Pinning the Order: The Nature of Quantum Criticality in the Hubbard Model on Honeycomb Lattice

In numerical simulations, spontaneously broken symmetry is often detected by computing two-point correlation functions of the appropriate local order parameter. This approach, however, computes the square of the local order parameter, and so when it is small, very large system sizes at high precisions are required to obtain reliable results. Alternatively, one can pin the order by introducing a local symmetry-breaking field and then measure the induced local order parameter infinitely far from the pinning center. The method is tested here at length for the Hubbard model on honeycomb lattice, within the realm of the projective auxiliary-field quantum Monte Carlo algorithm. With our enhanced resolution, we find a direct and continuous quantum phase transition between the semimetallic and the insulating antiferromagnetic states with increase of the interaction. The single-particle gap, measured in units of Hubbard $U$, tracks the staggered magnetization. An excellent data collapse is obtained by finite-size scaling, with the values of the critical exponents in accord with the Gross-Neveu universality class of the transition.

Popular Summary

The electrically insulating or metallic behavior of many solids can be very well understood based on quantum-mechanical theories that neglect the Coulomb interaction between their electrons. The so-called “Mott insulators,” however, defy such understanding in terms of noninteracting electrons and are insulating when the conventional theories expect them to be conducting. What’s more, this insulating behavior is often accompanied by an order of some kind—for example, spin order in magnetic Mott insulators—in fact, so often that the proclivity to order is often taken to be the very reason for the insulating behavior. But, is this necessarily the case? The answer to this question has been under considerable debate, in particular, in the context of the Hubbard model—the paradigmatic model for Mott insulators—on a honeycomb lattice. In this theoretical paper, we provide a compelling conclusion to the debate by revisiting the model with a new high-sensitivity quantum Monte Carlo simulation algorithm that is designed especially to pick up signals of the relevant order even when they are weak.

The Hubbard model we have investigated has, on average, one electron per site of the honeycomb lattice and is directly relevant to graphene. The electrons can hop from site to site and, when on the same site, interact with a repulsion. The limits of weak repulsion and strong repulsion are well understood: The former is a semimetallic state and the latter an insulating state with magnetic order. The unsettled debate concerns how one limit crosses over to the other: Is the crossover direct, or is there an intermediate insulating state without magnetic order? Our strong numerical results not only clearly point to a direct and continuous quantum transition between the semimetallic and magnetically ordered insulating phases, but also reveal the nature of the transition as a new universality class of quantum phase transitions—a consequence of the presence of Dirac cones in the electronic structure of the semimetallic state.

This universality class may therefore be relevant to a number of other electronic systems that feature Dirac points in their electronic structures, such as the surfaces of topological insulators and Weyl metals. The computational technique we have introduced may also prove useful in other problems where the order of interest is weak.

Figure 1

Comparison of the pinning-field and correlation-function approaches to determine the staggered moment at (a) $U/t=5$ and (b) $U/t=4$. The data sets at $h_0=0$ correspond to the correlations functions, and values of $\theta t=40$ are sufficient to converge to the ground state. For nonvanishing pinning fields, projection parameters $\theta t=320$ are required to guarantee convergence. We use $\Delta \tau t=0.1$, which for the symmetric Trotter decomposition yields converged results within our numerical accuracy. At $U/t=4$, comparison with the results of Ref. 2 shows excellent agreement. Lines corresponds to least-squares fits to the form $a+b/L+c/L^2$.

Figure 2

Single-particle gap. (a) The raw data at $U/t=3.8$. As is apparent, the systematic error stemming from the finite Trotter step is negligible within our accuracy. The data support a large imaginary time range consistent with a single exponential decay. Lines are least-squares fits of the tail of the imaginary time Green function to the form $Z{e}^{-{\Delta }_{\mathrm{sp}}\tau }$. Here, $Z$ corresponds to the single-particle residue and ${\Delta }_{\mathrm{sp}}$ to the single-particle gap. (b) Size dependence and extrapolation of the single-particle gap.

Figure 3

Magnetic moment at pinning field $h_0=5t$ as a function of $U/t$. Here, we have used $\theta t=320$ and $\Delta \tau t=0.1$. The solid lines are least-squares fits to the form $a+b/L+c/L^2$.

Figure 4

Staggered moment extrapolated to the thermodynamic limit (see Fig. 3) for two values of the pinning field. We have equally plotted the single-particle gap in units of $U$. The inset plots the staggered magnetization as obtained from a mean-field spin-density-wave ansatz.

Figure 5

Data collapse for the magnetization presented in Fig. 3. The exponents are taken for the $ϵ$ expansion of Ref. 6. (a) The crossing point pins down the value of $U_c$. (b) The data collapse, using $U_c/t=3.78$.

Figure 6

Data collapse for the single-particle gap. The exponents are taken for the $ϵ$ expansion of Ref. 6. (a) The crossing point pins down the value of $U_c$. (b) The data collapse again using $U_c/t=3.78$. For comparison, we have included the data for the magnetization.