Probing the Critical Behavior of a Sign-Problematic Model with Monte Carlo Simulations

TL;DR

This work probes how a sign problem in Monte Carlo simulations relates to phase transitions by studying the generalized Baxter-Wu model with complex couplings, which maps to a 1D quantum model and has exactly known critical properties. It compares the conventional average sign $\langle S\rangle_+$ and a modified sign $\langle\widetilde{S}\rangle^*_+$ as probes of criticality, finding that $\langle S\rangle_+$ shows a negative dip near Tc but can misfire, while $\langle\widetilde{S}\rangle^*_+$ tracks Tc in principle though with exponential sampling costs. The authors introduce and validate a sign-free reference model ${\cal Z}_+$ and demonstrate that GBW and ${\cal Z}_+$ share the same universality class (2D 4-state Potts) via finite-size scaling, enabling access to critical properties without the sign problem. This framework suggests simulating the reference model to infer universal critical behavior of other sign-problem plagued systems, offering a novel approach for studying phase transitions under sign problems.

Abstract

The sign-problematic generalized Baxter-Wu (GBW) model with asymmetric complex couplings is mapped onto a one-dimensional quantum model. Utilizing the model's exactly known critical properties, we study the relation between the conventional and the modified average signs and the phase transitions in the GBW model. We find that the average sign develops a negative peak near the critical point, but it is not a unique indicator of phase transition, as similar features can appear in non-critical regions. While the average modified sign provides a viable probe for the phase transition, the practical effectiveness of this method is limited by the exponential scaling of computational cost with the system's volume. We propose that the universal properties of the original model can be investigated through simulating the related reference model, based on the universality assumption. Using finite-size scaling analysis based on Monte Carlo simulations, we confirm the validity of this method, which thereby provides a novel framework for investigating phase transitions in systems plagued by the sign problem.

Figures (8)

• Figure 1: Triangular lattice with periodic boundaries.
• Figure 2: (a)The phase diagram of the generalized Baxter-Wu model with complex couplings. The critical lines dividing ordered and disordered phases are exactly known; The ordered phase "order1" is ferromagnetic, and another ordered phase "order2" is antiferromagnetic. The disorder phase exists between the two critical lines. The two red lines represent the line $K=K_c/T, \phi=\phi_c/T$ going through the critical point ($K_c=0.376165, \phi_c=0.3$) and the line $K=K_c/T, \phi=(0.3+\pi/4)/T$ going through a shifted point $(K_c=0.376165, \phi=0.3+\pi/4)$, respectively. (b)The RG flow diagram of the generalized Baxter-Wu model. $\Delta K^2<0$($\Delta K=i\phi$), corresponding to complex couplings, brings the model logarithmic corrections; $\Delta K^2>0$, corresponding to the real couplings, moves the model into the first-order range.
• Figure 3: A $\pi$ rotation about a lattice direction transforms the system configuration $\Gamma$(left) into a new configuration $\Gamma'$ (right). The up-triangle and down-triangle in the two configurations are exactly reversed. This relationship allows us to bundle the configurations together and eliminate the imaginary parts in their joint weight.
• Figure 4: The GBW model with $K=K_c/T, \phi=\phi_c/T$ where $(K_c, \phi_c)=(0.3, 0.440322696)$. The red dashed line denotes the critical point $T_c=1$ of the original model; The blue dashed line indicates the critical point $T_c^+=0.68083(7)$ of the ${\cal Z}_+$ model. (a)The average sign $\langle S \rangle_+$ as a function of the temperature $T$ for different system sizes. (b)The exactly calculated free energies $F$ of the GBW model, $F_+$ of the ${\cal Z}_+$ model, and the difference $\Delta F$ between them as functions of temperature $T$ for $L=6$.
• Figure 5: The 8 types of nearest neighbor configurations of a given site on the triangular lattice, each with 2,12,12,6,6,12,12, and 2 symmetric transformations. Flipping the center spin results in $|\Delta \left ( A\left ( \Gamma \right ) -B\left ( \Gamma \right ) \right )|= 0, 8, 0, 0, 0, 0, 8,0$. This means that the possible value of $|\Delta \left ( A\left ( \Gamma \right ) -B\left ( \Gamma \right ) \right )|$ must be multiples of 8.