology, Kawaguchi, Saitama 332-0012, Japan

    • Seiji Yunoki

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Contributions

S.S. developed the numerical codes; all authors performed the numerical simulations, analyzed the numerical data, and wrote the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Sandro Sorella or Seiji Yunoki.

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