Gross-Neveu Heisenberg criticality: Dynamical generation of quantum spin Hall masses

We consider fermions on a honeycomb lattice supplemented by a spin invariant interaction that dynamically generates a quantum spin Hall insulator. This lattice model provides an instance of Gross-Neveu Heisenberg criticality, as realized for example by the Hubbard model on the honeycomb lattice. Using auxiliary field quantum Monte Carlo simulations we show that we can compute with unprecedented precision susceptibilities of the order parameter. In O($N$) Gross-Neveu transitions, the anomalous dimension of the bosonic mode grows as a function of $N$ such that in the large-$N$ limit it is of particular importance to consider susceptibilities rather than equal-time correlations so as to minimize contributions from the background. For the $N=3$ case, we obtain $1/\nu =1.11\left(4\right), {\eta }_{\varphi }=0.80\left(9\right)$, and ${\eta }_{\psi }=0.29\left(2\right)$, respectively, for the correlation length exponent, and the bosonic and fermionic anomalous dimensions.

Figure 1

Schematic ground-state phase diagram with DSM, QSH, and SSC phases. The DSM-QSH transition belongs to the Gross-Neveu Heisenberg universality class. The QSH-SSC is an example of a monopole-free deconfined quantum critical point (DQCP) [25].

Figure 2

(a) Correlation ratio and (b) susceptibility of the spin-orbit coupling order parameter for different system sizes across the DSM-QSH phase transition.

Figure 3

(a) ${\lambda }_c$ as a function of system size is obtained from a crossing-point analysis of the correlation ratio of Eq. (7) for $L$ and $L+6$. (b) Correlation length exponent as a function of system size as obtained from Eq. (9). (c) Bosonic anomalous dimension as obtained from Eq. (11). From the fits (see text) we obtain ${\lambda }_c=0.0186\left(2\right), 1/\nu =1.11\left(4\right)$, and ${\eta }_{\varphi }=0.80\left(9\right)$ in the large system size limit.

Figure 4

As a cross-check for our determination of the critical point and exponents, we provide a data collapse with ${\lambda }_c=0.0186, 1/\nu =1.11$, and ${\eta }_{\varphi }=0.8$ for (a) the correlation ratio and (b) the QSH susceptibility.

Figure 5

(a) Monte Carlo estimate of $Z$ as defined in Eq. (13). (a) Size scaling of the fermionic anomalous dimension as obtained from Eq. (15). In the large system size limit, we obtain ${\eta }_{\psi }=0.29\left(2\right)$.

Figure 6

Spin-spin time displaced correlation function at the ordering wave vector. Here we consider the Hubbard model on the honeycomb lattice in the proximity of the GN O(3) critical point. We present data for different choices of the HS transformation where the field couples to the density (triangles) or to the magnetization (circles).

Figure 7

Green's function at the Dirac point for the same run as in Fig. 6.

Figure 8

Bin values for the spin (a),(b) and single particle (c),(d) susceptibilities for a $12\times 12$ honeycomb lattice at $U/t=4$ and $\beta t=12$. (a),(c) The HS field couples to the magnetization. (b),(d) The HS field couples to the density. Each bin consists of 2400 sweeps.

Figure 9

Time displaced spin-orbit correlation functions at $L=\beta =24$ and $\lambda =0.01875$.

Figure 10

Time displaced local Green's function for the same run as in Fig. 9.

Figure 11

Bin values for spin-orbital coupling susceptibility ${\chi }^{\text{QSH}}$ (d) and its three components ${\chi }^{{\text{QSH}}_x}$ (a), ${\chi }^{{\text{QSH}}_y}$(b), and ${\chi }^{{\text{QSH}}_z}$(c) for a $12\times 12$ honeycomb lattice at $\lambda /t=0.0186$ and $\beta t=12$. Each bin consists of 2400 sweeps.

Figure 12

Real space equal-time Green's function at zero temperature along different directions.

Figure 13

Here we plot the time displaced local Green's function, at $\tau =\beta /2$ for $\beta =L$ simulations. As apparent, this quantity is proportional to $L^{-2}$.