Preempting Fermion Sign Problem: Unveiling Quantum Criticality through Nonequilibrium Dynamics in Imaginary Time

TL;DR

This work tackles the fermion sign problem in quantum Monte Carlo simulations of quantum criticality by exploiting short-time imaginary-time nonequilibrium dynamics. The authors derive and employ a scaling framework, O(τ,g,L) = L^{-\ u} f(gL^{1/\\nu}, τL^{-z}), with z often equal to 1 for Dirac quantum critical points, to extract the critical point and exponents from nonequilibrium trajectories before the sign problem becomes severe. They validate the approach on two Dirac-fermion models and apply it to the sign-problematic SU(3) Hubbard model with staggered flux, uncovering a continuous Dirac-semimetal to λ8-antiferromagnetic transition that defines a novel chiral SU(3) Gross-Neveu universality class, distinct from conventional chiral Ising/XY/Heisenberg classes. The framework dramatically reduces computational cost relative to equilibrium PQMC and provides self-consistency checks via multiple initial states and τ values; it also outlines limitations, notably that the method is not a universal cure for the sign problem and may fail in regimes with particularly severe sign issues. The findings open a practical path to study sign-problematic fermionic quantum criticality and motivate further exploration of new universality classes beyond O(N) paradigms. The study suggests the potential for extension to other fermionic and bosonic sign-problematic systems and to experimental platforms probing Dirac-fermion quantum criticality.

Abstract

The notorious fermion sign problem, arising from fermion statistics, presents a fundamental obstacle to the numerical simulation of quantum many-body systems. Here, we introduce a framework that circumvents the sign problem in the studies of quantum criticality and its associated phases by leveraging imaginary-time nonequilibrium critical dynamics. We demonstrate that the critical properties can be accurately determined from the system's short-time relaxation, a regime where the sign problem remains manageable for quantum Monte-Carlo (QMC) simulations. After validating this approach on two benchmark fermionic models, we apply it to the sign-problematic Hubbard model hosting SU(3)-symmetric Dirac fermions. We present the first numerically exact characterization of its quantum phase diagram, revealing a continuous transition between a Dirac semi-metal and a SU(3) antiferromagnetic phase. This transition defines an unconventional Gross-Neveu universality class that fundamentally reshapes current understanding of Gross-Neveu criticality. Our work provides a powerful tool for investigating sign-problematic systems and quantum criticality.

Figures (22)

• Figure 1: Scheme of preempting sign problem to probe quantum criticality via short-time critical dynamics and the application in single-Dirac-fermion Hubbard model.a, With some typical initial states, scaling behaviors governed by the QCP are reflected in the short-time stage, in which the sign problem is still weak as shown in the inset. The white dashed line indicates the nonequilibrium critical region associated with the QCP. The red dashed line in the inset indicates the average sign at $\tau=0.3L^z$, where $z=1$ due to the nature of the Dirac QCP. b, Determination of QCP as $U_c = 7.220(37)$ via the intersection points of curves of the correlation length ratio $R_{\rm FM}$ versus $U$ for different sizes at $\tau=0.3L^z$ with DSM initial state. c, Determination of $1/\nu = 1.18(3)$ by scaling collapse of $R_{\rm FM}$ versus rescaled $(U - U_c)$, with a reduced chi-squared value $\chi_{\nu}^2=1.824$ indicating good quality. d, Determination of $\eta_\phi = 0.33(2)$ via scaling collapse of curves of the structure factor $S_{\rm FM}$ versus rescaled $\tau$ at $U_c$, with $\chi_{\nu}^2=0.606$. e, Determination of $\eta_\psi = 0.135(2)$ via scaling collapse of curves of the fermion correlation $G_{\rm f}$ versus rescaled $\tau$ at $U_c$, with $\chi_{\nu}^2=1.241$. The dashed lines in d and e indicate the boundary of the nonequilibrium scaling region; only the regions to the right are included in the scaling collapse analysis (see SM Sec. \ref{['Sec:S2-B']}).
• Figure 2: Probing quantum criticality via short-imaginary-time critical dynamics in spinless $t$-$V$ model with CDW initial state.a, Determination of QCP as $V_c=1.35(1)$ via the intersection points of $R_{\rm CDW}$ versus $V$ for different $L$ at $\tau=0.3L^z$, where $z=1$. Shown in Inset is the evolution of average sign with red dashed curve marks $\tau=0.3L^z$. b, Determination of $1/\nu=1.30(18)$ via scaling collapse of $R_{\rm CDW}$ versus rescaled $(V-V_c)$, with $\chi_{\nu}^2=1.518$ demonstrating the collapse quality. c-d, Determination of $\eta_\phi = 0.49(5)$ and $\eta_\psi = 0.073(4)$ via scaling collapse of the short-imaginary-time dynamics of $S_{\rm CDW}$ and $G_{\rm f}$ versus rescaled $\tau$, respectively, with $\chi_{\nu}^2=1.668$ and $0.963$ demonstrating the collapse quality in the nonequilibrium scaling region to the right of the dashed lines.
• Figure 3: Phase diagram and quantum criticality in $\rm SU(3)$ Hubbard model detected by short-imaginary-time dynamics with $\lambda_8$-AFM initial state.a, Phase diagram determined via short-imaginary-time dynamics. Insets show the energy spectra of DSM state (upper left) and sketch of $\lambda_8$-AFM order in which a fermion with one flavor (green "3") is situated at one sublattice and double fermions with the other two flavors (pink "1" and violet "2") are situated at the other sublattice (lower right). b-c, Critical point $U_c=1.10(5)$ for $\phi=0.075\pi$ and $1/\nu=0.68(5)$ determined via curves of $R_{\lambda_8\text{-AFM}}$ versus $U$ for different $L$ at $\tau=0.25L^z$, where $z=1$ and $\chi_{\nu}^2=1.239$ for the collapse quality. Shown in Inset of b is the evolution of average sign with red dashed curve marks $\tau=0.25L^z$. d-e, $\eta_\phi = 0.55(5)$ and $\eta_\psi = 0.15(3)$ determined via the scaling collapse of evolution curves of $S_{\lambda_8\text{-AFM}}$ and $G_{\rm f}$, respectively, with $\chi_{\nu}^2=1.382$ and $\chi_{\nu}^2=0.560$ demonstrating the collapse quality in the nonequilibrium scaling region to the right of the dashed lines.
• Figure S1: Schematic illustration of the three scenarios regarding the sign problem. The shaded region indicates where the sign problem is not severe and simulations are feasible. The green region corresponds to scenario 1 ($\tau_{\text{neq.}} < \tau_{\text{eq.}} < \tau_{\text{sign}}$), where equilibrium methods also work. The yellow region corresponds to scenario 2 ($\tau_{\text{neq.}} < \tau_{\text{sign}} < \tau_{\text{eq.}}$), where only the nonequilibrium method is feasible. The red region corresponds to scenario 3 ($\tau_{\text{sign}} < \tau_{\text{neq.}} < \tau_{\text{eq.}}$), where even the nonequilibrium approach fails. The boundaries define the maximum accessible system sizes $L_{\text{max}}^{(\text{eq.})}$ and $L_{\text{max}}^{(\text{neq.})}$ for each method.
• Figure S2: First-order phase transition characteristics in the $q=6$ quantum Potts chain observed during short imaginary time evolution.a, Binder ratio versus $g$ for different system sizes, where the curves do not intersect at a single point, and negative dips are observed. The emergence of negative dips in the Binder ratios is a hallmark of first-order transition. b, Imaginary time relaxation of the order parameter $M$ starting from the ordered phase at $g=1$. c, Failure of scaling collapse for b, with $a=1.0$ as an example. d, Fitting $M \propto L^{-a}$ for fixed $\tau/L$, where the value of $a$ does not converge as $\tau/L$ increases.