Thermal Ising transitions in the vicinity of two-dimensional quantum critical points

The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\nu$ and $\eta$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.

Figure 1

Finite size data for $B^Q$ (left panel) and $L^{{\eta }_{2\mathrm{D}}}{M}_2^Q$ (right panel) for the quantum Ising model at $\mathrm{\Gamma }/J=2.5$ in the critical region of the thermal Ising transition. The dashed line indicates the value of the critical Binder ratio for the classical Ising model $\left(\mathrm{\Gamma }=0\right)$ on the square lattice.

Figure 2

(Left) Successive crossing points for system sizes $L$ and $L+\mathrm{\Delta }L$ in the quantities $B^Q$ (circles) and $M_2^Q$ (squares) as functions of $1/L$ for $\mathrm{\Delta }L=16$, as obtained from simulations of the quantum Ising model at different values of $\mathrm{\Gamma }$. (Right) Closeup for $\mathrm{\Gamma }/J=3.0$.

Figure 3

Data collapse plot of the data for $M_2^Q$ of the quantum Ising model at $\mathrm{\Gamma }/J=2.5$ (full symbols). The line denotes the expanded scaling function. Also included are linearly rescaled data for $M_2^Q$ at $\mathrm{\Gamma }/J=3$ (open symbols), as detailed in the text.

Figure 4

Thermal phase diagram of the quantum Ising model on the square lattice.

Figure 5

Scaling of the thermal transition temperature $T_c$ in the vicinity of the quantum critical point. The dashed line indicates the asymptotic scaling according to Eq. (1), with $z=1$ and $\nu ={\nu }_{3D}$.

Figure 6

Deviation of the effective exponent $\nu$ to the value of ${\nu }_{3D}$ as a function of the relative size of the fit window for the critical line in the quantum Ising model close to the quantum critical point.

Figure 7

Data collapse of the finite temperature data for $B^Q$ (left) and $M_2^Q$ (right) of the quantum Ising model atop the quantum critical point. The lines denote the scaling functions ${\tilde{f}}_B$ and $\tilde{f}$ respectively, obtained from a Gaussian process regression.

Figure 8

Finite-size data for $B$ (left) and $L^{{\eta }_{2\mathrm{D}}}{M}_2$ (right) for the spinless fermion $t-V$ model at $V/t=1.75$ in the critical region of the thermal Ising transition. The dashed line indicates the value of the critical Binder ratio for the Ising model on the honeycomb lattice.

Figure 9

Data collapse plot of the data for $M$ of the $t-V$ model at $V/t=1.75$. The line denotes the expanded scaling function.

Figure 10

Thermal phase diagram of the spinless fermion $t-V$ model on the honeycomb lattice. The solid line is a fit of the numerical data to the scaling from Eq. (1), and the dashed line indicates the asymptotic scaling of the critical temperature $T_c=0.37966V$ in the large-$V$ limit.

Figure 11

Finite-temperature data for $B$ (left) and $M_2$ (right) for the spinless fermion $t-V$ model at $V/t=1.355$, atop the quantum critical point, for various system sizes.

Figure 12

Data collapse of the finite-temperature data for $B$ (left) and $M_2$ (right) for the spinless fermion $t-V$ model at $V/t=1.355$, atop the quantum critical point, for various system sizes. The lines denotes the scaling functions ${\tilde{f}}_B$ and $\tilde{f}$, obtained from a Gaussian process regression.

Figure 13

Attempted data collapse of the finite-temperature data $M_2$ for the spinless fermion $t-V$ model at $V/t=1.355$, atop the quantum critical point, for various system sizes with $\eta =0.3$ [18, 20] (left) and $\eta =0.45$ [19] (right), and $z=1$.