Self-learning quantum Monte Carlo method in interacting fermion systems

The self-learning Monte Carlo method is a powerful general-purpose numerical method recently introduced to simulate many-body systems. In this work, we extend it to an interacting fermion quantum system in the framework of the widely used determinant quantum Monte Carlo. This method can generally reduce the computational complexity and moreover can greatly suppress the autocorrelation time near a critical point. This enables us to simulate an interacting fermion system on a $100\times 100$ lattice even at the critical point and obtain critical exponents with high precision.

Figure 1

(a) Schematic phase diagram of the transverse-field Ising model coupled to Fermi surface. As a function of the transverse field, the system (both fermions and Ising spins) goes through a transition from ferromagnetic (FM) metal to paramagnetic (PM) metal. The black dot is the finite-temperature critical point $[T=1,h_c=2.774\left(1\right)]$ where we systematically demonstrate the superior performance of SLDQMC over DQMC. (b) Autocorrelation function $C\left(\tau \right)$ for $L=16$ system at the critical point in panel (a), for local update with DQMC, where the autocorrelation time is very long (larger than 600 sweeps). The autocorrelation function is defined as $C\left(\tau \right)=\left(\langle M\left(0\right)M\left(\tau \right)\rangle -{\langle M\rangle }^2\right)/\left(\langle M^2\rangle -{\langle M\rangle }^2\right)$ with $M\left(\tau \right)$ being the total magnetization of Ising spins for the $\tau \mathrm{th}$ sweep.

Figure 2

The coefficient of determination for multilinear regression ${\mathcal{R}}^2$ decides how many $J_{i,\tau ;j,{\tau }^{\prime }}\mathrm{s}$ (how many neighbor interactions) need to be considered in the $H^{\text{eff}}$. Purple line shows the $1-{\mathcal{R}}^2$ for the temporal neighbors while fixing the spatial neighbor nearest. Green line shows the $1-{\mathcal{R}}^2$ for the spatial neighbors while fixing the temporal neighbors. The temporal interaction is more long ranged (up to the tenth nearest neighbor) but the spatial interaction is short ranged (up to the second nearest neighbor).

Figure 3

Comparison of ${\tau }_L$ between DQMC and SLDQMC at the critical point. For DQMC, the critical slowdown with ${\tau }_L\sim L^{2.1\left(1\right)}$ is observed, while for SLDQMC, the critical slowdown has been complete cured, ${\tau }_L=1$, for all the system sizes up to $L=100$.

Figure 4

Uniform spin susceptibilities $\chi$ at the critical point as a function of system sizes, $\chi \propto L^{2-\eta }$ with $\eta =\frac{1}{4}$ as the anomalous dimension of 2d Ising universality. The linear system size is as large as $L=100$.