Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation

We discover a quantum Monte Carlo (QMC) method to solve the fermion sign problem in interacting fermion models by employing a Majorana representation of complex fermions. We call it the “Majorana QMC” (MQMC). MQMC simulations can be performed efficiently both at finite and zero temperatures. Especially, MQMC is fermion sign free in simulating a class of spinless fermion models on bipartite lattices at half filling and with an arbitrary range of (unfrustrated) interactions. Moreover, we find a class of $SU\left(N\right)$ fermionic models with odd $N$, which are sign free in MQMC but whose sign problem cannot be in solved in other QMC methods, such as continuous-time QMC. To the best of our knowledge, MQMC is the first auxiliary field QMC method to solve the fermion sign problem in spinless (more generally, an odd number of species) fermion models. We conjecture that MQMC could be applied to solve the fermion sign problem in more generic fermionic models.

Figure 1

The schematic quantum phase diagram of the $t-V_1-V_2$ spinless fermion model on the honeycomb lattice in the region of $V_1>0$ and $V_2<0$. In this region, MQMC simulations at zero and finite temperatures can be performed efficiently without a fermion sign problem.

Figure 2

(a) Finite-size scaling of the CDW structure factor $M_2$ obtained in the projector (zero-temperature) MQMC simulations for various $V_1$ and $L=9,12,15,18,21$. It is clear that the phase transition between the semimetal and the CDW phase occurs when $V_1$ is between $1.34$ and $1.38$. The error bars for measured quantities are show explicitly and they are negligibly small. (b) The Binder ratios $B\equiv M^4/M_2^2$ for various $V_1$, including $V_1=1.355$, and various $L=9–21$, are plotted. From crossing of the Binder ratios, we conclude that the critical value of $V_1$ for the CDW transition is $V_{1c}=1.355\pm 0.001$.