Detecting the Largest Correlations using the Correlation Density Matrix: a Quantum Monte Carlo Approach

TL;DR

The paper tackles the challenge of uncovering dominant correlations and potential unknown orders in quantum many-body systems without prior assumptions. It introduces a quantum Monte Carlo framework to measure the correlation density matrix $ρ^c_{AB}$ and uses its singular value decomposition to extract independent AB correlations via $X_i^{(A)}$ and $Y_i^{(B)}$ with weights $σ_i$, enabling bias-free identification of leading order parameters. The method is demonstrated on 1D and 2D transverse-field Ising models and the SU(2)-symmetric Heisenberg bilayer, providing accurate location of quantum critical points and consistent critical exponents, while also revealing the lattice-to-CFT operator content through operator decompositions. This approach offers a scalable tool for discovering unknown or exotic order parameters in large quantum systems, with potential extensions to nonlocal and topological orders using tailored cluster geometries and symmetry-guided analyses.

Abstract

We present a quantum Monte Carlo-based approach to detect and compute the most dominant correlations for many-body systems without prior knowledge. It is based on the measurement and analysis of the correlation density matrix between two (small) subsystems embedded in the full (large) sample. In order to benchmark this procedure, we investigate zero-temperature quantum phase transitions in one- and two-dimensional quantum Ising model as well as the two-dimensional bilayer Heisenberg antiferromagnet. The method paves the way for a systematic identification of unknown or exotic order parameters in unexplored phases on large systems accessible to quantum Monte Carlo methods.

Figures (14)

• Figure 1: Illustration of correlations between clusters $A$ and $B$ using operators $O_A$ and $O_B$.
• Figure 2: Schemes of the partition function sampling \ref{['eq:Z_SSE']}: (a) Normal SSE; (b) RDM SSE
• Figure 3: 1D TFIM: (\ref{['fig:1dTFIM_Layout']}) Layout of subsystems $A$ and $B$ chosen for the 4-site CDM, (\ref{['fig:1dTFIM_ScalingHarada']}) Non-linear scaling fit using 2-site CDM yields fitted parameters - $h_c^*=1.00152(97)$ and critical exponents $\nu^* = 0.955(10)$$\beta^* = 0.1266(49)$haradaBayesianInferenceScaling2011, (\ref{['fig:1dTFIM_ScalingKnown']}) Scaling collapse for 2-site CDM data using known exact values: $h_c = 1$, $\nu = 1$, $\beta=1/8$ ($\Delta_\sigma = 1/8$, $\Delta_\epsilon = 1$) difrancescoConformalFieldTheory1997, (\ref{['fig:1dTFIM_SV0vsLength']}) Log-log plot of the dominant singular value $\sigma_0$ from $\rho^c_4$ vs length at the critical point $h_c=1$. Estimated value of critical exponent is $0.2504(8)$. Hollow circles indicate finite temperature ED calculations at $T=1/L$ while filled circles indicate QMC results at the same temperature
• Figure 4: 1D TFIM: Singular values vs field ($L=60$) (a) $\rho_4^c$ (smaller singular values are not shown) and (b) $\rho_2^c$. The height of the shaded portion corresponds to the error in the singular values.
• Figure 5: Ising 1D: Physical correlations vs length for different field values for $\rho_4^c$ and $\rho_2^c$ in the 1D TFIM. The circles represent $\frac{1}{4}\braket{(\tau^z_0\mathbb{1}_1+\mathbb{1}_0\tau^z_1)(\tau^z_{L/2}\mathbb{1}_{L/2+1}+\mathbb{1}_{L/2}\tau^z_{L/2+1})}_c$ and triangles represent $\frac{1}{4}\braket{(\tau^z_0\tau^x_1+\tau^x_0\tau^z_1)(\tau^z_{L/2}\tau^x_{L/2+1}+\tau^x_{L/2}\tau^z_{L/2+1})}_c$