On the correlation between entanglement and the negative sign problem

TL;DR

Problem: whether entanglement in the ground state correlates with the negative sign problem in quantum Monte Carlo for strongly correlated quantum many-body systems. Approach: quantify the relation by comparing ground-state entanglement entropy computed by DMRG with the low-temperature average sign from AFQMC for a doped $4×16$ Hubbard system at $U/t=8$. Findings: both entanglement entropy and average sign exhibit a peak/dip around $h ≈ 0.2$, marking a region of maximal difficulty for both tensor-network and QMC methods, with half-filling sign-problem free. Significance: provides a new perspective on why simulations struggle in certain regimes and offers a basis for cross-method insights and future extensions to other methods and temperature ranges.

Abstract

In this work, we study the correlation between entanglement and the negative sign problem in quantum Monte Carlo for the simulation of low-dimensional strongly correlated quantum many body systems. Entanglement entropy characterizes the difficulty of many-body simulation with tensor network state related methods, while the average sign measures the difficulty in many-body simulation for a variety of quantum Monte Carlo methods. Although there exist cases where one type of method works better than the other, it is desirable to find the possible correlation between entanglement and average sign for general hard strongly correlated systems regarding computational complexity. We take the doped two-dimensional Hubbard model as an example and numerically calculate the doping evolution of both the entanglement in the ground state with Density Matrix Renormalization Group and the average sign in the Auxiliary Field Quantum Monte Carlo simulation at low temperature. The results show that they are indeed correlated. The entanglement entropy (average sign) shows a peak (dip) around 20% doping, indicating that it is the difficult region for both methods. The vicinity of 20% doping is also the most intriguing region in both the Hubbard model and cuprate high-Tc superconductors where competing states with close energy intertwine with each other. Recognizing the correlation between entanglement and average sign provides new insight into our understanding of the difficulty in the simulation of strongly correlated quantum many-body systems.

Figures (2)

• Figure 1: The evolution of (a) average sign in AFQMC, (b) entanglement entropy and (c) truncation error in DMRG with doping. In (a), average sign for $\beta = 2.5$, $3$ and $4$ are shown. In (c), the truncation errors in DMRG of different bond dimensions are shown. For the half-filling and the $h = 0.0625$ cases, small bond dimension already gives very small truncation error. We can find a correlation between average sign and entanglement entropy or truncation error. By doping away from the half-filling case, the average sign first decreases, reaching its minimum value at about $20\%$ doping, then starts to increase with the further increase of doping. The entanglement entropy shown in (b) are the values from a linear extrapolation with truncation error. The entanglement entropy and truncation error have opposite behavior as the average sign. Notice that region for small average sign and large entanglement/truncation error are difficult for both AFQMC and DMRG.
• Figure 2: The entanglement spectrum for the Hubbard model on a $4 \times 16$ cylinder with $U/t = 8$. Results for doping $h = 0, 0.0625, 0.1875, 0.375$, and $0.4375$ are shown. The entanglement spectrum for $h=0.1875$ decay most slowly which is consistent to the large entanglement entropy and truncation error shown in Fig. \ref{['fig1']}. The entanglement entropy at half-filling ($h = 0$) is steepest. The results for $h = 0.4375$ has a similar decay trend as $h=0.1875$ in the tail, but the leading entanglement spectrum has a steeper decay than the $h=0.1875$ case as shown in the inset.