Determinant Quantum Monte Carlo Simulations

• Determinant Quantum Monte Carlo (DQMC) is a numerically exact simulation method that maps quartic fermionic interactions to bilinear forms using auxiliary fields.
• It employs submatrix, delay, and Fock-state sampling updates to efficiently scale simulations for strongly correlated lattice systems.
• DQMC enables accurate measurement of observables such as Green's functions, entanglement entropies, and correlation functions in models like the Hubbard Hamiltonian.

Determinant Quantum Monte Carlo (DQMC) is a numerically exact approach for simulating strongly correlated lattice fermion systems. By mapping quartic interactions to bilinear fermionic forms coupled to auxiliary fields, DQMC enables efficient computation of finite-temperature and ground state properties in models such as the Hubbard and related electron-phonon systems. Systematic advances in DQMC algorithmics—including submatrix updates, efficient handling of multi-determinant wavefunctions, advanced stabilization, improved measurements of exponential observables, and explicit Fock state sampling—have driven increases in accessible system size, simulation fidelity, and relevance to experimental quantum simulation platforms.

1. Foundational Principles and Algorithmic Structure

DQMC begins with a Suzuki–Trotter decomposition of the partition function for a many-body Hamiltonian with quartic interaction terms, followed by a Hubbard–Stratonovich (HS) transformation. This maps the interacting Hamiltonian onto a path integral over classical auxiliary fields, rendering the fermion problem quadratic:

$$Z \approx \sum_s \int \mathcal{D}x\, e^{-\mathcal{S}_B(x)} \prod_\sigma \det M_\sigma(\tau)$$

where $s$ collects discrete (HS) and $x$ continuous (e.g., phonon) fields, $\mathcal{S}_B(x)$ is the bosonic action (if present), and $M_\sigma(\tau)$ is the fermion matrix for each spin sector. Monte Carlo sampling proceeds over the auxiliary field space, with observables derived via Wick’s theorem using calculated Green’s functions. Fast updates (e.g., Sherman–Morrison/Woodbury) exploit the low-rank nature of local field changes, but large-scale simulations demand further algorithmic innovations (Ahuja et al., 2010, Scemama et al., 2015, McDaniel et al., 2017, Sun et al., 2023, Sun et al., 15 Apr 2024).

2. Algorithmic Advancements and Performance Scaling

Efficient simulation of large fermionic systems with DQMC is achieved through several complementary advances:

• Submatrix Updates: Rather than sequentially updating the Green’s function after each accepted auxiliary field change, the submatrix scheme accumulates updates, restructuring them into block operations. The updated Green’s function after $i$ accepted updates is recursively given by

$$G^{(i)} = G^{(0)} + G^{(0)}\,P_{N\times ik}\,[\Gamma^{(i)}]^{-1}\,P_{ik\times N}\,G^{(0)}$$

where $P$ projects affected rows/columns and $\Gamma^{(i)}$ accumulates update information. This reduces computational scaling for acceptance calculation to $\mathcal{O}(\beta n_d^2 N)$ and achieves order-of-magnitude speedup over delay updates (Sun et al., 15 Apr 2024).

• Delay and Fast Updates: Delay updates accumulate accepted local/multi-site changes over $n_d$ steps before performing a full Green’s function update in CPU cache-efficient blocked operations, offering significant speedup, especially for larger matrices (Sun et al., 2023, McDaniel et al., 2017).
• Handling Extended Interactions and Multideterminant Wavefunctions: For models with extended interactions, the update algebra generalizes to higher-rank local changes (rank-$k$), and determinant calculations use Sylvester's theorem. Efficient multideterminant QMC for large CI/FN-DMC expansions leverages the Sherman–Morrison formula, determinant sorting, and norm-based truncation to reach linear (in determinant number) scaling (Scemama et al., 2015).
• Stabilization via Matrix Factorizations: Products of transfer matrices in DQMC become ill-conditioned at low temperature ($\beta\to\infty$). Stable computation of equal-time and time-displaced Green’s functions employs pivoted QR decompositions and structured inversion schemes, maintaining machine-precision accuracy and computational efficiency (Bauer, 2020).

3. Physical Observables, Measurement Algorithms, and Exponential Quantities

DQMC grants access to single- and multi-particle correlation functions, Green’s functions, susceptibilities, and more. Of particular importance are observables that involve exponential averages such as the free energy and Renyi entanglement entropies, where naive evaluation is dominated by exponentially rare events:

• Integral Algorithm for Exponential Observables: Observables of the form $\log \langle e^{\hat{X}} \rangle$ (e.g., Renyi entropy, free energy) are recast as integrals over an auxiliary parameter $t$:

$$\log \langle e^{\hat{X}} \rangle = \int_0^1 dt\, \langle \hat{X} \rangle_t$$

with $\langle \ldots \rangle_t$ computed under reweighted distributions. For the $n$-th Renyi entropy $S_A^{(n)}$, this transformation suppresses the dominant exponential fluctuations and reduces sampling variance without increasing overall computational complexity (Zhang et al., 2023).

• Efficient formulas and fast update schemes (using auxiliary matrices updated by Sherman–Morrison formulas) enable the computation of high-order Renyi entropies and free energies at large system sizes, with improved agreement to DMRG and other numerics (e.g., in the 2D Hubbard model) (Zhang et al., 2023).
• Self-learning and Wang–Landau Approaches: Alternative strategies for reducing critical slowing down and autocorrelation times include self-learned bosonic actions (cumulative update/global moves) and Wang–Landau flat-histogram sampling in configuration weight space, both offering exponential reductions in autocorrelation and enabling larger system or parameter exploration (Xu et al., 2016, Yao et al., 2021).

4. Explicit Fock-State Sampling and Connection to Quantum Simulation

Traditional DQMC samples in the auxiliary-field representation; observables are then computed as ensemble averages over such field configurations. The Fock-state DQMC (FDQMC) approach introduces explicit sampling over electronic Fock states $|\eta\rangle$, enabling generation of configurations that are directly analogous to measurement outcomes in cold-atom quantum gas microscopes:

• Each MC step samples a joint configuration $(x, |\eta\rangle)$ with weight $|Z_{x,|\eta\rangle}|$, where

$$Z_{x,|\eta\rangle} = \det(P_{|\eta\rangle}^T B_x P_{|\eta\rangle})$$

and $P_{|\eta\rangle}$ is a projection onto occupied sites in $|\eta\rangle$. Fock-state updates (occupation number flips) are realized efficiently with determinant ratio and Green’s function formulas (Sherman–Morrison updates), permitting sampling within canonical, spin-restricted, or no-doublon ensembles at no added cost.

• FDQMC enables accurate computation of high-order static correlations, direct benchmarking of experimental quantum gas microscope distributions, and precise evaluation of dynamical spectral functions after analytic continuation. It naturally extends to models with spin-fermion constraints, such as the Kondo lattice, without resorting to additional approximations (Ding et al., 27 Mar 2024).

5. Comprehensive Model Scope and Application Examples

DQMC has demonstrated versatility across a broad class of fermion lattice models, with technical variations tailored to the interaction structure:

• Correlated Electron Systems: DQMC has clarified the phase diagrams and excitation spectra of the Hubbard, extended Hubbard, three-orbital, and Kondo lattice models. For instance, submatrix updates in DQMC enabled finite-temperature simulation of the 3D Hubbard model on $8,000$ sites, accurately extracting $O(3)$ universality and mapping the Néel transition (Sun et al., 15 Apr 2024, Sousa-Júnior et al., 2023, Kung et al., 2016).
• Electron–Phonon Coupling: The framework extends naturally to models with electron–phonon interactions, including the Holstein and Su–Schrieffer–Heeger (SSH) variants. Hybrid Monte Carlo, Fourier acceleration, customized preconditioners, and flat-histogram updates address the associated “stiffness” and autocorrelation barriers. Comparison studies clarify essential distinctions between model variants (e.g., bond vs. acoustic/optical SSH), with sensitivity to coupling and phonon spectrum definitions (Cohen-Stead et al., 2022, Costa et al., 2023).
• Entanglement and Quantum Information: DQMC is uniquely positioned for measurements of entanglement entropy and spectrum even in interacting fermionic systems, leveraging mapping to free-fermion ensembles and integral algorithms. Frameworks exist for both ground-state (projector) and finite-temperature (thermal) calculations using general regions and Renyi index (Grover, 2013, Assaad, 2015, Zhang et al., 2023).

6. Software Ecosystem, Modularity, and Future Horizons

Open-source DQMC software (e.g. StableDQMC.jl, SmoQyDQMC.jl) written in high-level languages such as Julia incorporate flexible scripting, state-of-the-art stabilization, and interfaces for hybrid Monte Carlo and non-linear/anharmonic couplings. These frameworks enable large-scale, model-agnostic simulation, rapid development of custom workflows, and integration with Julia’s broader scientific stack (e.g., tensor networks, machine learning) (Bauer, 2020, Cohen-Stead et al., 2023).

The continual evolution of update schemes—culminating in submatrix and explicit Fock-state sampling—and the integration of advanced measurement, stabilization, and data analysis routines, underpin a computational platform poised for application to emergent quantum matter, quantum simulation, and fundamental studies in quantum statistical mechanics. Future research directions include extension to multi-band and moiré systems, nonequilibrium protocols, and more sophisticated finite-size scaling and error-control analysis, as well as further alignment with experimental program needs in ultracold atom and solid-state settings.