Abstract
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 {\it 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 semi-metallic and the insulating antiferromagnetic states with increase of the interaction. The single particle gap in units of the 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.
Highlights
Detecting spontaneous symmetry-broken phases in numerical simulations often relies on the measure of correlation function
Spontaneously broken symmetry is often detected by computing two-point correlation functions of the appropriate local order parameter
We focus on the Hubbard model on honeycomb lattice at the filling one-half, for which the presence of an intermediate spin-liquid phase has been controversial [1,2]
Summary
Detecting spontaneous symmetry-broken phases in numerical simulations often relies on the measure of correlation function. [2] shows that extrapolating from significantly larger system sizes would suggest almost complete disappearance of the spin liquid from the phase diagram The latter conclusion is reinforced here, where we find excellent data collapse and identical finite-size scaling of both the single-particle gap and staggered magnetization, with the distinct values of critical exponents, in accord with the Gross-Neveu universality class [5,6]. An excellent finite-size scaling of the data for both the staggered magnetization and the single-particle gap is found by assuming the values of the critical exponents 1⁄4 0:79 and 1⁄4 0:88 These values are the ones found in the first-order expansion for the Gross-Neveu-Yukawa field theory of this quantum phase transition [6], around its upper critical (spatial) dimension of three. The data strongly support the existence of a single quantum critical point separating the semimetallic and the insulating antiferromagnetic phases of the Hubbard model, with the quantum criticality belonging to the Gross-Neveu universality class [5]
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